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1 (C) :
Thus the multiplier Aa of a fixed point of
(oo). (c) J
=
alb E Q be a
h
with equality if and only if a is in the filled Julia set K( ¢>) of ¢>. 3.29. Let KIQ be a number field and let ¢>(z) E K(z) be a rational map of degree d 2: 2. For each finite place v E M�, write ¢>( z) = Fv ( z) IGv ( z) as a ratio of polynomials that are normalized for v, i.e., the coefficients of Fv and Gv are v-adic integers and at least one coef ficient is a v-adic unit. Let �¢',';'; be the local canonical height, as described in Theorem 3 .27, normalized using Gv in (3. 1 6). (a) Prove that �¢�';;' is well-defined, independent of the decomposition ¢>(z) as a ratio of v normalized polynomials. For nonarchimedean places v this serves to pin down a specific function �¢,';;' that depends only on ¢> and v (cf. Remark 3.28). (b) Give an example to show that it may not be possible to write ¢>(z) = F(z)IG(z) such that F and G are simultaneously normalized for all finite places v E M�. (c) More generally, prove that every ¢> E K ( z) can be written as F ( z) I G ( z) with F and G simultaneously normalized for all finite places v E M� if and only if K has class number 1. 3.30. Suppose that ¢>( z) E K ( z) has good reduction at a finite place v E MK . Prove that the function �
{
is a local canonical height by showing that it has the required properties. Section 3.6. Diophantine Approximation 3.31. Prove Dirichlet's Theorem 3.33, which says that for every a E many xI y E Q satisfying
�� - a l
lR "- Q there are infinitely
� y .
\
(Hint. Look at the numbers ya - Lyaj for 0 � y � A. They all lie in the interval [0, 1].
Divide the interval into A equal pieces and use the pigeonhole principle.) 3.32. Let B be a nonzero integer. (a) Prove that every solution (x, y) in integers to the equation
(3.50)
satisfies max lxl, IYI } � J4ii13. (Hint. Thepolynomial X 3 +Y3 factors in Q[X, Y] .) (b) Try to find an explicit upper bound for max{ lxl , IYI} for the similar-looking equation X 3 + 2Y 3 = B . The difficulty you face illustrates the fact, seen during the proof of Theorem 3.35, that the equation G(X, Y) = B is relatively easy to solve if G(X , Y) has distinct factors in Q[X, Y], but very difficult if it does not.
{
142
Exercises
(c) Find all solutions in integers x 2:: y to the equation (3.50) for the following values of B: (i) B = 2. (ii) B = 91. (iii) B = 728. (iv) B = 1729. Section 3.7. I ntegral Points in Orbits 3.33. Let ¢(z) = z + 1/z and write ¢n (1) = an /bn in lowest terms as in Example 3.50. (a) Prove by a direct computation that there are constants c, c' > 0 such that
c�
log an � c1 for all n 2:: 2. og n
1b-
(b) Try to prove directly that log an /log bn -+ 1 as n -+ oo . (c) Prove that ¢n (1) -+ oo as n -+ oo. (Notice that oo is a rationally indifferent fixed point, since ¢' ( oo ) = 1 .)
3.34. Let c E Z be a squarefree integer and let
# ( 0
for all n
2: 0.
(b) Let P be a fixed point of ¢ that is not totally ramified. Prove that for all n
2: 2.
(c) Generalize (b) to the case that P is a periodic point for ¢ of exact period m under the assumption that ¢= is not totally ramified at P. Use (a), (b), and (c) to deduce that if ¢ is not a polynomial map, then #¢ - n (P)
2: 3 if either
{ n 2:2: 3 n
4 and d = 2, and d 2: 3.
Exercises
143
3.38. Let KjQ be a number field, let S c MK be a finite set of absolute values on K, and let Rs be the ring of S-integers of K. Let ¢( z) E K ( z) be a rational function of degree d � 2 with ¢2 (z) .;_ K [z], and let o: E K be a wandering point for ¢. Prove that Oq, (o:) n Rs
is a finite set. (This exercise generalizes Theorem 3.43.) 3.39. Let K be a number field, let S C MK be a finite set of absolute values on K that includes all the archimedean absolute values, and let Rs be the ring of S-integers of K. Let ¢(z) E K(z) be a rational function of degree d � 2 satisfying ¢2 (z) .;_ K[z] . (a) Let n = 4 if d = 2 and let n = 3 if d � 3. Prove that the set
n
{o: E K : ¢ ( o: ) E Rs }
is finite. (b) Give an example with (n, d) = (3, 2) and oo nonperiodic for ¢ such that the set in (a) is infinite. (c) Same question as in (b) with n = 2 and d arbitrary. (d) Repeat (b) and (c) with oo a fixed point of ¢. 3.40. Let ¢1 , . . . , ¢r E Q( z) be rational functions of degree at least 2, and let be the col lection of rational functions obtained by composing an arbitrary finite number of ¢ 1 , . . . , ¢r. Note that each ¢i may be used many times. For example, if r = 1, then is simply the collec tion of iterates { ¢1}. Also note that in general, composition of functions is not commutative, so if r � 2, then is likely to be a very large set. (a) A rational map ¢ E K(z) is said to be of polynomial type if it has a totally ram ified fixed point. Prove that for such a map, there is a linear fractional transforma tion f E PGL2 ( K ) such that ¢f E K [z]. Further, if ¢ is not conjugate to zd , prove that one can take f to be in PGL2 (K). (b) Assume that contains no maps of polynomial type. Prove that there are finite sub sets 1 and 2 of satisfying
and (Here ¢ denotes the composition of ¢ with every map in .) (c) Let o: E Q. The -orbit of o: is the set O.p (o:) = ¢(o:) : ¢ E } . Continuing with the assumption that contains no maps of polynomial type, prove that the -orbit of o: contains only finitely many integers, i.e., prove that
{
Q.p ( o: )
n 1£ is a finite set.
3.41. Define the (logarithmic) height of a rational map ¢(z)
¢(z)
=
F(z) G(z)
=
E
Q(z) by writing d ao + a 1 z + · · · + adz d bo + b1z + · · · + bdz
with F(z), G(z) E Q[z] and setting h(¢)
=
h( [ao, a 1 , . . . , ad, bo , b 1 , . . . , bd] ) .
Prove the following quantitative version o f Proposition 3.46. For all d � 2 there is a constant C = C(d) such that for all integers N rational map ¢( z) E Q( z) of degree d with the following properties:
� 0 there exists a
144 • • • e
Exercises
q? (z) � C[z]. 0 is a wandering point for ¢. 0 , ¢(0) , ¢2(0) , ¢3 (0) , . . . , ¢N (0) E Z. h( r/J) :::; C · dN .
Try to do quantitatively better than this, for example, replace the height estimate with one of the form h(¢) :::; C · o N for some o < d. 3.42. Recall that a rational map ¢(z) E IQl(z) is affine minimal if its resultant Res(¢) cannot be made smaller via conjugation by an affine linear map f ( z) = az + b. If¢ is affine minimal and ¢2 is not a polynomial, we have conjectured (Conjecture 3 .47) that the size of 0 (a) n Z is bounded solely in terms of the degree of ¢. For each integer d � 2, let
C(d) = sup
{
# ( O¢(a) n z)
:
¢(z) E IQl(z), ¢2(z) � IQl[z], ¢ is affine minimal, and a E lP'N (IQl) ' PrePer( ¢)
}
.
Thus Conjecture 3.47 says that C(d) is finite. (a) Prove that for every integer N � 0 there exists an affine minimal rational map ¢(z) E IQl(z) such that ¢2 (z) � IQl[z], such that 0 is not a preperiodic point for ¢, and such that
are all integers. In particular, C(d) ---> oo as d ---> oo . (b) Prove that the function ¢(z) in (a) can be chosen to have degree l(N (c)
1)/2J . Hence C(d) � 2d + 2. Let ¢ (z ) be the function - 2565z4 + 9385z3 - 1 4955z2 + 1 2094z - 3720 ¢(z) 481z5 z 5 - 465z4 + 2185z3 - 6975z2 + 8254z - 3720 . Trace the orbit of 0. How many integers do you find? Find a rational function as in (a) of degree 2 and with the property that ¢; (0) is an integer for all 0 :::; i :::; 6. Hence C(2) � 7, which improves the bound from (b). Can you find infinitely many such functions? Can you prove that C(2) � 8? Find a rational function as in (a) of degree 3 and with the property that ¢; (0) is an integer for all 0 :::; i :::; 8. Hence C(3) � 9, which improves the bound from (b). Can you find infinitely many such functions? Can you prove that C(3) � 10? =
(d) (e)
Section 3.8. Integrality Estimates for Points in Orbits 3.43. Let Kv be a field complete with respect to the absolute value v, let ¢ : lP'1 ---> lP' 1 be a rational map of degree d � 2 defined over Kv, and let Pv : lP' 1 (Kv) x lP' 1 (Kv) ---> lR be the associated chordal metric. (See Sections 1 . 1 and 2. 1 for the definition of the chordal metric when v is archimedean and nonarchimedean, respectively.) Generalize Lemma 3.51 by showing that for every Q E lP' 1 ( Kv) there is a constant Cv = Cv ( ¢, Q) such that
min
Q' E- ' (Q)
Pv(P, Q'tQ() :::; Cv Pv (¢(P), Q)
Here eQ (¢ ) i s as defined in Lemma 3 .5 1 .
for all P E
lP'1 (Kv ).
145
Exercises
3.44. Let K be a number field and let ¢ : lP'1 --+ lP' 1 be a rational map of degree d � 2 defined over K. Prove that there is a finite set of absolute values S c MK such that for all v rf_ S and all Q E lP' 1 (K), the constant Cv(¢, Q ) in Exercise 3.43 may be taken equal to 1. 3.45. Let cp (z ) E Q(z) be a rational map of degree d � 2 with ¢2 (z ) rf_ Q [z], let a write cp n (a) = an /bn E Q as a fraction in lowest terms as usual. Prove that
E Q, and
3.46. Let K/Q be a number field, let v E MK be an absolute value on K, and let ¢( z) E K ( z) be a rational function of degree d � 2. Suppose that a E lP' 1 ( K) is a wan dering point for ¢ and that 'Y E lP' 1 ( K) is any point that is not a totally ramified fixed point of ¢2 . Prove that . - log Pv ( cpn (a) , "() hm = 0.
n ->oo
h ( cpn(a))
Taking first 'Y = oo and then 'Y = 0, explain how to use this result to generalize Theorem 3.48 to number fields. Section 3.9. Periodic Points and Galois Groups 3.47. Let P Prove that
E lP' N (K) be a point in projective space and let K(P) be its field of definition. K(P) = fixed field of { (]'
E Gal(K/ K) (J'(P) = P}. :
In mathematical terminology, this says that the field of moduli of P is a field of definition for P. We discuss fields of moduli and fields of definition for (equivalence classes of) rational maps in Chapter 4. Section 3.1 1 . Ramification and Units in Dynatomic Fields 3.48. Let cp (z ) =
z2 + c and let
be the polynomial whose roots are periodic points of period 3. (a) Prove that 3 (z) E Z[c, x] is a polynomial in the variables c and z and that it has integer coefficients. (b) Let a be a root of 3 ( z) and assume that the field Q( c, a) is an extension of Q( c) of degree 6 (i.e., 3 (z) is irreducible in Q( c) [z]). Theorem 2.33 implies that and are units. Compute these units explicitly as elements of Q[c, a]. (c) Prove that there is a field automorphism (]' : Q( c, a) --+ Q(c, a ) characterized by the fact that it fixes Q( c) and satisfies (]'(a) = ¢ ( a) = a 2 + c. (Note that in general, Q( c, a) will not be a Galois extension of Q(c) .) Prove that (]'3 is the identity map. (d) Compute the units (J'(u1 ) and (J'(u2 ). (e) Compute the units (J' 2 (u l ) and (J' 2 (u2 ). (f) Express (]'( u 1 ), (]'(u2 ), (]'2 ( ut), and (]'2 ( u2 ) in the form ±ul u�.
146
Exercises
3.49. Let ¢( z) = z2 + c and let qJ;; ( z) be as in the previous exercise. (a) Prove that qJ3 (z) factors into a product of two cubic polynomials in the ring Q(c) [z] if and only if c has the form c = - (e2 + 7)/4 for some e E Q(c). (b) Suppose that c = - (e2 + 7)/4. Show that there is a polynomial 9e (z) E Q[e, z] such that the factorization qJ3 (z) has the form ge(z)g-e(z). Compute the discriminant of ge (z) and verify that it is a perfect square in Q[e]. Conclude that if ge(z) is irreducible over Q(e), then its roots generate a cyclic Galois extension Ke of Q( e). (c) Let a be a root of 9e (z). Use the results of the previous exercise to construct units in Q( e) (a). Analyze these units for some small values of e, say e = 1, 5, 7, 9. (Note that we investigated the case e = 3 in Example 3.69.) 3.50. ** Let ¢( z) be a generic monic polynomial of degree d � 2, i.e., ¢( z) is a polynomial of the form zd + a1zd -l + · · · + ad , where a1 , . . . , ad are indeterminates. Let p be a prime and let a E Per;• ( ¢) . Theorem 3.66 says that the elements
Ui ,j =
¢i (a) -
� c/> (z), <�> ;, c/> (z)) = 0. They are points (polynomials) at which periodic points of different , , orders merge and break apart. For the quadratic polynomial
n (x, y) . Substituting this into the definition of Qq, gives (5.38). Similarly, if we let iil ( x, y) = ci!l ( x , y) , then homogeneity and an easy induction argument show that
set M .
167
4.2. Quadratic Polynomials and Dynatomic Modular Curves
It is clear that every Misiurewicz point is contained in the Mandelbrot set M, since preperiodic points certainly have bounded orbit. This suffices to prove an arith metic result. Proposition 4.22. The set ofMisiurewicz points is a set ofbounded (absolute) height in Q. More precisely, the height ofa Misiurewicz point 1 satisfies H ( 1) :::; 2. Hence there are onlyfinitely many Misiurewicz points defined over any given numberfield.
Proof A Misiurewicz point c = 1 is the root of a polynomial of the form
�1
� 1.
Z, ¢c,
and n for some m These are monic polynomials with coefficients in so not only is 1 an algebraic number, it is an algebraic integer. Further, the minimal polynomial of1 is a factor of Mm, n (c). On the other hand, every root of Mm, n (c) is in the Mandelbrot set M, since if c is a root of Mm, n (c), then 0 is preperiodic for so it certainly has bounded orbit. It follows from Proposition 4.19 that every Galois conjugate of 1 has absolute value at most 2. Let K = Q( 1) and let 11 , . . . , rr be the full set of Galois conjugates of I · Then r
Taking the
r1h root yields H (1) :::; 2.
D
We now describe an analytic characterization of Misiurewicz points. It depends on the following deep result giving an analytic uniformization of the complement of the Mandelbrot set. Theorem 4.23. (Douady-Hubbard) There is a conformal isomorphism/rom the ex terior of the unit disk to the complement of the Mandelbrot set,
() : {w E C : lwl
>
1 } ------- c " M.
z
Proof It is not hard to show that for all sufficiently large (depending on c) there is
a consistent way to choose square roots so that the limit
lim 2� n ---> oo converges and defines a holomorphic function '1/Jc on some region > Rc. (See Exercise 4.15.) If c � M, then one can prove that has an analytic continuation to C " M. The isomorphism () in the theorem is the inverse of the map
'1/Jc(z)
=
'1/Jc
See [ 142, 143, 141], [43, §9. 10], or [95, VIII §§3,4] for details.
lzl
D
Remark 4.24. An immediate consequence of Theorem 4.23 is the connectivity of the Mandelbrot set, since the theorem implies that IP'1 (C) " M is simply connected.
168
4. Families of Dynamical Systems
The uniformization map () from Theorem 4.23 can be used to give an analytic description of the Misiurewicz points.
x 2x 1}
"M
Theorem 4.25. Let () be the isomorphism {w E C : lwl > ---> C described in Theorem 4.23. Consider the doubling map 1-t on Q/7!... Let t E Q/7!.. be preperiodic, but not periodic, for the doubling map. Let m and n be the smallest positive integers for which we can write t in theform
(The fraction need not be in lowest terms. See Exercise 4.1 6.) Then the limit Ct = lim
r->1 + fJ(re21rit ) ) in M, although distinct values of exists and is a Misiurewicz point of type ( t E Q/7!.. may yield the same Misiurewicz point. m, n
Remark 4.26. For a given t E Q/7!.., the "spider algorithm" [213] can be used to com pute Ct numerically. The spider algorithm is mainly topological and combinatorial in nature, although the limiting process that yields Ct E C is analytic. 4.3
The Space Rat d of Rational Functions
{Az2 Bz
The set of quadratic polynomials + + C} has dimension three, since it may be identified with the set of triples (A, B, C) with A =1- In fancier language, the space of quadratic polynomials is equal to the algebraic variety { (A, B, C) E
0.
A3 : A =I- 0}.
We have seen in Section 4.2. 1 that every quadratic polynomial can be conjugated to a polynomial of the form + c, and that polynomials with different c values are not conjugate to one another. Thus the space of conjugacy classes of quadratic polynomials has dimension 1. It may be identified with the variety In the next few sections we study analogous parameter spaces for more general rational maps and their conjugacy classes. We begin in this section by explaining how the set of rational maps of degree d has a natural structure as an algebraic variety and how the natural action of the algebraic group on is an alge braic action. Then in Section 4.4 we discuss (mostly without proof) how to take the quotient of to construct the moduli space of conjugacy classes of rational maps of degree d. We continue in Section 4.5 by describing a natural collection of
z2
Ratd Ratd
A1 .
Md
PGL2 Ratd
algebraic functions on M d that are created using symmetric functions of multipliers of periodic points. These functions can be used to map Md into affine space. Finally,
A2 .
in Section 4.6 we use these functions to prove that M 2 is isomorphic to A rational map ¢ : of degree d is specified by two homogeneous polynomials
!fD1 ---. !fD1
4.3. The Space
Ratd of Rational Functions
169
G Res(F, G) [uP, uG] = [F, G], a= (X Y) =
such that F and have no common factors, or equivalently from Proposition 2.13, such that the resultant does not vanish. Thus a rational map of degree d is determined by the 2d + 2 parameters a0 , a1 , . . . , ad, bo , b1 , . . . , bd . However, if is any nonzero number, then so the 2d + 2 parameters that deter mine ¢ are really well-defined only up to homogeneity. This allows us to identify in a natural way the set of rational maps of degree d with a subset of projective space. To ease notation, for any (d + I)-tuple (a0 , . . . , ad ), let Fa , aoXd + a 1 Xd -l y + · + adYd
u
··
be the associated homogeneous polynomial. Similarly, if and b are ( d + )-tuples, we write b E lP'2d+ l for the point in projective space whose homogeneous coor dinates are
[a,[ao,] . . . , a , bo, . . . , ]. d bd
Ratd.
1
1 The set of rational functions ¢ lP' 1 lP' of degree d i s denoted 2 It is naturally identified with an open subset ofJID d+ 1 via the map b E JID2 d+ 1 Res( Fa , Fb) -:/:- � b
Definition.
by
a
{[a, ]
:
----+
[a, 0}] [FRat a, Ph]d , ·
:
f--+
Ratd,
which a priori is merely a set, thus has the The collection of rational maps structure of a quasiprojective variety. In fact, Rat d is an affine variety, since it is the complement of the hypersurface Res( Fa , Fb) in the projective space JID2 d+ l .
=0
an affine variety defined over Q. The ring of Q[Ratd] ofRatRatd isd isgiven explicitly by [ Res(�a�F�) d + · · + d + · · · + ]d = 2d . Q[Ratd] = Q l Equivalently, Q[Rat d ] is the ring ofrationalfunctions ofdegree 0 in the localization ofQ[ ao, d ] at the multiplicatively closed set consisting ofthe nonnegative powers ofRes(Fa, Fb). Proof We remind the reader that in general, if F E K[X0 , . . . , Xr ] is a homoge neous polynomial of degree then the complement of the zero set of F, Proposition 4.27. The variety regularfunctions
. . . ai d lJio b)t . . . bjd ai0o ah 1
:
io
·
i + Jo
a1 , . . . , ad, bo , b1 , . . . , b
n,
is an affine variety of dimension r. (See [ 1 98, Exercise 1.3.5].) Explicitly, each ratio nal function with io + i 1 +
· · · + ir =
n
is a regular (i.e., everywhere well-defined) function on V. There are functions, and together they define an embedding
(r�n) such
170
4. Families of Dynamical Systems
V
V
of into affine space. The affine coordinate ring of is the ring of polynomials in these fi ir , "0"1 · ·"· "r ] . + z. 1 + · · ·+z. r =n . o...
K[V] K [f• =
"
to
In the language of commutative algebra,
K[V] K[Xo, . . . , Xn , 1/ F] (o) is the set of rational functions of degree 0 in the localization of K[X0 , . . . , Xn ] at the multiplicatively closed set (Fi k�_ 0 . Applying this general construction to =
D
gives the results stated in the proposition.
(
Remark 4.28. The geometry ofRatd q, especially near its boundary, presents many interesting problems. See [121, 122, 369, 370].
.
Example 4 29 . Let
a0X2 + a1 XY + a2Y2 and b0X2 + b1XY + b2Y2 . Then the
be the resultant of collection of 84 functions
84 Rat2 Rat2 A .
gives an embedding of into Of course, this is not the smallest affine space into which can be embedded. Projecting onto appropriately chosen hy perplanes, there is certainly an affine embedding of the 5-dimensional space into see [ 198, Exercise IV.3 . 1 1]. Example 4.30. The set of rational functions of degree 1 is exactly the set of linear fractional transformations,
A11 ;
Rat2
Rah = PGL2 = {[o:X + (3Y, "(X + 8Y] : o:8 - f3"! f. 0} !P'3 . We note that PGL2 is not merely a variety, it is a group variety, which means that the maps PGL2 PGL2 PGL2 , and PGL2 PGL2 , ( h , h) !----+ h h f f- 1 ' c
x
______,
______,
!----+
defining the group structure are morphisms.
Ratd of Rational Functions 171 1 . Further, as we vary Each point of Ratd determines a rational map lP' 1 lP' the chosen point in Rat d , the rational maps "vary algebraically." We can make this vague statement precise by saying that the natural map ¢ lP'1 Rat d 1P'1 x Ratd , ([X,Y], [a,bl) ([Fa(X,Y),Fb(X,Y)j, [a,bl) , (4.20) 4.3. The Space
--->
x
:
______,
f-----*
is a morphism of varieties. The following definition is useful for describing families that vary algebraically.
Definition.
Let V be an algebraic variety. The projective line over V is the product
lP'� lP'�
A morphism 'ljJ over V is a morphism that respects the projection to V, i.e., the following diagram commutes, where the diagonal arrows are projection onto the second factor: :
--->
'\.
v
/
Then 'ljJ can be written in the form 'ljJ
=
[F(X, Y), G(X, Y)],
where K ( V , are homogeneous polynomials with coefficients that are rational functions on V. The degree of'l/J is the degree of the homogeneous polyno mials and Any morphism V W of varieties induces a natural morphism lP'� lP'�, which, by abuse of notation, we also denote by A. Thus
F, G E )[X Y] F G.4 A A(P, t) (P, A(t)) :
--->
=
--->
for
(P, t) E lP'� .
With this notation, the map (4.20) says that there is a natural morphism
Ratd.
over The next proposition says that this ¢ is a universal family of rational maps of degree d. Proposition 4.31. Let V be an algebraic variety and let
'1/J
:
lP'� ------t lP'�
be a morphism over V ofdegree d. Then there is a unique morphism
4More generally. if 'ljJ : II'� -+ II'� is a morphism of S-schemes. then 'lj!*OpN (1) 5 is the degree of 'ljJ.
�
OpN (d) and d s
172
4. Families of Dynamical Systems
..\ V :
such that the induced map ..\
:
JID�
--+
J!D l
--+
Ratd
J!Dkatd fits into the commutative diagram
v
-------+
lA
J!Dkatd
-------+
J!Dvl
lA
(4.21)
l d J!DRat
Remark 4.32. The commutative diagram (4.21) in Proposition 4.3 1 says that any algebraic family of degree d rational maps I}! : --+ factors through the family. Thus ¢ : J!Dkatd --+ J!Dkat d is a universal family of rational maps of degree d. It is an example of a.fine moduli space. See [322, 323, 335] for further information
JID� JID�
Ratd
about moduli spaces.
K[U]
Proof Let U
c V be an affine open subset and write for its affine coordi nate ring. The fact that 'ljJ is a morphism over V implies that it restricts to give a morphism 'ljJ J!Dh J!Dh over U. This restriction of 'ljJ to J!Dh has the form :
--+
adYd ,/3oXd f31 Xd� l y f3dYd], with ao, . . . , ad , f3o, . . . , !3d E K[U]. In other words, the coefficients of the polynomials defining 'ljJ are regular functions on the affine open set U. In particular, if there is a map ..\ V Ratd making (4.21) commute, then it must be given by ,\ = [a0, . . . , ad, /30 , . . . , !3d] at any points of V at which the ai and /3i are defined and the homogeneous polynomials in (4.21) have no common roots. given any point in V, we can find a neighborhood U of that point and ai, f3Now i E K[U] as above such that 'ljJ J!Dh J!Dh is given by '1/J([x, y], t) [ao(t)xd . . . + ad (t)yd , j30 (t)xd f3d (t)yd] for ([x , y], t) E J!Dh . (4.22) 'ljJ =
[aoXd + al xd� l y
+...+
+...+
+
:
:
=
--+
--+
+...+
+
Using a natural notation, we abbreviate this by writing
w (P, t) [Fa(t) (P), F,a(t) (P)] . In order to make the diagram (4.21) commute on J!Dh JID � , we are forced to define ..\ on U by =
c
Ratd.
Further, this ..\ will have the desired properties provided that its image lies in However, the fact that 'ljJ is given by (4.22) for every point in J!Dh implies that for every E U, the homogeneous polynomials have no common root and in It follows from Proposition 2.13(a) that their resultant
1JID . t
Fa(t) F,a(t)
Ratd of Rational Functions 173 Res(Fa, Ff3) E K[U] is nonzero at every point in U. Hence the image of the map A [a, b] defined by (4.23) is in Rat d . We have now proven that every point in V has a neighborhood U for which there is a map A U Rat d making (4.21) commute. Further, the maps on dif ferent U must agree on the intersection by the uniqueness discussed earlier. Fitting them together, we find that there is a unique map A V Ratd making (4.21) commute. 4.3. The Space
=
:
---+
---+
0
An ongoing theme of this text has been the observation that a pair of rational maps ¢ 1 , ¢2 E determine (arithmetically) equivalent dynamical systems ifthey are PGL2 -conjugate, i.e., if there is a linear fractional transformation f E PGL2 such that ¢2 ¢{ = f- 1 ¢f. The conjugation action ofPGL2 on is algebraic in the following sense.
Ratd
Ratd
=
Proposition 4.33. The map
(4.24)
Ratd
is an algebraic group action ofPGL2 on and is defined over Q. This means that the map (4.24) is both a morphism defined over Q and a group action. Proof The proof is mostly a matter of unsorting the definitions. Let
f= Then
+
+
[o:X {3Y, 1X oY] E PGL2
¢!
c
1P'3 and ¢ =
[Fa, Fb] E Ratd
c
lP'2 d+ l .
+
([X, Yl) [oFa(o:X + {3Y, 1X + oY) - {3Fb(o:X {3Y, 1x + oY), - 'YFa(o:X {3Y, 1X oY) o:Fb(o:X + {3Y, 1X + oY)].(4.25) The homogeneity of Fa and Fb shows that (4.24) at least gives a well-defined rational map =
+
+
+
Further, an elementary resultant calculation (Exercise 2.7) shows that the resultant of the two polynomials appearing in the righthand side of(4.25) is equal to
It follows that if f E PGL2 and ¢ E then ¢! is a well-defined point in This proves that the map (4.24) is a morphism. The fact that it is a group action is 0 then a straightforward, albeit tedious, calculation.
Ratd,
Ratd .
4. Families of Dynamical Systems
174
Example 4.34. In principle it is possible to explicitly write down the action ofPGL2
Ratd,
on but in practice the expressions become hopelessly unwieldy for even mod erate values of d. As illustration, we describe the action for d Let and set
f = [aX + {3Y, "(X + JY]
and
= 2.
Multiplying out both sides of (4.25) and equating the coefficients of yields the following formulas for the coefficients of Fa' and Fb' :
X2 , XY, and Y2
a� = a20ao + a"(Oa1 + loa2 - a2 f3bo2 - a{3"(b21 - {3"(2 b2 , a� = 2a{3oa0 + (ao2 + f3"f)Oai + 2"(0 a2 2- 2a{3 b02 - (ao + f3"t)f3b1 - 2{3"(0b2 , a� = {32 0ao + {3o a2l + 03a2 - {33bo - {3 ob2l - {30 b2 2, b� = -a2"(ao - a"( al -"(3a2 + a3bo2 + a "(bl +2 a"( b2 , - (f3"t + aJha2 1 - 2"(2 Ja2 + 2a f3bo + 2(ao + f3"t)ab1 + 2a'YJb2 , b� = -2af3"tao 2 b; = -{3 "(ao -{3"(0al - "(0 a2 + a{3 bo + a{3obl + ac5 b2 . 4.4
The Moduli Space M d of Dynamical Systems
The intrinsic properties of the dynamical system associated to a rational map ¢ de pend only on the PGL2 -conjugacy class of ¢, so it is natural to take the quotient of the space by the conjugation action of PGL 2 .
Ratd
Definition.
space
The moduli space of rational maps of degree d on
IP'1 is the quotient
M d = Ratd / PGL2 , where PGL2 acts on Rat d via conjugation ¢1 = f - 1 ¢! as described in Proposi tion 4.33. For the moment, the quotient space M d is merely a set, in the sense that for any algebraically closed field R, We denote the natural map from
Md Rat d Q[Ratd{] IP'Rat1 d .IP'1 } Q[Ratd]
Ratd to Md by
In order to endow with the structure of a variety, we observe that the action ofPGL2 on induces in the usual way an action ofPGL2 on the ring of regular is a function whose functions on Thus an element R E domain is the set ¢ --+ of rational functions on IP'1, and for E PG12 we define Rl E by the formula
Q[Ratd]
:
Rl (¢)
= R(¢1).
f
4.4. The Moduli Space
Md of Dynamical Systems
175
In this way we obtain a map
so it makes sense to talk about functions in action ofPGL2 . We write
Q[Ratd] that are invariant under the
Ratd .
for the ring ofPGL2 -invariant functions on Remark 4.35. For technical reasons, it is sometimes advantageous to replace PGL2 by a slightly different group. The special linear group is the subgroup of G12 defined by SL 2 =
{ (� �) aJ :
- �1
=
1} ,
and the projective special linear group PSL 2 is the image of SL2 in PGL2 • The via conjugation in the usual way. Geometrically, groups SL2 and PSL2 act on the actions of PSL2 and PGL2 on are identical, since it is easy to see that the natural inclusion SL 2 c GL2 induces an isomorphism
Ratd Rat
However, if
d
K is not an algebraically closed field, then the map
need not be an isomorphism; see Exercise 4.22. The following theorem provides the abstract structure of an algebraic variety.
quotient Ratd / PSL2 Theorem 4.36. There is an algebraic variety M d defined over Q and a morphism
with the
(4.26)
Q
defined over with thefollowing properties: (a) The map (4.26) is PSL2 -invariant, i.e., thefollowing diagram is commutative:
PSL2 x
(f, c/>)->q,J ---+ Ratd ----
f't Ratd
(-)
In terms ofelements, (¢/) = (¢) for all ¢ E
Ratd and all f E PGL2.
4. Families of Dynamical Systems
176
(b) The map on complex points is surjective and eachfiber is the full PSL2 (C)-orbit of a single rational map. Thus there is a bijection ofsets
(c) The variety M d is a connected, integral (i.e., reduced and irreducible), affine variety ofdimension 2d - 2 whose ring ofregularfunctions is the ring ofPSL2 invariantfunctions on Ratd, (d) Let VjC be a variety and let T : Rat d V be a morphism with the property that T(tjyf) T(tjy) for all tjJ E Rat d (C) and all E PGL2 (C). Then there is a unique morphism T : M d V satisfying t( (¢)) T( tjJ ) . ____,
=
____,
f
=
ProofSketch. A full proof of Theorem 4.36 (see [416]) uses the machinery of geo
metric invariant theory [322] and is thus unfortunately beyond the scope of this book. Geometric invariant theory tells us that there is a certain subset of!fD2 d+ l , called the stable locus, on which the conjugation action of PSL2 is well behaved. The main part of the proof is to use the PSL2 -invariance of the resultant to verify that
is a PSL2 -invariant subset of the stable locus of !fD2d+ l . Then the existence of the quotient variety M d with affine coordinate ring equal to the ring of invariant func tions in Ql[Ratd ] follows from general theorems of geometric invariant theory [322, Chapter 1]. Further, the fact that M d is connected, integral, and affine follows im mediately from the corresponding property ofRat d [322, Section 2, Remark 2]. The dimension of M d is computed as dim M d
=
dim Rat d - dim PSL2
=
(2d + 1) -
3
=
2d - 2.
This proves (a), (b), and (c). Finally, (d) follows directly from the description of M d and Ql[M d ] in (c), since the morphism T induces a map T* : C[Ratd ], and the assumption that T satisfies T( ¢f) = T( tjJ) implies that the image of T* lies in D the ring C[Rat d]PGL2 (1C) C[M d] .
Ov
____,
=
Remark 4.37. Theorem 4.36 says that the quotient Rat d / PSL2 is an algebraic va
riety. Milnor [301, 302) shows that Ratd (C)/ PGL2 (C) has a natural structure as a complex orbifold, which roughly means that locally it looks like the quotient of a complex manifold by the action of a finite group. Thus its singularities are of a fairly moderate type, although they can still be quite complicated. However, for rational maps of degree 2, we will see in Section 4.6 that not only is M 2 nonsingular, it has a particularly simple structure.
4.4. The Moduli Space
M d of Dynamical Systems
177
Md Q[RatdjP8L2• ofQ[Ratd] · Ratd Md k Ratd (k)
Remark 4.38. According to Theorem 4.36(c), the affine coordinate ring of
is the ring ofPSL2 -invariant functions However, this is the same as the ring of PGL 2 -invariant functions, since it suffices to check for invariance by the action of PGL2 (C) PSL2 (C) on an element by the conjugation ac is the quotient of Remark 4.39. The moduli space tion of the group PSL2 . In particular, if is any algebraically closed field, then the is exactly the collection of cosets of set of points by the conju gation action of PSL2 ( K ) PGL 2 (K). However, if K is not algebraically closed, then the natural map =
M d (K)
=
Md (K), at
is generally neither injective nor surjective. The correct description of least if K is a perfect field, is for every T E I K) there is an (K) (¢) E ( K ) ·. E PGL2 (K) such that T(¢) = ¢fT
} Md - { Md fr _Gal(k The field of moduli of a rational map ¢ E Rat d (k) is the smallest field L such that ( ¢) E M d ( L). Fields of moduli and related questions are studied in detail in Section 4. 1 0. In particular, see Example 4.85 for a map ¢ whose field of moduli is Q, yet ¢ is not PGL 2 (C)-conjugate to any map in Ratd (Q). The moduli space M d classifies rational maps up to conjugation equivalence, just as we want, but it has the defect that it is an affine variety. It is well known that if possible, it is generally preferable to work with projective varieties. How might we naturally complete M d by filling in extra points "at infinity"? Note that we should not do this in an arbitrary fashion. Instead, we would like these extra points to corre _
·
spond naturally to degenerate maps of degree d. One possibility is simply to start with all of IP'2 d+ l and take the quotient by the conjugation action of PSL2. Unfortunately, there is no natural way to give the quo tient JP>2 d+ 1 I PSL2 any kind of reasonable structure. For example, it is not a variety. So IP'2d+ l is too large. Ideally, we would like to find a subset S c JP>2 + l with the following properties: 1 . S contains 2. PSL2 acts on S via conjugation. 3. There is a variety T and a morphism S ----+ T that induces a bijection
d
Ratd .
4. The quotient variety T is projective. Geometric invariant theory gives us two candidates for S. The smaller candidate is the largest variety satisfying (3), but its quotient T may fail to be projective. The larger candidate has a projective quotient, but the map S(C)I PSL2 (C) ----+ T(C) in (4) may fail to be injective. The following somewhat lengthy theorem describes the application of geometric invariant theory to our situation, that is, to the conjugation action of PSL2 on IP'2 + 1 .
d
4. Families of Dynamical Systems
178 Theorem 4.40. There are algebraic sets
Ratd
C
Ratd
C
Ratds
C
lP'2d+ 1
with thefollowing properties: (a) The conjugation action ojPSL2 on Ratd extends to an action ojPSL2 on Ratd and Ratd8• (b) There are varieties M8 and M88 and morphisms ( · ) : Ratd -+ Md
(4.27)
and
that are invariantfor the action ojPSL2 on Ratd and Ratd8• The varieties M8 and M88 and morphisms (4.27) are defined over IQl. (c) Two points and in Ratd(C) have the same image in Md (C) ifand only if there is an f E PSL2 (C) satisfying
[a, b] [a', b']
[a', b'] = [a, b]f.
Thus as a set, Md(C) is equal to the quotient ofRatd (C) by PSL 2 (C).
[a, b] [a', b'] {[a, b] f : f E PSL2 (C) } {[a' , b'] f : f E PSL2 (C) } # 0. Equivalently, they have the same image if and only if there is a holomorphic map f : {t E C : 0 < l t l < 1} ---> SL2 (C) such that [a, b]f' = [a' , b'] . tlim --->0 (e) Md is a quasiprojective variety and Mds is a projective variety. (f) (Numerical Criterion) A point [a, b] E lP'2d+ l (C) is not in Ratds if and only if there is an f E PSL (C) such that [a', b'] = [a, b]f satisfies
(d) Two points and in Ratd8 (C) have the same image in Md8 (C) if and only ifthe Zariski closures oftheir PSL2 (C)-orbits have a point in common, n
2
d-1 d+1 and bi 0 for all (4.28) a� = 0 for all Similarly, [a, b] is not in Ratd ifand only if there is an f E PSL2 (C) such that [a', b'] = [a, b]f satisfies d+1 d-1 (4.29) and bi 0 for all i < ai = 0 for all < i ::;
2-
i
2-
I
=
=
i :S
2-.
2- .
(g) Md is isomorphic to Mds if and only if d is even.
Proof. The proof of this theorem is beyond the scope of this book. However, we note that half of (g), namely Md 2::: Mds for even d, follows directly from the numerical
criterion in (f), since for even d the criteria (4.28) and (4.29) are the same. See [416] for a proof of a general version of Theorem 4.40 over Z. See also [301, 302] for a similar construction over C. D
179
4.5. Periodic Points, Multipliers, and Multiplier Spectra
Remark 4.41. In Theorem 4.40, the set denoted by Rat;i is called the set of stable rational maps, and the set denoted by Rat;18 is called the set of semistable rational maps. Note that points in these sets need not represent actual rational maps of de
IP'1 .
Ratd
gree d on The intuition is that points in Rat;i and Rat;J8 that are not in correspond to maps that want to be of degree d but have degenerated in some rea sonably nice way into maps of lower degree. Remark 4.42. Theorem 4.40(c) says that the stable quotient M;l(C) has the nat ural quotient property, since its points correspond exactly to the PSL2 (C)-orbits of points in Rat;i(C). Quotient varieties with this agreeable property are called geomet ric quotients. The semistable quotient M;18 (C) has a much subtler quotient property. According to Theorem 4.40(d), points in Rat;18 (C) with distinct PSL2 (C)-orbits give the same point in M;is (q if their orbits approach one another in the limit. Quotients of this kind are called categorical quotients. As compensation for the less intuitive notion of categorical quotient, Theorem 4.40(e) tells us that M;18 is projec tive, so M;18 ( q is compact. Finally, Theorem 4.40(g) says that if d is even, then M;i and M;is coincide, so in this case, the moduli space M d has a projective closure with a natural (geometric) quotient structure. Remark 4.43. Applying the full machinery ofgeometric invariant theory to the action of PSL2 /Z on it is possible to prove versions of Theorems 4.36 and 4.40 over Z. In other words, there is a filtration of schemes over Z,
JP>id+ l ,
such that the group scheme PSL2 /Z acts on each of these schemes and such that the quotient schemes M C M;i C M88 exist in a suitable sense. In particular, Theorems 4.36 and 4.40 are true with Q replaced by the finite field lF The proof is similar to the proof over Q, but requires Sheshadri's theorem that reductive group schemes are geometricallly reductive. See [416] for details.
d
4.5
p·
Periodic Points, Multipliers, and Multiplier Spectra
The moduli space M d of rational functions modulo PSL2 -equivalence is an affine variety whose ring of regular functions
Ratd
consists of all regular functions on that are invariant under the action of PSL2 , or equivalently under the action of PGL2 (C); see Remark 4.38. Abstr�ct invariant theory, as described in the proof sketch of Theorem 4.36, says that there are many such functions. In this section we use periodic points to explicitly construct a large class of regular functions on Md .
4. Families of Dynamical Systems
180
be a rational function of degree d defined over the complex numbers. Associated to each fixed point P E Fix(¢) is its mul tiplier Ap ( ¢) E
Example 4.44. Let ¢ E
Ratd (q
>.1 - l (P) ( ¢1 ) =
and
Ap (¢) for all P E Fix(¢).
Md,
Thus in some sense the function ¢ >.q,(P) is a function on where P is a fixed point of¢. Unfortunately, this is not quite correct, since there is no way to pick out a particular fixed point P for a given map ¢. Recall that a rational map ¢ of degree d has d + 1 fixed points counted with appropriate multiplicities, say f-+
The points in Fix(¢) do not come in any particular order, so the set of multipliers for the fixed points, is an unordered set of numbers, but as a set, it depends only on {¢), the PGL2 equivalence class of ¢. Hence if we take any symmetric function of the elements in this set of multipliers, we get a number that depends only on { ¢). The elementary symmetric polynomials generate the ring of all symmetric func tions, so we define numbers
by the formula
d+ l T + Ap(¢) ) I�>·i (¢)Td+l - i . ( IT PEFix(c/>) i=O In other words, the quantity O"i ( ¢) is the elementary symmetric polynomial of the multipliers Ap1 ( ¢) , . . . , Apd+l ( ¢). From this construction, it is clear that O"i ( ¢! ) = O"i (¢) for all f E PG12 • Further, if we treat the coefficients of¢ [Fa, Fb] as indeterminates, then the fixed points Pi and the multipliers >.pi ( ¢) of ¢ are algebraic over Q( . . . , bd ) and form a com plete set of Galois conjugates, from which it follows that symmetric expressions in the A pi ( ¢ ), for example the functions O"i ( ¢ ), are in the field Q( . . . , bd ). With a bit more work, which we describe in greater generality later in this section, one can =
i1h
=
ao ,
ao ,
show that the multipliers Api ( ¢) are integral over the ring
Q[Md] ·
and hence that O"i(¢) E Thus symmetric polynomials in the multipliers of the fixed points of¢ are regular functions on the moduli space
Md.
181
4.5. Periodic Points, Multipliers, and Multiplier Spectra
Example 4.45. We illustrate the construction of Example 4.44 for rational maps of
degree 2. As usual, we write
The map rjJ has three fixed points P1 , P2 , P2 , and after much algebraic manipulation one finds that the first elementary symmetric function of the multipliers,
is given explicitly by the horrendous-looking formula
Notice that the denominator of a1 rjJ) is so a1 rjJ) is in It is far less obvious that this expression for a1 rjJ) is PGL2 -invariant. One can verify directly that a1 rjJ) = a1 rjJf ) by checking that a1 rjJ) does not change when a0, . . . , b2 are replaced by the expressions a� , . . . described in Example 4.34. We leave this task to the interested reader who has, we hope, access to a suitably robust computer algebra system. We have used the set of multipliers of the fixed points of a rational map rjJ to create PGL 2 -invariant functions on More generally, we can use the multipliers associated to periodic points of any order to create such functions. Recall from Section 4. 1 (page 149) that for any rjJ E we write
(
( ( Res(Fa, Fb), ( , b; (
(
IQ[Ratd].
Ratd.
Ratd r/Jn = [F¢,n (X, Y), G¢,n (X, Y)] with homogeneous polynomials F¢ , n , G¢ , n E K[X, Y] of degree dn . (See also Exer cise 4.9.) Then the set Pern ( rjJ) of n-periodic points of rjJ are the roots of the n-period polynomial ¢,n (X, Y) YF¢,n (X, Y) - XG¢,n (X, Y), and the set Per� ( rjJ) of formal n-periodic points of rjJ are the roots of the n1hdynatomic polynomial of rjJ, *,n (X' Y) = ll(YF,k (X' Y)-XG,k (X ' Y))11(n/k) = II ,k (X' Y)ll(n/k) kin kin where we proved in Theorem 4.5 that ¢ , n is a polynomial. The polynomial ¢ , n is homogeneous of degree dn + 1. For the purposes of this section, it is convenient to let Pern ( rjJ) be a "set with multiplicity" in the sense that a point appears in Pern ( rjJ) according to its multiplicity as a root of , n . Similarly, we denote the degree of ¢ , n by vd ( n) (see Remark 4.3) and we assume that points appear in Per� ( rjJ) according to their multiplicity as roots of ¢ , n . =
'
182
4. Families of Dynamical Systems
Definition.
Let ¢ E
Ratd. The n-multiplier spectrum of¢ is the collection ofvalues An (¢) = { Ap(¢) : P E Pern (¢) } .
The formal n-multiplier spectrum of¢ is the analogous set of values A�(¢) = { Ap(¢) : P E Per�(¢) } . In both sets, the multipliers are taken with the appropriate multiplicity. Example 4.46. Let ¢(z) = zd with d 2: 2. Then Per (¢) = U consists of the points and the ( dn - 1 ) 1h roots ofnunity. It is easy to check we have that >..o (¢) = Aoo (¢) = and for ( E
0, oo, 0,
{O,oo} J.t. dn _ 1
J..tdn _ 1
dn - 1
copies _____..._.
Hence
An (¢) = { O, O, dn, dn, . . . , dn . And if n 2: 2, then A� ( ¢) consists of rp( dn - 1) copies of dn, where rp (m) is the Euler totient function (not to be confused with the rational function ¢( z) ). Example 4.47. Let ¢(z) = z2 + bz. Then Per1 (¢) = Next we compute
{0, 1 - b,oo}
and
}
AI (¢) = {b, 2 - b,
0}.
cp2 (z) - z 2 -�."'' 2 = = z + (b + 1)z + b + l. ¢(z) - z *
The two points of formal period 2 are the roots of ¢ , 2 , *
(
Per2 ¢) -
{
}
- (b + 1) ± J(b + 1) 2 - 4 (b + 1) . 2
(4.30)
Letting o: and (3 denote these two values, we substitute them into to compute their multipliers, which tum out to be identical, Thus
A2 ( ¢) = { 4 + 2b - b2 , 4 + 2b - b2 } . The multiplier spectra An (¢) and A�(¢) depend only on the PGL2 -equivalence class of ¢, so we can use them to define functions on
M d.
4.5. Periodic Points, Multipliers, and Multiplier Spectra
183
Ratd and n 2:: 1. Define quantities O"in) (¢) for 0 :S i :S dn + 1 dLn+ l in) rjJ dn+ l -i .\) + (T = II i=O O" ( ) T . .A E An(
4>)
4>)
which gives
(jil l
0"�1 ) 2b 0"�1 ) 0. computed in (4.30), we find that
= b2 ' = Similarly, using the set A2 ( ¢) II (T + .X) (T + 4 + 2b - b2 ) 2 ' E ¢ =
2,
=
.A A; ( )
so
and
Theorem 4.50. For E n and i in the appropriate range, let and (;.� be the symmetric polynomials ofthe n-multiplier spectra of¢. (a) The/unctions
¢ Ratd, 2:: 1,
n) (¢)
and
Q[Ratd],
O"in) (¢) (4.31)
R ( ,. . )
are in i.e., they are rational functions in the coefficients ao , . , bd of the map ¢ = [Fa, Fb] with denominators that are a power of es Fa Fb . (b) Thefunctions (4.3 1) are PGL2 -invariant, and hence are in the ring of regular functions of the dynamical moduli space
Q[Md]
M d. �\ Proof We sketch the proof for O"in ) (¢) and leave ; n ¢) as an exercise for the reader (Exercise 4.26). (a) We write
4. Families of Dynamical Systems
184
using homogeneous polynomials of degree dn . Formal properties of the resultant and an easy induction imply that
(See Exercise 4.9.) Thus the rings are equal. This allows us to replace ¢ by
(
(
Q(ao,( [)F. a,. . Fb], bd), A1 ( L. ( [Fca, Feb] a0, , bd .
•
ao, . . . , bd Gal(LGal( L
( (
.
£ (0 ) ,
Fa, Fb)
a0, . . . , bd, and to simplify notation, r = Res(Fa, Fb) E A denote the resultant of Fa and Fb. With this notation, Theorem 4.27 says that the coordinate ring ofRatd is Q[Rat d], which equals A[r - 1 ] (0 ) , where again the 0 indi cates that we take rational functions of degree 0. We also note that the polynomial is irreducible in A. 5 We are first going to prove that O"i (¢) is in the ring A[b;t\ r- 1 ]. We dehomoge nize by setting [X, Y] [ z , 1] and write ¢(z) = Fa(z)l Fb(z). The fixed points of¢ are the roots of the polynomial ¢( z) - z, or equivalently, the roots of the polynomial zFb(z)-Fa(z) bdzd+ 1 +(bd- 1 -ad )zd+(bd- 2-ad_ I )zd- 1 + · + (bo-ai)z+ao. (4.32)
be the ring of polynomials in the indeterminates let
r
=
=
·
5Indeed, the resultant polynomial is geometrically irreducible, which means that it is irreducible in K[ao, . . . , bd] for any field K, see [436, §5.9].
185
4.5. Periodic Points, Multipliers, and Multiplier Spectra
A b;J 1 ] . For
The roots a 1 , . . . , ad+l ofthis polynomial in are integral over the ring [ any such root a, the corresponding multiplier is
L
F (a)Fa(a) - F� (a)Fb(a) ¢' (a) = £ . Fb(a) 2
A[b;J 1 ] B[r - 1 ]
Let be the integral closure of in the field L(a). It is clear that the numerator of¢' (a) is in We claim that its denominator is a unit in [ 1 ] . Suppose not. Then we can find a maximal ideal lfJ c with Fb(a) E lfJ. But a is a root of(4.32), so
B
B.
B r-
Thus z = a is a simultaneous root of Fa(z) = Fb (z) = 0 (mod lfJ),
so a standard property of resultants (Proposition 2. 13(a)) implies that
1 ], B[ r 1 B[r - ],
r
But this is a contradiction, since is a unit in so it cannot be an element of a maximal ideal. Hence Fb(a) is a unit in and therefore the multiplier ¢' (a) is in We have now shown that each of the multipliers ¢' ( a 1 ) , . . , ¢' (ad + 1 ) E L is integral over the ring Hence the symmetric polynomials in these quan tities, i.e., ( ¢) , d+ 1 ( ¢) , are in and are integral over [ 1 , ] . It fol lows that they are in since the ring is integrally closed in its fraction field L. A similar argument dehomogenizing [X, Y] = [1, w] shows that the are in the ring Therefore
B[r- 1 ]. 1 CJ1 . . . ,A[A[CJ bb;J;J\1 r-r-1 ]].(0) , £(0) , A[ad 1 , r- 1 ]C0l.
.
1 A[b;J 1 , r- 1 ] A b;J rCJ;
(b) Our earlier calculation (Proposition 1 .9) showed that for every complex point [a, b] E JP'2d+ 1 (C) with Res( Fa, Fb) -1- 0, the set of multipliers of ¢ = [Fa, Fb] is PGL2 (C)-invariant. Hence the same is true for the quantities ¢) for all i. This + J) be a linear fractional invariance says the following: Let z) = ( az + {, J and consider the difference transformation with indeterminate coefficients a,
f(
(3)(3, j ('-y
CJ;
z
(
(4.33)
Clearly (4.33) is a rational function in Q(a0 , . . . , bd, a, (3, { , J), and the PGL 2 (C) invariance of si(¢) says that (4.33) vanishes for all choices of � E GL2 (C) . It follows that (4.33) is identically zero. Finally, Theorem 4.36(c) says that Q[Md] is the subring ofPGL2 -invariant func tions in Q[Ra d] (see also Remark 4.38), so in particular the functions (4.31) are D in Q[Md] ·
( n
t
4. Families of Dynamical Systems
186
As noted earlier in Remark 4.43, the moduli space M d exists as a scheme over Z. Theorem 4.50 is also valid over Z, so in particular the affine co ordinate ring Z[M d] is the ring of PSL2 -invariant functions in Z[Rat d] . See [4 16, Theorems 4.2 and 4.5]. The functions u;n) and �� n) constructed in Theorem 4.50 are regular functions on M d , so they can be used to map M d to affine space. For example, in the next section we show that (0"* 1(l ) , 0"* 2(l) ) · M 2 --+ &A 2 is an isomorphism. In general, M d is not isomorphic to A.2d- 2 , but we might ask whether using a sufficient number of the u; n) or ��n) gives an embedding of M d into affine space. In particular, do the values of all u�n ) or ��n) (¢) determine the PGL2 (C) conjugacy class of ¢? The answer is no. The Latt(�s maps that we studied in Sec tion 1 .6.3 provide nontrivial families of rational maps whose multiplier spectra, and hence whose u� n) and ��n ) values, are all the same. Example 4.52. For each t E C * with t -1 - 2J, consider the rational map x4 - 2tx 2 - 8tx + t2
°
=
=
see Section 1 .6.3. Following standard notation, we write Et [m] for the points of Et of order m. Then the n-periodic points of ¢ are given by and it is not hard to compute the multipliers at these points,
a a a=
ll) and a -# oo, + ll) and oo,
if E x(Et [2n if E x (Et [2n if oo.
a -#
(For proofs of these statements, see Proposition 6.52 in Section 6.5.) For any m, the set E[m] of m-torsion points has order m2 , and for odd m the map x : E[m] --+ IP'1 is exactly 2-to-1 except at x- 1 ( oo ) , so we can use the listed values of Ao:(
-
>. E An ( t )
In particular, we see that every map
187
4.5. Periodic Points, Multipliers, and Multiplier Spectra
ain)
Ak
we use, the resulting map M 4 --+ always compresses all of many functions the maps
Ak .
is a curve in M4 that is compressed to a single point in An important theorem says that aside from Lath��s examples of this kind, the map on Md defined by the is finite-to-one. Theorem 4.53.
ain)
(McMullen [294, §2]) Fix d 2: 2, andfor each N 2: 1 let (4.34)
ain)
be the map defined using all ofthefunctions with 1 � n � N. IfN is sufficiently large, then the map ud, N is finite-to-one on Md(C) except for certain families of Lattes maps that it compresses down to a single point. In particular, it isfinite-to-one ifd is not a perfect square. Further, the same statement is true for the map .. e
*d,N . M d � defined using all ofthefunctions ; n ) with 1 � � N. (The flexible Lattes maps for which ud, and ud, • are notfinite-to-one are discussed in detail in Section 6.5.) (f'
---+ fl
n
N
N
·
McMullen's theorem says that aside from the flexible Lattes maps, the maps ud, N and ud, N * are finite-to-one onto their image. One might hope that they are actually injective if we avoid the flexible Lattes maps, but it turns out that this is far from true. Theorem 4.54. Define the degree ofu d, N to be the number ofpoints in d,� ( P) for a generic point P in the image u d, N (Md). One can show that the degree of ud, N stabilizes as N --+ oo. We write deg(u d)for this value. Then for every E > 0 there is a constant Cc such that u
for all d. In particular, the multipliers of a rationalfunction ¢ E Ratd determine the conju gacy class of¢ only up to De ( d� ) possibilities. -c
Proof We will prove this in Chapter 6 using Lattes maps associated to elliptic curves
with complex multiplication; see Theorem 6.62.
0
Following Milnor [300], we define the multiplier spectrum of a rational map ¢ to be the function A ( ¢) that to each positive integer n assigns the set An ( ¢). In other words, A(¢) is the set-valued function
Definition.
A(¢)
:
N
---+
Sets with Multiplicities,
4. Families of Dynamical Systems
188
Two rational maps are said to be isospectral if they have the same multiplier spec trum. Then another way to state McMullen's Theorem 4.53 is to say that aside from the flexible Lattes maps, the multiplier spectrum determines the rational map up to finitely many possibilities. The fact that the flexible Lattes maps form nontrivial isospectral families is proven in Section 6.5.
Remark 4.55. We show in the next section that the map
M2
z=x
is an injection, and in fact it maps isomorphically to the plane 2, so S:' On the other hand, the existence of the Lattes maps described in Example 4.52 shows that 4 , N cannot be finite-to-one on The degree of the map
M 2 A2 .
t7
M4 .
-
does not appear to be known. The Moduli Space M 2 of Dynamical Systems of
4.6
Degree 2
Theorem 4.50 tells us that symmetric combinations of the numbers in the multiplier spectra give well-defined functions on the moduli space of degree d dynamical systems on In this section we describe Milnor's explicit identification of with using two of these functions.
A2
Md
lP'1 .
M2
<J1 , <J2 , <J3 Q[Rat2]
be the three Theorem 4.56. (Milnor [30 1 ], see also [416]) Let E functions constructedfrom thefixedpoints ofa rational map ofdegree 2, i.e., 3
II (T + Ap(
----->
__::_____.,
189 M2 of Dynamical Systems of Degree 2 (c) Thefunctions IT 1 and IT2 are in Ql[M 2 ] and the map A2 (T = (0"1 , 0"2 ) : M 2 is an isomorphism of algebraic varieties defined over Ql. Equivalently, the in duced map
4.6. The Moduli Space
---7
is an isomorphism ofrings. and (d) All ofthefunctions can be expressed as polynomials in and with rational coefficients. (e) For anyfield extension the map : __, in (c) induces a bijection (Note that is not the same as see Remark 4.39 and Section 4. 1 0.) Remark 4.57. For some applications it is useful to have an explicit description of the
iTin) ;in) 1T1 1T2 2 K/Ql, M A 2 2 M2 (K) A (K). Md (K) Ratd (K)/ PSL2 (K); u
+-+
map
= (1T1 , 1T2 ) : Rat2 ---+ A2 described in Theorem 4.56(b). Let p(a, b) = a§b6 - a1 a2 b0b1 + a0a2 bi + aib0b2 - 2a0a2 b0b2 - a0a1 b1 b2 + a6b§ denote the resultant of Fa and Fb. Then, after some algebraic manipulation, one finds that IT 1 and 0"2 are given by the expressions p(a, b)1T1 (¢) = a{b0 - 4a0a1 a2 b0 - 6a§b6 - a0aib1 + 4a6a2 b1 + 4a1 a2b0b1 - 2a0a2 bi + a2b{ - 2aib0b2 + 4a0a2 b0 b2 - 4a2b0b1 b2 - a1 bib2 + 2a6b� + 4a1 b0 b§, p(a, b)IT2 () = -a6ai + 4a�a2 - 2aib0 + 10a0a1 a2 b0 + 12a§b6 - 4a6a2b1 - 7a1 a2 b0b1 - aibi + 5a0a2 bi - 2a2 bi + 2a6a1 b2 + 5aib0b2 - 4a0a2b0 b2 - a0a1 b1 b2 + 10a2 b0b1 b2 - 4a1 b0 b§ + 2a0b1 b� - bib§ + 4b0b� . ; Remark 4.58. According to Theorem 4.56(c), every function ; n) E Ql[M 2 ] is a polynomial in 0"1 and 0"2 . In practice, it can be quite challenging to find explicit expressions. Milnor [301] gives the examples 0"(;.i(2)1 = 20"1 + 0"2 , 3) = 0"1 (20"1 + 0"2 ) + 30"1 + 3, (;.�3) = (0"1 + 0"2 ) 2 (20"1 0"2 ) - 0"1 (0"1 + 20"2 ) + 120"1 + 28. Notice that these expressions are in Z[IT1 , IT2 ], rather than merely in Ql[IT 1 , IT2 ]. This reflects the fact that the map : M 2 A2 is actually an isomorphism of schemes over Z as described in Remarks 4.43 and 4.5 1 . The functions ;i n) are regular func tions on the scheme M 2 /Z, so they are in Z[IT 1 , IT2 ]. See [416] for details. u
*
+
u
__,
190
4. Families of Dynamical Systems
Rat2 (1C) A , A A1A, A2is, Aequal be the multipliers of 1 2 , 3 3 to 1, then we can
Proofof Theorem 4.56. (a) Let ¢ E and let the fixed points of¢. If we assume that none of apply Theorem 1 . 14 to deduce that 1
1 1 + + -= 1. 1 - A 1 1 - A2 1 - A 3
--
--
After some algebraic manipulation this becomes
0"1 (¢) = 0"3 (¢) +
¢ ¢ Rat2 (qRat2 (q
which completes the proof that 2 for all E whose multipliers are not equal to 1 . It is not hard to see that such are dense in (see Exercise 4.21), from which it follows that the function is identically zero. (b) The first property is a special case of Theorem 4.50, or more directly it is an immediate consequence of our earlier calculation (Proposition 1 .9) showing that the multipliers of a rational map are PGL2 -invariant. In order to prove the second property, it is convenient to show that every rational map of degree 2 is PGL2 -equivalent to a map of a particular shape. Lemma 4.59. (Normal Forms Lemma) Let ¢ E be a rational map of degree 2 and let be the multipliers of itsfixedpoints.
u(¢f) = u(¢)
Rat2 (1C) A , A , A 1 2 3 (a) /f)q>-..2 -1- then there is an f E PGL2 (C) such that ¢! (z) = zA22+Z +A11z . (4.35) Further, Res(z2 + A 1 z, >..2 z + 1) = 1 - A 1 A2 . (b) If A 1 A 2 = 1, then A 1 = A 2 = 1 and there is an f E PGL2 (1C) such that (4.36) ¢! (z) = z + � + �z . Proof We recall that if a E Fix(¢) has multiplier An, then the Taylor expansion of ¢ around a looks like cj;(z) = ¢(a) + cj;' (a)(z - a) + O(z - a) 2 = a + An(z - a) + O(z - a)2 . Hence ¢, 1 (z) = cf;(z) - z = (An - 1)(z - a) + O(z - a)2 , so ¢ has multiplicity 1 at the fixed point a if and only if An -1- 1 . We also note the formal identity (X - 1) 2 - (XY - 1 ) (XZ - 1) = X (X + Y + Z - 2 - XYZ) 1,
4.6. The Moduli Space
M2 of Dynamical Systems of Degree 2
191
and apply it using the relation
from (a). This yields the useful formulas
(>.1 - 1)22 (,\1 ,\2 - 1)(,\1 ,\3 - 1), (.-\2 - 1) = (,\2 ,\1 - 1)(.-\2 ,\3 - 1), (4.37) 2 (.-\3 - 1) = (,\3,\1 - 1)(,\3 ,\2 - 1). We now start with a rational map 2 a 1 z a2 ¢(z) aoz boz2 b1 z b2 and change coordinates in order to put ¢ into the desired form. ( a) For this part we assume that .-\ 1 .-\2 f- 1, so (4.37) tells us that .-\ 1 f- 1 and the fixed points associated to ,\ 1 and ,\2 have multiplicity 1 , so in .-\particular 2 f- 1. Thus they are distinct and we can find an element of PG 12 (q that moves them to 0 and respectively. After this change of variables has been made, the rational =
=
+
+
+
+
oo,
map ¢ satisfies ¢(0) = 0 and ¢( oo ) = oo, so it has the form with
ao
Since f- 0, we can dehomogenize by setting yields and
a0 1, and then a simple calculation =
Thus f has the form
¢(z) = zA2 2 Z b2 .-\b21 z with Finally, replacing ¢(z) by b2 1 ¢(b2 z) yields the desired form (4.35), and we calculate Res(z2 .A1 z, .A2 z + 1) = det .-\012 ,\11 001 1 - .-\1 .-\2 , ,\2 +
+
+
=
which completes the proof of (a). (b) The proof of this part is similar. We begin by moving the fixed point associated to to oo, so the map ¢ has the form
.-\ 1
with
.-\ 1 ,\ 2 1 ao. a0 1 b1
= combined with (4.37) tells us that The assumption that we have = Dehomogenizing = puts ¢ into the form
.-\1 = ,\2 = 1, so
4. Families of Dynamical Systems
192
Next we replace ¢(z) by ¢(z - b2 ) + b2 , so now ¢(z) looks like z 2 + a 1 z + a2 with ¢(z) z (Of course, the values of a 1 and a2 have changed.) Finally, replacing ¢(z) by ¢ ( y'a2 z) I y'a2 gives ¢ ( z) the desired form 1 ,�, z2 + a 1 z + 1 (4.38) '�-' ( z ) - z + a1 + - . z z We can compute the value of a 1 by observing that ¢ has a double fixed point at oo and that its other fixed point is at -a1 1 . (If a 1 = 0, then there is a triple fixed point at oo.) Thus the third multiplier is =
_
_
so we find that a 1 �. (This is also correct if a 1 of a 1 into (4.38) completes the proof of (b). =
=
0.) Substituting this value 0
We resume the proof of Theorem 4.56(b,ii). Let ¢1 and ¢2 be rational maps of degree 2 satisfying u ( ¢1 ) = u ( ¢2 ). Thus and and then the relation 0'3 0'1 - 2 from (a) implies that also 0'3(¢ 1 ) = 0'3(¢2 ). It follows that the set of multipliers of the fixed points of ¢1 and ¢2 are the same, say {>q , >.2 , >.3}, since they are the three roots (counted with multiplicity) of the polynomial T3 - 0'1 T2 + 0'2 T - 0'3. We consider two cases. Suppose first that >. 1 >.2 # 1. Then Lemma 4.59(a) says that ¢1 and ¢2 are 2-equivalent to the function ( z2 + >. 1 z) I ( >.2 z + 1), so they are PGL2 -equivalent to each other, and similarly if >. 1 >.3 # 1 or if >.2 >.3 # 1. We are left to consider the case >. 1 >.2 = >. 1 >.3 >.2 >.3 1. Then Lemma 4.59(b) tells us that >. 1 >.2 A3 1 and that ¢ 1 and ¢2 are both PGL2 -equivalent to the rational map z + z - 1 . This completes the proof of (b,ii), so we turn to (b,iii), the surjectivity of u. Given (8 1 , 82 ) E A2 (C), we set 83 = 8 1 - 2 and factor the polynomial =
=
=
=
=
=
Notice that the condition 83 = 8 1 - 2 gives the familiar relation (4.39)
193 M2 of Dynamical Systems of Degree 2 which in implies the usual formal identities (4.37) for -\ 1 , -\2 , -\ 3 . Suppose first that some Ai is not equal to 1, say ,\1 =/=- 1. Then ,\ 1 ,\2 =/=- 1 from (4.37), so we can set ¢(z) = zA22+Z +-\11z , since Res(z 2 + -\ 1 z, -\2 z + 1) = 1 - -\ 1 ,\2 from Lemma 4.59(a). The fixed points of ¢ are o: 1 = 0, o: 2 = oo, and 0:3 = i=�� . Its multipliers at o: 1 and o:2 are easily computed, 4.6. The Moduli Space tum
and
and the multiplier at
o:3 is a2 ( 82 . ,\1 = ,\2 = ,\3 1, a1 (¢) = 81 ¢(z) = z + z� 1
and ¢) = where we use (4.39) for the middle equality. Hence = which corresponds to the val We are left with the case ues 3. It is easy to check that the rational map has a triple-order fixed point at oo and that all three multipliers are equal to 1,
8 1 = 82 = so(c)u(¢)We briefl = (3,y3).sketch the proof, which uses basic methods from algebraic geometry. We refer the reader to [416, §5] for further details. The first step is to verify that the map u : M 2 ----> A2 is proper. This follows easily from the valuative criterion [ 198, 11.4. 7] using the fact proven in (b) that every fiber u� 1 ( t) consists of a single point. (Roughly speaking, a morphism of varieties over C is proper if its fibers are complete.) Next one checks that is finite, which can be proven using the fact that both M 2 and A2 are affine varieties (cf. [416, Lemma 5.6]). Alternatively, one can show in general that a proper quasifinite map ([299, that A2 and the bijectivity of u on complex points to prove that is an isomorphism (cf. [41 6, Lemma 5.7]). Finally, it is clear from the explicit formulas for a 1 and a2 in Remark 4.57 that u is defined over Q. (d) This is immediate from (c), which says that Q[M 2 ] = Q[ a 1 , a2 ], since we already know from Theorem 4.50(b) that ai n) and ;; n ) are in Q[M 2 ]. is a consequence of the fact that the isomorphism u : M 2 A2 in (c) is (e)definThis ed over Q. u
is finite
I, Proposition 1 . 1 0]). Then one uses the fact
is nonsingular
u
---->
0
Remark 4.60. Regarding the proof sketch of Theorem 4.56(c), we observe that a
W
morphism of irreducible varieties V may be bijective on complex points, yet not be an isomorphism. A simple example of this phenomenon is the cuspidal cubic map ---->
u : M 2 AA22. Itis does an isomorphism, it is crucial that u not appear to be known in general
Thus in the proof that ----> onto the nonsingular variety maps whether is nonsingular.
MM2 d
4. Families of Dynamical Systems
194
M2 2 A (a1 , a2 ) M2 M� M M�8, M2 • 2. Theorem 4.61. Let M 2 M� M�8 be the completion ofM 2 constructed using geometric invariant theory in Theorem 4.40. Then the isomorphism of dynamical systems The content of Theorem 4.56 is that the moduli space of degree 2 is isomorphic to and an explicit isomorphism is provided by the pair of functions created from the multipliers of the fixed points of a rational map. From our general theory (Theorem 4.40), the space sits naturally inside two larger spaces and but since d 2 is even, these two spaces coincide We conclude this section with a description of and will be denoted by =
=
extends to an isomorphism ii mutes:
=
:
M 2 IP'2 such that the following diagram comM 2 � A2 x (x t,y) 1 1 [ , , 1] -+
M2 (C) M2 (C) cPA,B (X, Y) [AXY, XY + BY2],
The points in that are not in correspond to degenerate maps of degree 2 that may be informally described as maps of theform
(4.40)
=
The point [A, B] is uniquely determined up to reversing A and B. D
Proof See [302] and [416, Theorem 6. 1 and Lemmas 6.2 and 6.3].
Remark 4.62. We expand briefly on what it means to say that the points in the
(M2 " M2 )(C)
set correspond to the maps cPA,B given by (4.40). Let U be a neighborhood of 0 and let
C
A1
be a rational map that is a morphism away from 0. We denote the image of t E U by 4>t to help remind the reader that 4>t is itself a map, i.e., 4>t Consider the composition :
IP'1 IP'1 . -+
which by abuse of notation we again denote by 4>. It is a rational map, and since U c and M2 is complete, we see that it is a morphism from U to In partic ular, the point 4>0 is a well-defined point in If 4>o E there is nothing to say, so we suppose that 4>0 � Then the second part of Theorem 4.61 means that possibly after choosing a smaller neighborhood of 0, there exists a morphism
A1
M2 (C). M2 (C).
M2 (C), M 2 .
4.7. Automorphisms and Twists f
195 :
U
------+
PGL2
such that the conjugate map ¢! has the form with a0 , . . . , b2 regular functions on U and satisfying ao (O)
= a2(0) = bo (O)
=
0,
b 1 (0)
=
1,
and
In other words, the degeneration of ¢! at t 0 has the form [AXY, XY + BY 2] . Further, the map ¢ determines the point [A, B ] = a 1 (0), b2 (0)] E lP'1 (
[A, B]
[
c---+
[AXY, XY + BY2 ] ,
where the symmetric group on two letters s2 acts on lP' 1 by interchanging the coor dinates. (It is an exercise to show that lP' 1 /S2 is isomorphic to lP'1 .) 4. 7
Automorphisms and Twists
As we have repeatedly seen, from a dynamical perspective the geometric properties of a rational map ¢( z ) and its conjugates ¢! = f- 1 ¢f are the same, since conjuga tion by f E PGL2 (
and similarly the set of rational maps that are K-equivalent by Remark 4.63. In some sense we already have a notation (¢) for the PGL2 (R) equiv
Ratd M d
alence class of¢. However, we generally view ( · ) as a map and (¢) as a point in the moduli space while we think of [¢] as a set of rational maps. This is the reason for the notational distinction.
M d,
--t
196
4. Families of Dynamical Systems
Not all conjugates qi of ¢ need be distinct. For example, the rational map b - ¢( = satisfies =
-z) ¢(z), so ¢! = ¢ for the linear fractional transformation f(z) = -z. The set of transfor mations f E PGL (K) that fix ¢ is an interesting group whose properties play an ¢(z) az + -z
2 important role both geometrically and arithmetically. Definition. Let ¢( E be a rational map. The automorphism group of¢ is the group E PGL2 (K) : (Another common name for Aut(¢) is the group ofselfsimilarities of¢.) Remark 4.64. It is easy to check that Aut(¢) is a subgroup of PGL 2 (K) and that for any h E PGL2 (K) the map
z) k ( z) Aut(¢) = {f
¢1 (z) = ¢(z)}.
is an isomorphism (see Exercise 4.32). Thus as an abstract group, Aut(¢) de pends only on the PGL 2 -conjugacy equivalence class of ¢; more precisely, the k equivalence class of ¢ determines Aut(¢) in PGL2 up to conjugation. Proposition 4.65. Let E K(z) be a rational map ofdegree d 2': 2. Then Aut(¢) is a.finite subgroup ofPGL2 (K), and its order is bounded by afunction ofd. Proof Let E Aut ( ¢). Then for any point E lP' 1 ( K) and any n ;:::: 1 we have
¢(z)
P
f
¢n (P) (¢ft (P) u- 1
=
n,
n.
n.
n , n3
n
for i = 1, 2, 3, the action of ¢ on the sets of primitive periodic points yields a homomorphism (4.41 ) from into a product of three symmetric groups. We claim that the homomorphism (4.41) is injective. To see this, we observe that any il1 the kernel of (4.41) fixes each Per�: ( ¢); hence fixes at least three points in lP' 1 K); hence is the identity map. Thus the homomorphism (4.41 ) is injective, D which clearly implies that Aut ( ¢) is finite.
Aut(¢) f( f
f
4.7. Automorphisms and Twists
197
Remark 4.66. Proposition 4.65 tells us that the automorphism group Aut(¢) of a rational map is a finite subgroup of PGL 2 (K). For K C, or more generally for any field of characteristic 0, the classical description offinite subgroups ofPGL2 (C) =
says that every such subgroup is conjugate to either a cyclic group, a dihedral group, or the symmetry group of a regular three-dimensional solid (i.e., the tetrahedral, octahedral, and icosohedral groups). See, e.g., [414]. Example 4.67. Let ( be an nth root of unity, let 7./J(z) E K(z) be any rational map, and let ¢(z) z?jJ(zn ). Then Aut(¢) contains a cyclic subgroup of order n gener ated by the map f(z) = (z. Example 4.68. The map ¢(z) = (z 2 - 2z)/( -2z + 1) has an automorphism group Aut(¢) that is isomorphic to the symmetric group on three letters. More precisely (see Exercise 4.36), the automorphism group of¢ consists of the following six linear fractional transformations: 1 z-1 1 z ,1-z � Aut(¢) = z, -- , -- , z z 1-z z-1 =
S3
{
- ,
--
Example 4.69. Consider the rational map
} S3 .
rPb(z) = z + b , z whose automorphism group Aut(¢b) = {z, -z} has order 2. These maps are all PGL2 -equivalent, since the linear fractional transformation f(z) = zJbfC gives ¢c = ¢{. Thus the geometric dynamical properties of rPb are the same for all b. How ever, the arithmetic properties of rPb may change quite substantially depending on the value of b, since the change of variable involves a square root. So although rPb and r/Jc are always PGL2 (K)-equivalent, they are not PGL2 (K)-equivalent unless b/ c is a -
perfect square in K. The underlying reason for the existence of these "twists" is the
fact that Aut( ¢ 1 ) is nontrivial.
Definition.
set
Let ¢( z) E K ( z) be a rational map. The set of twists of¢ over K is the
}
{
�e �lasse� of maps 7./J such . K) = K-equivalen that 7./J IS K-eqmvalent to ¢ Remark 4.70. As noted earlier, the geometric dynamical properties of the maps in Twist ( ¢/ K) are identical, but their arithmetic properties may be significantly different. For example, if two rational maps are K-isomorphic, then the field exten sions of K generated by their periodic points are the same. This is clear from the fact that Pern (¢1) = E Pern (¢) } , so if E PGL2 (K), the fields generated by Pern(¢) and Pern(¢f) are identical. However, if ¢ and 7./J are only K-isomorphic, these fields may well be different. This often provides a convenient method for proving that two maps are not K-isomorphic, as in the following example. wist( ¢/
T
f
{f- 1 (P) : P
198
4. Families of Dynamical Systems
Example 4.71. Continuing with Example 4.69, for each b E K* we let
¢b(z)
=
z
+ -.zb
k
We saw that these maps are all -isomorphic, which gives a map of sets K*
-----+
Twist(¢ 1 / K),
(4.42)
Note that if bIc is a square in K, say bIc a2 , then ¢b and ¢c are K-isomorphic, since ¢c ¢{ for the map f(z) az E PGL2 (K). Thus (4.42) induces a well defined map K* I K* 2 -----+ Twist( ¢1 / K), (4.43) We can use Remark 4.70 to prove that the map (4.43) is injective. (We as sume that K does not have characteristic 2.) A quick computation shows that bY2 , so the primitive 2-periodic points of ¢b are ± Fbf2. Y) Hence if ¢b and ¢c are K-isomorphic, then the fields K( FFJ72) and K( FC/2) are the same, so blc must be a square6 in K. Notice that we could not use Per 1 (¢b) to prove this result, since the only fixed point of ¢b is oo. These quadratic twists of ¢1 ( z) are analogous to quadratic twists of elliptic curves (cf. [410, §X.5]). Thus fix E K and, for each D E K*, let Ev be the elliptic curve Ev : DY2 Then all of the Ev are isomorphic over but ED1 and Ev2 are isomorphic over K if and only if the ratio D 1 ID2 is a square in K. This gives a natural map K* I(K*) 2 ---.. Twist(EI/ K). Remark 4.72. How does the theory of twists fit in with the moduli spaces Md con structed in Section 4.4? The answer is that if ¢ and 't/J are twists of one another, then their corresponding points (¢) and ('t/J) in the moduli space Md are equal. This is true because points in Md ( ) classify rational maps of degree d modulo PGL ( ) conjugation, so points in M d ( ) do not detect whether the conjugation is defined over K. In other words, there is a natural map =
=
¢b ,2 (X, k,
=
=
2X2 +
=
X3A,B + AX + B.
kk
2k
(4.44) but this map is not one-to-one. It fails to be injective precisely for those maps in Ratd(K) that have nontrivial twists; cf. Remark 4.39. On the other hand, the next proposition and Exercise 4.38 imply that the map (4.44) is injective on a Zariski open subset of Ratd. We now prove that a rational map with no automorphisms has no nontrivial twists. Later, in Theorem 4.79, we prove a much stronger result. Proposition 4.73. Let ¢( z) E
K ( z) be a rational map ofdegree d 2 2 and assume that its automorphism group Aut(¢) is trivial. Then ¢ has no nontrivial twists, i.e., Twist (¢I K) has only one element, the K-equivalence class of¢ itself 6 It is an easy exercise to prove that K (VA) and K ( vB) are isomorphic if and only if A/ B is a square in K, assuming that K does not have characteristic 2.
199
4.8. General Theory of Twists
'lj; K(z) a G l(k f(z) f ¢(z)PGL2 (K) a(¢) = ¢ a('lj;) = 'lj;, a 'lj; ¢! = 'lj; = a('lj; ) = a(cpf) = a(fcpf- 1 ) = a(f)a(cp)a(f- 1 ) = a(f)cpa(f)- 1 = cpa(fl. Hence a(f)f - 1 E Aut ( ¢) . But Aut(¢) is trivial by assumption, so a(!) = f. This is true for all a E Gal(k IK), so we conclude7 that f E PGL2 (K). Hence 'lj; = ¢! is K-isomorphic to ¢, so it represents the trivial twist. D
Proof Suppose that
E is a twist of ¢, so there is an E such that We let an element E a I K) act on and in the natural way by applying to each ofthe coefficients. Notice that and since the coefficients of ¢ and are in K. Hence
'lj; = ¢f.
Remark 4. 74. As we have seen in Section 4.3, the set Ratd of all rational maps of
degree d is a Zariski open subset oflP'2dH . It is the complement of the hypersurface described by the vanishing of the resultant Res( Fa, Fb) 0. One can show that the set of rational maps ¢ E Ratd having nontrivial automorphism group forms a proper Zariski closed subset of Ratd; see Exercise 4.38. Thus most rational maps have no nontrivial automorphisms, and those that do, fall into finitely many irreducible alge braic families. Further, since for all E the automorphism groups of¢ and are isomorphic as abstract groups, there is a proper Zariski closed subset of Md de termined by the conjugacy classes of rational maps with nontrivial automorphisms. It is an interesting geometric problem to describe the irreducible subvarieties mak ing up this set and an interesting arithmetic question to describe their rational and integral points. See Exercises 4.28 and 4.4 1 .
=
f PGL2
4.8
cpf
General Theory of Twists
In this section we develop the basic theory of Galois twists in an abstract setting. Although we apply this material only to rational maps, in Section 4.9, and to lP'1 , in Section 4. 10, we develop the theory in some generality in order to clarify the underlying principles. Let X and Y be objects defined over the field K. For example, X and Y might be curves or algebraic varieties or rational maps. 8 We consider X and Y to be equivalent if they are isomorphic over K. However, it may happen that they are isomorphic over but not over K. This leads us to consider the following set.
k,
Definition.
the set
Let X be an object defined over the field K. The set of twists ofXIK is
7The proof that PGL2(K) is the subgroup ofPGL2 (K) fixed by Gal(K I K) is not hard, although it does use Hilbert's Theorem 90. 8Formally, X and Y should be objects in a category on which the Galois group Gal( KI K) acts in an appropriate way. We do not concern ourselves with such formalism and leave it to the reader either to formulate the correct abstract concepts or to restrict attention to those situations, namely algebraic varieties and rational maps, in which the Galois action is clear.
200
4. Families of Dynamical Systems
}
{
K-isomorphism classes of objects Y K) such that Y is defined over K �nd . Y is isomorphic to over K In other words, an element of K) is an object Y that is defined over K such that there is an isomorphism i ----+ Y, but the isomorphism i might be defined only over an extension of K. Two elements Y and Y' in K) are considered to be equivalent if there is an isomorphism j Y ----+ Y' that is defined over K. The following examples should help to clarify this definition. Example 4.75. Our first example deals with twists ofrational maps. Let ¢(z) E K(z) be an odd rational map, i.e., a rational map satisfying ¢(-z) = -¢( z). Then for each b E K* we can define a new rational map
Twist(XI
=
X
Twist (XX I :
:
1
/b
Twist(XI
The odd parity of ¢(z) implies that
=
I ----+ Twist(
t
Ca
:
cz.
=
c2 ,
lP'1 .
x2 y2
K
All of these curves are isomorphic over via the explicit isomorphism
(4.46) x , y) = (x �, y �) . Further, if bI is a square in K, say bIa = c2, then Ca and Cb are isomorphic over K via the isomorphism i(x, y) (ex, cy) . Thus exactly as in Example 4.75, we obtain i(
a
=
a natural map
K* K* 2
Twist(cl IK) ) where [Ca] K denotes the K-isomorphism class of Ca. Notice that iftwo curves C and C' are K-isomorphic, then the K-isomorphism i : C ----+ C' identifies their K-rational points C(K) ----+ C'(K). This suffices to prove that the curves c1 and c_ 1 are not isomorphic over Q, since and c_ 1 (Q) = 0. Hence C_ 1 represents a nontrivial element of Twist ( C1 IQ), and indeed a nontrivial element of Twist( CI/lR.). More generally, a famous theorem of Fermat says that ifp I
----+
i :
is an odd prime number, then
201
4.8. General Theory of Twists
p = 1 (mod
if and only if
4).
Hence if p = 3 (mod 4), then Cp represents a nontrivial element of Twist( Cl/Q) . It is not hard to show that different primes p = 3 (mod yield distinct elements of Twist(Cl /Q); see Exercise 4.44. Returning to the general situation, let X be an object defined over K and let Y represent an element of Twist(X/ K). This means that there is a K-isomorphism
4)
i : y ------+ X. We wish to determine whether X and Y are K -isomorphic. If i is itself a K-isomor phism, then we are done; but even if is not a K-isomorphism, it may be possible to modify i and tum it into a K-isomorphism. In order to measure the extent to which i fails to be a K-isomorphism, it is natural to make use of Galois theory, since i is defined over K if and only c:r( i) = i for every cr E Gal(K/K). Thus for each element cr E Gal(K/K), we consider the composition of maps a-(i - 1 ) y i X g : X The map 9a- = icr(i - 1 ) is a k-automorphism of X, i.e., it is an isomorphism from X to itself defined over K. If is already defined over K, then 9a- (x) = x is the identity map, but in general 9a- will be a nontrivial automorphism.
i
0"
-------+
-------+
.
i
Proposition 4.77. Let X be an object defined over K, let Y be a twist of X/ K, choose a K-isomorphism i : Y X, and define a map __,
g :
(a) The map
Gal(K/K) ------+ Aut(X), Gal(K/ K)
------+
9a- (x)
=
(ic:r(i- 1 ))(x).
Aut(X) ,
satisfies
for all cr, T E Gal(K / K). Maps satisfYing 9a-T ga- ( gT) are called 1-cocycles. (b) Y is the trivial twist of X ifand only ifthere is an f E Aut(X) satisfYing =
cr
for all E Gal(K / K). cr
Maps of theform fcr(f - 1) are called 1-coboundaries.
i o (crT)( i - 1 ) i o cr( T(i) - 1 ). Consider the following commutative diagram of maps: o- ( T ( i )- 1 ) y
Proof (a) We have 9a-T
=
=
X
� X
o- ( i )
1
i
r
X � y
4. Families of Dynamical Systems
202
The top row is 9ar. but if we travel around the diagram the long way, we get i o a(C 1 ) o a(i) o a(T(i) - 1 ) (b) Suppose first that 9 is a coboundary, say 9a We verify that the isomorphism f-1i Y a E Gal(kI K) we have :
a(f-1i)
=
a(f-1 )a(i)
=
f-19aa(i)
=
=
--+
=
9a o a(9r)·
fa(f- 1 ) for some f E Aut(X).
X is defined over K. For any
r1ia(i-1 )a(i)
=
r1i.
This proves that J-1i is defined over K, so Y is K-isomorphic to X. Next suppose that Y is K-isomorphic to X, say j Y --+ X is a K-isomorphism. Then a(j) j for every a E Gal(kI K), so we have :
=
9a
i o a(i-1)
=
=
i o r1 o a(j) o a(C 1 )
=
(i o r 1 ) o a((i o r 1 ) - 1 ) .
Thus if we let f ir1 E Aut(X), then 9a fa(f- 1 ) , which proves that 9 is a coboundary and completes the proof of the proposition. D =
=
As the terminology suggests, there is a cohomology theory underlying Proposi tion 4.77. Definition.
is a map
Let r be a group that acts on another group 9:r
_____. A
satisfying gaT = gaa(gT ) for all a, T E
and a 1-coboundary (from r to r
A. A 1-cocycle (from r to A)
_____. A,
A) is a map ofthe form
a f----+ fa(r1 )
for some f E
A)
r,
A.
A
The cohomology set H1 (r, is defined to be the collection of 1-cocycles r modulo the equivalence relation that two 1-cocycles 91 and 92 are cohomologous if the map 9]"192 is a 1-coboundary. --+
A
Remark 4.78. If the group is abelian, then H1 (r, A) is itself an abelian group, and in this situation it is possible to define cohomology groups Hn (r, for all n 0.
A)
�
For the general theory of group (and Galois) cohomology, with many important ap plications to class field theory, Diophantine equations, and arithmetic geometry, see for example [97, 396, 410]. A slight elaboration of the proof of Proposition 4.77 shows that if Aut(X) is abelian, then there is a well-defined one-to-one map Twist(XI K)
_____.
H1 ( Gal(k I K), Aut(X)) ,
[Y]K
f----+
(a 1---t iya(i¥1 )) ,
(4.47)
where iy Y X is a k-isomorphism; see Exercise 4.42. In some cases, for example when X is an algebraic variety, the map (4.47) is an isomorphism. :
--+
4.9. Twists of Rational Maps
4.9
203
Twists of Rational Maps
According to Proposition 4.77, every twist corresponds to a 1-cocycle, and a twist is trivial if and only if its 1-cocycle is a 1-coboundary. The natural question that arises is whether every 1-cocycle comes from a twist. It turns out that the answer depends on the category from which the objects are being chosen. For example, in the category of algebraic varieties, every 1-cocycle does come from a twist. We will not prove this general result, but in the next section we treat the case of twists of IP'1 . In this section we describe what happens for rational maps.
Gal(KIK) PGL2 (K) a(a)z + a(b) a(!) = a ( azcz ++bd ) = a(c)z +a(d) az + b for f = -- PGL2 (K) and a Gal(K IK). cz + d The automorphism group Aut(
We define an action of the Galois group on the group of linear fractional transformations in the natural way, thus
Definition.
E
E
g :
E
g :
g
g.
g
ga
204
4. Families of Dynamical Systems
PGL2 (K) f E PGL2 (K). PGL2 (K) g f EAut(¢) 1) PG 1 ( K) 9 = f (fa 2 E Gal(KIK). ¢ K, c;bIKE K(z), g. g E Gal(KIK). since we are given that 9a E Aut( c;b) , c;i>9a = c;/> c;/>ga = 9a ¢ by definition of c;/>9", 1 1 since 9a = fO'(f - 1 ) by assumption, c;/>jO'(f- ) = jO'(f- )¢ 1 r ¢f = (J'ua( -) 1 )¢0' (!) multiplying1by r 1 and by O'(f), c;/>f = c;/> f since O'(f - ) = O'(f) - 1 , ¢! = O'(c;b) "(f) = O'(¢f ) since c;b(z) E K(z), so O'(c;/>) = ¢. We have proven that ¢! = ( ¢f ) for all E Gal( K IK), which shows that ¢! ( z) E K(z)In andorderthusto completes the proof that (b) implies (a). prove the cohomological description of Twist( ¢1 K), we first use Proposition 4.77, which says that there is a natural embedding of Twist( ¢1 K) into the cohomology set H 1 (Gal(KI K), Aut(¢)) (cf. also Remark 4.78). Then the equivalence of(a) and (b) tells us that an element of H 1 ( Gal(KIK), Aut( c;b)) comes from an element of Twist( ¢1 K) if and only if it becomes trivial when mapped to D H1 (Gal(KIK), PGL2 (K)).
Hence is the 1-coboundary associated to the element This proves that (a) implies (b). Next suppose that is an 1-cocycle that is a 1-coboundary. This means that there is an with the property that for all We claim that ! is a twist of with 1-cocycle What we need to check is that ¢! is defined over i.e., that ¢! since once we know that, it is clear from the definitions that is its associated 1-cocycle. Let Then
g
(J
0'
0'
0'
0'
Remark 4.80. A formal argument with commutative diagrams shows that
Br(K)
K.
where is the Brauer group of The Brauer group plays an important role in class field theory and many other areas of number theory and arithmetic geom etry. For example, so has only two elements, and the same is true for finite extensions of !Qp. The Brauer group of a number field is more complicated. Example 4. 81. Let be a field of characteristic not dividing n, and let be a rational map whose automorphism group is
Br(!Qp) = IQIZ, Br(!Qp)[2]
K
¢( z) E K( z)
Aut(¢) = { (z : ( E JLn }, where we recall that JLn K denotes the set of nth roots of unity. Then there is an isomorphism (4.49) K*IK* n Twist(c;I>IK), c
�
4.9. Twists of Rational Maps
205
(- 1 ¢((z) ( J.tn · ¢(z) 1 z---+ (z, ¢(z) z'lj;(zn )¢(z) z'lj;(bz'lj;n( z)), K(z) K(z). z -
Note that by assumption we have for all E In particu lar, the function so it has the is invariant under the substitution form for some E (Exercise 4.37). Thus the twist of given in (4.49) is equal to so it is indeed in Since we are assuming that the automorphism group of ¢ is isomorphic to not merely as an abstract group, but also in terms of the way in which acts on Aut(¢) and on a standard result in Galois cohomology9 says that there is an isomorphism =
=
J.tn ,
¢(z) Gal(RIJ.tK)n ,
(en--+ ) a ( V'b
) . (4.50) V'b This allows us to identify Twist (¢I with a subset of In order to show that they are isomorphic, Theorem 4.79 says that we must show that every cocy cle in (4.50) becomes a coboundary when we consider it as a cocycle with values in But this is clear, since with our identification of with Aut(¢), the cocycle in (4.50) is equal to
b
1----t
K* IK* n . J.tn z with f(z) V1J E PGL2 (K).
K)
PGL2 (K).
=
Note that this example generalizes Example 4.75, which dealt with the case 2. Example 4.82. Let E be a rational map with automorphism group Aut(¢) (See Exercise 4.35.) The Galois group acts triv ially on Aut(¢), so we have n =
=
z,{ z- 1 }.¢(z) K(z)
Gal(KI K)
The isomorphism is given explicitly by associating to any b E (j 1-----)
{ zz-1
if a ( v'b ) if ( v'b) a-
= =
v'b, -v'b.
K* / K * 2
the cocycle
(4.5 1)
To ease notation, we let (3 v'b and let g be the cocycle described by (4.5 1 ). Then Theorem 4.79 says that g comes from a twist of ¢ if and only if there is some E satisfying 9a ). Using the fact that g E we are looking for an E satisfying 9 This statement is equivalent to H1 (Gal( K / K), K * ) 0, which is a version of Hilbert's Theo
a (z)
=
f PGL2 (K) PGL (K) fo-(f- 1 f 2 =
=
rem 90. Using this and taking Galois invariants of the Kummer sequence 1
----+
1-Ln
----+
yields the cohomology long exact sequence
K*
0: �----+ 0: n
------+
K*
----+
1
{ z, z - 1 } ,
206
4. Families of Dynamical Systems
if a( j3) j3 , a(f)(z) { f(z) 11 f(z) if a(j3) -j3. It is not hard to construct such an f, for example f(z) -j3j3zz++11 Note that det ( !!3 � ) 2j3 =1- 0, so f is in PGL2 (K). Hence every gives a twist of ¢, and indeed we can write the twist associated to b j32 E K* IK* 2 explicitly using f, ) ¢ + j3 j3 ( !;:: 1 ¢f (z) ( )+1 For example, let Md ( z) z d be the t:fh-power map. Then a judicious use of the binomial theorem yields a formula for the b-twist Mdb} of Md, " " Mj \ z) � C�) b' z �� ( � ) b' z + ' 2k 1 In particular, the first few b-twists of Md are 1 + 3bz2 ' (b) M3 (z) 3z + bz3 =
=
=
=
g
=
=
=
_d.
'I'
.
f]z+l -f]z+l
=
"
�
·
_
4.10
Fields of Definition and the Field of Moduli
If¢( z) is a rational map whose coefficients lie in a field K, then it is interesting to study the orbits of points whose coordinates lie in K or in finite extensions of K. The smallest field K containing the coefficients of ¢ can generally be determined by inspection. For example, the map ¢(z) z 3 + 1 is clearly in Q(z) and the map ¢(z) z 3 i is just as clearly in Q(i)(z). However, if we make a change of variables f ( z) iz in the latter map ¢( z) z3 i, we find that ¢! (z) r 1 (¢(f(z))) -i¢(iz) - i( -iz3 i) - z 3 + 1 E Q(z). Thus the map ¢(z) z 3 + i is really a Q(z) map that has been altered by an injudicious change of variables. Definition. Let K be a field of characteristic 0, let ¢( z) E k ( z) be a rational map with coefficients in an algebraic closure of K, and let K' IK be an extension field. We say that K' is afield ofdefinitionfor ¢ ifthere is a linear fractional transformation f(z) E PGL2(K) such that =
=
+
=
=
=
=
=
+
+
=
=
In other words, K' is a field of definition for ¢ (or an FOD for short) if, after a change ofvariables, ¢ has coefficients in K'.
4.10. Fields of Definition and the Field of Moduli
207
As in Section 4.9, we can use Galois theory to investigate the possible fields of definition of a given rational function ¢( z) E k ( z) . We let Gal( kI K) act on k ( z) in the natural way by applying CJ E Gal ( k IK) to the coefficients of ¢ E k ( z), CJ ( ¢) = CJ
(
ao + a 1 z + · · · + adz d bo + b1z + · · · + bdz d
)
=
CJ(ao) CJ(a 1 )z + · · · + CJ(ad)zd CJ(bo) + CJ(b l )z + · · · + CJ(bd)zd ·
+
Then Galois theory (and Hilbert's Theorem 90) tell us that
(
)
¢ E K' (z) for the field K' = fixed field of { CJ E Gal(k I K) : CJ( ¢) = ¢} . Suppose instead that ¢ is merely equivalent to a map whose coefficients are in some field K'. That is, suppose that there is a linear fractional transformation f E PGL2 (k) such that ¢! E K'(z). Then CJ(¢f) = ¢! for all CJ E Gal(KIK'), so f- 1 ¢! = ¢! = CJ(¢f) = CJ( j - 1 ¢! ) = CJ( j- 1 )CJ(¢)CJ(f) = CJ( j ) - 1 CJ(¢)CJ(f). Solving for CJ( ¢) yields CJ(¢) = CJ(f)f - 1 ¢fCJ(f - 1 ) = (f(J(f- 1 )r 1 ¢(JCJ(f - 1 )) = ¢fa(r 1 J . Thus CJ(¢) is equal to ¢ conjugated by the map jCJ(f - 1 ) E PGL2 (K), so in partic ular CJ( ¢) and ¢ are equivalent. We have proven that
(defi K' is a field of ) (afor9aevery CJ Q_al(k IK') there exists ) nition for ¢ E PG 12 ( K) such that CJ ( ¢) = ¢g" E
===}
·
(4.52)
Turning this around, we start with an arbitrary rational map ¢(z) E K(z) and study the automorphisms CJ E Gal( k IK) for which u( ¢) is equivalent to ¢. Definition. Let rjJ(z) E K(z) be a rational map. We associate to rjJ a subgroup Gq, of Gal(k IK) and a field Kq, defined by Gq, = {CJ E Gal(KIK) : u(¢) = ¢9" for some ga E PGL2 (K) }, Kq, = fixed field of Gq, = { a: E k : CJ(o:) = a for all CJ E Gq,}. The field Kq, is called the field ofmoduli (FOM) of¢. Remark 4.83. The group PGL2 (K) acts on the space of rational maps via the usual conjugation action, ¢! = f - 1 ¢f. If we consider as usual the equivalence class consisting of all maps that are conjugate to a particular map ¢, then Gq, is the sub group of Gal(kI K) consisting of elements that map the set [¢] to itself. Equiv alently, if ¢ has degree d, let (¢) be the image of ¢ in the moduli space M d = Ratd I PSL 2 that we defined and studied in Section 4.4. The space M d is defined over Q (Theorem 4.36) and the field of moduli of ¢ is exactly equal to the field generated by the coordinates of the point (¢) E Md.
208
4. Families of Dynamical Systems
We begin by proving some elementary properties ofFOM and FOD, in particular, the important fact that the field of moduli is contained in every field of definition. Proposition 4.84. Let
¢(z) E ofGal(K K(z). I
(a) The set Gq, is a subgroup K). (b) Let K' be afield of definition for ¢. Then Kq, FOM <:;;; FOD.
<:;;;
K'. Informally, we say that
Proof The proof of this proposition is simply a matter ofunsorting definitions. Thus
let
a, T E Gq,. Then
Thus
(aT)(¢) is equivalent to ¢, so aT E Gq,. Similarly,
a- 1 E a- 1 (¢)Gal(kI E Gal(kI E PGLGal( 2 (K)kI E a(¢) a(¢) ¢9". a Gal(kI Gal(kI
Gq,. This proves which shows that is equivalent to ¢, and hence that K). that Gq, is a subgroup of Next let K' be a field of definition for ¢. Under this assumption, we proved earlier (4.52) that for every such that K') there is a 9a Gq, = Kq,) by In other words, is equivalent to ¢, so = definition. This proves that K') c Kq,), and hence by Galois theory, that Kq, c K'. (We are also using the fact that Kq, is a finite extension of K.) D a
It follows from Proposition 4.84 that the smallest possible field of definition for ¢ is the field of moduli of ¢, but it is not clear whether the field of moduli is always a field of definition. The following example shows that we can have FOM =/=- FOD. Example 4.85. Let
¢(z) = i (:�� r Clearly Q( i) is a field of definition for ¢. Let be complex conjugation, so Gal(Q(i)IQ) {1, a}, and let g (z) -11 z. Then we obtain 1 3 -1lz - 1 ) i ( -1lz +1 3 3 -1lz + 1 ) -i ( � ) a(¢)(z). i ( -1lz z+1 -1 This shows that E Gq,, so Gq, {1, a} and Kq, = Q. In other words, Q is the field of moduli of ¢. Now suppose that Q is actually a field of definition for ¢. This means that we can find some f E PGL2 (QJ) such that ¢! E Q(z). In particular, letting a denote complex conjugation, we have a
=
=
=
a
=
=
=
4.10. Fields of Definition and the Field of Moduli
209
=
It is not hard to verify (see Exercise 4.39) that Aut(¢) 1, so we deduce that -11 this equation says that Using the fact that
g (z) = az b cz d
f = ga(f).
z, - azcz + Jb '
+ +
+
where we use an overscore to denote complex conjugation. The fact that these two linear fractional transformations are equal means that there is a E such that
A Q* a = -Ac, b = -Ad c = Aa, d = Ab. ,
Multiplying the first and third equations gives
This is a contradiction, since of. 0 and we cannot have 0. Hence Q is not a field of definition for ¢, so we have an example of a map with FOM of. FOD. In order to investigate more closely the question of when the field of moduli of a rational map ¢ is a field of definition, we make the simplifying assumption that
A
a=c=
(4.53)
Aut(¢) = 1 .
Replacing K by Kt>, we may assume that K is the field of moduli of ¢ This means that for every E Gal(K I K) there exists a ga E PGL2 (K) satisfying .
a
Note that the assumption (4.53) that Aut(¢) 1 implies that ga is uniquely de termined by The next proposition describes some of the properties of the map a I-t
=
ga . a.
z)=
d
K ( be a rational map of degree 2 2 with field of moduli K and satisfying Aut(¢) 1, andfor each E Gal( K I K) write ¢9rr as above. (a) The map
Proposition 4.86. Let ¢ E
a
a(¢) =
is a 1-cocycle, i.e., it satisfies for all a, T E Gal(K I K).
(b) K is afield ofdefinition for ¢ ifand only if is a 1-coboundary, i.e., ifand only
f
ifthere is an E PGL2(K) such that
g
a
for all E Gal(K I K).
4. Families of Dynamical Systems
210
Proof
a,
(a) Let T E Gal(KI K). Then
a ( (¢)) = a (rjJ9T ) a(r/Jt(gr ) = rjJ9
=
r
=
=
=
=
=
=
The criterion for FOM FOD given in Proposition 4.86 may seem complicated, but it represents a tremendous simplification. If we try to use the definition of FOD directly, we need to search for an E PGL2 (K) that makes the coefficients of ¢1 lie in K. The substitution =
f
rjyl (z)
=
(az+b) cz+d b ' (az+b) cz+d + a _
drjJ - c A. 'f'
even for a rational function ¢( z) of small degree, has coefficients that are very com plicated expressions in the quantities a, b, d. It is thus difficult to determine whether there is some choice of a, b, d that makes the coefficients lie in K. On the other hand, the cocycle-coboundary criterion c
c
,
,
FOM = FOD
{:=::}
g(T has the form fa(r 1 )
in Proposition 4.86 involves only linear functions, i.e., elements of PGL2 (K), so it is often considerably easier to apply. The 1-cocycles that arise in the FOD = FOM question have the form
g : Gal(KIK) --+ PGL2 (K).
The group PGL2 (k) is the automorphism group of the projective line lP'\ so these cocycles should be associated to twists oflP' 1 . We saw in Section 4.8 that every twist gives rise to a 1-cocycle, but in general it is a delicate question to determine whether every 1-cocycle comes from a twist. It turns out that this is true for algebraic varieties, but since we do not need the most general result, we are content to construct the twists of lP' 1 that are needed to answer our question about FOD FOM. (See also Example 4.87.) =
211
4.10. Fields of Definition and the Field of Moduli Proposition 4.87. Let
g : Gal(KIK) ---+ PGL2 (K) be a 1-cocycle, and assume that g has the property that there is a finite extension LI K such that grr = 1 for all a E Gal( k I L ). Then there is an algebraic curve C defined over K and an isomorphism i C JP> 1 defined over k such that the 1cocycle Gal(KIK) ---+ PGL2 (k), a � ia(i- 1 ), is equal to the 1-cocycle g. Hence C is a twist ofJP> 1 IK, and C is the trivial twist ofJP>1 IK ifand only if g is a 1-coboundary. Proof We construct the curve C by describing its field of rational functions. Note that the field of rational functions for the curve JP>1 is the field k ( z) and that the Galois group Gal( kI K) acts naturally on k ( z) by acting on k and leaving z fixed. Now consider another field of rational functions in one variable K(w) , but this time with a "twisted action" of Gal(kI K). We define this twisted action by let ting Gal(k I K) act on k as usual, but setting a(w) = g;; 1 (w). In other words, if g17(z) = (az + b)l(cz + d), then a(w) = g;; 1 (w) = -dwcw-+ba , 10
:
---+
and for any rational function
we have
+ a(at)g;; 11 (w) + a(a2 )g;;11 (w)22 + + a(ad )g;;11 (w)dd . a(F)(w) = a(ao) a(bo) + a(bt)g;; (w) + a(b2 )g;; (w) + + a(bd)g;; (w) It is clear that a(F + G) = a(F) + a( G) and that a(FG) = a(F)a( G). However, to be an action, we must also have (ar )(F) = a( r (F) ). We use the cocycle relation to verify this condition, a(r(w)) = a(g; 1 w) = a(g; 1 )a(w) = a(g; 1 )g;; 1 (w) = g;;) (w) = (ar)(w). We now look at the subfield of k (w) that is fixed by the twisted action, = {F E K(w) : a( F) = F for all a E Gal(KIK)}. · ··
· · ·
K
We prove that the field K has the following properties:
100ne says that g is a continuous 1-cocycle for the profinite topology on Gal( K/ K) and the discrete topology on PGLz (K).
4. Families of Dynamical Systems
212
(i) K n k = K. (ii) K has transcendence degree 1 over K. (iii) kK
=
k(z).
To verify (i), let a E K n k . Then a E k , so the action of Gal( k I K) on a is the usual Galois action. On the other hand, the fact that a E K means that the Galois action is trivial. Hence a E K. This shows that K n k c K, and the opposite inclusion is obvious. Next let Ll K be a finite Galois extension with the property that g7 1 for every T E Gal( k I L ) . This implies that the action of an element (} E Gal(k I K) on w depends only on the image of(} in the quotient group =
Gal(LIK)
=
Gal(KIK)I Gal(KIL) ,
since ifT E Gal(KIL), then ((J T ) ( ) W
= (}
(gT ( )) W
=
(} ( ) . W
It follows that the polynomial P(T)
=
II
(T - A(w))
=
II
(T - g): 1 (w)) E k (w) [T] (4.54)
.XEGal(L/K)
.XEGal(L/K)
is well-defined. Further, we claim that the coefficients of P(T) are in K. To see this, for any (} E Gal(kI K) we note that (J(P) (T)
=
II
(T - (JA(w))
=
P(T) ,
.XEGal(L/K)
since the effect of replacing A with (}A is simply to permute the order of the factors in the product. We have now constructed a polynomial P(T) E K[T], and we observe that w is a root of P(T). Hence by definition, the extension K( w) I K is an algebraic extension. Since the element w is transcendental over K, this proves that K has transcendence degree at least 1 over K. On the other hand, K is contained in the field k (w), and it is clear that k ( w) I k has transcendence degree 1 over K. Therefore K has tran scendence degree exactly 1 over K, which proves (ii). Finally, consider the splitting field .C over K of the polynomial P(T). As already noted, we have w E .C. The definition of P(T) shows that there is a natural surjection Gal(LI K) -----* Gal(.CIK), which implies in particular that .C LK. Hence w E LK, which proves that K(z) c KK. This gives (iii), since the other inclusion is true from the definition of K. We now have the tools needed to complete the proof of Proposition 4.87, but we pause briefly for an example illustrating the general construction. =
213
4.10. Fields of Definition and the Field of Moduli
Example 4.88. Define a map by the rule
g
= i, ga (z) = { z- 11z ifcr(i) . f (T ( z ) - -z. 1
"
.
It is easy to check that is a 1-cocycle, and indeed it is the 1-cocycle described in Example 4.85. Fix an embedding of Ql into C and let Gal(QliQ) denote complex conjugation. The twisted action on Ql( is given by
g
p(a) =
pE
w)
a E Ql and p(w) = -1lw. The field is the subfield of Ql( w) consisting of elements that are fixed by the twisted action. The coefficients of the polynomial P(T) defined by equation (4.54) /(
a
for
give us elements in JC,
P(T) = (T - w)(T - p(w)) = (T - w) (T + � ) = T2 - (w - � ) T - 1. This yields one interesting element in JC , namely u = w - w - 1 , and we observe that the quantity v = i( w + w - 1 ) gives another element in JC. It is not hard to show that and v generate JC, = K(u, v) = K (w - �' i (w � )) . (See Exercise 4.45.) Of course, u and v are not independent, since 2 u2 v 2 = (w - �) + (i (w + �)) = -4. The field is the function field of the curve C u2 + v 2 = -4. Notice that C is defined over Q, and C is Ql-isomorphic to JP 1 , but C is not «;))-isomorphic to JP 1 , since C(Q) = 0. From our general theory, the fact that C is a nontrivial twist of JP1 IQ is equivalent to the fact (proven by a direct calculation in Example 4.85) that the 1-cocycle g is not a 1-coboundary. u
/(
+
+
2
J(
:
Resuming the proof of Proposition 4.87, we have constructed a field J( that is the function field of a curve CI K, and we have an isomorphism
= k(w) � k(z), w z, that induces a K-isomorphism C ---+ JP1 . In other words, the functions w on C and z on JP 1 are related by the formula w = z i. The curve C is a twist of lP 1 I K, and its associated cocycle is given by cr ---+ icr(i- 1 ). We compute KJC
+------+
i
:
o
214
4. Families of Dynamical Systems
since a(z) = z, z o a(i) = a(z o i) = a(w) since w = z o i, 1 by definition of the twisted action on k ( w), = g;; (w) dw - b where 9a (z) = az + b E PGL2 (K),-cw dz o +i -a b since w = z o i,cz + d -cz o i + a = Z 0 9a- 1 0 z. Thus a(i) = g;; 1 o i, which proves that 9a = ia(i - 1 ) is the 1-cocycle associated to the k-isomorphism C IP'1 . This completes the proof that the algebraic curve C is a twist of IP' 1 IK whose associated 1-cocycle is Finally, Proposition 4.77 tells us that C is the trivial twist --
.
i
:
--+
g.
D
if and only if its associated 1-cocycle is a 1-coboundary.
=
K(z)
Returning to the question of FOD FOM, let ¢(z) E be a rational func tion with field of moduli K and trivial automorphism group. We have constructed a 1-cocycle Ga K) PGL2 (K) that is associated to ¢ (Proposition 4.86) and a twist C,p of K that is associated to the 1-cocycle (Proposition 4.87). We have also proven the following chain of equivalences:
g : l(kI
--+
IP'1 I
g
K is a field of definition for ¢ ¢==} is a 1-coboundary ¢==} Cq, is K-isomorphic to IP' 1
g
(Proposition 4.86), (Proposition 4.87).
It remains to find a way of determining whether a twist of IP' 1 is K-isomorphic to We will use the Riemann-Roch theorem to provide two sufficient conditions for resolving this problem. For the convenience of the reader, we recall the general statement of the Riemann-Roch theorem, although we will need it only for curves of genus 0.
IP'1 .
I
Theorem 4.89. (Riemann-Roch Theorem) Let C K be a smooth projective curve ofgenus defined over K. (a) There is a divisor on C ofdegree that is defined over K. (b) Let D be a divisor on C that is defined over K and assume that the degree ofD satisfies deg(D) 2: 1. Then there is afunction f E K(C) satisfying
g
2g -
2g - 2
div(f)
+ D 2: 0.
Corollary 4.90. (Riemann-Roch in Genus 0) Let C K be a smooth projective curve ofgenus 0 defined over K. (a) There is a K-rational divisor ofdegree on C.
2
I
215
4.10. Fields of Definition and the Field of Moduli
(b) Let D E Div(C) be a divisor defined over K and satisjjling deg(D) there is afunction f E K(C) with div f D 2 0.
( )+
2
1. Then
Proof The proof of the Riemann-Roch theorem over an algebraically closed field
is given in most introductory texts on algebraic geometry, such as [198, IV. 1 .3], or see [255, Chapter I] for an elementary proof due to Weil and [41 0, II, §5] for an overview. The divisor of degree 2g - 2 in (a) is a canonical divisor, i.e., the divisor of any K-rational differential form such as df for any nonzero f E K (C). In particular, for curves of genus 0 we get a K-rational divisor D of degree -2, so -D is a K rational divisor of degree 2. D Proposition 4.91. Let C be a twist ofiP'1 I K. Thefollowing are equivalent:
(a) C is the trivial twist of IP' 1 I K, i.e., C is K-isomorphic to IP'1 . (b) C(K) is nonempty, i.e., C has a point with coordinates in K. (c) There is a divisor D ofodd degree such that D is defined on IP'1 =
I: ni (Pi )
(R) I:n (a(P )) I:n (P ) as a i i i i
over K, i.e., for all a E Gal(KIK) we have formal sum ofpoints.
=
Proof If C is the trivial twist of!P'1 I K, then there is a K-isomorphism j C IP' 1 . In particular, j C K) IP' 1 ( K) is a bij ection, so C K) is certainly nonempty.
( This proves that (a) implies (b). :
--+
P
:
(
--+
(P)
Next, it is clear that (b) implies (c), since if E C(K), then the divisor D has odd degree (one is an odd number!) and is clearly defined over K. Finally, suppose that the degree deg(D) of D is odd and that D is defined over K. We also use the fact that C and IP' 1 are -isomorphic, so in particular C is a curve of genus zero. It follows from the Riemann-Roch the orem (Corollary 4.90(a)) that there is a K-rational divisor on C having degree -2, say
n=
=
= I: ni
= I:Rni (Pi )
Consider the divisor E
= D + -n -2-1 D' = (PI ) + (P2 ) + + (Pn ) - -n -2-1 ((QI ) + (Q2 )). ···
The divisor E is defined over K and has degree 1, so the Riemann-Roch theorem (Corollary 4.90(b)) tells us that there is a rational function 'if; on C such that 'if; is defined over K and 'if; has degree In other words, 'if; is a map 'if; C IP' 1 of degree 1 defined over K, and hence C is K-isomorphic to IP'1 . This shows that (c) implies (a) and completes the proof of the theorem. D
1.
:
--+
We have now assembled all of the tools that we need to prove the main theorem of this section. We state the theorem in full generality, but give the proof only for rational maps with trivial automorphism groups. The general case is proven similarly, but there are many additional technical complications. Theorem 4.92. Let ¢( z ) E
z ) be a rational map of degree d 2 2. Then the field of moduli of¢ is afield ofdefinition for ¢ in the following two situations:
R(
216
(a) (b)
4. Families of Dynamical Systems
¢(z) has even degree. ¢( z) is a polynomial.
Proof We prove the theorem under the assumption that Aut(¢) = 1. See [414] for
a proof in the general case. Without loss of generality, we may assume that K is the field of moduli of¢. Let Gal( R I K) PGL2 K) be the 1-cocycle associated to ¢ (Proposition 4.86) and let Cq, be the twist of I K associated to the 1-cocycle (Proposition 4.87). This means that there is a R -isomorphism
g
:
--+
lP'1 (
g
such that Using many of the results proven in this chapter, we have the following chain of equivalences: K is a field of definition for ¢ {=:::}
g is a 1-coboundary
{=:::}
Cq,(K) is not empty
{=:::}
{=:::}
'1/J
:
Cq, is K-isomorphic to lP' 1
(There is a divisor on Cq, (K) of) odd degree and defined over K
(Proposition 4.86), (Proposition 4.87), (Proposition 4.91 ) , (Proposition 4.91 ).
We are going to produce divisors and points on the curve Cq, using the map Cq, Cq, defined by the composition --+
- 1 ¢i is defined over K. To verify this, we i '1/J a ( a('l/J) a(i- 1 ¢i) = a(C 1 )a(¢)a(i) = a(i- 1 )¢9" a(i) a(i- 1 )g;/ ¢gaa(i) = a(i- 1 )(ia(C 1 )) - 1 ¢ia(i- 1 )a(i) C 1 ¢i '1/J. Hence '1/J Cq, Cq, is defined over K. We also note that ¢ and '1/J have the same degree, since i is an isomorphism. (a) Let D,p be the divisor of fixed points of '1/J, that is, the collection of fixed points of '1/J counted with appropriate multiplicities. In the language of algebraic geometry, D,p is the pullback by the diagonal map P 1------t ( P, P) , ofthe graph { (x, 'ljJ(x)) : x E Cq,} of'ljJ. A map of degree d has exactly d + 1 fixed points (counted with multiplicities), and if the map is defined over K, then the divisor of fixed points is defined over K,
We begin by checking that the map let E G al KI K) and compute
=
=
=
:
=
=
--+
i.e., it is fixed by the Galois group. Thus D,p is a divisor of degree d + 1 on Cq,
4.10. Fields of Definition and the Field of Moduli
217
and D,p is defined over K. By assumption, d is even, so D,p has odd degree. Hence by the chain of equivalences derived earlier, the field K is a field of definition for ¢. (b) The map ¢ lP'1 lP' 1 is a polynomial by assumption, so it has a totally ramified fixed point P. In other words, ¢( P) = P and the ramification index of¢ at P satisfies p ( ¢) d. The map i C
e
=
____,
:
is a totally ramified fixed point of 7/J. Suppose first that P is the only fixed point of ¢ in lP'1 ( K) with ramification index d. Then Q is the only fixed point of 7/J in C
7/J
since is defined over K. In particular, taking R Q to be the given fixed point of ramification index d, we see that 7/J(O"(Q)) = O"(Q) and d = = so ( Q) is also a fixed point of 7/J of ramification index d. Hence O" ( Q) Q, and since this holds for every O" E Gal(K / K), we conclude that Q E C¢(K). Hence C¢(K) is not empty, so we again conclude that K is a field of definition for ¢. We are left to consider the case that ¢ has a second fixed point of ramification index d, say ¢( P') = P' and P ' ( ¢) = d. (Note that the Riemann-Hurwitz formula, Theorem 1 . 1 , precludes more than two such points.) Choose some linear fractional transformation h E PGL2 (K) satisfying h(oo) P and h(O) P'. Then ¢h satisfies =
eq(7/J) er7(Q) (7/J),
O"
=
e
=
=
so ¢h must have the form ¢h ( z) = czd for some c E K*. But any rational map of this form has a nontrivial automorphism group. Indeed, its automorphism group is a dihedral group of order 2 ( d - 1) generated by the maps z
f---+
and
(z
z
f---+
a z,
/
where cd -1 = 1 and ad- 1 = 1 I c. In any case, we have ruled out this case by our assumption that Aut(¢) = 1, which completes the proof of Theorem 4.92. D Remark 4.93. The distinction between the field of moduli and fields of defini
tion is important in the study of abelian varieties; see for example the work of Shimura [398]. More recently, the FOM FOD question has been investigated for the collection of of covering maps ¢ X B up to automorphism of the base B. In particular, this is much studied for B = lP' 1 (cf. Grothendieck's "dessins d'enfant"). If also X B JP' l , then one studies the set of rational maps ¢( z) E ( ) un der the left composition equivalence relation ¢ rv f¢ for f E PGL2 (K). This bears =
:
=
=
____,
k
z
218
4. Families of Dynamical Systems
considerable resemblance to the material in this section, where we use instead the equivalence relation ¢ "' f -1 ¢f, but there are significant differences. For example, Couveignes [I l l] shows that using the relation equivalence ¢ "' f¢, there are poly nomials in Q[z] with FOM =1- FOD, in direct contrast to Theorem 4.92. For further results, see for example work of Debes and Douai [ 1 17, 1 18, 1 19] and Debes and Harbater [120]. 4.1 1
Minimal Resultants and Minimal Models
Let P(X, Y) = 2: ai1 x i y1 E K[X, Y] be a polynomial and let p be a prime of K. We define the order of P at p to be
Notice that ordp (P) = 0 if and only if all of the coefficients of ¢ are p-integral and at least one coefficient is a p-unit. In Section 2.4 we defined the resultant of a rational map ¢ (with respect to p) by writing ¢ = [F, G] using homogeneous polynomials F, G E K[X, Y] satisfying min { ordp (F), ordp (G) } = 0 and setting Resp (¢) = Res(F, G).
The resultant Resp ( ¢) is well-defined up to multiplication by the 2tfh power of a p-adic unit, so in particular ordp (Resp ( ¢)) depends only on ¢ and p. We also recall
that ¢ has good reduction at p if and only if its resultant is a p-adic unit. Example 4.94. Even if¢ has bad reduction at p, it may be possible to change coordi nates and achieve good reduction. In other words, there may be some f E PG L2 ( K) such that Res(¢!) is a p-unit. For example, the map ¢(z) z + p2 z -1 has bad re duction at p, since =
However, if we let f ( z) = pz, then ¢! ( z) = z + z -1 , which has good reduction at p. Our aim in this section is to study this phenomenon. In particular, we study the extent to which we can eliminate, or at least ameliorate, bad reduction in ¢ by con jugating ¢ with a linear fractional transformation in PGL 2 (K). Let ¢ = [F, G] be a rational map given by homogeneous polynomi als F, G E K[X, Y], let f E PGL2 (K), and choose a matrix
Definition.
representing f. We define polynomials FA, GA E K [X, Y] by the formulas
4.11. Minimal Resultants and Minimal Models
FA(X, Y) = 8F(aX + (3Y, "(X + 8Y) - (3G(aX + (3Y, "(X + 8Y), GA(X, Y) = -"(F(aX + (3Y, "(X + 8Y) + aG(aX + (3Y, "(X + 8Y).
219
(4.55) (4.56)
It is often convenient to write this in matrix notation as
[��] = A [� : �] , adj
Aa = ( .!'"'� -!) is the adjoint matrix to A. Note that the conjugate cpf = f-1 o cjJ o f of cp by f is equal to ¢1 (X, Y) = [FA(X, Y), GA(X, Y) j . Proposition 4.95. Let cjJ = [F, G] be a rational map ofdegree d described by homo geneous polynomials F, G E K[X , Y] and let p be a prime ideal. (a) The valuation of the minimal resultant of cp is given by the formula ordp (Resp (cjJ) ) = ordp (Res(F, G)) - 2d min{ ordp (F), ordp (G)}. (4.57) Note that there is no requirement that the coefficients ofF and G be p-integral or that some coefficient be a p-adic unit. (b) Let A E GL2 (K). Then with FA and GA defined by (4.55) and (4.56), ordp (Res( FA, GA)) = ordp (Res(F, G)) + (d2 +d) ordp (det A), min{ ordp(FA), ordp(GA) } 2 min{ ordp (F), ordp (G) } + (d + 1) ordp(A), where ordp (A) denotes the minimum of the order of the coordinates of the ma trix A. (c) Inparticular, ifU E GL2 (Rp). then ordp (Res(Fu, Gu)) = ordp (Res(F, G)), min{ ordp(Fu ), ordp( Gu) } = min{ordp (F), ordp (G)}. Proof (a) Choose a constant c E K* satisfying ordp(c) = min{ordp(F),ordp(G) } . where
d
j
Then
where we have made use of the homogeneity property of the resultant (Proposi tion 2.13(d)). This gives the desired result (4.57). (b) An elementary calculation (see Exercise 2.7(c)) shows that
Res(FA,GA) = (detA)d2 +d Res(F,G), so taking ordp gives the first part of (b). For the second part, we observe that every coefficient ofFA and GA is a sum of terms of the form
220
4. Families of Dynamical Systems
( ofcoefficient ) ( homogeneous polynomial of ) degree d + 1 in Z[a, (3, 8] F or G x
Hence
'J,
·
{ ordp (FA), ordp (GA)} 2: min{ ordp (F), ordp (G)} + (d + 1) ordp (A). ( ) The assumption that U E G1 2 ( Rp) is equivalent to the two conditions ordp(U) 2: 0 and ordp ( det U) = 0, i.e., the coefficients of U are p-adic integers and the determinant is a p-adic unit. Applying (b) with A = U gives ordp (Res(Fu, Gu)) = ordp (Res(F, G)), min { ordp(Fu ), ordp(Gu)} 2: min { ordp(F), ordp(G) }. (4.58) This almost completes (c). We next apply (b) to the polynomials Fu and Gu and to the matrix A u - 1 E GL 2 (Rp). Using the fact that (Fu )u- 1 = F and similarly for G, we find from (b) that min { ordp (F), ordp (G)} = min { ordp ( (Fu )u-1 ), ordp (( Gu )u-1)} 2: min { ordp (Fu ), ordp ( Gu)}. This gives the opposite inequality to (4.58), which completes the proof of (c). 0 min
c
=
Definition. Let K be a number field and let ¢(z) E each prime p of K, define a nonnegative integer by
K(z) be a rational map. For
Ep(¢) = /EPGL2 min K ordp Resp(¢1 ). ( )
In other words, ¢) is the exponent of the power of p dividing the resultant of the conjugate ¢! that is closest to having good reduction at p. Then the (global) minimal resultant of¢ is the integral ideal
Ep (
(4.59)
G (F, G)
if¢ = [F, G], the coefficients of F ordp Res(F, G) = ordp 9'\.p.
We say that is a minimal mode/for ¢ at p and are p-integral and
(See Section 6.3.5 for the analogous definition of mimimal models of an elliptic curve.) Remark 4.96. The product (4.59) defining 9'\.p makes sense, since 0 for all but finitely primes p. To see this, write ¢ for any and having coefficients in the ring of integers of K. Then there are only finitely many primes p with > 0, and it is clear that ¢) 0 for all other primes.
ordp Res( F, G)
= [F, G] Ep ( =
F Ep(¢)G=
4.1 1. Minimal Resultants and Minimal Models
221
The minimal resultant of a rational map is clearly invariant under PGL2 (K) conjugation. It measures the extent to which the conjugates of cj; have bad reduction, so provides a convenient measure of the arithmetic complexity of the conjugacy class of ¢. A coarser way to measure arithmetic complexity is simply to take the product of the primes with bad reduction, which we denote by (4.60)
1)1<1>
91¢ ,
It is clear that divides and it is tempting to conjecture an inequality in the opposite direction that would be a dynamical analogue of Szpiro's conjecture [205, F.3.2] for elliptic curves. Conjecture 4.97. Let Kj
NK/Q 91¢ ::; (NK/Q 1)1r.
The minimal resultant gives one way to measure the arithmetic complexity of a rational map, but note that there are infinitely many PGL2 (K)-inequivalent rational maps of a given degree whose minimal resultants are the same. For example, the minimal resultants of the polynomials are the same as u ranges over all units in the ring of integers of K. The moduli space Md provides an alternative way to measure the arithmetic complexity of the conjugacy class of a rational map. If we fix a projective embed ding Md JPN , then cj; E K(z) determines a point (¢) E Md(K) and we can '----7
take the height of the corresponding point in J!DN (K) . However, this way of measur
ing arithmetic complexity is also not entirely satisfactory, since twists of a rational map give the same point in Md, yet are arithmetically quite different. Note that the same situation arises in the theory of elliptic curves, where curves with the same j invariant need not be arithmetically identical. This suggests combining the primes of bad reduction with the height coming from moduli space. We do this and formulate a dynamical version of a conjecture of Lang (cf. [202], [254, page 92], or [410, VIII.9.9]). Recall that the canonical height of a point P satisfies h¢(P) = 0 if and only if P is preperiodic for cj; (Theorem 3.22). The following conjecture says that the height of nonpreperiodic points grows as cj; becomes more arithmetically complicated. Conjecture 4.98. Fix an embedding ofthe moduli space Md in projective space and let hMd denote the associated heightfunction. Let K be a numberfield and d � 2 an integer. Then there is a positive constant c = c( K, d) such thatfor all rational maps cj; E K(z) ofdegree d and all wandering (i.e., non-preperiodic) points P E J!D 1 (K),
222
4. Families of Dynamical Systems
In the theory of elliptic curves, the notion of global miminal Weierstrass equation is extremely useful; see the discussion in Section 6.3.5 and [41 0, VIII, §8] for further details. We briefly discuss a dynamical analogue. Definition. Let K be a number field and let ¢(z) E K(z) be a rational map. Then ¢ has a global minimal model if there is a linear fractional transforma tion f E PGL2 (K) and homogeneous polynomials F and G satisfying ¢1 = [F, G] with the property that the coefficients ofF and G are in the ring of integers of K and
ordp (Res(F, G)) = ordp ( ryt¢ )
for every prime p.
In other words, the pair ( F, G) is simultaneously a minimal model for ¢ at every prime p of K. In the remainder of this section we develop some tools that help to determine whether a given rational map has a global minimal model. Let K be a number field, let ¢(z) E K(z) be a rational map of degree d, and write ¢ [F, G] with polynomials F and G as usual. (a) There is a (fractional) ideal ap, c ofK satisfYing
Proposition 4.99.
=
ryt¢ =
{
Res(F, G)a}�c ifd is odd, Res(F, G)a'J:., c ifd is even.
(4.61)
(b) If d is odd, then the ideal class of aF,G depends only on ¢, independent of the choice ofF and G.
(c) Ifd is even, then aF,G depends only on ¢ up to multiplication by the square ofa principal ideal.
E PGL2 (K) with corresponding A E GL 2 (K), we use Proposi tion 4.95(a,b) to compute
Proof For any f
ordp (Resp (¢1)) = ordp (Res( FA, GA)) - 2dmin{ ordp (FA), ordp ( GA ) } = ordp (Res(F, G)) + (d2 + d) ordp ( det A) (4.62) - 2d min{ ordp (FA), ordp (GA) } .
For each prime ideal p we choose a linear fractional transformation /p E PGL2 (K) and corresponding matrix Ap so as to minimize the resultant of ¢1P . In other words, (4.63) Combining (4.62) and (4.63) yields ordp ( ryt¢ ) - ordp (Res( F, G)) = d [(d + l) ordp (det Ap) - 2 min{ordp (FAp ) , ordp (GAp ) }] . (4.64)
223
4.11. Minimal Resultants and Minimal Models
Hence if we define an ideal
aF,G by the rule
d odd, d even,
aF,G aF G G. c K* , F, G] [ F ', G'] F' cF G' cG. c2d Res(F, G)a�'fc,cc = Res(cF, cG)a�'fc,cc 91¢ = Res(F, G)a}�c · This is an equality of ideals, so by unique factorization of ideals we conclude that CClcF,cG = ClF,G · Hence the ideals ClcF,cG and ClF,G differ by a principal ideal. Next suppose that d is even. Then by a similar calculation we find that c2d Res(F, G)a�F,cG Res(cF, cG)a�F,cG = 91¢ Res(F, G)a'fc,c · Hence (c2 )acF, cG = aF, G , so ClcF,cG and ClF,G differ by the square of a principal Definition. Let K be a number field, let ¢( z) E K ( ) be a rational map of degree d, and write ¢ [F, G] with polynomials F and G as usual. If d is odd, we write 0.¢1 for the ideal class of aF, G in the ideal class group of K. If d is even, we write a¢1 for the image of aF,G in the group of fractional ideals modulo squares of principal ideals. In both cases, by analogy with the theory of minimal equations of elliptic curves (cf. [410, VIII §8]), we call a¢; the Weierstrass class of¢ over the.field K. By Proposition 4.99 the Weierstrass class a¢; depends only on ¢, independent of the chosen lift [F, G].
then (4.64) says that satisfies the desired formula (4.61 ). It remains to determine the extent to which the ideal depends on the choice ofF and Let ¢ = [ be two lifts of ¢. Then there is a constant E sat = isfying = and Suppose first that d is odd. We use (4.61) to compute =
=
=
=
��
D
z
=
K
K
K
K
The triviality of the Weierstrass class gives a necessary condition for the exis tence of a global minimal model.
K
K,K(z) PGL2 (K), assume that ¢ = [F, G] with polynomials F and G having coefficients in the ring of integers of K and satisfying
Proposition 4.100. Let be a number field and cp(z) E a rational map of degree d � 2. If ¢ has a global minimal model over then its Weierstrass class 0.¢1 K is trivial. Proof Replacing ¢ by cpf for an appropriate choice of f E we may
for every prime j:l .
It follows from the defining equation (4.61) of
aF,G that
for every prime j:l. Hence = (1), so its image in the ideal class group (if d is odd) or in the group of ideals modulo squares of principal ideals (if d is even) is also trivial. D
aF,G
Exercises
224
Exercises Section 4. 1 . Dynatomic Polynomials 4.1. Let ¢ ( z ) = z + 1/z, or in homogeneous form, ¢ ( [X, Y]) = [X 2 + Y 2 , XY]. (a) Compute the first few dynatomic polynomials il>¢ , n (X, Y) for ¢, say for n = 1, 2, 3 , 4. (b) Prove that for all n 2:: 2, the dynatomic polynomial il>¢, n satisfies il>¢ , n (±X, ±Y) = il>¢, n (X, Y). Deduce that il>¢, n (X, Y) E Z[X 2 , Y 2 ] . (c) Prove that the field 1Ql2 ,q, generated by the points of exact period 2 is the field IQl (A) . (d) Prove that the field 1Ql3 ,q, generated by the points of exact period 3 is an S3 extension of!Ql. (Hint. Show that the roots of il>q,, 3 ( 1, fo, ) E Z[w] generate a cyclic cubic extension of!Ql.) (e) Prove that ¢ , 4 (X, Y) factors into a polynomial of degree 4 and a polynomial of de gree 8. (See Exercise 4.40 for a more general result.) Describe the fields generated by the roots of each factor. 4.2. Prove the following elementary properties of the Mobius function. (See (4.2) on page 148 for the definition of the Mobius function.) (a) Let n 2:: 1 be an integer. Then
L JL( d) di n
ifn = 1, ifn 2:: 2 .
g
=
(b) Let g( n) be a function whose domain is the positive integers, and define a new func tion f (n) by f ( n) = L d in g ( d) . Prove that g(n)
=
L f (d)JL G) . di n
This is called the Mobius inversionformula. (c) Prove that the n1h cyclotomic polynomial
�(zk ki n
_
1) �(n/k)
is indeed a polynomial. (Hint. Use (b) and the fact
zk
-
1 has distinct complex roots.)
4.3. Let vd (n) = deg(¢, n ) be the number of formal n-periodic points of a rational map of degree d, counted with multiplicity (see Remark 4.3). (a) Prove that lld (1) d + 1 and =
vd ( n)
=
L JL (I) dk ki n
for n 2:: 2.
(b) Make a table of values of vd(n) for some small values of n and d. (c) Prove that formally as power series. For what range of x
> 0 do the series converge?
Exercises
225
I Type of Point
I Char K I aj,(n) values
Multiplier
aj, (n) = 0 for all n
Wandering Periodic Periodic
not root of unity root of unity
0
Periodic
root of unity
p>O
aj, (n) > O for exactly one n aj, (n) > O for exactly two n's a? ( n) > 0 for n = tpk ,
exactly two t's and all k ::0: 0
Table 4.4: Values of aj,(n). 4.4. Let ¢(z) E K (z) be a rational function of degree d ::0: 2 and suppose that z = 0 is a fixed point of ¢ with multiplier A = 1. Write ¢(z) = z + z e 'lj;(z) with e ::0: 2 and '!j;(O) =!= 0, where e = ao (¢, 1) in the notation of Theorem 4.5. (a) Prove that ¢n (z) z + nz e'lj; (z) + O (z2e- 1 ) . =
(b) Assume further that K has characteristic p > 0 . Prove that
for all k ::0:
1.
4.5. Verify that the description of the values o f aj, (n) given in Table 4.4 i s correct. 4.6. Let ¢( z) be a rational function, let n ::0: 1, let p be a prime with p f n, and let P E (a) Prove that a ? (n , rjJP ) = a ? (n , ¢) + a ? ( np , ¢) . th (b) Deduce that the n dynatomic polynomial for cjJP factors as
(c) More generally, if gcd(n , k)
=
1P' 1 .
l, prove that
*n k ,¢ - Il (n* ,¢J. ) p( k fj)
·
jjk
4.7. * Let
a _ 1 zd-l + + a 2 z2 + a1z + ao d be a monic polynomial of degree d and let �. ¢ (z) be its nth dynatomic polynomial. Each root a of � , ¢ (z) has an associated multiplier A ¢ ( a ) = ( ¢n ) ' (a) . ¢(z) = zd
+
· · ·
(a) Prove that there is a (unique) monic polynomial whose nth power satisfies
IT
;,,¢ ( a ) =O
(x - A ¢ (a) ) = Resz ( �.¢ (z) , x - (¢n )' (z) ) ,
where Resz means to take the resultant with respect to the z variable. (Hint. All n of the points in the orbit 0¢ ( cle ) have the same multiplier.)
Exercises
226 (b) Let Ck (x) E Z[x] denote the m < n, define
kth cyclotomic polynomial, and for integers min with
Prove that the discriminant of �, (x) is given by the formula
Disc �, ¢ = Hn ,¢( 1t (c) Let min with m
II !:J.;;_,r;:, .
m in m
< n. Prove that the resultant of �, and ;,, ¢ is given by the formula Res( �,¢' ;,,¢) = ±!:l. :, m .
Conclude that 1> has a point of formal period n whose exact period m is strictly smaller than n if and only if there is a point of formal period m whose multiplier is a primi tive (n/m)1h root ofunity. 4.8. This exercise describes an analogue of Theorem 4.5 for automorphisms of projective space. Let 1> : IP'N _, IP' N be an automorphism defined over a field K, i.e., 1> E PGL N +l (K). We say that 1> is nondegenerate if the equation 1>( P ) = P has only finitely many solutions in IP'N (K). (a) Let A E GL N+ 1 (K) be an invertible matrix with coefficients in K representing the map ¢>. Prove that 1> is nondegenerate if and only if every eigenspace of A has dimen sion 1. (b) Assume that 1>n is nondegenerate, let rn c IP'N X IP'N be the graph of 1>n ' and denote the diagonal map by f:J. : IP'N _, IP' N x IP'N . Following Remark 4.4, we define the 0-cycle of n-periodic points of 1> to be the pullback
¢, n
= !:J. * (f¢) = L ap( cj>, n) ( P) PE'f' N
and the 0-cycle of formal n-periodic points to be
L ll (�) kin
¢,n =
L af, (cj>, n)(P).
PEP N
IP'N be a point of primitive period m for ¢>. Prove that
a p ( n) =
{ ap(m) 0
ifmln, ifm f n,
{
ap(m) ( a*p n ) _
and
0
ifm = n, if m =1- n.
In particular,
Fn + l = Fn (F(X, Y), G(X, Y))
and
Gn + l = Gn (F(X, Y), G(X, Y)) .
(a) Prove that Fn and Gn have no common factors.
227
Exercises (b) More precisely, prove that the resultant of Fn and Gn is given by Res(Fn, Gn) = Res(F, G) (2n - l )dn - l . (Hint. Use Exercise 2. 12.) (c) Prove that ¢n = [Fn, Gn]. (d) For all n , m � 0 , prove that
Fn+m(X, Y) Fn (Fm(X, Y), Gm(X, Y)) , Gn+m(X, Y) = Gn (Fm(X, Y), Gm(X, Y)) . =
4.10. With notation as in Exercise 4.9, we define the (generalized) (m, n )-periodpolynomial of 1> to be the polynomial
m,n(X, Y) n (Fm(X, Y), Gm(X, Y)) , where n(X, Y) = YFn - XGn is the usual n-period polynomial of ¢. =
(a) Prove that
m,n(X, Y) = Gm(X, Y)Fn+m(X, Y) - Fm(X, Y)Gn+m(X, Y). (b) Let P E lP' 1 ( K). Prove that
m,n(P) = 0 if and only if ¢m+n (P) ¢m (P) . Thus P is a root of m,n if and only if P is a preperiodic point with "tail" of length at =
most m and with period dividing n. (c) Prove that for all m, n � 1, the quotient
m,n m-l,n is a polynomial. 4.11. We continue with the notation from Exercises 4.9 and 4.10. Let �(X, Y) be the n1h dy natomic polynomial of ¢. Then for m, n � 1 we define the (generalized) (m, n)-dynatomic "polynomial" of¢ to be
� (Fm(X, Y), Gm(X, Y)) . m ' n (X, Y) (X, Y) ) ;', ( Fm-l(X, Y), Gm l Prove that ;, ,n (P) = 0 if and only if 1>m (P) has formal period n. Points satisfy ing ;, ,n (P) = 0 are called preperiodic points withformal preperiod ( m, n). ** Prove (or disprove) that ;, ,n (X, Y) is a polynomial. *
(a) (b)
=
Section 4.2. Quadratic Polynomials and Dynatomic Modular Curves 4.12. We continue with the notation from Exercise 4.7. Thus for any monic polynomial ¢(z), we define a monic polynomial on ,q,(x) by
4>n.q,(<>)=O
4> n .q,(<>)=O
and for min with m < n we set
�n,m = Res(Cn;m (x), 8m, q, (x)), where Ck (x) is the k1h cyclotomic polynomial. We now specialize to 1>c(z) write 8n(c, x) 8n, ¢c (x) and �n,m (c) to indicate the dependence on c. =
=
z2 + c and
228
Exercises
{a) Prove that Dn (c, x) E Z[c, x] and that � n ,m (c) explain the powers that appear in Table 4.3. (b) Prove that
E Z[c]. Then use Exercise 4.7(c) to
deg z
=
T
=
(c)
=
and deduce that the set
{c E Q : �n , m(c)
=
0 for some m <
n with m i n }
is a set of bounded height. Conclude that the set of bifurcation points in the Mandelbrot set M is a set of bounded height. (Hint. The formula � n ,m (c) = W n , m(4c) bounds the denominator of c, then mimic the proof of Proposition 4.22.) (d) Prove that �n ,l (c) and �2 n , 2 (c) are irreducible in IQI[c] for all n 2: 1. (e) ** Prove the following conjecture o f Morton-Vivaldi [3 14]: The polynomial �n ,m(c) is irreducible in IQI[c] for all m i n with m < n. 4.13. Let ¢c(z) = z2 + c and write �(c, z) for the n1h dynatomic polynomial of cPc· Con tinuing with the notation from Exercise 4. 1 2, we let 8n (c, x) E Z[c, x] be the polynomial defined by (4.65) and Jn (c, x) = T deg
Z(n) : 8n (y, x)
=
0.
Prove that there is a morphism
(b) Define an action of i
E Z/n:Z on Y1 ( n) by
(y, z)
=
(y , ¢� (z)) .
(See Section 4.2.3.) Prove that the map F in (b) is invariant for this action, i.e., prove that F (y, ¢� (z)) = F (y, z) for all i E Z/n:Z. Deduce that there is a unique morphism Y0(n)
---> Z(n) such that the composition
Y1 (n) ---> Yo(n) ---> Z(n) is the map F. (c) Prove that the map Yo (n) ---> Z ( n) in (b) has degree 1, so the equation 8n ( y, x) = 0 gives a (possibly singular) model for Y0 (n). (d) Prove that Z(1) and Z(2) are nonsingular and that Z(3) and Z(4) are singular. Resolve the singularities of Z ( 3) and Z (4) and check directly that Yo (3) and Yo (4) are curves of genus 0.
229
Exercises
4.14. Let ¢c(z) = z2 + c and let
1) , ( 2, 1), (3, 1) ,
(4, 1), ( 1 , 2 ) , ( 2, 2) , (3 , 2 ) , (4, 2 ) , (1, 3), ( 2, 3) , ( 1 , 4) } . (b) From your list, it should be apparent that many of the leading terms of <1>;',, 1 and <1>;',, 2
{ ( 1,
coincide. Prove that
deg(m , 2 - <1>m , 1 ) *
*
=
1 ( * ) 2 deg <1>m ,2
=
1
2 deg ( <1>m ,1 ) *
for all m � 1.
(Note that this is a special property of the generalized dynatomic polynomials for ¢c.) 4.15. Let c E C and let holomorphic function
¢c(z) 1/Jc
satisfying
z2
=
:
{
Z
+
c. Prove that there is a number Rc > 0 and a
E C : lzl > Rc }
--+
C
and (Hint. Show that there is a consistent way to choose square roots so that 1/Jc can be defined as
limn�oo 2� .)
4.16. Consider the doubling map
D
:
Q/71..
--+
Q/71.. ,
D(t)
=
2t mod 71.. .
Fix t E Q/71.. and let m � 0 and n � 1 be the smallest integers such that the denominator oft divides 2 m ( 2 n - 1). (a) Prove that t is periodic for D if m = 0 and that t is strictly preperiodic for D if m � 1. (b) Prove that m � 0 and n � 1 are the smallest positive integers such that t can be written as a (not necessarily reduced) fraction of the form t -
a
- 2m ( 2n - 1) . -=---:-c:-----:-;-
(c) If t is preperiodic for D, prove that the preperiod of t is equal to m and that Dm ( t) is periodic with exact period n. (d) For t as above, we say that t is of type (m, n) for the doubling map D. For a given pair ofpositive integers (m, n) , how many t E Q/71.. are of type ( m, n)? The map D is used in the analytic characterization of Misiurewicz points; see Theorem 4.25. However, note that (d) does not count the exact number ofMisiurewicz points of type (m, n ) , because distinct t may give the same Misiurewicz point. 4.17. ** Let ¢c(z) (of c) by
Fm n (c) '
=
IT kin
=
z2 + c as usual, and for integers m, n � 1, define a rational function
( ¢�¢-�+l+kk (O)(O) -- ¢¢��-(0)1 (0) ) J.t(n/k)
Then set
Gm ' n (c)
=
(c) { FFmm,n,n(c)/ F1,n(c)
=
n fl ( "'f'mc -1+k (O) + "'f'mc - 1 (0) ) J.t( / k) . kin
ifm '/= 1 (mod n) or m = 1, if m = 1 (mod n ) and m =/=- 1.
230
Exercises
(a) Prove that Gm,n(c) is a polynomial in c. More precisely, prove that Gm,n(c) is in Z[c] . (b) If n ;:::: 2, prove that the roots of Gm,n are the Misiurewicz points of type (m, n). (c) Prove that Gm,n(c) is irreducible in Q[c]. 4.18. Let V be a variety and suppose that the points of V algebraically parameterize a family of quadratic polynomials 'ljJ together with a marked point .X of formal period n. Theorem 4. 1 1 says that there is a unique morphism TJ : V __.. Y1 ( n) satisfying
TJ (P) = ('lj;p (z), .X( P )) E Form al (n ) . Note that by construction, each point 7 ( c, ) E Y1 ( n) is identified with a quadratic =
a
polynomial ¢-r and point J-t ("/) of formal period n via ¢-r ( z )
= z2 + c
and J-t("/)
= a.
Prove that there is a unique morphism of varieties
f V :
____,
PGL2
such that the following identities are true for all P E (a)
( fp 1 '1/Jp jp )( z ) = ¢TJ(P) (z). ( TJ r 1 .X)( P ) = (J-t TJ)(P). 0
V(K):
0
(b) o o o In other words, prove that the following two diagrams commute: .pf
f - l o )>.
JP'�
----+
JP'�
v
----+
JP'�
JP'�l (n)
----+
.p
JP'1Y1(n)
Y1 (n)
I" ----+
JP'Y1(n) 1
lT}
lT}
lT}
lT}
(Hint. During the proof of Theorem 4. 1 1 we defined a linear fractional transformation fp ( z). Prove that fp is uniquely determined by P. Deduce that the map P >---> fp is a morphism.)
4.19. In Example 4.14 we gave an explicit description of the quotient map Y1 (2) Perform a similar analysis and describe the map Y1 (3) __.. Yo ( 3).
__..
Y0(2).
4.20. We proved that the dynatomic modular curves X1 ( 1), X1 ( 2), and X1 ( 3) are isomorphic to JP' 1 . This exercise asks you to investigate X1 (n) for other small values of n. (a) Prove that X1 (4) is a curve of genus 2 and Xo(4) is a curve of genus 1. (b) Prove that Xo(4)(1Ql) is finite, and Yo(4)(1Ql) is the empty set. (c) Prove that X1 (5) has genus 14 and Xo(5) has genus 2. (d) * Prove that X0(5)(1Ql) is finite, and Yo(5)(1Ql) is the empty set. (e) Compute the genera of X1 (6) and Xo(6). (f) ** Find all rational points in Y0(6)(1Ql). Section 4.3. The Space Ratd of Rational Functions 4.21. Let ¢ = [Fa, Fb] E Ratd be a rational map of degree d. Prove that the following are equivalent: (a) At least one of the fixed points of ¢ has multiplier equal to 1 . (b) The polynomial Y Fb X Fa has a multiple root. -
Exercises
231
Deduce that there is a nonzero homogeneous polynomial D(a, b) E Q[ao , . . . , bd] such that (4> has a fixed point whose) . . 11er equals 1 muItip
-¢==?
D(a, b) = 0.
Hence
{ 4> E Ratd : 4> has a fixed point whose multiplier equals 1 } is a proper Zariski closed subset ofRatd. Section 4.4. The Moduli Space Md of Dynamical Systems
Let GLn be the group of n x n matrices with nonzero determinant, let SLn be its subgroup of matrices with determinant 1, let PGLn be the quotient ofGLn by its subgroup of diagonal matrices, and similarly let PSLn be the quotient of SLn by its subgroup of diagonal matrices. (a) Let K be an algebraically closed field. Prove that the natural map from PSLn (K) to PGLn (K) is an isomorphism. (b) More generally, prove that for any field K there is an exact sequence 1 PSLn (K) PGLn (K) � K*/K* n ----+ 1 . 4.22.
----+
----+
4.23. We say that a separable rational map c/>(z) E K(z) is very highly ramified if there is a point P E lP'1 such that the ramification index of 4> satisfies P ( 4>) ;::: 3. Let e
V
=
{ 4> E Ratd : 4> is very highly ramified} .
(a) Prove that V is a proper Zariski closed subset ofRatd. Hence "most" rational maps are not very highly ramified. (One says that a generic map of degree d is not very highly ramified.) (b) Prove that V is invariant under the conjugation action of PGL2 . (c) Prove that the quotient V/ PSL2 is a Zariski closed subset of Md = Ratd / PSL2 . 4.24. Let c/>(z) E K(z) be a nonconstant rational function. The Schwarzian derivative Sc/> of 4> is the function � c/>"(z) 2 (Sc/>)(z) = c/>"'(z) c/>' (z) 2 c/>' (z) It measures the difference between 4> and the best approximation to 4> by linear fractional transformations. (a) If f(z) (az+ b)/(cz+d) is a linear fractional transformation, prove that (Sf)(z) = 0. (b) Let f be a linear fractional transformation. Prove that 4> and f o 4> have the same Schwarzian derivative. (c) Let c/>(z), 'lj;(z) E K(z) be nonconstant rational functions. Prove that
( )
=
(S(c/> o '¢)) (z)
=
(Sc/>) ('¢(z)) · ('¢' (z)) 2 + (S'Ij;)(z).
(d) In particular, iff is a linear fractional transformation, prove that and deduce that the quadratic differential form
232
Exercises
wq, (z)
=
(S¢)(z) (dz) 2
is invariant under the substitution ( ¢, z) f-+ ( ¢1, f - 1 z ) . Thus the map ¢ a natural map Md (quadratic differential forms on lP' 1 ). (e) Suppose that ¢(z) has a multiple zero or pole at z = a , say
f-+
wq, induces
---->
¢(z) a(z - a) m + =
with a =/= 0 and lml
· · ·
2: 2.
Prove that S¢ has a double pole at a. More precisely, prove that the Laurent series expansion of S¢ around a looks like
(S¢)(z)
=
1 - m2
-- (z - a ) - 2 + · · · . 2
(f) Prove that the map
(¢, P)
,___..
(S¢)(P),
is a morphism.
** An algebraic variety V is called unirational if there is a rational map lP'N ---+ V whose image is Zariski dense in V, and the variety V is called rational if there is a rational map lP'N V that is an isomorphism from an open subset of lP'N to an open subset of V. It is clear that every moduli space Md is unirational, since Ratd is an open subset of lP'2 d 1 and the map Ratd Md is surjective. We also know that M 2 is rational, since Theorem 4.56 says that M2 � A 2. For which values of d is Md a rational variety? In particular, is M3 rational? 4.25.
---+
+
---+
Section 4.5. Periodic Points, Multipliers, and the Multiplier Spectrum
This exercise asks you to prove the part of Theorem 4.50 that was left undone in the text. Prove that the map
4.26.
defines a function in Q[Ratd] . Prove that it is PGL2-invariant and deduce that it defines a function in Q[Md] . 4.27.
** What is the degree of the map
U3 ,N : M3 ----> A
k
for sufficiently large N? Similarly, what is the degree of 4 ,N on M4 away from the Lattes locus on which it is constant? u
Section 4.6. The Moduli Space
M2 of Dynamical Systems of Degree 2
4.28. Prove that ¢ E Rat2 is conjugate to a polynomial if and only polynomial maps in M2 � A2 trace out the line 2. x =
cr1
( ¢)
=
2. Thus the
4.29. Let ¢ E Rat 2 ( q be a rational map of degree 2 and suppose that one of its fixed points has a nonzero multiplier .-\.
233
Exercises (a) Prove that there are an f E
PGL2(1C) and a c E C such that
q/(z) � (z + c + �) . =
(4.66)
z + +
/
This generalizes the normal form given in Lemma 4.59. (b) Verify that the multiplier of ¢/ at the fixed point = oo is A00 ( ¢ ) A. (c) Let A 1 , A2, A3 be the multipliers of the fixed points of ¢ with A1 = A. Prove that the number in (4.66) satisfies c2 = 4 - A 1 (2 A2 A3).
c
=
4.30. Fix d 2: 2 and consider the subset of Ratd defined by
Bi Critd For d
=
ofRatd.
=
{¢ E
2 we have BiCrit2
=
Ratd
:
¢ has exactly two critical points}.
Rat2, but for larger d the set BiCritd is a proper subset
(a) Prove that BiCritd is an algebraic variety of dimension 5. (b) Prove that conjugation induces a natural action of PGL2 on morphism of varieties
(c) Suppose that ¢ E
BiCritd has critical points at 0 and
oo.
BiCritd,
Prove that ¢(
i.e., there is a
z) has the form (4.67)
(d) Let ¢ E BiCritd and apply a conjugation to move the critical points of ¢ to 0 and oo, so ¢ has the form (4.67) described in (c). Prove that the following quantities depend only on the conjugacy class of ¢:
(e) Let ¢, 'lj; E
BiCritd and suppose that w (¢) w ('lj;), II(¢) = u('lj;) , =
T (cp)
=
T ('lj;) .
Prove that ¢ and 'lj; are PGL2 conjugate (working over an algebraically closed field). (f) Prove that the three quantities (4 . 68) described in (d) satisfy the relation
+
UT wd - 1 (w l ) d+ 1 2 and no other relations. Conclude that M�icrit is isomorphic to A . (This generalizes The iCri orem 4.56, since M� t M 2.) =
=
(g) ** Describe the stable and semistable completions of M�icrit coming from geometric invariant theory. Section 4.7. Automorphisms and Twists 4.31. In Section 4.7 we defined two rational maps ¢ and 'lj; to be equivalent if there is an f E PGL2 (K) such that 'lj; = ¢! , and similarly to be K-equivalent if there is an f E PGL2(K) such that 'lj; = ¢! .
234
Exercises
(a) Prove that these definitions do indeed define equivalence relations on the set of rational maps. (b) More generally, suppose that a group G acts on a set X. Define a relation on X by setting y if there exists a E G such that y = ( ) Prove that this is equivalence relation. 4.32. Let ¢(z) E K(z) be a rational map. (a) Prove that Aut(¢) is a subgroup ofPGL2(K). (b) Let h E PGL2(K). Prove that Aut(¢h) = h -1 Aut(¢) h, so Aut(¢) and Aut(¢h) are conjugate subgroups ofPGL2(K). "'
x
a x .
a
an
4.33. Let ¢(z) E K(z), let f E Aut(¢), and let a be a critical point of ¢ (i.e., ¢'(a) = 0). Prove that f (a) is also a critical point of ¢. More generally, prove that a and f (a) have the same ramification index. 4.34. Describe all polynomials ¢(z) E K[z] whose automorphism group Aut(¢) is nontriv ial. (Hint. Iff E Aut(¢), what does f do to the totally ramified fixed point(s) of¢?) 4.35. Let ¢( z) E K ( z) be a rational map of degree d and write ¢( z) as a quotient of polyno mials d o ¢(z) abo + ab 1 z + .· .· ·· ++ bad zd · + 1z + dz Prove that f(z) = z -1 is in Aut(¢) if and only ifb; = ad - i for all i. 2 2z)/( -2z + 4.36. Let ¢(z) = ( z (a) Prove that Aut(¢) contains the maps z-1 z , =
1 ). { z, -,1 --, -1-, 1 --1 , 1 - z }
-
z
z
- z
z -
and that they form a group isomorphic to Ss . Prove that # Aut ( ¢) 6, so in fact Aut(¢) � Ss . (Hint. Find the fixed points of¢.) (b) Compute the values of a1 ( ¢), a2 ( ¢), and ( ¢). 4.37. Let ( E K be a primitive nth root of unity and let ¢(z) E K(z) be a nonconstant rational map. (a) Prove that Aut(¢) contains the map f(z) = (z if and only if there is a rational map 'lj;(z) E K(z) such that ¢(z) z'lj;(zn ). (b) Prove that Aut(¢) contains both of the maps f(z) (z and g(z) = 1/z if and only if there is a polynomial F(z) E K[z] such that the rational map 'lj;(z) E K(z) in (a) has the form 'lj;(z) F(z)/(zd F(z - 1 )), where d deg(F). Verify that the group generated by f and g is the dihedral group of order 2n. (c) Let ¢(z) = z d . Prove that Aut ( ¢) is a dihedral group of order 2d - 2. 2 4.38. We identify the set of rational functions Ratd of degree d with an open subset of !P' d + 1 as described in Section 4.3. (a) Let f E PGL2. Prove that the set {¢ E Ratd : qyf = ¢} is a (possibly empty) Zariski closed subset ofRatd. If f(z) i- z, prove that it is a proper subset ofRatd . (b) Let A C PGL2(K) be a nontrivial finite subgroup. Prove that the set a3
=
=
=
=
{ ¢ E Rat d ¢1 = ¢ for all f E A} is a proper Zariski closed subset ofRatd. :
235
Exercises
(c) Prove that up to conjugation, PGL2 ( K) has only finitely many distinct finite subgroups of any given order. (d) Prove that the set { ¢ E Ratd : Aut(¢) -1- 1} is a proper Zariski closed subset of Rat d. (Hint. Note that the order of Aut(¢) is bounded by Proposition 4.65.) (e) Prove that the set {¢ E Ratd : Aut(¢) #- 1} is PGL2-invariant and defines a proper Zariski closed subset of Md. (Hint. The groups Aut(¢) and Aut(¢1) are conjugate subgroups ofPGL2; see Remark 4.64 and Exercise 4.32.) 4.39.
Let a E C* and d � 1 and consider the rational function 1 d ¢(z) = a z +
( ) z-1
Prove that Aut(¢) = {f E PGL2(1C) : ¢1 = ¢} is trivial. 4.40. Let ¢( z) E K ( z) be a rational map of degree d � 2 and assume that Aut ( ¢) contains the element h(z) = -z oforder 2. (a) Suppose that d is even. Prove that at least one of the fixed points of ¢ ( z) is defined over K. (b) Suppose further that d = 2. Prove that there is an f E PGL2(K) such that h1 = h and such that ¢1 has the form ¢1 ( z) = az + b . z 1 (c) Write <Pa,b(z) = az + bz- • Prove that for all E K*, the maps <Pa ,bc2 and <Pa , b are PGL2(K)-equivalent. (d) In homogeneous form we have <Pa ( [X, Yl ) = [aX 2 + bY2 , XY]. Let .P�(a , b; X, Y) be the associated dynatomic polynomial. Prove that .P�(a, b; X, Y) E Z[a, b, X, Y]. (e) * If n � 4 is even, prove that .P� (a , b; X, Y) is reducible. More precisely, prove that there are nonconstant polynomials \lin , An E Z[a , b, X, Y] such that -
c
,
b
.P:(a, b; X, Y) = \lin(a , b; X, Y)An(a, b; X, Y). (Hint. Divide the set of points P of formal period n into two subsets depending on whether the involution h permutes the orbit O¢ (P) or sends it to a different orbit.) This result is due to Manes [286].
Let ¢ = az + bz- 1 E Rah (a) Prove that (171 ( ¢) , 172 ( ¢) ) depends only on a. (b) We saw in Exercise 4.40 that Aut(¢) contains { ±z }, so it has order at least 2. Find all values of a for which Aut(¢) is strictly larger than { ±z }. (c) As a varies, prove that the set of points
4.41.
describes the curve
C : 2x3 - 8xy + x2 y - 4y2 - x2 + 12x + 12y - 36 = 0
in .A.? � M 2 . (d) Prove that the curve C in (c) is singular at the point ( - 6, 12).
236
Exercises
(e) Move the singular point to the origin and perform a further change of variables to prove that the curve C is isomorphic to the curve
y2 = 4x3 + x4 . Thus the singularity of C is a cubic cusp. (f) The point ( -6, 12) is the unique singular point of the curve C, so the rational map ¢ satisfying (0"1 ( ¢) , 0"2 ( ¢) ) = ( -6, 12) should be special in some way. How is it special? (Hint. Compare with the answer to (a). See also Exercise 4.36.) Section 4.8. General Theory of Twists 4.42. Let X be an object defined over K, and for each twist phism iy : Y ----> X. Assume that Aut( X) is abelian. Prove that Twist (X/ K)
_____.
H1 (
G
Y of X, fix a K-isomor
l( K / K), Aut( X)) ,
a
is a well-defined one-to-one map of sets. (See Remark 4.78.) Section 4.1 0. Fields of Definition and the Field of Moduli 4.43. Let ¢ E K(z) be a rational function. We know from Proposition 4.84 that the field of moduli K.p is contained in every field of definition of ¢. Prove that K.p is equal to the intersection over all fields of definition of ¢. 4.44. Let Ca : x2 + y2 = a be the family of curves studied in Example 4. 76. (a) Let p be a prime number satisfying p = 1 (mod 4). Prove that Cp is Q-isomorphic to C1 . (b) Let p be a prime number satisfying p = 3 (mod 4). Prove that Cp(Q) = 0. Deduce that Cp is not Q-isomorphic to C1, and hence that [Cp]K represents a nontrivial element of Twist (Cl /Q). (c) Let p and q be distinct prime numbers that are congruent to 3 modulo 4. Prove that Cp and Cq are not Q-isomorphic. Deduce that Twist(C1/Q) is an infinite set. 4.45. Let g be the cocycle 9u
( )={ Z
z if 0" ( i) -1/z if O"(i)
= i, = -i,
described in Example 4.88, and let K be the associated fixed field in Q(w ) . (a) Prove that K = Q(u, v ) , where u = w - 1 /w and v = i ( w + 1/w). Deduce that C : u2 + v2 -4 is the twist of !P' 1 /Q associated to the cocycle g. (b) Let ¢(z) = i((z - 1)/(z + 1)) 3 be the rational map from Example 4.85 satisfying O"(f) = ¢g" . Our general theory says that K is a field of definition for ¢ if and only if C(K) =1- 0 , where C is the curve in (a). For example, C(Q ( H ) ) =1- 0, so Q ( H ) must b e a field of definition for ¢(z). Find an explicit linear fractional transformation f E PGL 2 (Q) such that ¢1 (z) E Q( H ) (z). =
Section 4.1 1 . Minimal Resultants and Minimal Models 4.46. ** This exercise raises some natural questions concerning global minimal models. (a) Is it true that every rational map ¢(z) E Q(z) of (odd) degree d 2: 2 has a global minimal model over Q?
237
Exercises
(b) Let K be a number field, let R be the ring of integers of K, let S be a finite set of primes, and let ¢(z) E K(z) be a rational map of degree d 2: 2 . We say that ¢ has a global S-minimal model if there is a linear fractional transformation f E PGL2 (K) and homogeneous polynomials F and G satisfying ¢f = [F, G] with the property that the coefficients of F and G are in the ring of S-integers Rs and ordp (Res(F, G))
=
ordp (9\q,)
for every prime p r:J. S .
Suppose that R s i s a principal ideal domain. Is i t true that every ¢ o f (odd) degree has a global S-minimal model? (c) Let K be a number field and ¢( z) E K ( z) a rational map of (odd) degree d 2: 2 . Suppose that the Weierstrass class iiq,; K is trivial. Is it then necessarily true that ¢ has a global minimal model? (This is true in an analogous situation for elliptic curves; see [410, VIII.8.2].) 4.47. ** Let K be a number field and let ¢ E K ( z) be a rational map of degree d 2: 2. Let S be a finite set of primes of K. Prove that
is a finite set.
{
1/1
E Twist(¢/ K) : 1/1 has good reduction at all p r:J. S }
4.48. ** Fix an embedding of the moduli space Md in projective space and let hM d denote the associated height function. Let K/Q be a number field, let d 2: 2 be an integer, and let B 2: 1 be a number. Prove that the set
contains only finitely many PGL2 (K)-conjugacy classes of rational maps. 4.49. * Let K be a number field, let R be the ring of integers of K, let S be a finite set of primes of R, and let d 2: 2 be an integer. Prove that there is a finite set of rational maps l3K,S,d C Ratd(K) such that if 4> E Ratd(K) is a rational map satisfying ( I ) ¢ has three or more critical points, (2) the critical points of ¢ remain distinct modulo all primes not in S, (3) the critical values of ¢ remain distinct modulo all primes not in S, where we recall that a critical value of¢ is the image of a critical point of ¢, then there are automorphisms f, g E P GL 2 ( Rs) such that
f o e/J o g E BK ,S,d · (Note that when we say that the critical points or values are distinct modulo a prime p, we really mean that they are distinct modulo \J3 for all primes \J3 lying above p in a suitable finite extension of K.)
Chapter 5
Dynamics over Local Fields : Bad Reduction In this chapter we return to the study of dynamical systems over complete local fields such as Qlp. We saw in Chapter 2 that if a rational map ¢(z) E Qlp(z) has good reduction, then its Julia set is empty, in which case considerable information about the dynamics of¢ on IP' 1 (Qlp) may be deduced by studying the dynamics of the reduction ¢ on IP' 1 (JFP). But if ¢( z) has bad reduction, then the situation is far more complicated. Indeed, since "interesting" unpredictable dynamics occurs only in the Julia set, we might say that the good reduction scenario studied in Chapter 2 is the uninteresting situation. This chapter is devoted to the interesting case! The field Qlp and its finite extensions have the agreeable property that they are complete, but they are not algebraically closed, so they are more analogous to lR than they are to
C.
This suggests that we work instead with an algebraic closure QP o f !Qp ,
but unfortunately we then lose the completeness property! Going one step further, we take the completion ofQp, and it turns out that this field, denoted by reP = the completion of the algebraic closure ofQlp, is both complete and algebraically closed. For proofs of the basic properties of Qlp and reP, see for example [8 1, 1 75, 1 83, 249, 382]. The fields re and reP share many common properties, but they also differ in crucial ways. In particular, the field of p-adic complex numbers reP is not locally compact! However, it is often essential to work in reP, rather than in a finite extension ofQlp, for example if we want to guarantee the existence of large numbers of periodic points. Table 5.1 compares some of the properties of the complete fields Qlp, QP, reP, IR, and C. The subject of p-adic and more general nonarchimedean dynamics is relatively new. After some early articles [50, 201, 278, 433, 438] in the 1980s, there was an explosion of interest that put the subject on a firm footing with a body of significant theorems and, just as importantly, an array of fascinating conjectures. In this brief 239
240
5. Dynamics over Local Fields: Bad Reduction
Nonarchimedean metric Algebraically closed Complete Locally compact Totally disconnected
.; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .;
Table 5.1: Comparison of complete fields.
chapter we can provide only a glimpse into this active area of current research, with many important topics omitted entirely in order to keep the chapter at a manageable length. For example, we do not touch on the important concept of local conjugacy, nor do we describe Rivera-Letelier's classification ofFatou domains [373, 375]. For the reader desiring further information, we mention the following articles on p-adic dynamics that are listed in the references: [4, 5, 13, 14, 22, 23, 26, 30, 29, 50, 63, 53, 54, 56, 57, 58, 59, 60, 62, 70, 71, 72, 73, 82, 84, 104, 1 16, 1 70, 168, 169, 1 85, 1 88, 1 89, 201 , 206, 208, 220, 222, 236, 237, 238, 239, 240, 241, 242, 243, 244, 245, 262, 266, 267, 268, 271, 272, 273, 274, 278, 280, 282, 283, 320, 3 1 8, 3 19, 334, 337, 338, 339, 340, 344, 348, 352, 355, 372, 373, 374, 375, 376, 378, 389, 427, 433, 438, 449]. 5.1
Absolute Values and Comp letions
We recall from Section 2. 1 that a valued field is a pair ( K, I · I ) consisting of a field K and an absolute value I · I on K. In this section we briefly remind the reader of the construction and basic properties of the completion of a valued field. Definition. Let (K, I · I ) be a valued field. A sequence a1, a2 , a3, . . . E K is Cauchy if for every E > 0 there exists an N = N (E) such that for all
m, n
� N.
In other words, the sequence an is Cauchy if an - am � 0 as m, n � oo . It is clear that every convergent sequence is Cauchy. A valued field K is said to be complete if every Cauchy sequence in K converges.
{ }
i
i
Example 5.1. The real numbers lR and the complex numbers C are complete with
respect to their usual absolute values.
Theorem 5.2. Let ( K, I · I K ) be a valuedfield. Then ( k, I · I k), unique up to isomorphism of valuedfields,
ties:
(a) There is an inclusion K C k as valuedfields. (b) k is complete.
there exists a valuedfield with the following proper
241
5.1. Absolute Values and Completions
(c) K is a dense subset ofk. Further, k is the smallest valuedfield satisfYing (a) and (b) in the following sense: Let ( L, I · I L) be any complete valuedfield containing K. Then there is a unique inclusion k L ofvaluedfields that respects the inclusions of K into k and L. Thefield k is called the completion of K with respect to the absolute value I · I K · <---+
Proof (Sketch) The field k may be constructed as follows. Let C be the set of all Cauchy sequences in K and make C into a ring by setting
and We define an absolute value on C by setting
and we define an equivalence relation "' on C by
Then one checks that these definitions are consistent and that the quotient
is a complete field with the desired properties. Note that K is identified with the subfield of k consisting of constant sequences. For further details on completions, see for example [190, 4 §7] or [217, IV § 1 ]. D Let (K, I · IK) be a completefield and let L/ K be afinite extension. Then there is a unique absolute value I · I L on L extending the absolute value on K. Thefield L is complete with respect to I · I L · Proof (Sketch) One checks that I · I L is given by Theorem 5.3.
LK lai L = INL;K ( a ) I �[ : ] . D
See [259, Proposition XII.2.6] or [382, II §4]. Example 5.4. The following table gives some examples of completions.
Field
Absolute Value Completion lR l a loo = max { a, -a} c
Q Q(i) l a + bi l = va2 +b2 Q I alp =
p- ord p ( a)
Qp
242
5. Dynamics over Local Fields: Bad Reduction
Remark 5.5. Theorem 5.3 says that for a finite extension L/ K, there is a unique absolute value on L extending the absolute value on K, and further that L is complete
k
with respect to this extended absolute value. If we go to the algebraic closure of K, then we still get a unique absolute value, since is the compositum of the finite extensions of K, but unfortunately may not be complete. (See, e.g., [249, III.4, Theorem 12] for a proof that Qp is not complete.) So we use Theorem 5.2 to form the completion of the valued field as in the next theorem. Theorem 5.6.
k
Let
k
k
Lemma 11.4.2]. 5.2
A Primer on Nonarchimedean Analysis
Throughout this section we take K to be a field that is complete with respect to an absolute value I · I satisfying the nonarchimedean (ultrametric) triangle inequality Ia + !31
::;
max{ Ia I, 1 !3 1 } for all o:, {3 E K.
(5. 1)
We recall from Lemma 2.3 that if Ia I f. 1!3 1, then (5. 1) is an equality. We set the following notation for the ring of integers of K, its unit group, and its maximal ideal: R = RK = {o: E K : Io:I :S: l}, R* = R'K = {o: E K : Io:l = l}, 9J1 = 9J1K = {a E K : lo:l < 1}.
We define "open disks," "closed disks," and "circles" in K by the usual formulas: D(a, r) = {u E K : l u - ai < r} = open disk of radius r at a, D(a, r) = { u E K : l u - al ::; r} = closed disk of radius r at a, S(a, r) = {u E K : lu - al = r } = circle ofradius r at a.
243
5.2. A Primer on Nonarchimedean Analysis
r) However, it is important to note that in the nonarchimedean setting, the disks r ) and the circle r ) are simultaneously open and closed sets! To see and this, let E S( r . Then for any s < r we have S( r , since any E s satisfies
D(a, D(a,x a, ) S(a, D(x, s) c a, ) y D(x, ) so I Y - a l = max{I Y - x l , l x - al} = IY - xl This shows that S (a, ) is open, and it is clearly closed by definition. It follows that ) = D(a, ) S(a, ) is open and that D(a, ) = D(a, ) "- S(a, ) is closed. D(a,Notice that D(O, ) and D(O, ) are groups under addition, i.e., they are closed under addition and negation, with a nontopological meaning of the word "closed." More precisely, the closed unit disk D(O, 1) is the ring of integers R of K and the open unit disk D(O, 1) is the maximal ideal 9J1 of R. Recall that the chordal distance between two points P1 = [X1 , Yl] and P2 = [X2 , Y2 ] in 1P' 1 (K) is the quantity (cf. Section 2.1) X2 Y1 1 p(P1 ' p.2 ) -- max{I X1IXI , IY1 Y12I }-max { I X2 I , IY2 I } . The chordal metric p satisfies 0 p(P1 , P2 ) 1, so it is bounded. In particular, if K is a finite extension of Qlp, so K is locally compact, then lP'1 ( K) is compact, since it is a locally compact bounded metric space. On the other hand, the field Cp is not locally compact, so although lP'1 (Cp) is a bounded space, it is not compact, nor even locally compact. We can define open and closed disks in lP'1 ( K) using the chordal metric. We will use the same notation as above: D( Q, ) = {P E lP'1 (K) : p(P, Q) r} = open disk of radius in lP'1 (K), D( Q, ) = {P lP'1 (K) p(P, Q) = closed disk of radius in lP' 1 (K). It should be clear from context whether we are working in K or in lP'1 (K). We observe that lP'1 ( K) is equal to the union of the two unit disks lP'1 (K) = {[x , 1] : lx l 1} {[1 ,y] : I Y I 1}, < s < r,
r U
r
r
r
r.
r
r
r
::::;
r r
r
::::;
r
<
E
::::; r }
:
r
:S
r
U
:S
each of which is metrically isomorphic to Rx . Remark 5. 7. It is sometimes convenient to make the assumption that the radius r of a disk is equal to the absolute value of some element of In this case we say that the disk has rational radius. We note that if is a finite extension of ramification degree e, then the value set : n E Z}. Similarly, the value has the form r E set is equal to Thus rational radius really means that the radius is a rational power of p. The nonarchimedean nature of the absolute value implies that a sequence in is Cauchy if and only if
IC; I
K
{pr : I KQl}.* I
K/Qlp{pnfeK*.
{aih>o
lim l ai+l - ail = 0.
,_,(X)
244
5. Dynamics over Local Fields: Bad Reduction
This follows immediately from the inequality
This observation and the completeness of K tell us that a power series ¢(z) converges if and only if
00
=
L ai (z - a) i E K[z] i =O
lim i ai (z - a) i l = 0.
t---> 00
In other words, the "nth term test" from elementary calculus becomes both necessary and sufficient in the nonarchimedean setting. A function ¢ : D(a, r) sented by a power series
Definition.
00
----t
K is holomorphic (or analytic) if it is repre
¢(z) = L ai (z - a) i E K[z - a]
i =O
(5.2)
that converges for all z E D(a, r). The order of¢ at a, denoted by orda (¢), is the smallest index i such that ai =1- 0. A meromorphic function on fJ (a, r) is a quotient ¢ = ¢1 /¢2 of functions ¢1 and ¢2 =/=- 0 that are holomorphic functions on D(r, a).1 A meromorphic func tion ¢ ¢1 /¢2 induces a well-defined map ¢ fJ ( a, r) lP'1 (K) , =
:
------7
The order of¢ at a is the difference We say that ¢ has a zero (respectively a pole) at z = a if orda ( ¢) > 0 (respectively iforda(¢) < 0). The next proposition describes some elementary properties of nonarchimedean holomorphic and meromorphic functions. Proposition 5.8. (a) Let ¢(z) be a holomorphicfunction on the closed disk D( a, r) and let b E D(a, r). Then ¢(z) is a holomorphicfunction on D ( b, r), i.e., ¢(z) is given by a convergent power series in K[z - b]. (b) Let ¢(z) be a nonzero holomorphicfunction on D(a, r). Then the zeros of ¢(z) in fJ (a, r) are isolated. This means that if¢( b) = 0, then there is a disk fJ ( b, E ) such that ¢(z) =/=- 0 for all z E D(b, E ) "- {b }. 1 More precisely, a meromorphic function is an equivalence class of pairs (
5.2. A Primer on Nonarchimedean Analysis
245
(c) Let rj;(z) be a meromorphicfunction on D(a, r) and suppose that the only pole of rj;(z) in D(a, r), if any, is z Laurent series,
=
a.
Then
rj;(z) is represented by a convergent
CXl
¢(z) = L ai (z - a) i E K[z] for a!! z E D(a, r) "'- {a}. (5.3) i= - m (d) Let rj;(z) be a meromorphic function on D(a, r). Then for every b E D(a, r) there is an s such that ¢( z) is represented by a convergent Laurent series on D(b, s). Proof (a) Let ¢(z) = I >i (z - a) i , and for k � 0, define coefficients bk by the formula
(These values are not mysterious. If K has characteristic 0 then they are the usual Taylor coefficients bk = (1/k!)(dk¢/dzk)(b).) The series defining bk converges since the convergence of ¢(z) on D(a, r) implies that
l ai (b - a) i l ::; l ail ri __, 0 as i __, oo. Further, we have the estimate so Hence l bk lr k 0 as k __, oo, so the power series .I.:�= O bk (z - b)k converges on D(b, r). Finally, we check that the series represents ¢ by computing __,
� bk (z - b) k � � G) ai (b - at-k (z - b) k � ai � G) (b - a)i-k (z - b) k =
=
L ai (z - a) i . i =O (b) Let b E D(a, r) be a point with ¢(b) = 0. We will find a deleted neighbor hood of b on which ¢(z) is nonvanishing. We use (a) to expand ¢ as a power se ries ¢(z) = .2.::.: bi (z - b) i centered at b and converging on D(b, r). The assumption =
that ¢ =I= 0 means that some coefficient is nonzero; we let j be the smallest index such that bj =I= 0. The fact that ¢ converges on [) ( b, r) imp lies that I bi I ri 0 as i __, oo , so there is a constant C such that l bil ri < C for all i. Let E = r1 + 1 l bj I /2C. We claim that the only zero of ¢(z) on D(b, E) is z = b. To see this, we take any z E D(b, E) and estimate __,
246
5. Dynamics over Local Fields: Bad Reduction
(
z max lbi l · l z - b l i :S max C l - b l •2:J + 1 r •2:J + 1
) i :S C ( l z -r bl ) j+l
Clz - bl . (z - b)l C E . I (z - b)l. 1 . 1 :=:; rJ+ l r1 + 1 I Hence for all z E D (b, E ) with z # b, the first term in the series =
¢(z) bj (z - b) 1 + =
=
1
2 l bj (z -
b)l. 1 .
00
L bi (z - b)i
i=j + l
has absolute value strictly larger than any of the other terms, so the nonarchimedean triangle inequality (Lemma 2.3) implies that for all z E D(b, E) . In particular, ¢(z) # 0 for all z E D(b, E ) with z # b. ( c ) Write ¢( z) = ¢ 1 ( z) /¢2 ( z) as a quotient of functions that are holomorphic on D( a, r). We give the proof of (c) in the case that ¢2 ( z) is a polynomial, which is the only case that we will need. For the general case, see [81 , 1 75] or Exercise 5.7. Let b E D( a, r) with b # a and ¢2 (b) 0. Taking the Taylor series of ¢1 and ¢2 around b, we can write =
and where 'lj; 1 and 'I/J2 are nonvanishing holomorphic functions on D(b, r). Then the as sumption that ordb(¢) � 0 implies that we can write ¢(z) as a quotient
ofholomorphic functions on D(a, r) such that the denominator does not vanish at b. Repeating this process for each of the zeros of ¢2 in D(a, r) other than z a, we find that ¢(z) is a quotient of holomorphic functions on D(a, r) such that the denominator does not vanish except possibly at a. Further, this cancellation process must stop, since we have assumed that ¢2 ( z) is a polynomial, so it has only finitely many zeros. By abuse of notation, we again write ¢(z) ¢ 1 (z)/¢2 (z), where we may now assume that the polynomial ¢2 (z) has no zeros in D(a, r) other than z a. We assume for the moment that K is algebraically closed and factor ¢2 ( z) as =
=
=
n
¢2 (z) c(z - a) e IT (z - ai ) ei . i= 1 By assumption, the roots satisfy ai � D(a, r), or equivalently lai - ai > r, for all =
1
:=:; i :=:;
n.
The reciprocal 1/¢2 ( z) has a partial fraction expansion
247
5.2. A Primer on Nonarchimedean Analysis
n "' B . . A . + "' � (z - a)J {:;t � (z - ai )j
e 1 = "' (z) ¢2
J
e;
tJ
for certain coefficients Aj , Bij E K. (See [259, IV §5] or [436, §5. 10] .) The terms with negative powers of z - a form part of the desired Laurent series. We claim that all of the other terms are holomorphic on D(a, r ). To verify this, we let a E K with Ia - a l > r, and for each 2 0 we consider the function 1/ ( z - a )k + l . If 0 we get a geometric series
k
k= 1 = 1 -1 � ( z - a ) n - (a - a) - 1 = = 1 z - a (z - a) - (a - a) 1 - (a - a)- (z - a) a - a f;:a a - a
(5.4)
This series converges for all z satisfying l z - a l < I a - a l , so in particular for all z E D(a, r), since I a - al > r. Hence 1/(z - a) is holomorphic on D(a, r). More generally, the same argument works using the identity2
1 (z - a)k + 1
k+ 1 00 ( + k ) ( z - a ) n = ((z - a) - 1(a - a))k+ 1 = (a( --1)a)k+ 1; k a-a n
This completes the proof that 1/7P2 (z) is holomorphic on D(r, a) in the case that K is algebraically closed. If K is not algebraically closed, we let L be the completion of the algebraic closure of K and use the above argument to write ¢( z) as a power series in L [z - a] that converges on D(a, r) for L. We now observe that when we add up the series for the different terms in the partial fraction expansion, the coefficients of each power of z - a are symmetric expressions in the roots of 7/J2 ( z). Thus the resulting power series actually lives in K[z - a]. We have shown that 1/¢>2(z) is represented by a Laurent series on D(a, r). Mul tiplying that Laurent series by the holomorphic function ¢> 1 ( z) gives a convergent Laurent series (cf. Exercise 5.3) representing ¢(z) on D(a, r). (d) Write ¢(z) ¢ 1 (z)/¢2 (z). Then (b) says that there is a neighborhood D(b, s ) of b such that ¢2 ( z) =/=- 0 for z E D ( b, s ) with z =1- b. Hence ¢ has no poles other than b on D(b, s ) , so (c) says that it is represented by a Laurent series on D(b, s ) D
=
.
Let ¢ D (a, r) K be a holomorphic function represented by a power series as in (5.2). The norm of¢, relative to the disk D( a, r ), is the quantity
Definition.
:
-+
Remark 5.9. The assumption that ¢ converges on D(a, r) implies that l ai lri -+ 0, so there will be at least one, and at most finitely many, indices i satisfying 1 1 ¢ 1 1 lai l r i .
=
2 0ver a field a characteristic 0, this identity can be derived by taking the k1h derivative of (5.4) and dividing both sides by k!. This proof fails in characteristic p if k 2: p, but one can either prove the formula directly by algebraic manipulations, or else one can clear denominators, observe that this yields an identity in the two-variable polynomial ring Z[a, z], and then reduce modulo p.
248
5. Dynamics over Local Fields: Bad Reduction
Further, if D(a, r) has rational radius, then r E IK* I, so there is a b E K * satisfy ing 11 4> 11 l b l . The nonarchimedean nature of the absolute value gives immediately the inequal ity
=
l l
i �O
We now show that 11 ¢ 11 is more or less a Lipschitz constant for a holomorphic map ¢. Proposition 5.10.
Let ¢(z) E K[z] be a power series converging on D(a, r). Then
l lltJl
I
I � ai ( (z - a) i - (w - an I � lz - wl � � "' � (z - a) ' - ' - ' ( w - a)' I
I ¢ ( z) - ¢( w) I =
:::; l z - w l sup sup l ai l · l z - ai i - l - j lw - a l j 0
i �O 5o j < i :::; l z - w l sup l ai l ri - 1 , i �O l z - wl llt! r l.
since l z - a!, lw - a l :::; r,
= 5.3
0
Newton Polygons and the Maximum Modulus Principle
A powerful tool in complex analysis is the maximum modulus principle,3 which as serts that a holomorphic function ¢( z) on an open set U c C has no maximum on U. Equivalently, if D C U is any closed disk in U, then i¢(z) i attains its max imum value on the boundary aD of D. In this section we prove a nonarchimedean analogue of the maximum modulus principle that is of similar fundamental impor tance in the theory of nonarchimedean analysis. However, in the nonarchimedean setting we cannot prove the maximum modulus principle using path integrals and Cauchy's residue theorem. In their place we substitute the powerful method of the Newton polygon. 3 Indeed, Ahlfors [ I ) says that "because of its simple and explicit formulation it is one of the most useful general theorems in the theory of functions. As a rule, all proofs based on the maximum principle are very straightforward, and preference is quite justly given to proofs of this kind."
5.3. Newton Polygons and the Maximum Modulus Principle
249
¢(z)
The Newton polygon of a nonarchimedean power series is very easy to describe, and it can be used to give a precise description of the distribution of zeros of the power series. To ease notation, we let
= -logP l z l for z E Cp. Notice that the valuation v is a surjective homomorphism v c; Q, since we have normalized I I so that IPI = p- 1 . Definition. Let ¢(z) I: an zn E Cp[z] be a power series. The Newton polygon of¢ is the convex hull of the set of points { (n, v(an )) : n = 0, 1, 2, . . . } , where by convention we set v(O) = oo . Informally, the Newton polygon is created as follows: take a vertical ray starting at the point 0, v( a0)) and aiming down the y v(z)
:
---*
·
=
(
axis. Then rotate the ray counterclockwise, keeping the point (0, v( a0)) fixed, until it bends around all of the points (n, v(an )). The Newton polygon consists of a set of line segments that connect the dots required to create the convex hull. A typical Newton polygon is illustrated in Fig ure 5.1 (a). It has a segment from (0, 5) to (2, 1 ) , a segment from (2, 1) to ( 4, - 1 ) , a segment from ( 4, -1) to (7, -1 ) , etc. A Newton polygon may have infinitely many line segments, or it may terminate with an infinite ray. A fundamental theorem says that the Newton polygon of an analytic function
contains a tremendous amount of information about the roots of the function. It pro vides a very powerful tool for studying nonarchimedean power series. Theorem 5.11. Let
¢(z) E Cp[z] be apower series. Suppose that the Newton poly N.
gon of ¢ includes a line segment of slope
m
whose horizontal length is
i.e., the
Newton polygon has a line segment runningfrom
(n, v(an )) whose slope is
to
(n + N, v(an+ N ))
v(an + N ) - v(an ) . N
m = --'---'---'------'-----'-
Suppose further that ¢ converges on the closed disk of radius pm. Then ¢( exactly N roots o:, counted with multiplicity, satisfYing pm.
l o: l =
z) has
See [249, IV.4, Corollary to Theorem 14]. We observe that the proof of this result for polynomials or rational functions is quite easy. For power series, one first proves a p-adic version of the Weierstrass preparation theorem saying, roughly, that factors into the product of a polynomial and a nonvanishing power se ries such that the initial parts ofthe Newton polygons of and coincide. Then the theorem for power series follows immediately by applying the elementary result for polynomials to g D Proof
¢(z) 7/J(z)
g (z)
(z).
¢(z) g(z)
5. Dynamics over Local Fields: Bad Reduction
250
•
•
• • •
•
¢ has no roots in 15(0, (a) Typical Newton polygon
1)
(b) Positive-slope Newton polygon
Figure 5. 1 : Examples ofNewton polygons. Example 5.12.
The Newton polygon ofthe power series
¢(z) = p5 +p4z+pz2 + pz3 +p- 1 z4 +p- 1 z5 +p3z6 +p- 1 z7 +p2 z8 +p3z9 + . . . is illustrated in Figure 5.l(a). The leftmost line segment has slope -2 and width 2, so ¢(z) has exactly 2 roots o: satisfying i o: l = p- 2 (assuming that ¢(z) converges on the appropriate disk). Similarly, ¢(z) has exactly 2 roots satisfying i o: l = p-I , exactly 3 roots satisfying i o: i = 1, and exactly 2 roots satisfying i o: i = p2 . Our first application of the Newton polygon is a nonarchimedean version of the
classical maximum modulus principle from complex analysis.
Theorem 5.13. (Maximum Modulus Principle) Let that converges on a disk 15( a, r) ofrational radius. (a) There is a point E 15 (a, r) satisfying
(3
¢(z) E Cp [z] be apower series
l ¢((3) 1 = syp l ¢(z) l = 11¢11 · zED(a, r ) (b) If¢ does not vanish on 15( a, r ) , then l ¢(z) l = 11¢11 for all z E 15(a, r). In other words, either ¢ has a zero in 15(a, r ) , or else it has constant magnitude on 15(a, r). Remark 5.14.
For many applications, it suffices to know that the maximum modulus principle is true for rational functions E . In this case, both (a) and (b) are quite easy to prove. We do (b) and leave (a) for the reader. Replacing by a)/c for some c with = r, we may assume that is well-defined and nonvanishing on the disk D(O, We factor as
¢(z) Cp(z) ¢(z) l e1i). ¢(z) ¢(z) = zk (1( 1 -- o:fJ11z)(z)(l1 -- o:fJ22z)z) ((11 -- f3O:srz)Z) ··· ···
z (z -
5.3. Newton Polygons and the Maximum Modulus Principle
251
z= k= J,Bil 1 J l - ai z l = J l - ,Bi z l = 1 for all z E D(O, 1), and hence l ¢ (z) I = 1 for all z E D(O, 1). Proof of Theorem 5.13. Write ¢(z) = I: ai (z-a) i and choose constants b, c E K* with J c J = r and J b J = 1 ¢ 1 (cf. Remark 5.9). Consider the series i = L a�ci zi . (cz) a '1/J ( z) = b- 1 ¢(cz +a) = b- 1 L i i=O i=O The convergence of ¢ on D (a , r) clearly implies the convergence of 'ljJ on D(O, 1), and we have aici I J aiJ ri llt)l II'I/JII = supo I b = supo b = b = 1 . i:: i:: J J J J Replacing ¢ by '1/J, we are reduced to proving the theorem under the assumptions that ¢ converges on the unit disk D(O, 1 ) and that 1 ¢ 1 = 1. (a) The condition 1 ¢ 1 = 1 says that every coefficient of ¢ lies in R, and the fact that ¢ converges on D( 0, 1 ) implies that all but finitely many coefficients lie in the maximal ideal 9Jl of Cv. So when we reduce ¢(z) modulo 9Jl, we get a nonzero polynomial
The assumption that ¢ has no zero or pole at 0 implies that 0. Further, we have < 1 for all i, since otherwise ¢ would have a zero at < 1 and E D(O, 1 ) or a pole at ,6; E D(O, 1). It follows that
a; 1 J ail
00
00
a 1 , . . . , ar
z)
E R be representatives for the distinct roots of ¢( mod 9Jl in the Let residue field R/9Jl. The residue field is infinite, since K is algebraically closed, so we can find a ,6 E R satisfying
for all 1 :::; i :::;
r.
Then ¢( ,6 ) ¢- 0 (mod 9Jl), so 1 ¢( ,6 ) 1 1 (b) Write I: as usual and consider the Newton polygon of The orem 5. 1 1 asserts that if some line segment of the Newton polygon of ¢( were to have slope m :::; 0, then ¢(z) would have at least one root satisfying pm :::; 1, and hence would have a root in D(O, 1). Thus our assumption that has no roots in D(O, 1) implies that every line segment of the Newton polygon has strictly positive slope. (An illustrative Newton polygon is given in Figure 5.l(b).) In partic ular, directly from the definition of the Newton polygon, this implies that
= = J ¢J .
¢(z) = aizi ¢(z)
a
v(ai) v(ao) Equivalently, we have J ail < J ao J for all >
i 2:
¢(z). =z) J a J¢(z)
for all i 2: 1. 1. We first observe that this gives
252
5. Dynamics over Local Fields: Bad Reduction
Second, we note that it implies that the constant term a0 in the series with z E D(O, 1) has absolute value strictly larger than any of the other terms, so the ultrametric in equality (Lemma 2.3) says that l¢(z) l
= \ao \ = \\¢\\
for all z E D(O, 1 ) .
D
Remark 5. 15. The maximum modulus principle (Theorem 5.13(a)), which we stated over Cp, is true more generally as long as K has infinite residue field, but otherwise
it need not be true. For example, let
= z - zP on D (O, 1). Clearly \\¢\\ = 1, but for every (3 E Zp = D(O, 1), Fermat's little theorem tells us that (3 (JP (mod p). Hence l¢((3) I 5: p - 1 < \\¢\\ for all (3 E D(O, 1). K
= Qp
and ¢(z)
=
We conclude this section with some useful consequences of the maximum mod ulus principle, including the fundamental fact that p-adic holomorphic and rational functions send closed disks to closed disks. Let ¢(z) E Cp [z] be a nonconstant power series that converges on a disk D (a, r) ofrational radius. (a) ¢(D(a, r) ) is a closed disk. (b) Write ¢( D (a, r)) = D (¢(a), s) . Then l ¢' (a) l 5: s /r. (c) If¢' (a) -=J 0, then there exists a radius t > 0 such that
Proposition 5.16.
l ¢(z) - ¢(w) l
= l ¢' (a) l · \z - w\
for all z, w E D(a, t).
Proof Replacing ¢(z) with ¢(cz + a) - ¢(a) for some c E Cp satisfying \c\ = r, we may assume that ¢(z) E Cp[z] converges on D(O, 1) and that ¢(0) 0. Write ¢(z) = :Ln?: l an zn as usual (note that a0 = 0 since ¢(0) 0), and let =
=
s ( a)
=
s�up l ¢(z) l .
zED(O,l)
The maximum modulus principle (Theorem 5. 13(a)) says that s = \\¢\\ = sup \an \ · n ?:O
Let j 2: 1 be the smallest index such that s = \ a1 \. The definition of s clearly implies that ¢(.D(O, 1)) <;;: D (O, s ) , so we are reduced to proving the opposite inclusion. Let (3 E D(O, s) . Consider the Newton polygon of the power series
5.3. Newton Polygons and the Maximum Modulus Principle
The fact that I ,B I
:::;
s = l aj I and I an I :::; l aj I for all n
and
253
� 1 implies that
so the Newton polygon of ¢(z) - ,B includes one or more line segments connect ing ( 0, v(,B)) to (j, v( aj )). Further, since the point ( 0, v(,B)) is no lower than the point (j, v(aj )), at least one of those line segments has slope rn :::; 0. The funda mental theorem on Newton polygons (Theorem 5.1 1) then tells us that the power series ¢(z) - ,B has a least one root a satisfying la l pm :::; 1 . This proves that there is a point a E D(O, 1) satisfying ¢(a) = ,8, and since ,B E D(O, s ) was arbi trary, this completes the proof that ¢(D(O, 1)) = D(O, s ) . (b) As in (a), the maximum modulus principle (Theorem 5 .13(a)) gives =
s = sup l ¢(z) l = 11¢11 = sup l an l n 2:0
zED(O,l)
� la1 l ·
This is the desired result, since r 1 and a 1 = ¢' (0) . ( c ) Continuing with the notation from (b), we compute =
For all n � 2, all 0 :::; i < n, all t :::; 1, and all z, w E D(O, t) we have Hence if we choose a value oft satisfying 0 < t < 1¢'(0) 1/ 11¢11, then the double sum in the righthand side of(5.5) has absolute value strictly smaller than 1¢' (0) I , and (5.5) reduces to the desired inequality l ¢(z) - ¢(w) l :::; lz - w l · l ¢'( 0 ) 1 .
This completes the proof of (c) and provides an explicit value for t.
0
Corollary 5.17. In each ofthefollowing situations, the indicated map ¢ is both open and continuous: (a) ¢ : D(a, r ) --.. Cp is a nonconstant analytic map. (b) ¢ : D(a, r ) -. JP'1 (Cv ) is a nonconstant meromorphic map. (c) ¢ lP'1 ( Cp ) --.. lP' 1 ( Cp ) is a nonconstant rationalfunction. :
254
5. Dynamics over Local Fields: Bad Reduction
(a) The continuity is a consequence of Proposition 5.10, which gives the stronger assertion that ¢ is Lipschitz. The openness of ¢ follows easily from Propo sition 5 .16, since the collection of "closed" disks (which is also a collection of open sets) { D(b, t) : t > 0 and b E Cp } forms a base for the topology of Cp. (b) Let b E D(a, r ) . Proposition 5.8(d) says that we can write ¢ as a Laurent series in some neighborhood D(b, s ) of b. If¢ does not have a pole at b, then (a) completes the proof. If ¢ does have a pole at b, we consider instead the meromorphic func tion 1/¢(z). It too can be written as a Laurent series in some neighborhood D(b, s ) , and it has no pole at b, so again we are done using (a). (c ) A rational function is clearly everywhere meromorphic, since it is the ratio of two power series (i.e., polynomials) that converge on all ofCp. (For the point at oo, D change coordinates and use 1/¢(z).) Hence we are done from (b). Proof
5.4
The Nonarchimedean Julia and Fatou Sets
In this section we recall some basic notions of convergence for collections of func tions, define the Fatou and Julia sets of a rational map ¢(z) E K(z) over a field K with an absolute value, and use a formula from Chapter I to show that in the nonar chimedean setting, the Fatou set is always nonempty. We begin with three definitions. Definition. Let U be an (open) subset of IP' 1 ( K) and let be a collection of func tions ¢ (K). (a) is equicontinuous on U if for every P E and every E > 0 there exists a 8 > 0 such that :
U JID1 ___.
U
D ) D (¢(P) , ) for every ¢ E . (b) is uniformly continuous on U if for every > 0 there exists a 8 > 0 such that ¢ (D (P, 8) n U) D ( ¢( P) , �:) for every ¢ E and every P E U. (c) is uniformly Lipschitz on U if there is a constant C such that ¢ ( ( P, 8)
c
c
E
c
p(¢(P), ¢(Q)) 5o C · p(P, Q)
for every ¢ E and every P, Q E U.
In the case that { ¢n} is the collection of iterates of a single function, we say simply that ¢ is equicontinuous, uniformly continuous, or uniformly Lipschitz.
=
It is important to understand that equicontinuity is weaker than uniform continu ity, because equicontinuity is relative to a particular point, while uniform continuity is uniform with respect to all points in In particular, uniform continuity is an open condition, whereas equicontinuity is not. Similarly, the uniform Lipschitz property
U.
255
5.4. The Nonarchimedean Julia and Fatou Sets
is an open condition, and indeed it is even stronger than uniform continuity. The following implications are easy consequences of the definitions: uniformly Lipschitz
=:::}
uniformly continuous
=:::}
equicontinuous at every point
As we will discover throughout this chapter, in a nonarchimedean setting it is often just as easy to prove that a family of maps is uniformly Lipschitz as it is to prove that it is equicontinuous. Definition. Assume first that K is algebraically closed. Then the Fatou set F( ¢) is the union of all open subsets ofJID1 ( K) on which ¢ is equicontinuous, i.e., F( ¢) is the largest open set on which ¢ is equicontinuous. The Julia set .:J ( ¢) is the complement of the Fatou set. In general, the Fatou set of¢ over K, which we denote by F( ¢, K), is the intersection of F(¢, with JID1 ( K). Similarly, the Julia set .:J ( ¢, K) is the complement ofF(¢, K) in JID1 ( K).
K)
Proposition 5.18. For every integer n 2:: 1,
and
We proved this over
Proof
We recall Theorem 1 . 14, which says that the multipliers of the fixed points of ¢ satisfy ""'
L
PEFix(¢)
1 1 - .Ap (.+.) 'f'
1
provided that
Ap (¢) =I= 1 for all P E Fix(¢).
The following corollary of this formula has the useful consequence that nonar chimedean Fatou sets are never empty, a fact that is false in the archimedean setting (cf. Theorem 1 .30, Example 1 .3 1 , and Theorem 1 .43). Corollary 5.19. Let K be an algebraically closedfield of characteristic 0 that is complete with respect to a nonarchimedean absolute value. Let ¢( z) E K ( z ) be a rationalfunction ofdegree d 2:: 2. Then ¢ has a nonrepellingfixed point. Proof If some fixed point P has multiplier ¢) = 1, then P is nonrepelling and we are done. Otherwise, we can use Theorem 1 . 14 to estimate
Ap(
Hence there is at least one fixed point Q satisfying
256
5. Dynamics over Local Fields: Bad Reduction
It follows that 0
so Q is nonrepelling.
Proposition 5.20. Let K be an algebraically closedfield of characteristic 0 that is complete with respect to a nonarchimedean absolute value. Let ¢( E K be a rationalfunction ofdegree d :::=: 2. (a) Let P E lP' 1 (K) be a nonrepelling periodic pointfor ¢. Then P is in the Fatou set F(¢). (b) Let P E lP' 1 ( K) be a repelling periodic point for ¢. Then P is in the Julia set :1(¢). (c) The Fatou set F( ¢) of¢ is nonempty.
z) (z)
Making a change of variables, we can move P to 0, and then Proposition 5 . 1 8 lets us replace ¢ by ¢ , so we may assume that 0 is a fixed point. This puts ¢ into the form
Proof
n
F(z) = .Xz + z2G(z) with G(O) -=/= 0 and .X = .Ao (¢) E K. Thus I G (z) l is bounded away from 0 if we stay away from the roots of G, and clearly F ( z) is bounded on any disk around 0, so there are a disk D(O, r) and a constant C such that I F (z)/G(z) l C for all z E D(O, r). Let 5 = min{r, 1 / C} . (a) The assumption that 0 is a nonrepelling fixed point means that I .XI 1 . Hence for z E D(O, 5) we have l ¢(z) l = I .Az + z�;�) I :::; l z l max { I .X I , I z��) I } :::; l z l . This proves that ¢ is nonexpanding on the ball D(O, 5), so its iterates are uniformly Lipschitz on that disk (with Lipschitz constant 1). Hence 0 is in the Fatou set. (b) For this part the assumption that 0 is a repelling fixed point means that I>• I > 1. Hence for z E D(O, 5 ) with z -=/= 0 we have I z�;�) I :::; l z l < I .Xz l . ¢(z)
:::;
:::;
The strict inequality allows us to conclude that
l ¢(z) I = I .Xz + z�;�) I = I .Xz l for all z E D(O, 5) . Suppose now that 0 E F( ¢). This implies that there is some > 0 such that if a E D(O, ) then every iterate satisfies ¢n(a ) E D(O, 5). But then as l ¢n(a)l = I .X I n l a l contradicting ¢n(a) E D(O, 5). Hence 0 E :1(¢). ( ) Corollary 5 . 1 9 says that ¢ has a nonrepelling fixed point and (a) tells us that E
t: ,
----4
c
oo
n ____. oo,
nonrepelling periodic points are in the Fatou set. Hence F( ¢) -=/= 0.
0
5.5. The Dynamics of ( z 2
- z) Ip
257
Remark 5 . 2 1 .
Corollary 5.19 is also true in characteristic p, since it follows directly from the fixed point multiplier formula (Theorem 1 . 14), which can be proven in char acteristic p either by reduction modulo p from the characteristic-0 case or by using the abstract theory of residues (see Exercise 5.1 0). It follows that Proposition 5.20 is also true in characteristic p. 5.5
The Dynamics of
( z2 - z) /p
In this section we illustrate the general theory by studying the p-adic dynamics of the map z2 - z for a prime p 2: ¢(z) = p Note that ¢ has bad reduction at p. An important observation is that the Julia set sits inside two disjoint disks. Proposition 5.22. Let ¢( z) = ( z2 - z) Ip. I I (a) lim l¢n(z) l = oo. l z l > - and l z - I I > ===? p p n ---> oo (b) J (¢) c D(O, Ilp) u D(I, I lp).
3.
Proof
(a) We consider two cases. First, if l z l > I, then l z l = l z - I I , so we find that
l
I
l¢(z) l = z(z - I) = p · l z l · l z - I I = P · I z l 2 · p In particular, I¢( z) I > I, so we can apply this inequality again. Repeating the process shows that Next suppose l zl :::; 1 . Then max{ l z l , lz - I I } = I , so using the assumption that l z l > Ilp and l z - I I > Ilp, we compute l¢(z) l =
l z(zp- I) = P · max{ lzl , l z - I I } · min{ l z l , l z - I I } > 1 . I
Now we can apply the earlier result to ¢(z) to conclude that I ¢n ( z) I ----) oo . (b) From (a) we see that every point outside ofthe two disks D(O, IIp) U D(I, Ilp) is attracted to the superattracting fixed point at oo, so such points are in the Fatou 0
�
The proposition tells us that the Julia set is contained in D(O, Ilp) U D(I, Ilp), but not every point in these disks is in the Julia set. For example, ¢(pa ) = pa2 - a = - a (mod p),
5. Dynamics over Local Fields: Bad Reduction
258
so if - o: is not in D(O, 1/p) U D(1, 1/p), then ¢n (po:) is attracted to oo, hence it is in the Fatou set. It is not easy to predict which points in the two disks are attracted to oo. However, since only points with bounded orbit can be in .:J (¢), we let
A = A(¢) { z E Cp I ¢n ( z) I is bounded for n 0}. =
�
:
It is clear that A is a completely invariant set, and Proposition 5.22 tells us that A is contained in two disjoint disks, which for notational convenience we henceforth denote by 10 and h :
A c lo U h,
where 10 = D(O, 1/p) and h = D(1, 1/p).
The orbit of a point z E A is contained within A, so each iterate ¢n ( z) is contained in one of the two disjoint disks 10, h . The itinerary of z is the se quence [{30 {31 {32 . . . ] of numbers f3n E {0, 1 } determined by the condition Definition.
for n = 0, 1, 2, . . . . In other words, the itinerary [f3n ] specifies how the points ¢n ( z) jump back and forth between the disks 10 and h . Our goal is to show that the dynamics of the itineraries of the points in A ac curately reflects the actual dynamics of ¢ on the set A. In particular, we will prove that A .:J ( ¢). In order to do this, we make a brief digression to discuss symbolic dynamics. =
5.5.1
Symbolic Dynamics
Symbolic dynamics is a tool for modeling seemingly more complicated dynamical systems. It has a long history and is used in many areas of mathematics. See [276] for a thorough introduction to the subject. In this section we briefly develop enough of the theory to complete our analysis of the dynamics of ( z 2 - z) /p. Let S = { a1 , a2 , . . . , as} be a finite set of symbols and let SN
=
{ elements sequence� of } S m
be the set of sequences [{30{31 {32 . . . ], where each f3n E S. (Here N {0, 1, 2, . . . } denotes the set of natural numbers.) We put a metric on SN by fixing a number p > 1 and setting =
p( 0: , {3) p-(smallest n with an #- f3n) . Thus the more that the initial terms of o: and {3 agree, the closer they are to one another. It is easy to check that p is a metric, and indeed a nonarchimedean metric: 1 p( o:, {3) 0. p( o:, {3) = 0 if and only if o: = {3. =
•
•
�
�
5.5. The Dynamics of ( z 2 •
- z ) Ip
259
p (a, !') :S max{ p (a, /3 ), p(/3 , !') } .
Symbolic dynamics is the study of the dynamics of continuous maps sN sN. An important map on sN is the left shift map L : sN sN' defined by ___,
___,
In other words, L simply discards the first term in the sequence and shifts each of the remaining terms to the left. More formally, the sequence L(/3) is defined by L(f3) n = f3n+1 . The next proposition describes some elementary properties of the map L. Proposition 5.23. Let SN be the space of S-sequences with associated metric as above and let L : sN ___, sN be the left shift map. (a) If p(a, /3) < 1, then p( L(a), L(/3)) p · p(a, /3) . (b) L is continuous (indeed Lipschitz), and it is uniformly expanding on each ofthe "disks " {a E SN : ao = O"i } , i = 1, 2 , . . . , s .
=
(c) The set SN contains exactly sn points satisfYing Ln (a) = a. In other words,
Pern (SN, L) has sn elements. (d) The periodic points ofL are dense in SN . (e) There exists a point ')' E SN whose orbit OL (!') = { Ln (!') : n 2': 0} is dense in SN . (Maps with this property are called topologically transitive.) Proof (a) The condition p ( a, /3) < 1 is equivalent to a0 = /30. It is then clear from the definition that p ( L( a), L(/3) ) = p · p( a , /3), since L( a) n =I L(f3) n if and only if an+ 1 =/= f3n+1 · (b) The metric p satisfies O :::; p :::; l, so (a) implies that p ( L(a) , £(,6) ) S p· p ( a , (3) for all a , /3. Hence L is Lipschitz. Further, (a) says that L is expanding by a factor
of p on each of the disks. ( c ) A sequence a E SN satisfies Ln ( a ) = a if and only if its first terms repeat, i.e., it has the form n
a = [aoa 1 . . . an-1 aoa 1 . . . an - 1 aoa 1 . . . an-1 . . . ..._____..., _____..., _____..., same n terms same n terms initial n terms
]
There are s choices for each of a0 , a1 , . . . , an _ 1 , so sn possible elements. (d,e) We leave the proof of these elementary results as exercises for the reader (Ex D ercise 5. 1 2). 5.5.2
The Dynamics of ( z
2
- z
) /p
Recall that we are studying the p-adic dynamics of the map ¢( z ) = ( z 2 - z) Ip for a prime p 2': and that we have defined
3,
260
5. Dynamics over Local Fields: Bad Reduction
and h = D ( 1, 1 /p) , n A = { z E Cp : ¢ (z) is bounded for all n 2: 0} c !0 U h , (3(z) = [f3of31 f32 . . . ] = (Itinerary of z E A), where f3n is defined by ¢n (z) E If3n ·
!0 = D ( O, 1/p)
The sequence (3(z) is an element of the space ofbinary sequences {0, 1 } N , so we obtain a map (of sets)
(3(z) = itinerary of z. Proposition 5.24. With notation as above, the itinerary map (3 : A the following properties: (a) (3 is injective. (b) (J(A n Qp) = {0, l } N, i.e., (3 restricted to A n Qp is surjective. (c) (3 respects the metrics on A and {0, l } N, i.e.,
Iz - wI
= P (f3 ( z) , f3 (w))
for all z, w
--+
{0, 1 } N has
E A.
(d) Let L : {0, 1 } N --+ {0, 1 } N be the left shift map on {0, l } N . Then (3 ¢ = L (3. o
o
In other words, the following diagram is commutative:
Proof (a) We begin with the following observation. Let u be 0 or 1.
Ifz, w E Iu, then I
(5.6)
To verify (5.6), we use the assumption that z, w E Iu to write z = u + px and w = u + py with lxl :::; 1 and IYI :::; 1 . Then
¢(z) - ¢(w) = (x - y) ((2u - 1) + p(x + y)) . The quantity 2u - 1 is ±1, so the final factor is a unit and we have
z-w l ¢(z) - ¢(w) l = lx - yl = - = p · lz - wl . p
l
l
Now suppose that z , w E A have the same itinerary (3 . Then ¢n (z) and ¢n (w) are in the same If3n for every n 2: 0, so applying (5.6) to these two points, we find that + for all n 2: 0. l ¢n+ l (z) - ¢n l (w) l = p · I
(5.7)
5.5. The Dynamics of ( 22 - 2)
/p
261
The lefthand side of (5.7) is bounded as oo, since 2 and w are in A. Hence we must have l2 - wl = 0. (b) Let w E J0 U h . We claim that ¢- 1 (w ) consists of two points {20, 2 1 }, one of which is in 10 and one of which is in h , and further, if w E Qlp, then 20 and 21 are in Qlp. To see why this is true, we fix w and solve or equivalently, solve ¢(2) w, 22 - 2 - pw = 0.
n ____,
=
The quadratic formula, the binomial theorem, and some elementary algebra gives the two solutions explicitly as ( 1) 1 ± + 4pw) 1 1 2 = � ± � � (4pw ) k = � ± � � pw ) k .
k 1 (1 1 2 2 2k - 1 (2kk ) ( 2 2 2 ( k/2) Note that the series converge for all lwl 1. The plus sign gives a solution satisfying 1 1 (mod p) and the minus sign gives a solution satisfying 2 = (mod p), so one solution is in Io and the other is in h . Further, if w E Qlp, then 2 and 2 are L.....
L....,
k=O
2
-
-
k=O
::;
2
=
0
2 1 also in Qlp. We are going to apply this observation to (nonempty) sets U C 10 U h . The inverse image ¢-1 ( U) of such a set thus consists of two nonempty disjoint pieces, one in Io and one in h . And since the inverse images of Qlp points in J0 U h are again in Qlp, we have
In particular, if U contains a point in Qlp, then each of the pieces of ¢- 1 (U) contains a point in Qlp. Let 10 10 n Qlp and 11 = h n Qlp, or equivalently, 10 and 11 are the open unit disks in Zv centered at and 1, respectively. For any binary sequence of o:o o: 1 0:2 . . . O:n with 2: define a set la0a1 . . . an = { 2 E Qlp : z E lao and ¢(z) E la 1 and ¢2 (2) E la2 . . . and ¢n (2) E la n } n Qlp lao n ¢ - 1 (laJ n ¢-2 (laJ n ¢- ( laJ· =
n 1, 0
n
=
n n A, then the initial n terms in the itinerary of 2 °
0
0
Notice that if z E la0a1 . . . an are o:oo: 1 . . . O:n. The sets la0 .. . are closed in Qlp, since ¢ is continuous and 10 and 11 are closed, and they are nested in the sense that an
We claim that they are also nonempty. To prove this by induction, suppose that we know that 1f3of3 1 . .. f3n- ! is nonempty for all sequences fJofJ1 . . . f3n - 1 of length We use the equality
n.
262
5. Dynamics over Local Fields: Bad Reduction
and apply the inductive hypothesis to see that la 1 a2 ... an is nonempty. Then from our earlier remarks we know that ¢- 1 ( la1a2 ... a n ) consists of two pieces, one in J0 and one in J1 , so its intersection with lao is nonempty. . . ] be any binary sequence. Then Let
a = [o:0o:1 o:2
.
la
�
nn20
lao a l ··· an
is the intersection of a nested sequence of nonempty closed bounded subsets of Qp. By compactness, the set Ja is nonempty, and by construction, any point in Ja has bounded orbit and itinerary This proves that the itinerary map
o:.
is surjective. We also note that since every point in la has itinerary and since we know from (a) that (3 is injective, it follows that la consists of a single point. (c ) If there is nothing to prove. Assume that -I- and let be the first index at which their itineraries diverge, so by definition, p (f3 ( (3( This means that for each 0 ::::; < the points ¢n z) and ¢n are in the same disk (either J0 or J1 ), but ¢ and ¢ are in different disks. Repeated application of (5.6) tells us that
z = w,
n
k ( z)
k, (w) k
a z wz), w)) k= p- . k (w)
(
for all 0 ::::; On the other hand, the assumption that ¢k ( z) and ¢k ( in J0 and one in Jl ) implies that
n
::::;
k.
(5.8)
w) are in different disks (one (5.9)
Combining (5.8) and (5.9) yields
i z - w i = j ¢k (z) p-/k (w)j = plk = p (f3(z),(3(w)). (d) It is clear that the itinerary of¢( z) is the left shift of the itinerary of z. D Proposition 5.24 allows us to identify the dynamics of the polynomial map ¢( z) = ( z 2 - z) /p with the dynamics of the shift map on the space of binary se quences. It then becomes an easy matter to read off a great deal of information about the dynamics of¢( z) from elementary properties of the shift map. Corollary 5.25. Let p 2 3 be a prime and let ¢(z) = z2 p- z and A = {z E Cp ¢n ( z) is boundedfor all 2 0}. (a) .:1(¢) = A Qp. :
c
n
263
5.6. A Nonarchimedean Montel Theorem
#periodic Pern ( ¢)point= 2of¢n +is1 repelling. for all n ;:::: 1, and aside from thefixedpoint at every (c) The repelling periodic points are dense in .:J ( ¢ ). (d) There exists a point w E .:J ( ¢) such that the orbit 0 (w) is dense in .:J ( ¢ ) . (Thus ¢ is topologically transitive on .:J ( ¢) . ) (b)
oo,
.p
Proof Let f3 : A
{0, 1 }N be the itinerary map. Proposition 5.24 tells us that f3 is injective, and further that it is surjective even when restricted to A n Qlp. It follows that A c Qlp, which proves one part of(a). For the other part, we note that Proposition 5.24 says that f3 is an isomorphism of metric spaces --+
(The proposition says that f3 is bijective and respects the metrics, so its inverse also respects the metrics.) The proposition also tells us that this isomorphism transforms the map ¢ into the left shift map i.e., f3 o ¢ o f3 . Hence the dynamical properties of¢ acting on A are identical to the dynamical properties of acting on the space of binary sequences. We proved earlier that ¢) c A, so
L,
.:J(
=
L
L
/3 : .:J(¢) � .:J(L).
However, the shift map is uniformly expanding on each of the disks f3(Io) and f3(h) (Proposition 5.23(b)). A uniformly expanding map is nowhere equicon tinuous, so = from which we deduce that .:J ¢ A (b) The points not in A are attracted to oo, so the only periodic point in lP'1
.:J(L) L,
L
/3 : Pern (¢)
()
=
.
(
Pern (L), and we proved (Proposition 5 .23(c)) that # Pern ( L) = 2 n . Further, we know that the shift map L is uniformly expanding (by a constant factor p) on each of /3(10) and f3(h), so via the metric isomorphism f3 and the iden tification of ¢ with L, we find that I ( ¢n ) ( ) I pn for every E Pern ( ¢) except In particular, every periodic point other than is repelling. (c,d) The density of the (repelling) periodic points in .:J(¢) and the topological transitivity of ¢ on .:J( ¢) A follow from the corresponding facts for L acting on {0, 1 }N (Proposition 5.23(d,e)). z
=
n
A�
'
oo.
z
=
oo
z
=
5.6
D
A Nonarchimedean Montel Theorem
In this section we give an important characterization of the Fatou and Julia sets and use it to draw a number of conclusions. The analogous results over
5. Dynamics over Local Fields: Bad Reduction
264
Theorem 5.26. (Hsia [208]) Let be a collection ofpower series that converge on D( a, r ), and suppose that there is a point a E K such that
a�
U ¢(D(a, r)) .
E if>
Then is uniformly Lipschitz on D(a, r) with respect to the chordal metric p, so in particular, it is equicontinuous on D( a, r ) .
c E K with l ei = r (see Remark 5.7) and replace each function ¢(z) E by ¢(cz + a). This changes the chordal metric by only a bounded amount (Theorem 2. 14), so we are reduced to the case of power series converging on the unit disk D(O, 1). By assumption, the function ¢(z) - a does not vanish on D(O, 1), so Theorem 5 . 1 3(b) tells us that
Proof We choose some
(5 . 10) l¢ (z) - al = II ¢ - all is constant for all z E D(O, 1). Suppose that a =/= 0 (the case a = 0 is similar) and consider the following
decomposition of the set :
1 = {¢ E : II ¢ - all � l al } , 2 = {¢ E : II ¢ - all > lal} .
If ¢ E 1 and z, w E D(O, 1 ) , then
p ( 1 , since z, w E D(O, 1). = lalp ( z, w) This shows that the functions in 1 satisfy a uniform Lipschitz inequality. Next let ¢ E 2 • Note that the definition of 2 and (5. 1 0) imply that l¢(z) - al = II ¢ - a ll > l a l for all z E D(O, 1 ) . (5. 1 1) Using (5. 1 1) and Lemma 2.3, and then applying (5. 1 0) again, we find that
I
p(¢ (z) , ¢(w)) =
(5. 1 2)
5.6. A Nonarchimedean Montel Theorem
265
<- II ¢ - a ll · l z - w l2 from (5. 12) and Proposition 5 . 1 0, max 1 , II ¢ - a ll } regardless of the value of II ¢ - a l l , :::; l z - wl since z , w E iJ(O, 1). = p(z , w )
{
Thus functions in the set 2 are nonexpanding, so in particular they satisfy a uniform Lipschitz condition. Finally, we note that if a = 0, then the same argument works using the sets
:
It is now a simple matter to use Theorem 5.26 to prove a nonarchimedean version ofMontel's theorem for rational functions. Theorem 5.27. (Nonarchimedean Montel Theorem, Hsia [208]) Let be a collec tion of rational, or more generally meromorphic, functions D( a, r) --+ lP' 1 ( K), and suppose that the union (5. 1 3) U ¢(D(a, r))
¢E if>
omits two or more points of lP' 1 ( K). Then satisfies a uniform Lipschitz inequality on D( a, r ) , so in particular is an equicontinuous family offunctions on D( a, r ). Proof Let a = [a 1 , a 2 ] and (3 = [(31 , fJ2 ] be two points oflP' 1 (K) that are not in the
union (5. 1 3). Consider the family of rational (or meromorphic) functions
{
}
"' = a2 f (z) - a 1 g(z) : f (z) E · (32 f(z) - fJ1 g(z) g(z)
,T,
By construction, we have
'1/J(z) -1- oo for all '1/J E \jf and all z E D(a, r),
so Proposition 5.8(c) tells us that the functions in \jf are holomorphic on D(a, r) (i.e., they are given by convergent power series). We also know that '1/J(z) -1- 0 for all z E iJ( a, r ), so the functions in \jf omit at least one point in K. It follows from Theorem 5.26 that there is a constant C1 such that
p('1/J(z), '1/J( w )) :::; C1 p(z, w) for all '1/J E \jf and all z , w E D(a, r). Finally, let A ( z) be the linear fractional transformation
(5. 14)
The assumption that a and (3 are distinct implies that A is invertible. Then a very special case of Theorem 2. 14 (for the rational map A - 1 ) says that there is a constant C2 = C2 (A) = C2 (a, (3) > 0 such that
5. Dynamics over Local Fields: Bad Reduction
266
for all z, w E JP>1 (K). Further, by construction we have ¢ E z , w E D(a, r ) ,
�
if and only if A ¢ E
C1 p(z, w) ;::: p (A(¢(z)), A(¢(w)) ) ;::: c:; 1 p ( ¢(z), ¢(w) )
o
(5. 15) w,
so for any
from (5.14), from (5.15).
This completes the proof of the nonarchimedean version ofMontel theorem.
D
Remark 5.28. The proof of Theorem 5.27 is fairly straightforward. Later, in Sec tion 5.10.3.4, we describe a deeper p-adic version ofMontel's theorem on Berkovich space; see Theorem 5.80. In the classical setting, there are a number of important properties of the Julia set that follow more or less formally from Montel's theorem. We conclude this section with a few instances. Proposition 5.29. Let ¢ JP' 1 ( K) JP'1 ( K) be a rational map of degree d ;::: 2 and let U c JP' 1 ( K) be an open set such that U n .J ( ¢) # 0. In particular, we are :
-+
assuming that the Julia set of¢ is nonempty. (a) The set Un:::: o ¢n (U) omits at most one point of lP' 1 (K). (b) Suppose that the set in (a) does omit a point. Then ¢ is a polynomial function and the omitted point is the totally ramifiedfixedpoint. (In other words, there is a change ofvariables f E PGLz(K) such that ¢! ( z ) E K[z] and the omitted point has been moved to oo.)
D(a, r ) , which are both open and closed, so it suffices to prove the proposition under the assumption that U = D( a, r ) . If the union omits two or more points oflP' 1 (K), then Montel's theorem (Theorem 5.27) implies that ¢ is equicontinuous on U, contradicting the assumption that U contains a Julia point. This proves (a). If the union omits a:, then ¢ - l (a:) = {a:}, so a: is a totally ramified fixed point of ¢. Hence ¢ is a polynomial map by definition (page 1 7), and after a change of variables it becomes a polynomial in K [z] (Exercise 1 . 9(c)). D Let ¢ JP' 1 ( K) -+ JP' 1 ( K) be a rational map of degree d 2 2. Recall that a subset E of JP' 1 ( K) is said to be completely invariantfor ¢ if it is both forward and backward invariant, ¢- 1 (E) = E = ¢(E). Proposition 1 .24 says that the Fatou set F( ¢) and the Julia set .J( ¢) are completely invariant. (The proof in Chapter 1 is over C, but the proof works, mutatis mutandis, for any complete field.) In Chapter 1 we used the Riemann-Hurwitz formula to characterize all finite completely invariant sets (see Theorem 1 .6). More precisely, we showed that a finite completely invariant set E has at most two elements. Further, if #E = 1, then after a change of variables, ¢(z) E K[z] is a polynomial and E = { oo }, and if #E = 2, Proof The set U is covered by disks
:
5.6. A Nonarchimedean Montel Theorem
267
then again after a change of variables, ¢( z ) zd or ¢(z) = z- d and E = {0, oo }. We now show that except for these trivial cases, the Julia set is the smallest closed completely invariant subset of lP'1 ( K) . =
Proposition 5.30. Let ¢ : lP' 1 ( K) ----+ lP' 1 ( K) be a rational map of degree d 2 2, and let E � lP' 1 ( K) be a closed completely invariant subsetfor ¢ containing at least three points. Then E is an irifinite set and E :2 J ( ¢).
Proof Theorem 1 .6 tells us that a finite completely invariant subset contains at most
3
two points, so our assumption that #E 2 implies that E is infinite. Notice that the complete invariance of the closed set E implies the complete invariance of its com plement U, which is an open set. It follows that the union Un>o ¢n (U) omits at least two points, since it in fact omits the infinite set E. Montel's theorem (Theorem 5.27) tells us that U � :F( ¢). Hence E ;;2 J( ¢ ). D
Remark 5.3 1 . Proposition 5.30 tells us that if the Julia set J( ¢) is nonempty, then it
is the smallest closed completely invariant set containing at least two points. (Notice that the case of exactly two points is ruled out by the fact that if ¢ has a completely invariant subset containing exactly two points, then ¢ is conjugate to either z d or z- d , in which case its Julia set is empty.) Corollary 5.32. Let ¢ : lP' 1 ( K) assume that J ( ¢) -:/:- 0. (a) J(¢) has empty interior. (b) Let P E J(¢) and let
----+
lP' 1 ( K) be a rational map of degree d 2 2, and
O¢ (P)
= nU2:0 ¢-n (P)
be the backward orbit ofP. The Julia set J( ¢) is equal to the closure ofO¢ (P) in lP'1 (K).
(c) J( ¢) is a perfect set, i.e.,for every point P E J( ¢), the closure ofJ( ¢) "'- { P} contains P. (d) J ( ¢) is an uncountable set. Proof (a) Let 8J(¢) denote the boundary of the Julia set J(¢). Theorem 1 .24
tells us that :F( ¢) and 8:1 ( ¢) are completely invariant, so the same is true of their union 8J(¢) U :F(¢). This union is also closed, since its complement is the interior of J ( ¢). Proposition 5.20 says that the Fatou set :F( ¢) is always nonempty, and since it is open, it must contain infinitely many points. Hence the union 8:1 ( ¢) U :F( ¢) is an infinite, closed, completely invariant set, so Proposition 5.30 tells us that J(¢)
�
8J(¢) u :F(¢) .
=
But J(¢) and :F(¢) are disjoint by definition, which proves that J(¢) 8:1(¢), i.e., the Julia set has empty interior. (b) We know that J ( ¢) is completely invariant, so in particular 0¢ ( P) c J ( ¢) for any point P E J( ¢ ).
5. Dynamics over Local Fields: Bad Reduction
268
Next let U be any open set with U n .:J( ¢) -1- 0. Then Proposition 5.29(a) tells us that Un>o q;n (U) omits at most one point, and Proposition 5.29(a) says that if it does omit a point, that point is a totally ramified fixed point, hence is in the Fatou set. In particular, the possible omitted point cannot be P, since P E .:!(¢) by assumption. This proves that P E Un ->o q;n (U), or equivalently, that there is some 2:: 0 such that U n q;- n (P) -1- 0. This proves that every open set U that intersects .:J ( ¢) nontrivially also intersects the backward orbit o; ( P) nontrivially. Hence .:J ( ¢) is contained in the closure of 0¢ (P). ( c) Let Po E .:J ( ¢). We claim that the backward orbit o; (Po) must contain a non periodic point. To see this, suppose instead that 0¢ (Po) consists entirely of periodic points. Then ¢- 1 (Po) consists of a single point, so Po is a totally ramified periodic point and hence in the Fatou set, contrary to assumption. Therefore we can find a nonperiodic point P1 E 0¢ (Po). The point P1 is in .:J (¢), since .:J ( ¢) is completely invariant, so (b) tells us that n
Po E closure of 0¢ (PI ) .
On the other hand, Po i s not in 0¢ ( P1 ) , since otherwise P1 would be periodic. Hence Po E closure of ( .:!(¢) " {P} ) . (d) The Baire category theorem [387, §5. 1,5.2] implies that a nonempty perfect subset of IP'1 ( K) is uncountable. D 5. 7
Periodic Points and the Julia Set
Our goal in this section is to show that the Julia set .:J ( ¢) of a rational map ¢ is contained in the closure of the periodic points of ¢. We begin with an elementary lemma that is obvious in the classical setting by a compactness argument, but which requires a different proof over a non-locally compact field such as Cp . Lemma 5.33. Let ¢ 1 (z) and ¢2 (z) be power series that converge on suppose that ¢ 1 ( D(a, r )) n ¢z (D( a, r )) 0. Then
D(a, r ) , and
=
i_nf p ( ¢ 1 (z), ¢z (z) ) > 0.
z ED(a,r)
Proof Let
M1 = s_up l ¢ 1 (z) l
and
z E D (a,r)
M2
=
sup l ¢z(z) l .
z ED(a,r)
The maximum modulus principle (Theorem 5.13(a)) says that there are points z1 , zz E D(a, r ) such that ¢ 1 (zi ) M1 and ¢z (zz) Mz. In particular, M1 and M2 are finite, since ¢ 1 and ¢2 are power series that converge on D(a, r ).
=
=
269
5.7. Periodic Points and the Julia Set
Let M = max{M1 , M2 , 1}. Then for any z E D(a, r ) we have
On the other hand, the function (PI - ¢2 does not vanish on D( a, r ) by assumption, so Theorem 5.13(b) tells us that for all z E D(a, r ) . Hence
D
The next lemma is used in conjunction with Lemma 5.33 to move a varying set of pairs of points { a , ,8 } to the specific pair { 0, 1 }. Lemma 5.34. Let A, B C Cp be bounded sets that are at a positive distance from one another. In other words, there are constants �' > 0 such that
sup i al ::::; � ' aEA For each (a, ,8) E A
sup I,BI ::::; � ' (3EB x
and
o
inf ( a , ,8) = > 0. (5. 1 6) aEA , (3 EB p
o
B, define a linearfractional transformation
La,(3 (z) = ( ,8 - a)z + a. Then there is a constant C > 0, depending only on � and 8, such that
p(La ,(3 (z), La',(3' (z') ) ::::; C · m ax {p(a , a') , p (,8, ,8') , p (z, z') } for all a , a' E A, all ,8, ,8' E B, and all z , z' E JID1 (Cp). Remark 5.35. Although Lemma 5.34 appears somewhat technical, it is not saying
anything mysterious. The linear fractional transformation La,(3 is determined by the three conditions La ,(3 (0) = a ,
La,(3 (oo) = oo.
The lemma is asserting, roughly, that if we take two nearby ( a , ,8) values, then the associated transformations are close to one another, where we use the chordal sup norm
p ( L , L') =
sup {p (L(P) , L' (P) ) } PEIP'1 (ICp) to measure the closeness of two maps. Thus Lemma 5.34 is equivalent to the assertion that the map
( a , ,8 , z)
is Lipschitz.
�
La,(3 ( z) = (,8 - a)z + a
270
5. Dynamics over Local Fields: Bad Reduction
Proof ofLemma 5 .34. To ease notation, for x, y E Cp we write
lx, y l
= max{lxl, IYI } .
We also assume (without loss of generality) that Ll 2: 1 and 8 :::;; 1 . Then for any o:, o:' E A and /3, /3' E we have lo: - o:' l p(o:, o:' ) · lo:, 1 l · lo:', 1 1 :::;; Ll 2 p(o:, o:') , (5. 1 7) l/3 - /3'1 p(/3, /3') · 1/3, 1 1 · 1/3', 1 1 :::;; Ll2 p(/3, /3').
B = =
Let o:, o:' E A, let f3 , /3' E and let z, z ' E lP'1 ( Cp ). Directly from the definitions of L a ,(3 and the chordal metric, we have
B,
I ( (/3' - o:' )z' + o:') - ( (/3 - o: )z + o:) I max { 1 (/3 - o: )z + o:l, 1 } · max{ 1 (/3' - o:' )z' + o:' l , 1 } " (5. 1 8) Assuming for the moment that z -=1- oo and z ' -=1- oo, we multiply out the numerator and estimate it using the triangle inequality: p(La ,f3 (z), L a' ,!3' (z' ) ) =
1 ((/3' - o:')z' + o:' ) - ((/3 - o:)z + o:) l l/3'z' - f3z - o:' z' + o:z + o:' - o: l = l!3'(z' - z) + (/3' - f3)z - o:'(z' - z) + (o: - o:') z + (o:' - o:) l :S: max{l/3' - /3l · lzl, lo:' - o:l · lz, 1 1 , l z' - zl · lo:', /3' 1 } :::;; max{Ll2 p(/3, /3') · lzl, Ll2 p(o:, o:') · lz, 1 1 , l z' - zl · lo:', /3' 1 } from (5.17), :::;; max{Ll 2 p(/3, /3' ) · lzl, Ll2 p(o:, o:') · lz, 1 1 , Ll · lz' - z l } from (5. 16), :::;; max{ Ll2 p(/3, /3' ) · lzl, Ll 2 p(o:, o:') · lz, 1 1 , Llp(z, z' ) · lz, 1 l · l z', 1 1 } definition of p, :::;; Ll2 · max{p(/3, /3') , p(o:, o:') , p(z, z') } · lz, 1 l · lz', 1 1 . =
Substituting this into (5 . 1 8) and doing a little bit of algebra yields max{p(/3, /3' ) , p(o:, o:' ) , p(z, z ' ) } lz', 1 1 -' z '-.1-'-.., l < Ll 2 . -----;-:-,---1--' (5. 1 9) max{ l (/3 - o:)z + o:l, 1 } max{ I (/3' - o:' )z' + o:' l , 1 } · We are left to show that the righthand side is bounded in terms of 8 and Ll. By symmetry, it suffices to bound the first fraction. We consider two cases. First, if lzl :::;; Ll j8, then we have the trivial estimate -----,-�
lz, 1 1 < iz 1 1 < Ll _ max{ l (/3 - o:)z + o:l, 1 } - ' - 8
271
5.7. Periodic Points and the Julia Set
Second, suppose that lzl > fl/ 8 . Then the fact (5. 1 6) that 1/J - al � 8 implies that f (/J - a)z f > !:1
Hence
� lal,
1 (/J - a)z + al = f (/J - a)z f
so
� 8 lzl .
lz, l l = lz, 1 1 < � < !:1 max{ I (/J - a)z + al, l } max {8 lzl, l} - 8 - 8 ' Thus fl/ 8 serves as an upper bound in both cases, and substituting this bound into (5 . 1 9) yields the estimate
�
4
p(La,(3 (z) , La ' ,(3' (z' ) ) :::; 2 max{p(!), !)') , p(a, a' ) , p(z, z ' ) } .
This completes the proof of Lemma 5.34 with explicit dependence on 8 and !:1 in the case that z -1- oo and z ' -1- oo. The remaining cases are similar and are left to the �cr
0
We next prove a version ofMontel's theorem in which the two omitted points are allowed to move. Lemma 5.34 is the key technical tool that allows us to uniformly replace the two moving points with two particular points, thereby reducing the proof to our earlier result. Theorem 5.36. (Montel Theorem with Moving Targets, Hsia [208]) Let ¢ 1 , ¢2 be power series that converge on D( a, r ) , and suppose that
Further let be a collection ofrational, or more generally meromorphic, functions on D(a, r) such that for all ¢ E
and all z E D(a, r). Then satisfies a uniform Lipschitz inequality on D( a, r ) , so in particular is an equicontinuousfamily offunctions on D(a, r).
Proof The proof is very similar to the proof of Theorem 5 .26, but somewhat more
elaborate. First we note that since ¢ 1 , ¢2 : .D( a, r) K have disjoint images, there is at least one point a omitted by both of them. Making a linear change of variables z z - a , we may assume that ¢ 1 and ¢2 omit the value 0. Then Theorem 5 . 13(b) tells us that ---->
f--+
for all z E D(a, r). We are going to apply Lemma 5.34 to the disjoint bounded sets ¢1 ( D(a, r) ) and ¢2 ( D ( a , r ) ) . Thus for each point w E D (a , r ) , if we let
272
5. Dynamics over Local Fields: Bad Reduction
then Lemma 5.34 says that there is a constant C such that p(Lw (z), Lu(z') ) :::; C max{p( ¢1 ( w ) , ¢1 (u) ) , p (¢2 (w ) , ¢2 ( u) ) , p(z, z') } for all w , u E D(a, r) and all z, z' E lP'1 (Cp ). (5.20) (In the notation of Lemma 5.34, we have set a = ¢1 (w), (3 = ¢2 (w ) , a' = ¢1 ( u) , (3' ¢2 ( u).) We next use the fact that ¢1 and ¢2 themselves satisfy a Lipschitz condition. More precisely, Proposition 5.10 says that
=
for all w , u E D(a, r) and i Since w and u are bounded, this implies that there is a constant C'
2
for all w , u E D(a, r) and
p (tPi (w) , rPi (u) ) :::; C' p(w , u)
Substituting this into the inequality (5.20) yields
=
1 , 2.
1 such that
i = 1 , 2.
p(Lw (z) , Lu(z') ) :::; CC' max{p( w , u), p(z, z') } for all w , u E D(a, r) and all z, z' E 1P' 1 (Cp ) . (5.21)
We are now ready to prove Theorem 5.36. The idea is that we know that each ¢ E if> omits at least two values, but the omitted values may vary with ¢, so we use the linear transformations Lw to move the omitted values to two specific points. Thus for each ¢ E
and
¢
for all w E D(a, r).
To see why this is true, note that contradicting the assumption on ¢, and similarly for 'l/;¢ ( w ) 1. Hence the family of maps \[1 on D( a, r) omits at least the two values 0 and 1, so Mantel's theorem (Theorem 5.27) tells us that \[1 satisfies a uniform Lipschitz inequality,
=
p('l/;¢ ( u) , 'l/;¢ ( w) ) :::; C" p( u , w)
for all ¢ E if> and all u , w E D(a, r). (5.22)
273
5.7. Periodic Points and the Julia Set
Using this and our earlier estimates, we compute, for ¢ E and u, w E p ( cj; (u ) , cj; (w ))
= p( Lu ('¢¢ (u)) , Lw ('¢¢ (w)))
D(a, r),
by definition of'¢ ¢, from (5.21), from (5.22).
:::; CC' max{p (u , w ) , p ('¢¢ (u ) , '¢
This completes the proof that the family of maps is uniformly Lipschitz.
D
We now have the tools to prove the main theorem of this section. Theorem 5.37. (Hsia [208]) Let cj;(z) E K(z) be a rational/unction of degree d with d 2: 2. Then
.J ( ¢)
C
Per(¢) ,
i.e., the closure of the periodic points of¢ contains the Julia set of¢;. Proof We may clearly assume that .J ( ¢) is not empty. Take any open set U having nontrivial intersection with .J ( ¢). We must show that U contains a periodic point. The Julia set is a perfect set (Corollary 5.32), so the open set U actually inter sects .J ( ¢) in infinitely many points. In particular, there is a point E U n .J ( ¢) that is not the image of a ramification point of ¢, since ¢ has at most 2d - 2 ramification points. This implies that there is a neighborhood c U of such that
P P
D(P, r)
consists of d disjoint open sets with the property that the maps ¢ : v;
___,
for 1
D(P,r)
:::; i :::;
d
are bijective. In particular, they have inverses ¢i :
D(P,r)
� v;
given by convergent power series. (This is a p-adic version of the one variable inverse function theorem. See Exercise 5.5.) We take i = 1 and i 2 and consider the maps ¢; 1 and ¢2 and the disjoint sets V1 and V2 as illustrated in Figure 5.2. We now examine the effect of applying the iterates q;n of ¢ to the disk fJ The assumption that fJ ( contains a point of the Julia set of ¢ means that ¢ is not equicontinuous on ) , so Theorem 5.36 tells us that the iterates of¢ cannot avoid both of the moving targets described by the power series ¢ 1 and ¢2 . Hence there exists an iterate q;n of¢ and a point Q E fJ such that either
=
P, r)r D(P,
or
(P, r).
(P, r)
Applying ¢ to both sides and using the fact that ¢ ¢ 1 and ¢ ¢2 are both the identity map on yields
D(P, r) so Q E D(P, r)
o
q;n +l (Q)
c
U is a periodic point.
= Q,
o
D
274
5. Dynamics over Local Fields: Bad Reduction
Figure 5.2: Inversion of ¢ over a critical-point-free neighborhood. Remark 5.38. In the classical setting over C, one can further show that the Julia set is equal to the closure of the repelling periodic points. This follows from the complex
analogue of Theorem 5.37 combined with the fact that a rational function over C has only finitely many nomepelling periodic points. Unfortunately, a rational function over Cp may well have nonempty Julia set and infinitely many nomepelling periodic points. However, one still hopes that the classical result is true in the nonarchimedean setting.
Conjecture 5.39. (Hsia [208]) Let ¢( z) E K ( z) be a rational function of degree d 2 2. Then the Julia set J ( ¢) is equal to the closure ofthe repelling periodic points
of¢.
Some evidence for Conjecture 5.39 is provided by the following result ofBezivin. It says that if the conjecture is false, then there are maps with nonempty Julia set containing no periodic points. Theorem 5.40. (Bezivin) If a rational function ¢( z) E Cp ( z) has at least one re pelling periodic point, then J ( ¢) is the closure of the repelling periodic points of¢. In particular, one repelling periodic point implies irifinitely many repelling periodic points.
Proof See [71] for the first assertion. The second then follows immediately from
Corollary 5.32(d), since an uncountable set cannot be the closure of a finite set. D
However, some evidence against Conjecture 5.39 is provided by Benedetto [58, Example 9], who shows that it is possible for a rational function to have a sequence of attracting periodic points whose limit is a repelling periodic point! Further, a slight variation of [57, Example 3] shows that for every > 0 there is a polyno mial ¢( z) E Cp [z] of degree p + 2 that has no repelling periodic points of period smaller than yet ¢( z) does have repelling periodic points of higher periods. Example 5.41. Consider the polynomial map
n
n,
¢(z) =
zP - z
p
.
5.7. Periodic Points and the Julia Set
275
, then l aP I > a > 1 1 I ial, l ¢(a) l = l aP; a l =p · l aP - a l =p· l aP I > pi a l. Hence ¢n (a) --> so a E F(¢), since a is attracted to the attracting fixed point at infinity. We also observe that if a E D(O, 1) n Qlp = Zp, then Fermat's little theorem tells us that aP = a (mod p), so ¢(a) E Zp. Thus Zp is a completely invariant subset of ¢. Hsia [206, Example 4. 1 1] (see also [449]) explains how to identify the dynamics of ¢ on Zp with a shift map on p symbols, similar to the example studied in Section 5.5, from which one deduces the following facts: .J(¢,Qlp) = Zp. .J ( ¢, Qlp) contains all of the periodic points of ¢ (other than ) so in partic ular all of the periodic points of ¢ are defined over Qlp, and all except are It is clear that the Julia set of¢ is contained in D(O, ), since if I so oo,
•
oo
•
•
,
oo
repelling. .J( ¢, Qlp) .J ¢ Cp ), since Theorem 5.37 tells us that .J ¢ Cp ) is contained in the closure of the periodic points of ¢. Thus .J( ¢, Cp) is compact. (See also [73].)
= (
,
(,
3 + az2 + b with a, b E z;. p ¢(z) = pz We first consider the fixed points of ¢, which are the roots of the equation pz3 + az2 - z + b = 0. The assumption that a, b E z; implies that the roots satisfy l a 1 i = p and l a2 i (Look at the Newton polygon!) We also observe that paf and aai have a3 1 =p21, .while inorm a 1 - b has norm p, so a1 must have the form a1 = - p-a + c for some c with l ei = 1. Example 5.42. Let � 5 be a prime, and let
This allows us to compute
l ¢'(al ) l = j 3pai + 2aa1 - 1 1 = l al (3pal + 2a) - 1 1 = l al (-a +3pc) - 1 1 = i a 1 i = p. Thus a 1 is a repelling fixed point of¢. A similar, but more involved, calculation can be used to show that there are re pelling periodic points of higher orders. Alternatively, we can invoke Bezivin's The orem 5.40, which says that the existence of the single repelling fixed point a 1 implies that ¢ has infinitely many repelling points whose closure is .J ( ¢). In order to study the periodic points in the Fatou set, we observe that ¢ is non expanding on D(O, 1 ). To see this, note that ¢ maps the disk D(O, 1 ) to itself, so in particular 11¢11 :::; 1. Applying Proposition 5.10 yields l ¢(z) - ¢(w) l :::; i z - wl for all z, w E D(O, 1).
276
5. Dynamics over Local Fields: Bad Reduction
Hence 4> is uniformly Lipschitz on D(O, 1), so D(O, 1) c F( 4>) . Notice that the nth it erate of 4> has the following form (think about the reduction of 4>n modulo p, which is the same as the nth iterate of the reduction ¢(z) = iiz2 + b):
n (z) = Az3n +
·
· ·
+
Bz2 n + l + Cz2 n + Dz 2n - l +
coefficients in p'llp
i
c E z;
· ·
·
+ Ez + F .
coefficients in 'llp
Again using the Newton polygon, we see that the polynomial 4>n (z) - z has ex ::; 1. These roots are actly 2n roots (counted with multiplicity) satisfying points of period dividing n for 4> and they are in the Fatou set, since we showed that D(O, 1) c F(¢) . One can prove that this gives infinitely many distinct periodic points in D(O, 1). See [206, Example 4. 1 1].
o:
5.8
l o: l
Nonarchimedean Wandering Domains
We first recall a famous theorem from complex dynamics (see Theorem 1 .36). Theorem 5.43. (Sullivan's No Wandering Domains Theorem [426]) Let ¢( z) E C( z) be a rational function of degree d ;:=: 2 over the complex numbers and let U C F( 4>) be a connected component ofthe Fatou set of¢. Then U is prepe riodic in the sense that there are integers n > m > 0 such that
In other words, the connected component U does not wander, whence the name of the theorem.
The first obstacle to translating Sullivan's theorem to the nonarchimedean setting is the fact that Qp and Cp are totally disconnected. Thus with the classical definition of connectivity, there is no good way to break up the Fatou set into "connected" com ponents. Various alternatives have been studied, including disk connectivity, rigid analytic connectivity, and the use of Berkovich spaces. In this section we consider disk connectivity, which is the simplest of the three to describe. We state a theorem of Benedetto to the effect that a large class of rational maps in Qp ( z) have no wandering "disk domains" and illustrate the result by proving it under the somewhat stronger hypothesis that there are no critical points in the Julia set of ¢. We also note that Benedetto has shown that the statement is false over Cp , that is, there are rational maps over CP that do have wandering disk domains. Much work has been done on this problem, especially in a series of papers by Benedetto [54, 56, 57, 58, 59, 63, 62] and Rivera-Letelier [372, 373, 375, 376, 378], but there are still many open questions. The material that we cover in this section is due to Benedetto.
5.8. Nonarchimedean Wandering Domains 5.8.1
277
Disk Domains and Disk Components
The ordinary definition of connectivity is not useful for studying the totally discon nected spaces QP and Cp , nor is the notion of path connectedness helpful. We begin with an abstract notion of connectivity that uses a chain of "disks" in place of a path. Let X be a topological space and let V be a collection of open subsets of X. (In practice, V will be a base for the toplogy of X.) For convenience, we refer to the sets in V as disks. Let U c X be an open subset and let P E U . The disk component of U con taining P (relative to V) is the set of Q E U with the property that there is a finite sequence of disks D 1 , D2 , . . . , Dn C U such that Definition.
Di Di+ l =1- 0 n
for all l � i < n.
In other words, Q is connected to P by a path of disks, as illustrated in Figure 5.3, although as we shall see, in the nonarchimedean world, Figure 5.3 is somewhat mis leading. Note that we define disk components only for open subsets of X. It is easy to see that U breaks up into a disjoint union of disk components (Exercise 5.22).
Figure 5.3: A path of disks from P to Q. Example 5.44. Let X = C and let V be the usual collection of open disks in C. Then the disk components of an open set U c X are the same as the usual path connected components. This is clear, since if r is a path from P to Q, then r can be covered by open disks contained in U, and the compactness of r shows that it suffices to take a finite subcover. Thus the definition of disk components and the related notion of disk connectivity (see Exercise 5.23) are reasonable. For example, Sullivan's theorem may equally well be stated in terms of the disk components of the open set :F( ¢ ) . For the purposes of this section, we modify slightly our definition of disks in lP' 1 ( Cp ) . The resulting topology is the same, and indeed the disks contained within the unit disk D(O, 1) are the same, but the alternative definition is more convenient for working with disks that may contain the point at infinity. Definition.
given by
The standard collection ofclosed disks in lP' 1 (Cp ), denoted by Vclosed , is
278 Vclosed
=
{
5. Dynamics over Local Fields: Bad Reduction
all closed disks D(a, r) and the complement JP>1 (Cp) " D(a, r) of all open disks D(a, r).
Of course, despite the name, all of the disks in Vc1osed are both open and closed sets. Note that Dclosed includes all such disks, not only the disks of rational radius (cf. Remark 5.7). One can show (Exercise 5.24) that the disks of rational radius in Vc1osed are exactly the images of the unit disk D(O, 1) via elements ofPGL2 (Cp). Similarly, the standard collection of open disks in JP>1 (Cp), denoted by Dopen , is given by Dopen
disks D(a, r) and = { alltheopen complement JP>1 (Cp) " D(a, r) of all closed disks D(a, r).
In the nonarchimedean world, every disk component has a very simple form. More precisely, it is either a disk, the complement of a single point, or all of JP>1 ( Cp). Proposition 5.45. Let Dopen and Vc1osed be, respectively, the collections ofstandard open and closed disks in JP> 1 ( Cp) as defined above. (a) Let D1 , D2 E Dclosed· Then one of thefollowing is true:
(iii) D1 � D2 . (i) D 1 n D2 = 0. 1 (iv) D2 � D 1 . (ii) D 1 U D2 JP> (Cp). (b) Let U C JP>1 ( Cp) be an open set and let V be a disk component of U relative to Vclosed· Then V has one ofthefollowingforms:
=
(iii) V E Vclosed U Dopen · D2 = JP>1 (Cp), we are done. Otherwise, choose any point in the complement of D 1 U D2 and use a linear fractional transformation to move that point to oo. This reduces us to the case that neither D1 nor D2 contains oo, so they have the form D 1 D(a 1 , rl ) and D2 D(a2 , r2 ). If D 1 and D2 are disjoint, we are done, so we may assume that there is a point Proof (a) If D 1
U
=
=
and switching D 1 and D2 if necessary, we may also assume that r 1 ::; r2 . Let /3 E D 1 . Then so /3 E D2 . Hence D 1 � D2 . (b) If V = JP> 1 (Cp), we are done, so we assume that V -=1- JP> 1 (Cp) · Using a linear fractional transformation to move a point not in V to oo, we are reduced to the case that oo � V.
279
5.8. Nonarchimedean Wandering Domains
Let D 1 , . . . , Dn E 'Dc1osed be any path of disks contained in V. Each pair of adjacent disks (Di , DH 1 ) has nonempty intersection, so (a) tells us that one of them is contained within the other. Applying this reasoning to each pair, we see that the union U�=l Di is itself a closed disk, i.e., the union is in 'Dc1osed· This shows that every disk path in V actually consists of a single disk. Fix some point a E V and let R = sup {r 2: 0 : D(a, r) C V}. Note that R > 0, since a E V and V is open. If R = oo, then V done, so we assume that R < oo. We claim that D(a, R)
s;::
=
V s;:: D(a, R).
Cp and we are
(5.23)
The lefthand inclusion is clear from the definition of R. For the righthand inclusion, suppose that b E V and consider a disk path from a to b lying within V. From our previous remarks, this disk path consists of a single disk D( a, s). The definition of R and the fact that D (a, s) s;:: V tell us that s :s; R, and then the fact that b E D (a, s) tells us that b E D( a, R). This gives the other inclusion. But we get even more. If there is even a single b E V satisfying lb - ai = R, then s = R and D(a, R) D(b, s) s;:: V, so we find that V = D(a, R) E 'Dclosed · On the other hand, if l b - ai < R for every b E V, then V c D(a, R), so (5.23) tells us that V D( a, R) E 'Dopen· This completes the proof of Proposition 5.45. D =
=
5.8.2
Hyperbolic Maps over Nonarchimedean Fields
In this section we prove that the Julia set of a rational map ¢ contains no critical points if and only if¢ is strictly expanding on its Julia set. This result is used later to prove that such maps, which we call (p-adically) hyperbolic, satisfy a p-adic version of Sullivan's no wandering domains theorem. On first reading, the reader may wish to omit the somewhat technical proof of the main theorem in this section and proceed directly to the application of the theorem in proving Theorem 5.55 in Section 5.8.3. Theorem 5.46. (Benedetto [56]) Let K/Qp be a.finite extension ofp-adicjields and let ¢(z) E K(z) be a rationalfunction ofdegree d 2: 2. Proposition 5.20(c) tells us that F( ¢) =/= 0, so changing variables if necessary, we may assume that oo E F( ¢ ) . Then thefollowing are equivalent: (a) There are no critical points in .7( ¢ ) . (b) For everyfinite extension L / K there exists an integer m 2: 1 such that
for all a E .7(¢) n L. In other words, q;m is strictly expanding on .7 ( ¢)
n
L.
Definition. Let K/QP be a finite extension. A rational function ¢ E K(z) will be called p-adically hyperbolic if it satisfies the equivalent conditions of Theorem 5.46.
See Exercise 5.25 for the relationship with the classical definition ofhyperbolicity.
280
5. Dynamics over Local Fields: Bad Reduction
Remark 5.47. The classical analogue of Theorem 5.46 over C is much weaker. It
¢
says that some iterate of is strictly expanding on the Julia set if and only if the closure of the postcritical set is disjoint from the Julia set. (The postcritical set is the union of the forward orbits of the critical points.)
(¢), we can take L = K (a) and observe(a)thatis clear, since if a is a critical (¢m )'(a) m-iIT=O1 ¢'(¢i (a)) 0 (since ¢' (a) = 0). Thus the existence of a critical point in J ( ¢) implies that ( ¢m )' has a root in J ( ¢) for every so (b) cannot hold. The other implication is more difficult. Using the assumption that ¢' does not vanish on J ( ¢) , we can apply Proposition 5. 16(c) to every point a in J ( ¢) n L to find a disk D( a , r ) with the property that j ¢(z) - ¢(w)j j ¢'(a) j · l z - w l for all z , w E D(a, ra ) · These disks form an open covering of the compact set J ( ¢) n L, so we can find a finite subcover. Let E be the smallest radius of the disks in this finite subcover. We then consider a finite covering of J ( ¢) n L by disks of radius E, say ProofofTheorem 5.46. The implication (b)
point in J
=
=?
=
m,
a
=
The (nonarchimedean) triangle inequality implies that each disk in this covering is
contained in one of the disks of the previous covering, so we conclude that ¢ stretches by a constant factor on each In other words, there are constants such that for each 1 ::::: i ::::: n,
D( ai, E).
Si
z, w E D(ai, E). next step is to show that for any fixed a E J ( ¢) n L, the set of derivatives n 1 , 2, 3, . . . , (5.24) for all
The
=
is unbounded. We prove this by contradiction, using the following claim: Claim 5.48. Let J n L and suppose that there is an > 1 such that
a E (¢) R j(¢n )'(a)j :::; R for all n � 1. Then ¢n (lJ(a,E/R)) D(¢n (a),E) for all n 1. (5.25) ProofofClaim 5.48. We verify the claim by induction on n. The inclusion (5.25) is clearly true for n 0, since R > 1. Suppose the inclusion (5.25) is known for all 0 n N. Our choice of E ensures that for any f3 E J( ¢) n L, the map ¢ stretches C
:::::
�
=
<
by a constant factor on the disk [) (/3, E), so in particular each of the maps
281
5.8. Nonarchimedean Wandering Domains
0 stretches by a constant factor. It follows that
:S n
<
N,
stretches by a constant factor. In other words, there is a constant S such that for all z, w E D(o:, E/ R). Taking w = o: and letting z -+ o:, we see that S = I (
n
This shows that the inclusion (5.25) is true for
n
=
n
N, hence for all by induction.
0
Recall that we are in the midst of a proof by contradiction that the deriva tives (5.24) are unbounded. Under the assumption that the derivatives are bounded, we have proven the inclusion (5.25), which in tum certainly implies that
U
C
U D (¢n (o:, E)) n ::O:l
C
U
D(/3, E).
{3 EJ(¢)nL
However, the Julia set is bounded (since we assumed that oo E F( ¢) ), so an E-neighborhood of the Julia set is also bounded. In particular, the above union omits at least two points, so by the nonarchimedean Montel theorem (Theorem 5.27), the map ¢ is equicontinuous on D(o:, E/ R). This is a contradiction, since o: E J( ¢) by assumption, which completes the proof that the derivatives (5.24) are unbounded. We now know that for each point o: E J ( ¢) n L there is some integer ma such that I ( ¢m"' ) ' ( o:) I 2 2. (There is no significance to the number 2; any number strictly larger than 1 would suffice.) By continuity, there is a disk D ( o:, sa ) such that I ( ¢m"' ) ' I 2 2 at every point in the disk. Taking a finite subcovering, we can cover J(¢) n L by disks with the property that there are integers m 1 , . . . , mt for all l
:::;
i
:::;
2 t
1 such that
and all z E jji ·
For convenience, we may assume that the disks D 1 , . . . , Dt are pairwise disjoint, since if two closed disks have a point in common, then one is contained in the other and may discarded. It remains to show that there is a single iterate ¢m that is expand ing on all of the disks. In fact, we show that this is true for all sufficiently large m .
282
5. Dynamics over Local Fields: Bad Reduction
'1/Ji
To ease notation, we let = ¢m; . Given any point a E :1 ( ¢) n L, we map out an orbit and an itinerary for a by applying whenever we land in fh More formally, define inductively an orbit a0, a 1 , . . . and a sequence of indices io, i 1 , . . . by the following procedure:
'1/Ji
io = (index so that ao E Di0 ), i 1 = (index so that a 1 E i2 = (index so that a2 E
and and and
a0 = a a 1 = 'l/Ji0 (ao) a2 = 'l/Ji 1 (a i )
Dh), Di2 ),
and
Di , then an+ is obtained by applying '1/Ji = ¢m; to an . for all = 0, 1 , , (5.26) n since by construction, the point ai n is in Din and '1/Ji n = ¢m; . We next observe that the derivative ¢' ( is continuous and nonvanishing on the compact set .J( ¢) n L. (Continuity follows from the assumption that � :1( ¢) and nonvanishing is our assumption (a) that :1(¢) contains no critical points.) It follows that there is a constant > 0 such that for all ,B .J(¢) n L. (We may assume that fl 1 .) Using the chain rule and applying this repeatedly yields In other words, if an is in We note that
I
2, . . .
n
z)
oo
f1
E
:::;
l (¢n ) ' (.B) I =
n- l
IT 1¢' (¢i.B) I 2:: fln
i= O
for all ,B E .J( ¢) n L and all n 2:: 0. (5.27)
M = max{m1 , ... , mt} and let N be any integer satisfying > fl- M2 . (5.28) We claim that I (¢N)' I > 1 on .J(¢) L, which will complete the proof of the theorem. Let
2N
4M
n
To verify this claim, let a E :1 ( ¢) n L and let a0, a 1 , . . . and i0, i 1 , . . . be the orbit and itinerary of a as described earlier. We use the chain rule to compute
N -mio -mi1 - · · -mik first be M. M > N-m· -m· - ··· -m·'l.k -> N- (k+ 1)M so k + > N/M. We have thus found integers k and satisfying •
We continue this process until the exponent comes smaller than Notice that this implies that to
2
21
'
r
5.8. Nonarchimedean Wandering Domains
283
(¢N) ' (o:) = 1/J�0 (o:o ) 7/( (o:I )'lj;�2 (o:2) · · · 1/J�k (o:k ) · (¢r ) ' (o:k+ l )
with > N/M - 2 and r < M.
k
Hence
I (¢N) ' (o:) l = I1/J�o (o:o ) 1/J�l (o:I ) 'lj;�2 (o:2 ) · · · 1/J�k (o:k ) l · l (¢r ) ' (o:k+I ) I 2k 2: 2 / M - 2 . /-LM since k > N/M - 2 and M, N �------�--� � 2 J.'r from (5.27) 2 from (5.26) r <
>1
from the choice (5.28) of N.
This shows that I ( ¢N) ' ( o:) I > 1 for all N > which completes the proof of Theorem 5.46. 5.8.3
M2 log2 (J-L- l ) and all o: E .7( ¢)
n L,
D
Wandering Disk Domains
If U is a disk component of the Fatou set F(¢) of a rational map ¢ E Cp(z), then ¢(U) may not be a full disk component of F(¢). This situation, which does not occur in the complex case, prompts the following definition. Definition.
field, and let
Let ¢( z) E Cp ( z) be a rational map defined over a nonarchimedean 'DC
(¢)
=
(
{ disk components of the Fatou set F ¢) }
be the collection of disk components of the Fatou set of ¢. Then ¢ induces a map of the set 'DC ( ¢) to itself according to the rule
(¢), U (disk component ofF(¢) containing ¢( U)). (5.29) We say that U E 'DC ( ¢) is periodic, preperiodic, or wandering according to the 'DC
(¢)
------>
'DC
1-------t
behavior of U under iteration of the map (5.29).
Example 5.49. Let p be an odd prime and let
¢(z) =
z2 - z p
be the function that we studied in Section 5.5. We proved (Proposition 5.22) that the Julia set of¢ is contained in the union of two open disks,
D(O, 1) U D(1, 1), and that :T (¢) contains the repelling fixed points 0 and 1. .7(¢)
c
For o: E Cp, let B ( o: ) denote the disk component of o: in F(¢). We claim that B ( -1) = D ( - 1, 1). To see this, we observe that B ( - 1 ) cannot contain any larger disk, since it does not contain 0. On the other hand, D(-1, 1 ) is in F(¢), since it is disjoint from D(O, 1) U D ( 1, 1). Hence B ( - 1) = D ( -1, 1).
284
5. Dynamics over Local Fields: Bad Reduction
Now consider the image point ¢( - 1) = 2p- 1 and the image of the associated disk component ¢ (B ( - 1 ) )
=
¢ ( D( - 1 , 1)) = D(2p- 1 , p).
The disk D(2p-I , p) is contained in :F( ¢) , but it is certainly not the largest disk around 2p- 1 contained in :F(¢) . Indeed, 2p-l E JP 1 (Cp) " D(O, 1)
c
:F(¢).
It is not hard to check that JP1 (Cp) " D(O, 1) is the full disk component of :F(¢) containing ¢ (B( - 1) ) . (No Wandering Disk Domains Conjecture) Let K/Qp be a finite extension and let ¢( z) E K ( z) be a rational map of degree d :::: 2. Then the Fatou set :F( ¢) has no wandering disk components. Conjecture 5.50.
Benedetto proves Conjecture 5.50 for a large class of rational maps. In order to state his result, we give four definitions (some of which we already know): Definition. Let ¢ E Cp ( z) be a rational map of degree d :::: 2 and let P E JP1 ( Cp). We say that P is:
(i) (ii) (iii) (iv)
Julia if P E .J(¢). critical if e p ( ¢) :::: 2. wildly critical if ep(¢) 0 (mod p). recurrent if there is a sequence of integers i.e., if P is in the closure of the set { ¢n ( P) =
n: in----+:::: oo1}.such that c/Jni (P) ----+ P,
Theorem 5.51. (Benedetto [54]) Let KjQp be a finite extension of p-adic fields and let ¢( z) E K ( z) be a rational map ofdegree d :::: 2. Assume that ¢ has no wildly critical recurrent Julia points (defined over Then the Fatou set :F( ¢) has no wandering disk components.
K).
We make three short observations concerning Theorem 5.5 1 . Remark 5.52. If p is odd, then Theorem 5.51 is true for "almost all" rational maps in Cp(z). This is true because if ¢ ( z) has a wild critical point P, then in particular it has a point whose ramification index satisfies e p ( ¢) :::: p It is not hard to show that all of the critical points of a "generic" rational map of degree d have ramification index equal to 2. (See Exercise 4.23 for a more precise statement.) Remark 5.53. For a fixed degree d, Theorem 5.51 is true for all primes p > d, since p > d rules out the existence of wild critical points. Remark 5.54. If a recurrent critical point P is in the Fatou set :F( ¢) , then one can show that P is in fact periodic (Exercise 5.26). On the other hand, if P is critical, then locally around P the map ¢ looks like ¢ (z) = ¢(P) + a(z - z(P))k + for some a E CP and some :::: 2. Thus if Q is sufficiently close to P, then
:::: 3.
k
· ·
·
5.8. Nonarchimedean Wandering Domains
285
p ( ¢(Q) , ¢(P) ) = p(P, Q) k , so ¢ is highly contractive near P. And if P is also recurrent, then ¢n (P) gets very close to P infinitely often, so it receives the highly contractive effect of ¢ infinitely often. This should cause points near P to remain near P, and thus force P into the Fatou set. The critical recurrent condition and the Julia condition are thus in opposition to one another, which means that nonperiodic recurrent critical points should be quite rare. On the other hand, Rivera-Letelier [378] has shown that there are maps having wildly critical recurrent points (which are then necessarily in the Julia set) in lP' 1 (Qp). It is not known whether Rivera-Letelier's examples have wandering disk domains. In the other direction, it is known that there are rational maps defined over Cp that have wandering disk domains [63, 59, 62, 378]. In these examples the critical points are in the Fatou set, so Theorem 5.55 implies that the maps cannot be defined over a finite extension of QP . We now use Theorem 5.46 and a simple compactness argument to prove that the Fatou sets of p-adically hyperbolic maps over finite extensions of Qp have no wandering disk domains. This is not as strong as Theorem 5.5 1, but still covers an important class of maps. The proof of Theorem 5.51 uses similar ideas, but is more complicated; see [54]. Theorem 5.55. (Benedetto [56]) Let K/QP be a.finite extension ofp-adicfields, let ¢( z ) E K ( z ) be a rational map ofdegree d 2: 2, and assume that the Julia set .:J ( ¢) contains no critical points of¢, i.e., assume that ¢ is p-adically hyperbolic. Then the Fatou set F( ¢) has no wandering disk components. Proof Proposition 5.20 assures us that F( ¢) is nonempty, and it is open, so it con
tains an algebraic point. (Note that Qp is dense in Cp.) Replacing K by a finite extension and changing coordinates, we may assume that oo E F( ¢), and indeed we may even assume that .:J(¢) c D(O, 1). Equivalently, we may assume that the disk component of oo contains lP'1 (Cp) ' D(O, 1 ) . We suppose that U c F (¢ ) is a wandering disk component of F(¢) and de rive a contradiction. Replacing U with the disk component containing ¢n (U) for a sufficiently large n, we may assume that the orbit of U does not include the disk component at oo. In particular, ¢n (U) c D(O, 1) for all n 2: 0. Again taking a finite extension of K if necessary, we can find a point a0 E K and a radius r0 such that D(ao, ro) C U. At this stage we fix the field K and we use Theorem 5.46 to find an integer m such that I ( ¢m )'I > 1 on .:J ( ¢) n K. Replacing ¢ by ¢m , it suffices to consider the case that W I > 1 on .:J(¢) n K. For each n 2: 0, the image ¢n (iJ(a0 , r0 )) is a (closed) disk centered at the point an = ¢n (ao), say In particular, ¢ (D(an , rn ) ) = D(an+l, rn + d, so applying Proposition 5 . 16(b) yields
286
5. Dynamics over Local Fields: Bad Reduction
(5.30)
We also know that the disks D (O:n , rn ) are disjoint, since D (O:n , rn ) is contained in ¢n (U), and further, each disk D(o:n , rn ) contains a point of D(O, 1)nK. It follows that the radii must satisfy limoo rn = 0, (5.3 1) n---+ since the set D(O, 1) n K has finite volume, so can contain only finitely many nonempty disjoint disks of any fixed radius. It follows from (5.30) and (5.31) that there are infinitely many with the property that
n
i.e., since rn of points
----t
0, we must have rn+ 1 < rn infinitely often. Consider the infinite set
{ o:n : l ¢' (o:n ) l < 1, n = 1,2,3, . . } . It is contained in the compact set D(O, 1) n K, so it contains an accumulation point f3 E K. The continuity of ¢' implies that l ¢'(!3) 1 :::; 1, which shows that f3 E F(¢), since we used Theorem 5.46 to ensure that I ¢'1 > 1 on :J ( ¢). .
Let V be the disk component of F(¢) containing {3. Then by construction V contains infinitely many of the iterates O:n = ¢n ( o:0). Since the radii of ¢n ( U) and ¢n (V) shrink to 0 as oo, it follows that some iterate ¢n (U) is contained in V and that some (nontrivial) iterate of ¢n (V) is contained in V. Therefore U is not wandering, contradicting our original assumption. D
n
5.8.4
----t
Wandering Disk Domains Exist in CP
We have proven that p-adically hyperbolic maps defined over Qp have no wandering disk domains. More generally, Theorem 5.51 shows that rational maps ¢( z) E Qp ( z) with wandering disk domains are very rare, if they exist at all. And of course, Sul livan's theorem 5.43 says that rational maps ¢(z) E C (z) defined over the complex numbers never have wandering domains. It is thus somewhat surprising to discover that there are very simple polynomial maps defined over Cv that have wandering disk domains. Theorem 5.56.
(Benedetto [59]) For c E Cv, let ¢c(z) be the polynomial cPc (z)
=
(1 - c)zP+ l + czP .
Then there exists a value a E Cv such that: (1) :J(c/Ja) -::/- 0, (2) F( cPa) has a wandering disk domain, (3) F( cPa) contains every critical point of cPa·
Proof See [59] for a proof of this specific theorem, and see [63, 62, 373, 378, 380]
for generalizations and related results.
D
5.9. Green Functions and Local Heights
5.9
287
Green Functions and Local Heights
The canonical height h¢ associated to a morphism ¢ JID1 --+ JID 1 is defined as the limit of d - n h ( ¢n ( P)) as n --+ oo. The utility of h1; lies in the two formulas :
( ) + 0( 1)
h¢(P) = h P
and
( ))
h¢ (¢ P = dh¢(P),
where the first says that h1; contains arithmetic information and the second says that h1; transforms canonically. It is tempting to try a similar construction locally and define (say) 1 (v-adic local height of a) = lim (5.32) log max{ l ¢n (a ) I 1 } . -+oo n d n It is clear that if the limit (5.32) exists, then it transforms canonically, and indeed if ¢(z) is a polynomial, then the limit does exist and everything works quite well (see Exercise 3.24). Unfortunately, for general rational maps the limit (5.32) may not exist. Rather than working directly with a rational map ¢ JID1 JID1 , it turns out to be easier to develop a theory oflocal heights by first lifting ¢ to a map A2 --+ A2 and then constructing a real-valued function g on A2 that satisfies the canonical trans formation formula g ( = dg In this section we construct the Green function g, prove some of its basic properties, and then use g to construct local canonical height functions on JID1 as described in Theorem 3.27. A point in projective space y] E JID 1 is given by homogeneous coordinates, so We make explicit the natural projec it is really an equivalence class of pairs tion map v
:
(x, y))
(x, y).
[x,
that sends a point (x, y) we write
E
--+
(x, y).
,
:
(x, y) � [x, y],
A2 to its equivalence class [x, y] A; = A2 {0, 0}
E
JID 1 . To ease notation,
"-
for the affine plane with the origin removed. Let ¢ JID1 JID 1 be a rational map of degree d. Then ¢ can be written as usual in the form ¢ = with homogeneous polynomials and of degree d having no common factors. The polynomials and then define a map :
--+
[F, G]
F G F G (x,y) = (F(x,y),G(x,y)),
that fits into the commutative diagram A 2 .p A 2 &* ------+ &* JID l � JID l
We call a lift of¢. By homogeneity, if = is one lift of¢, then every other lift of ¢ has the form for some constant c E K*.
c [cF, cG] [F, G] =
5. Dynamics over Local Fields: Bad Reduction
288
Definition. Let K be a field and let v be an absolute value on K. We denote the absolute value (or sup norm) ofa point E A2 ( K) by
( x, y)
Similarly, the absolute value (or sup norm) of one or more polynomials is the maxi mum of the absolute values of their coefficients. (We have already made use of this convention in the proof of Theorem 3 . 1 1 .) We begin by recalling how a map of a point.
:
A2(K)
-t
A2(K) affects the v-adic norm
Proposition 5.57. Let K be afield with an absolute value v. Let
be given by homogeneous polynomials F, G E ofdegree 2 1, and assume that F and G have no common factors in K (a) There are constants c1 , c2 > 0, depending only on and v, such that
[x, K[y]. x, y]
d
I (x,y) l v Cl - I (x,y) I v - C2 forall (x,y) E A; (K). (5.33) (b) If v is nonarchimedean and satisfies I I v = 1, then (a) is true with the explicit constants
d
<
< 1
'
Proof (a) We proved inequality (5.33) for general morphisms lP'N
lP'M during the course of proving Theorem 3.1 1 . More precisely, see (3.6) on page 92 for the upper bound with an explicit value for c2 (, v), and see (3. 7) on page 93 for the lower bound. (b) By homogeneity, it suffices to prove (5.34) for points satisfying = 1. Then the upper bound is obvious from the triangle inequality, and the lower bound was proven during the course of proving Theorem 2. 14, see (2.5) on page 57. D -t
I (x, y) l v
The next result describes a kind of v-adic canonical height associated to a map A2 A2. The construction is the same as the one that we used to construct canonical heights in Section 3.4. Proposition 5.58. Let K be a field with an absolute value v, let ¢ lP' 1 lP' 1 be a morphism ofdegree 2 2, and let = F, G) A2 A2 be a lift of¢. (a) For all E A; K) the following limit exists: (5.35) = }!..,. � 1
:
-t
(x, y) d (
We call
(
Qq, (x, y)
Qq, the Green function of .
:
:
-t
dn log l
-t
5.9. Green Functions and Local Heights
289
(b) The Green function is the unique function A; ( K) ----+ IR having the following two properties:
for all (x , y) E A; (K). (5.36) Qq, (ll>(x , y) ) = dQq,(x, y) Qq,(x, y) = log II (x, y) llv + 0( 1 ) for all (x, y) E A; (K). (5.37) (c) Ifv is nonarchimedean and satisfies \\i!l\\v = 1 and I Res ( F, G ) lv = 1, i.e., if the map ¢ = [F, G] : JP'1 JP' 1 has good reduction at v, then Qq,(x , y) = log ll (x , y)ll v for all (x, y) E A; (K). iil
----+
(The converse is also true. See Exercise 5.27.) (d) For all (x, y) E A; (K) and all c E K*, the Greenfunction Qq, has thefollowing homogeneity properties:
(5.38) Qq,(cx , cy) = Qq,(x , y) + log \ c \ v · 1 (5.39) log \c l v · Yc (x, y) = Qq,(x , y) + d-- 1 (e) The Green function Qq, : A; ( K) ----+ IR is continuous. (In fact, Qq, is Holder continuous, but this is more difficult to prove. See Exercise 5.28.)
Proof We consider the two functions
and Proposition 5.57(a) tells us that they satisfy
loglliil(x, y) ll v dlog ll (x , y) llv + 0( 1 ) =
for all (x , y) E A; (K).
(5.40)
This is exactly the situation needed to apply Theorem 3.20, from which we conclude that the limit (5.35) exists and satisfies (5.36) and (5.37). Further, Theorem 3.20 says that Qq, is the unique function satisfying (5.36) and (5.37). This completes the proof of (a) and (b). ( c) The assumptions \\1>\\v = 1 and I Res ( F, G ) Iv = 1 combine with Proposition 5.57(b) to tell us that I I i!l (x, y ) ll v = II (x , y) II � for all points (x, y) E A; (K). Hence by induction we obtain
Then the definition of Qq, immediately gives Qq,(x, y) = log ll (x , y) ll v ' which proves (c). The map
(d)
c
290
5. Dynamics over Local Fields: Bad Reduction
Hence
1 logi i
:
----+
:
:
----+
l 9
m
m
The Green function allows us to decompose the canonical height of local terms.
hq, into a sum
lP' 1 lP' 1 be a rational function of degree d defined over K, and let <J? be a fixed lift of¢. For each absolute value v E MK, let be the associated Green function as described in Propo sition 5.58, where we now include v in the notation so as to distinguish between different absolute values. Then the (global) canonical height decomposes as a sum Theorem 5.59. Let K be a number field, let ¢
2: 2 9
:
for all P
----+
= [x , y] E lP'1 (K).
Proof Let
ry (x , y) = L nv 9
for (x, y) E A; (K),
so a priori the function TJ is a function on A; ( K). However, applying the Green func tion homogeneity property (5.38) from Proposition 5.58(d) and the product formula (Proposition 3.3), we see that
ry(cx, cy) = L nv (9
291
5.9. Green Functions and Local Heights
so ry gives a well-defined function on IP'1 (K). Next we use the transformation property (5.36) from Proposition 5.58(b) to com pute
Similarly, we use the normalization property (5.37) from Proposition 5.58(b) to esti mate
=
L nv (log II P II v + Ov(l)) = h(P) + L nv0v(1) ,
(5.41)
where h(P) is the usual height of the point P E IP'1 (K) and the Ov(l) are the bounded functions appearing in (5.37). We further observe that Proposition 5.58(c) says that we may take Ov (1) = 0 for all but finitely many v E MK. More precisely, we may take Ov(1) = 0 for all v satisfying (i) v E M� ,
(ii) II II v = 1 ,
and
(iii) I Res( ) I v = 1.
Hence the final sum in (5.41) is a bounded function. We have now proven that ry IP'1 ( K) lR satisfies :
ry ( ¢(P)) = dry(P)
--+
and
ry(P) = h(P) + 0(1).
It follows from the uniqueness ofthe canonical height (Theorem 3.20) that ry is equal 0
� �-
In Section 3.5 we described a decomposition of the canonical height he/> as a sum of local canonical height functions �c/>,v• but we deferred the proof. The intuition in Section 3.5 was that the local canonical height should measure
�¢,v(P) = - log( v-adic distance from P to oo ) . More generally, it is convenient to define a local canonical height that measures the v-adic distance from P to a collection of points. In the following theorem we de scribe a set of points, possibly with multiplicities greater than 1, by specifying the homogeneous polynomial E E K[x , y] at which they vanish. In slightly fancier ter minology, we are identifying positive divisors in D iv(IP'1 ) with homogeneous poly nomials in K[x, y] . IP'1 --+ IP'1 be a rationalfunction of degree d 2: 2 defined over K. Fix a lift = ( F, G) of¢ and let Qcp be the associated Green function. Theorem 5.60. Let K be afield with an absolute value v and let ¢
:
5. Dynamics over Local Fields: Bad Reduction
292
(a) For any homogeneous polynomial E ( x, y) E K [x, y] ofdegree e 2': 1 we define
�q,, e ([x, yl ) eQq,(x, y) - logiE(x, y) l v for [x, y] E lP' 1 (K) with E(x, y ) -1- 0.
=
Then �q,,E is a well-definedfunction on lP'1, i.e., the value of�q,, e (P) does not depend on the choice ofhomogeneous coordinates [x, y]for P. Thefunction �¢,E is called a local canonical height associated to ¢ and E. In the special case that E(x, y ) y, we drop E from the notation and refer simply to a local canonical height associated to ¢. (b) For all P E lP'1 (K) with E(P) -1- 0 and E(if.>(P)) -1- 0 we have
=
A
A
I (E o if.>)(P) d I
. v (Note that the homoegeneity of E and if.> ensures that the ratio ( E o if.>)/ Ed is a >.. q, ,E ( ¢(P)) d>..q, , e (P) - log
=
well-definedfunction on lP'1.) (c) Thefunction
E(P)
E(P) (5.42) >.. q, ,e(P) + log I liP I v , II � which a priori is defined only at points P satisfYing E(P) -1- 0, extends to a P
A
f-------7
bounded continuous function on all oflP'1 (K). (d) Given the particular choice of lift if.>, the function unique real-valuedfunction
�q,,E defined in (a) is the
satisfYing (b) and (c). If cif.> is a different lift, then with the obvious notation,
In particular, any two local canonical heights differ by a constant.
Qq,(cx, cy) Qq,(x, y) + log l clv, while the ce E(x, y), so the difference
Proof The Green function satisfies
polynomial E satisfies E(cx, cy)
=
=
eQq,(x, y) - log I E(x, y) lv defining �¢,E does not change ifwe replace (x, y) by (ex, cy). This proves (a). (b) We compute
�q, , e (¢(P)) = eQq, (if.>(x, y)) - logiE( if.> (x, y)) l v edQq,(x, y) - log i E(if.>(x, y)) l v from Proposition 5.58(b), = d�q,, e (P) + d log I E(x, y) lv - log i E(if.>(x, y)) l v ·
=
5.9. Green Functions and Local Heights
293
( c ) Directly from the definition of �.E we see that the righthand side of (5A2) is equal to (5,43) e(Qq,(P) - log II P II v) · The boundedness of(5A3) is exactly (5.37) in Proposition 5.58(b). Further, we know from Proposition 5.58(e) that Qq,(x, y) is a continuous function on A;(K), and it is clear that logl l (x, y) ll v is also a continuous function on A; (K), so the differ ence (5,43) is a continuous function on A;(K). Further, the difference is invariant under ( x, y) (ex, cy) , so it descends to a continuous function on lP' 1 ( K). Suppose that � and �� both satisfy (b) and (c), and let ��� = � - �1 • Writing
(d)
r--+
(
Iv �" (P) = �(P) + log I E(P) P II II �
Iv ) ) (�'(P) + log I E(P) P II II � ' _
we see from (c) that ��� extends to a continuous bounded function on all of lP'1 ( K). Let C be an upper bound for I �"I · From (b) we find that ��� ( ¢(P)) = d�" (P) for all P E lP' 1 (K) with E(P) =/- 0 and E ( ¢(P)) =/- 0, and iterating this relation yields �11 ( ¢n ( P) ) = dn �11 (P ) provided E ( ¢i ( P)) -1- 0 for all 0 ::; i ::; Hence (5,44) for all points P E lP' 1 ( K) satisfying E ( ¢i ( P)) -1- 0 for all 0 ::; i ::; n. But each equation E ( ¢i ( P)) = 0 eliminates only finitely many points, so the inequality (5,44) is true for all but finitely many points oflP' 1 (K). Then the continuity of ��� tells us that I � �� ( P) I ::; Cd-n is true for all P E lP'1 ( K). Since is arbitrary, this proves that ��� ( P) = 0, so � = �'. Finally, the effect of replacing by c
n
Finally, as promised in Section 3.5, we prove that the global canonical height is equal to the sum of the local canonical heights. Theorem 5.61. Let K be a number field, let ¢ lP' 1 ----+ lP' 1 be a rational func tion of degree d 2: 2 defined over K, and fix a lift = (F, G) of ¢. Choose a :
homogeneous polynomial E(x, y) E K[x, y], andfor each absolute value v E MK, let �,E,v be the associated local canonical height described in Theorem 5.60, where we now include v in the notation so as to distinguish between different absolute values. Then the (global) canonical height has a decomposition as a sum of local canonical heights, for all P E
lP'1 (K) with E(P) -1- 0.
294
5. Dynamics over Local Fields: Bad Reduction
Proof We use the definition of �c/>,E,v in terms of the associated Green function Q�, v
from Theorem 5.60(a) to compute
�
�
( de de E L nv �cf> ,E , v (P) = de E L nv ( g E)Q�, v (P) - logiE(P ) IJ v EMK vEMK
Theorem 5.59 says that the first sum is equal to h¢(P), while the product formula (Proposition 3.3) tells us that the second sum is 0. (Note that this is where we use the D assumption that E(P) -=1- 0.) Remark 5.62. If v E M� is nonarchimedean and ¢ has good reduction at v, then the
Green function and the local canonical height are given by the simple formulas
(
)
max { lxl v , I Y i v } Q� ,v (x, y) = log max { lxl v , I Y i v } and >.., c/>, E,v ( [x, y] ) = log · I E(x, y) lv Thus it is only for maps with bad reduction that the Green and local canonical height functions are interesting. This should not come as a surprise to the reader, since bad reduction is the situation in which dynamics itself becomes truly interesting. Of course, this is said with the understanding that every rational map over C has "bad reduction," so the dynamics of holomorphic maps on ( are always interesting and complicated. Remark 5.63. For additional material on dynamical Green functions and dynamical local heights, see [21 , 88, 234, 233].
lP'1 q
5.10
Dynamics on Berkovich Space
We have seen that Cp is a natural space over which to study nonarchimedean dy namics, since it is both complete and algebraically closed. However, the field Cp has various unpleasant properties: Cp is totally disconnected. Cp is not locally compact, so the unit disk { z E Cp : l zl ::; 1 } and projective line ( Cp ) are not compact. The value group IC;I = l� consists of the rational powers of p, so it is not discrete in IR>0, yet neither is it all of!R> O · This list suggests that it might be better to work in some larger space. There is a general construction, due to Berkovich [64, 67], that solves these problems for Cp and other more complicated spaces. The study of dynamics on Berkovich spaces started during the 1990s and is an area of much current research. In this section we briefly describe the Berkovich disk and associated affine and projective lines and •
•
•
lP'1
295
5.10. Dynamics on Berkovich Space
discuss some very basic dynamical results. In a final subsection we state without proof some recent results. For further reading, see [26, 5 1 ] for an introduction to dynamics on Berkovich space and see [373, 375, 376, 379, 377, 380, 381] for Rivera Letelier's fundamental work in this area. 5.10.1
The Berkovich Disk over CP
The unit Berkovich disk [JB is a compact connected metric space that contains the totally disconnected non-locally compact unit disk in Cw We describe two construc tions of [JB, the first an explicit description as the union of four types of points and the second as a set ofbounded seminorms on Cp[z]. 5.10.1.1 The Four Types of Berkovich Points
The most concrete description of [JB is as the union of the following four sets of points: Each point a in the standard unit disk D(O, 1) of Cp is associated to a point of the Berkovich disk, which we denote by
Type-I Berkovich Points.
�a, o
Each closed disk D(a, r ) contained in D(O, 1) with ra
Type-11 Berkovich Points.
dius r E I C; I denote by
=
E D- s .
pifl is associated to a point of the Berkovich disk, which we �a, r
E D- s ·
Type-III Berkovich Points. Similarly, each closed disk D( a, r ) C D(O, 1) with positive radius r ¢. I C; I = pQ is associated to a point of the Berkovich disk, which we naturally also denote by �a,r·
These are the trickiest points in [JB. They are associ ated to nested sequences of closed disks
Type-IV Berkovich Points.
with the property that
n D(an , Tn ) = 0.
n2: 1
We denote these points by �a ,r, where, as the notation suggests, the vectors a and r are a = (a 1 , a2 , . . . ) and r = (r1 , r2 , . . . ) . Remark 5.64. Note that Berkovich points �a, r of Types II and III are disks D(a, r ) ,
so different values of a may yield the same Berkovich point. Indeed, we have if and only if
r = s
and I a - bl :::; r,
5. Dynamics over Local Fields: Bad Reduction
296
since these are the conditions for the disks D( a, r) and D ( b, s ) to coincide. Similarly, two Berkovich points of type IV are the same if their sequences of disks can be suitably intertwined. See Exercise 5.40 for details. Remark 5.65. A point �a, r of Type-I, II, or III corresponds to a disk (possibly of radius 0), so we define the radius of �a, r to be r. The radii ro, r 1 , r2 , . . . of a Type IV point �a,r are nonincreasing, so the limit r = limi_,oo ri exists and is called the
radius of�a,r·
We claim that the radius of a Type-IV point is strictly positive. To see this, sup pose that �a,r has radius 0. Then the sequence of points a 1 , a2 , . . . is a Cauchy se quence in Cp, so it converges to a point a E Cp. Let
i5i = jnf ,ri zED(ai )
lz - al
be the distance from a to the ith disk. Notice that 0 � <51 � <52 � <53 � · · since the disks form a decreasing sequence. On the other hand, i5i � I a - ai I ----+ 0 as i oo . Hence i5i = 0 for all i, so a E D(ai, ri ) for all i and the intersection is nonempty, contradicting the assumption that �a,r is a Type-IV point. ·
,
----+
5.10.1.2 The Berkovich Disk as a Set of Seminorms
The description of the Berkovich disk fJB as a union of points of Types I, II, III, and IV is very concrete, but it can be awkward to apply, since one must deal with four different kinds of points. There is an alternative description of fJB as a collection of seminorms on the ring CP [z] that is sometimes easier to use and that also naturally generalizes to other rings and other spaces.
4
Definition.
Fix R > 0. The Gauss norm II
·
II R on Cp[z ]
is defined by
The maximum modulus principle (Theorem 5.13(a)) tells us that the Gauss norm is equal to the sup norm, II JII R = sup
l. i f (z) zED(O,R)
Definition.
A (nontrivial) II ll wbounded seminorm on Cp[z] is a nonconstant map ·
with the following properties: 1. 1! 1 2: 0 for all E Cp[z] . 2. = I · for all J, E Cp[z].
l fgl I f l g l
f
g
40ne can unify the four types of points by defining all of them as equivalence classes of nested se quences of closed disks. See Exercise 5.40.
297
5.10. Dynamics on Berkovich Space
3. If + g l ::; max I f I , l g l } for all J, g E Cp [z] . 4. lfl ::; IIJ I I R for all f E Cp[z] . (This is the boundedness condition.) Thus a seminorm has all of the properties of an absolute value except that there may be nonzero elements f E Cp[z] with lfl = 0. Definition. The (closed unit) Berkovich disk [JB is the set
{
[JB = I · ll 1 -bounded seminorms on Cp [zl } . More generally, the closed Berkovich disk ofradius R is the set
{
D� = II
{
·
I wbounded seminorms on Cp [zl } .
The definition of fJ B as a set of bounded seminorms is quite unintuitive, but its utility becomes clear when one examines Table 5.2 and sees how each of the four types of Berkovich points naturally defines a seminorm on Cp[z] . A funda mental theorem of Berkovich, whose proof we omit, says that every bounded semi norm on Cp[z], and more generally on certain power series rings containing Cp[z], comes from one of the four types of Berkovich points. (See [64, page 1 8] or [26, Proposition 1 . 1].) Notice that the seminorm I · l o corresponding to the Berkovich point �o , 1 is exactly the Gauss norm l f l o, = 11! 11 � , so following Baker and Rumely, we call �o, 1 the Gauss point. 1
I
Seminorm Point I Type-I �a , O lfl a,O = lf( a ) l Types-II & III �a, r lfla , r = lf( z ) l zED(a, r) Type-IV �a,r IJia,r limoo IJian , Tn n Table 5.2: Seminorms on Cp [z] associated to Berkovich points.
sup
=
--+
Remark 5.66. It is easy to see that the seminorms associated to points of Type-II, III,
and IV are actually norms. (See Exercise 5.35.) However, the seminorm associated to a point �a, o of Type-I is not a norm, since lfla , o = 0 if and only iff vanishes at a. For example, l z - al a, o = 0. This explains why the Berkovich disk is defined using seminorms, rather than norms. Remark 5.67. In order to properly develop function theory on the Berkovich disk and to glue disks together to create larger spaces, it is advantageous to use sets of seminorms on more general rings, such as power series rings. For example, let 11' R be the ring of power series in Cp[z] that converge on the closed disk D(O, R) of radius R, i.e.,
{
11'R = f(z)
=
I >n Zn E CCp [z] : J�moo l cn iRn n 2:0
=
0
}·
298
5. Dynamics over Local Fields: Bad Reduction
The ring 1l'R is a Banach algebra over Cp using the Gauss norm, which the maximum modulus principle tells us is the same as the sup norm. The Berkovich disk D� is often defined to be the set of bounded seminorms on the ring 1l'R - One can show, using the Weierstrass preparation theorem, that this leads to the same set of points and the same topology as taking the I · llwbounded semi norms on the polynomial ring Cp [z]; see [26]. The ring 1l'R is an example of a Tate algebra. Applying the same construction to more general Tate algebras allows one to construct Berkovich spaces for the associated rigid analytic spaces. 5.10.1.3 Visualizing the Berkovich Disk
In order to visualize the Berkovich disk, we place the Gauss point �0, 1 at the top of the page and observe that there is a line segment running from any point � -=1- �0, 1 up to the Gauss point. If� = �a,r is of Type-I, II, or III, then this line segment is the set of points La,r = { �a , t : r S t S 1}. Notice that any two line segments La,r and Lb,s merge with one another at the point �a, t = �b,t determined by t = max{r, s, l b - a l } . This is the smallest allowable value of t for which the disks D(a, t) and D(b, t) acquire a common point, hence for which they coincide. Thus one can imagine the various line segments continually merging as they run upward toward the Gauss point at the top of the tree. The line segment running up from a Type-IV point �a,r is slightly more compli cated. It is the set of points 00
00
i= 1
i=1
(5.45)
(Note that ro = 1 by definition.) The Type-IV point �a,r is included in the Berkovich disk precisely to provide an endpoint for the union of line segments U La; ,r; . Now imagine starting at the Gauss point and moving downward through the tree. We claim that at any instant, there are three possible scenarios: 1 . If you have reached a point �a,o of Type I or a point �a,r of Type IV, then you have hit the end of a segment and cannot proceed further. 2. If you have reached a point �a,r of Type II, then r > 0 is in the value group of Cp and there are countably infinitely many branches along which you can move down the tree. 3. If you have reached a point �a,r of Type III, then r > 0 is not in the value group of Cp and there is only one direction to move down the tree.
5.10. Dynamics on Berkovich Space
299
The picture for points of Type I is clear. In order to understand Types II and III, suppose that we fix a point of Type II or III. Each b E D(a, r) gives a line segment that runs up from the Type-I point and through the point Two such line segments and if and only if merge before reaching l b - b'l < r, so it is really each open disk D(b, r) inside the closed disk D( a, r) that gives a line segment running up to If is of Type III, then D(b, r) = D( a, r) for any b E D( a, r ) , so there is only one segment running downward from The situation is much more interesting, and complicated, if is of Type II. In this case D( a, r) is covered by a countable union of open disks D(b, r ) , so there is a countable set of branches downward from A convenient, although noncanon ical, way to describe the branches is as follows. Let lfJ = { z E Cp : I z I < 1} denote the maximal ideal in the ring of integers of Cp and fix some c E Cp with l ei = r. Then the open disks of radius r are in one-to-one correspondence with the residue field lFP via the map
�a,r Lb,o Lb' ,o �a,r · �a,r ·
Lb,o
�a,r
�b,o
�a,r
�a,r
�a,r ·
{ disks D(b, r) inside D(a, r) }
-+
JFP ,
�r,a ·
D(b, r)
f-------7
(b/c) mod \l}.
The surjectivity of this map is clear, and the injectivity follows from the fact that D(b, r) = D(b' , r) if and only if l b - b'l < r = l eiIn order to fit the Type-IV points into the picture, let
be any sequence of nested disks and consider what happens as we move down the line segments The fact that the disks D(ai , ri ) are nested implies that the line segments extend one another downward as i increases. If the intersec tion n D (a i , ri ) is nonempty, then it is equal to D (a, r) for some a E Cv and some r 2:: 0, so the intersection corresponds to a point of Type I, II, or III forms a closed line segment. and the union together with the endpoint However, if the intersection n D ( ai , ri ) is empty, then there is no actual disk D(a, r) sitting at the bottom of the union of the line segments Thus as already noted, the points of Type IV exist precisely to remedy this situation and to ensure that every downward path has a termination point. Further, this explains why we defined by (5.45) to be the line segment running from to (See Exercise 5 .41 .) The Berkovich disk [JB is "illustrated" in Figure 5.4. Of course, Figure 5.4 is at best a pale imitation of the true glory of the Berkovich disk, since in a complete picture of [JB, every line segment contains a countable number of Type- II points, off of each of which there is a countable number of downward branches. Unfortunately, despite advances in modem technology, there are still no computer packages capable of fully rendering a (countably) infinitely branched broomstick.
LaLa,,,,rrii · U La,,ri
�a,r �a,r
La,r
5.10.1.4 The Gel'fond Topology on the Berkovich Disk
The next step is to put a topology on the Berkovich disk.
ULa,,r, · �a,r �0, 1 .
300
5. Dynamics over Local Fields: Bad Reduction
�0, 1 � Gauss point I
I Type-I point I
-
�0 , 0
�a ,O
�b,O
Figure 5.4: The Berkovich disk D 8 . Definition. The Gel'fond topology on D 8 is the weakest topology such that for every f E Cv [ z] and every B > 0, the following sets are open:
U(f, B) = {x E D8 : lfl x < B}
V(f, B) = {x E D8 : lfl x > B}.
and
Theorem 5.68. (Berkovich) The Berkovich disk fJB with the Gel 'fond topology is a compact path-connected Hausdorffspace.
Proof See [64, Theorem 1 .2.1] or [29, Appendix D) for the proof that [JB is compact
and Hausdorff and [64, Corollary 3.2.3] for the proof that it is path connected.
D
Basic open sets for the Gel ' fond topology on JJ!3 , viewed as a tree, can be de
scribed by taking (and deleting) branches of the tree as described in the following definition. The closed branch of [JB rooted at (a, r), denoted by (r.. a , r , consists of all points �b,s such that �a , r is on the line segment Lb,s running from �b,s to �0,1, together with whatever Type-IV points are needed to finish off the bottom of any open line segments. Thus
Definition.
{
lr-.. a , r = �b,s s :
:S r and l b - a l :S r }
U
{ appropriate Type-IV points}.
The open branch rooted at (a, r), denoted by �r-..: , r is obtained by starting with the closed branch lr-.. a ,r • removing the point �a , r. and then taking the connected com ponent that contains �a,O · If �a,r is of Type III, this is simply the closed branch at �a , r with the single point �a,r removed; but if �a,r is of Type II, then there are countably many branches at �a , r and we select the one that includes the point �a ,O · It is not hard to see that �r-..: , r = �b,s : s < r and l b - a l
{
<
r } U { appropriate Type-IV points}.
Then a base of open sets on [JB for the Gel' fond topology consists of all sets of the following three types:
301
5.10. Dynamics on Berkovich Space • • •
Open branches. Open branches with a finite number of closed subbranches removed. The entire tree with a finite number of closed branches removed.
Remark 5.69. There is a natural inclusion -
D(O , 1 ) D- s , a �---+ �a, o, that identifies the unit disk D(O , 1) as the set of Type-1 points in the Berkovich disk D8. It is not hard to check that the Gel'fond toplogy on D8, restricted to D ( O , 1 ) , is the usual topology induced by the metric on Cp. See Exercise 5.44. 5.10.2
'------+
The Berkovich Affine and Projective Lines
It is relatively easy to construct the Berkovich affine line A8 and the Berkovich projective line lP'8 as topological spaces, which we do in this section. It is more difficult to construct them as ringed spaces with sheaves of functions appropriate for doing analysis. See Remark 5.74 for a brief discussion and references. 5.10.2.1 The Berkovich Affine Line A8
Recall that the Berkovich disk J)B consists of four types of points. Each may be described in terms of disks that are contained in the closed unit disk D ( O , 1 ). Equivalently, J)B is the collection of I · 1 1 1-bounded seminorms on Cp [z]. More generally, we define the Berkovich disk D� ofradius R to be the collection of II · llwbounded seminorms on Cp [z]. It is given the Gel'fond topology and has its own Gauss point �o ,R corresponding to the seminorm
It is clear how to define points of Type I, II, III, and IV in D� using closed disks contained in D (O, R), just as we did for J)B. Further, there is an inclusion This is clear from the definition of D� as a set of seminorms. In terms of the picture of D� as a branched tree, we see that D�1 is the closed branch of D� lying below 2 the Gauss point �o ,R1 of D�1 . The Berkovich affine line AP is the union of the increasing collection of Berkovich disks,
Definition.
AB = u D�, R>O
with the topology induced by the direct limit topology on the individual Berkovich disks. It suffices, of course, to take the union over any increasing sequence of radii, for example, over R = pk with oo.
k ----+
5. Dynamics over Local Fields: Bad Reduction
302
Thus every point a E Cv, every disk D( a, r) c Cv, and every nested sequence of disks with empty intersection gives a point in the Berkovich affine line A. 8 , and A. 8 is composed of exactly this collection ofpoints. In particular, there is a natural inclusion ofCv as the set ofType-1 points in A. 8 , We also note that A. 8 inherits a tree structure from the natural tree structure of the Berkovich disks D�. However, the tree A.8 extends infinitely far upward; there is no Gauss point sitting at the top of A. 8 . 5.10.2.2 The Berkovich Projective Line JP' 8
The easiest way to construct the Berkovich projective line JP'8 as a topological space is to glue together two copies of the Berkovich disk [JB along their annuli AnnB = { �a,r E D- B l a l = 1 } :
using the map We note that the map f is induced from the inversion map f ( z) = 1/ z, since it is easy to check (Exercise 5.37) that ifO rJ_ D(a, r), then
f ( D(a, r)) = {z - 1 : l z - ai :S r} = D(a- 1 , r/ia l 2 ). In particular, i f Ia I = 1, then f ( D(a, r)) = D (a- 1 , r ) , so f(�a , r) = �a -l, r ·
The full Berkovich disk is the disjoint union of the annulus and the open branch containing �o ,o , D- B = AnnB U lf\ 0, 1 . o
Thus when we glue two copies of J)B along their annuli, the only parts of the two disks that are not identified are the two open branches A\� 1 . Hence another way to construct JP'8 is to take one copy of [)B and attach one extra copy of A\� 0 running vertically upward from the Gauss point �0, 1 . The result is illustrated in Fi�e 5.5. It is natural to denote the extra vertical branch by A\:0 and to label its points using the reciprocals of the points in D(O, 1),
A\:0 = {�a , r : l a l > 1 and r < 1} U {�oo , o}.
The open and closed branches in A\:0 are defined using the natural identification �
Ao Ao /1\0, 1 f-------> /1\ oo •
For example, a basic open neighborhood of the Gauss point �o, 1 is obtained by re moving from JP'8 a finite number of closed branches, some of which may be in the vertical branch A\"oo at infinity.
303
5.10. Dynamics on Berkovich Space
Extra branch "at infinity"
Figure 5.5: The Berkovich projective line
JP'8 .
Remark 5.70. There is a natural embedding of A.S into lP'8 . However, the inversion map f ( z ) = 1/z used to glue together the two pieces of can cause notational
8 lP' confusion regarding the "radius" of points of Types II and III. For example, the point �p-2 ,p-3 in the branch fr..:O of lP'8 would be denoted by �p-1 ,p when viewed as a point in A8 . Thus it might be wiser to denote the points in fr.. using some :O
alternative notation, for example ta, r, but we will not do so. Remark 5.71. A useful alternative construction oflP,s mimics the construction of the scheme lP'i as the set of homogeneous prime ideals. It starts with the set ofbounded seminorms on the two-variable polynomial ring Cp [x , y] that extend the usual ab solute value on Cp and that do not vanish on the maximal ideal (x, y). Two semi norms I 1 1 and I · are considered equivalent if there is a constant c > 0 such that for all homogeneous f E Cp[x, y]. Then is the set of equivalence classes of such seminorms. For details of this construction, see [66]. Let P E lP' 1 (Cp). To create a seminorm from P, choose homogeneous coordi nates P = [a, b] and set I f I = lf(a, b) 1. Notice that a different choice of homoge neous coordinates for P gives an equivalent seminorm. This embeds lP' 1 ( Cp) into
12
lP'8
5.10.2.3
Properties of
A8 and JP'8
JP'8 .
As repayment for the effort required to construct them, Berkovich spaces have many nice properties that Cp lacks.
304
5. Dynamics over Local Fields: Bad Reduction
(a) The Berkovich disks D� are compact, Hausdorff, and uniquely path connected. (b) The Berkovich affine line A.6 is locally compact, Hausdorff, and uniquely path connected. (c) The Berkovich projective line JID6 is compact, Hausdorff, and uniquely path con nected.
Theorem 5.72.
5
D
Proof See [26] and [64].
Remark 5.73. As noted earlier, the Berkovich affine line A6 contains a copy of A 1 (Cp) = Cp, since each a E Cp gives an associated Type-1 point in .D�
�a,o
provided R ;::: l aJ. Similarly, the Berkovich projective line JID6 contains a copy of the classical projective line JID1 ( Cv) via the map a f-----+
E
{ �a�a,,oO �oo,O
D (O , 1) if lal :::; 1,
E /r..:O E
/r..:O
if 1 < Ia I < oo , if a = 00.
One can show that the restriction of the Gel'fond topology on A6 and JID6 to A 1 (Cp) and JID1 ( Cv), respectively, gives the topology induced by the usual metric on A 1 ( Cp) and the chordal metric on JID1 (Cv) · See Exercise 5.44. This explains why the Gel'fond topology is the "right" topology to use on Berkovich spaces. Remark 5.74. We have constructed A6 and JID6 purely as topological spaces. It is more difficult, but very important, to refine this construction and make A6 and JID6 into ringed spaces with structure sheaves built up naturally from rings of functions. There are two approaches, both due to Berkovich. The first takes unions of open Berkovich disks, which have a natural structure as analytic spaces, and glues them along open annuli; see [26, 64]. The second glues affinoids (which are closed) using nets; see [65]. This second construction is less intuitive, but it allows one to functo rially attach a Berkovich analytic space to any reasonable rigid analytic space. 5.1 0.3
Dynamics on Berkovich Space
Having constructed the Berkovich spaces .06, A 6, and JID6, we are finally ready to study iteration of maps on these spaces. 5.10.3.1
Polynomial and Rational Maps on Berkovich Space
Let ¢(z) E Cp[z] be a polynomial. There is a natural way to extend the map ¢ : A 1 (Cp) A 1 (Cp) to a map on Berkovich affine space ¢ : A6 � A.6. In terms of seminorms, the action of ¢ is simply given by composition, �
lfi
=
=
305
5.10. Dynamics on Berkovich Space
However, it is perhaps easier to understand the map ¢ A8 A8 by looking at the action of ¢ on points of Types I-IV. Recall (Proposition 5.16) that ¢ maps disks to disks, say :
¢( D(a, r)) = D(¢(a), R)
---+
for some R = R(¢, a, r) .
Then for points of Types I, II, and III we define ¢(�a,r )
=
�¢(a) ,R(¢ , a , r) •
and for points of Type IV we take the usual limit
Remark 5.75. The maximum modulus principle (Theorem 5.13) allows us to explic
itly describe the radius R(¢, a, r) of the image ¢(D(a, r)). First expand ¢(z) as a polynomial in powers of z - a, say d
¢(z) = l: c;(¢, a)(z - a) i . i=O Then
R(¢, a, r) = Imax l c;(¢, a) l r i = ::; t ::; d
sup
zED(a, r )
l ¢(z) - ¢(a) l .
A rational function ¢( z) E Cp ( z) similarly induces a map on the Berkovich projective line JID8 extending the usual map on JID1 (CP ). If ¢ has no zeros or poles on the disk D( a, r ) then it is relatively easy to describe the value of ¢( �a , r ) . We know in this situation that ¢ ( D( a, r)) is a disk, say equal to D (¢(a), s ) . Then ,
{
¢ (�a, r ) = �¢(a),s
�¢(a) -l ,s/l¢(a) 12
if l ¢ (a) l ::; 1, if 1¢( a) I > 1 .
Note that tlle assumption tllat ¢ does not vanish on D(a, r) is equivalent to the in equality 1¢( a) I > s, so the indicated points are in D8. The description of ¢(�a, r ) when ¢ has zeros and/or poles on D(a, r) is more complicated. An explicit description in terms of open annuli is given by Rivera Letelier [373, 375, 376]. (See also [26, Section 2].) A succinct, but less explicit, way to specify the induced map ¢ JID8 JID8 is to use the construction of JID8 as a space of homogeneous seminorms as described in Remark 5. 7 1 . Then for a given seminorm � E JID8, the seminorm ¢( � ) is determined by writing ¢ = [F, G] using homogeneous polynomials F and G and setting :
---+
if(x, y ) l ¢ (� ) = if(F(x, y), G(x, y)) i
for all homogeneous f E Cp[x, y].
306 5.10.3.2
5. Dynamics over Local Fields: Bad Reduction The Julia and Fatou Sets in Berkovich Space
A natural way to put a metric on the Berkovich spaces [JB, A 8, and !P'8 is to use the underlying tree structure and measure distances along line segments. Unfor tunately, this path-length metric does not give the Gel'fond topology, and as we have observed, it is the Gel'fond topology that extends the natural metric topolo gies on D (O , 1), A 1 (Cp), and IP' 1 (Cp) . (See Exercises 5.42 and 5.44.) It is possible to define a metric that does yield the Gel'fond topology, but the definition of the "Gel'fond" metric is quite indirect. See [26, Corollary 1 .3]. So rather than using a metric, we instead characterize the Fatou and Julia sets in Berkovich space using an abstract topological version of equicontinuity. Let X and Y be topological spaces and let be a collection of con tinuous maps X Y. The set is (topologically) equicontinuous at x if for every point y E Y and every neighborhood V c Y of y there are neighborhoods U c X of x and W c Y of y such that for every rjJ E , the following implication is true: Definition.
-+
rjJ(U) n W -1- 0
==?
rjJ(U) c V.
Intuition: is equicontinuous at x if for each y E Y, whenever rjJ E sends some point close to x to a point that is close to y, then rjJ sends every point close to x to a point that is close to y. One can show that if Y is a compact metric space, then topological equicon tinuity agrees with the usual metric definition of equicontinuity. (See [26, Proposi tion 7 . 1 7].) We say that is (topologically) equicontinuous on X if it is topologically equicontinuous at every point of X.
Let rjJ(z) E Cp(z) be a rational map. The (Berkovich) Fatou set of rjJ is the largest open subset of lP'8 on which rjJ is equicontinuous, or more precisely, on which the set of iterates { rjln } is equicontinuous. The (Berkovich) Julia set ofrjJ is the complement of the Berkovich Fatou set. We denote these sets by :F8 ( ¢ ) and .:J8( rjJ ), respectively. Definition.
Remark 5.76. Recall that the classical points in !P'8, i.e., the points of Type-1, form
a copy of IP' 1 ( Cp) sitting inside !P'8. As noted earlier in Remark 5. 73, the restric tion of the Gel'fond topology on !P'8 to the classical points gives the same topol ogy on IP'1 (Cp) as that induced by the chordal metric. Using this one can show that equicontinuity at a classical point of!P'8 using the Gel'fond topology is equivalent to equicontinuity using the chordal metric. Hence the classical Fatou and Julia sets sit within their Berkovich counterparts: and
rjJ(z) E Cp(z) be a rational map of degree at least 2. Various authors have shown that there is a unique probability measure 1-l¢ on !P'8 satisfying
Remark 5.77. Let
¢ * 1-l¢ = d . 1-l¢
and
5.10. Dynamics on Berkovich Space
307
(Recall that a probability measure is a nonnegative measure of total mass 1 .) We call f-l
Theorem 5.78. Let rjJ( z ) E Cp(z) be a rational map of degree at least 2. The sup port of the canonical measure f-l
Proof This theorem is an amalgamation of results due to Baker, Rumely, and Rivera
Letelier. We refer the reader to [26, Section 7.5] for the construction of the canonical measure and to [26, Theorem 7. 1 8], [27, Theorems 8.9 and A.7], and [38 1 ] for the proof that f-l
Example 5.79. Let ¢( z ) E Cp(z) be a rational map of degree at least 2 and sup
pose that ¢ has good reduction. We know (Theorem 2.17) that the classical Julia set .J ( ¢) c lP'1 ( Cp) is empty. Using the construction of the canonical measure, it is not hard to show [26, Example 7.2] that for a map of good reduction, the canonical measure is entirely supported at the Gauss point, i.e., f.l
and
f.l
Thus .18(¢) = {�0, 1 }, so the nonempty Julia set guaranteed by Theorem 5.78 is not very interesting, since it consists of a single point. Hence even in Berkovich space, the most interesting dynamical behavior occurs for maps of bad reduction. On the other hand, if the conjugates ¢! of ¢ have bad reduction for every f E PGL2 (Cp), then .J(¢) is a perfect set, and hence uncountable. (See Theorem 5.82.) 5.10.3.3 The Map ¢ ( z )
=
z 2 on Berkovich Space
To conclude our brief foray into Berkovich space, we illustrate Berkovich dynamics by studying the simplest possible map, namely ¢( z ) z 2 • For any a E Cp, we expand ¢( z ) - ¢( a ) = z 2 - a2 = 2a ( z - a ) + ( z - a ) 2 . =
Assuming henceforth that p 2: and using our convention that all points in !P'8 have radius satisfying r ::; 1, we find that
3
r/J (�a,r) = �(a),s
with s = max{ l2a l r, r 2 } = r · max{ lal, r }.
(5.46)
This explicitly gives the action of ¢ on points of Types I, II, and III in fJB, and the action of ¢ on Type-IV points is given by the appropriate limit.
308
5. Dynamics over Local Fields: Bad Reduction
The formula (5.46) allows us to compute many orbits O (�a ,r ) . For example, suppose that ia l < 1 and r < 1. Then I
In other words, the open branch /t\� , 1 is in the attracting basin of the fixed point �0,0 . (We leave it to the reader to make this informal argument rigorous using the Gel'fond topology on D 8 . ) What are the fixed points of ¢? For a point �a, r E JJB of Type I, II, or III in the Berkovich disk, we have
r = r · max{ iai , r} and D(a, r) = D(a2 , r) r = 0 and a2 , or max{ l ai, r} = 1 and la - a2 i ::; r. There are three cases to consider. First, if r = 0, then a a2 , so we see that �o ,o and 6,o are the only fixed points of Type I in D 8 . Second, if r = 1, then �a ,r is equal to the Gauss point �o , 1 , which is clearly fixed by ¢( z). Finally, suppose that 0 < r < 1. Then �a , r is fixed if and only if
{
tP (�a, r) = �a,r
a =
=
ial 1 =
and
Thus is fixed if and only if a E D(1, r ) , in which case �a,r 6,r· This exhausts the Type-1,-II, and-III fixed points in D 8 . A similar analysis on the branch leading up to �oo,o yields one more fixed point, namely the Type-! point �oo,o at the top of the tree. Hence aside from Type-IV points, the fixed-point set of ¢ on JP'5 consists of two (attracting) Type-I fixed points �o , o and �oo,o and the line segment running from the Gauss point �0 ,1 down to the (neutral) Type-I fixed point 6, 0 ,
�a,r
=
Fix (¢ , lP'5) = { �o,o, �oo , o } U
{6, r : 0 :::; r :::; 1 } .
We leave it as an exercise to show that ¢( z) = z 2 has no fixed points of Type IV (Exercise 5.36). We have seen that every point in /t\� ,o is attracted to �o,o, and the Gauss point �o, 1 is fixed. We next show that every other point in J)B of Type II or III is preperiodic. Let �a ,r E JJB be such a point, which means that
0
and
ia l = 1.
Then the disk D (a, r) has positive radius, so it contains points that are algebraic over Qp (note that Qp is dense in Cp). Replacing a with such a point, we may as sume that a is algebraic over Qp. Next we observe that (5.46) combined with the assumption that ia l = 1 implies so by iteration
5.10. Dynamics on Berkovich Space
309
Hence Let K = Qp ( a) and notice that
U fJ (q;n (a) , ) n K c D(O, 1) n K. n 2': 0 r
The disk D(O, 1) n K cannot contain infinitely many disjoint disks of radius r, so there must exist m > n such that Then q;m(�a,r) q;n(�a ,r ), so �a , r is preperiodic. We conclude this section by using the definition of topological equicontinuity to directly demonstrate that the Gauss point �0, 1 is in the Julia set of¢. The intuition is that any neighborhood of �0, 1 contains points �o,r on the line segment connecting �o, 1 to �0,0. If < 1, then the iterates q;n (�o,r) approach �o,o, but the Gauss point �o, 1 is fixed. Hence q;n(�0, 1 ) does not remain close to q;n(�o,r ). We now make this argument ngorous. We suppose that x = �0, 1 E :F8 ( ¢) and derive a contradiction. Let y �0,0, and let V /1-..� ,1 ; 2 be our chosen neighborhood of �0,0. The definition of equicontinuity says that there are neighborhoods �0, 1 E U and �o,o E such that for all n ;::: 0,
=
r
=
=
W
The implication (5.47) remains true if we replace U and hoods, so we may assume that U
=
JP'B "-
k
U /1-..ai ,ri
with 0 <
i= 1
r;
(5.47)
W by smaller neighbor 1
< for all i, and
with 0 < < 1. r
Choose a value of s satisfying max
r;
1 �i�k
< s < 1.
Then �o,s E U, since �o,s is not on any of the closed branches ffi. ai , ri · On the other hand, q;n (�o, s) �o,sn E W /1-..�,r for sufficiently large n, since we just need to ensure that sn < This proves that =
=
r.
so the assumption that ¢ is equicontinuous at �0, 1 implies that
310
5. Dynamics over Local Fields: Bad Reduction
But this inclusion is clearly not true, since, for example, �0, 1 E U is fixed by ¢, but �o , 1 � A\� , 1 12 . Therefore ¢ is not equicontinuous at �o, 1 , so �0, 1 E .J8(¢). A similar case-by-case argument using the explicit description (5 .46) of the ac tion of ¢ shows that every other point in lP'8 is in the Fatou set. We leave the details for the reader. 5.10.3.4 Further Results
We briefly describe, without proof, some deeper results on Berkovich dynamics. Our exposition follows [27], and the author is grateful to Baker and Rumely for making their preprint available. Theorem 5.80. (Strong Montel Theorem on lP'8) Let ¢ E Cp(z) be a rational map ofdegree at least let � E lP'8, let U C lP'8 be an open neighborhood of�, and let V be the union of
2,
"
"
3
Proof This theorem is due to Baker and Rumely [27, Theorem 7.1] for maps ¢ defined over a finite extension of Qp, and to Rivera-Letelier in the general case;
0
see [27, Theorem A. l ] and [381].
The proof is based on Rivera-Letelier's classification of periodic components in the Fatou set. In order to describe this classification and its applications, we need to define what it means for a periodic point in Berkovich space to be attracting or repelling. For Type-I points, i.e., for points in lP'1 (Cp), we use the usual definition. It turns out that all attracting periodic points in lP'8 are Type-I points. (See Exercise 5.45 for an explanation of why this is reasonable.) The definition of repelling periodic points is more complicated. Repelling periodic points are all of Type I or II [376, Proposition 5.5], where we use Rivera-Letelier's definition that a Type-11 periodic point is repelling if its residual degree (see [376, Section 5]) is at least
2
2.
Definition. Let ¢ E Cp ( z) be a rational map of degree at least and let � E lP'8 be an attracting periodic point of period n. The basin ofattraction of� is the set
The connected component of this set is called the immediate basin ofattraction of�. Recall that a point P is called recurrent for ¢ if it is in the closure of {
:
2:
We are now ready to state Rivera-Letelier's strong Montel theorem for rational maps on the Berkovich projective line.
5.10. Dynamics on Berkovich Space
311
Theorem 5.81. (Rivera-Letelier) Let Uq, be the set ofpoints � E !P'8 with the property that there is a neighborhood U of� such that
IP'l (Cp ) "
u ¢n(U)
n 2': 1
3
contains at least points. (a) Every periodic connected component ofUq, is either an immediate basin ofat traction of¢ or a connected component ofthe domain ofquasiperiodicity of¢. All such components are in :F8 ( ¢). (b) Every wandering component ofUq, is contained in :F8 ( ¢). Proof The proof of (a) is given in [27, Theorem A.2] and the proofof (b) is in [27,
Corollary A.5].
D
As in the classical setting, Montel's theorem has a large number of important consequences, some of which we state here. Theorem 5.82. Let ¢
E Cp ( z ) be a rational map ofdegree at least 2.
(a) Ifthere is some f E PGL2 (Cp ) such that the conjugate ¢! has good reduction, then the Julia set .]"8 ( ¢) consists ofa single point.
(b) If every conjugate ¢! of ¢ has bad reduction, then the Julia set .]"8 ( ¢) is a
perfect set, and hence in particular it is uncountable. (c) Let � E .]"8 ( ¢ ), let U E !P'8 be an open neighborhood of �' and let V be the union of cj;n(U) for all n 2': 1. Then (i) V contains IP'8 " IP'1 (Cp ); (ii) V contains .]"8( ¢); and (iii) IP' 1 (Cp ) " V consists ofat most two points. (d) The Julia set .]"8 ( ¢) is either connected or else it has infinitely many connected components. Further, it has empty interior. (e) Let � E .]"8 ( ¢ ). Then the backward orbit of� is dense in .]"8 ( ¢). (f) Let V c IP'8 be a closed completely ¢-invariant set containing at least points. Then V :::2 .]"8 (¢). (g) The Julia set .]"8 (¢) is exactly equal to the closure of the repelling periodic points in IP'8.
3
Proof (a) See [27, Lemma 8.1]. (b) See [27, Corollary 8.6]. ( c ) See [27, Theorem 8.2].
(d) See [27, Corollary 8.3 and Corollary 8.7]. See [27, Corollary 8.5].
(e) (f) See [27, Corollary 8.8].
(g) See [27, Theorem A.7].
D
Exercises
312
Exercises Section 5. 1 . Absolute Values and Completions 5.1. Prove that up to equivalence, the only nontrivial absolute values on Q are the usual archimedean absolute value and the p-adic absolute values. (This result is known as Os trowski's theorem.) Section 5.2. A Primer on Nonarchimedean Analysis 5.2. (a) Let ¢(z) be a holomorphic function on D( a, r ). Prove that the Taylor series coeffi cients of ¢(z) are uniquely determined by ¢. (b) Let ¢( z) be a function that is represented by a convergent Laurent series on the punc tured disk D( a, r) " {a}. Prove that the coefficients of its Laurent series are uniquely determined by ¢. 5.3. (a) Let ¢(z) and 1/J(z) be holomorphic functions on D(a, r). Prove that the prod uct ¢(z)1jJ(z) is a holomorphic function on D(a , r). (b) Let ¢(z) and 1/J(z) be meromorphic functions represented by Laurent series on D(a, r). Prove that the product ¢(z)1jJ(z) i s a meromorphic function and is represented by a Laurent series on D( a, r ). 5.4. Let ¢( z) E Cp [z ] be a nonconstant polynomial. (a) Prove directly (i.e., without using Newton polygons) that ¢ sends a closed disk D(a, r) to a closed disk [J ( ¢( a) , s) . (b) Prove that ¢ maps D( a, r) bijectively to [J (¢(a), s ) if and only if
l
¢ (z) - ¢( a) I
= �r l z - a l
for all z
E
D(a, r).
5.5. This exercise outlines a prove of a p-adic inverse function theorem. Let ¢ E Cp ( z ) be a rational function of degree d ;::: 1 and let P E JID 1 ( Cp) be a point that is not a critical value of ¢, i.e., P is not the image of a critical point of ¢. (a) Prove that there is a disk [J ( P, r) centered at P such that ¢ - 1 ( [J ( P, r)) consists of d disjoint open sets Show that the Vi are disks. (b) Let ¢i : Vi ---+ [J ( P, r) be the restriction of ¢ to Vi for each 1 :::; i :::; d. Prove that ¢i is bijective. (c) Possibly after reducing r, prove that the inverse maps ¢; 1 : D(P, r) ---+ Vi are given by convergent power series. (If oo E ¢ - 1 ( P), either change coordinates or instead use a convergent Laurent series.) 5.6. (a) Let K be an algebraically closed field that is complete with respect to a (nonar chimedean) absolute value and let ¢ : JID 1 (K) ---+ JID 1 (K) be a rational map. Prove that ¢ is an open map, i.e., the image of an open set is an open set. (b) Consider the map ¢ : JID 1 ( Q3 ) ---+ JID1 ( Q3) defined by ¢( z) = z2 . Prove that ¢ is not an open map by showing that the image of the open disk D ( 0, 1) in JID1 ( Q3) is not open. Thus rational maps over complete, but not algebraically closed, fields need not be open.
Exercises
313
5.7. Let ¢(z) E Cp[z] be analytic on D(a, r) and assume that ¢(a) -1- 0. Prove that 1/¢(z) is analytic on some closed disk centered at a. More precisely, prove that 1/¢(z) is analytic on D(a, t) for any t < rl ¢(a) l / ll ¢ 11· Section 5.3. Newton Polygons and the Maximum Modulus Principle 5.8. This exercise generalizes Proposition 5 . 1 6(b) by asking you to prove a general Cauchy estimate for p-adic analytic functions. Let ¢(z) E Cp[z] be a power series that converges on D(a, r) and let ¢ ( D(a, r)) = b ( ¢(a) , s ) . Prove that for all n ;::o:
1.
(Note that a s n increases, the estimate becomes worse, since In! I --+ 0 a s n --+ oo.) Section 5.4. The Nonarchimedean Julia and Fatou Sets 5.9. Prove the implications between uniform Lipschitz, uniform continuity, and equicontinu ity stated in Section 5.4. More generally, give a definition of what it means for a family of functions to be equi-Lipschitz and prove the following implications:
I uniformly Lipschitz I
===}
I equi-Lipschitz at every point I
I uniformly continuous I
===}
I equicontinuous at every point I
5.1 0. This exercise develops the abstract theory of residues (on lP' 1 ) in order to prove Theo rem 1 . 14 in arbitrary characteristic. Let ¢( z) E K ( z) and a E K. The function ¢( z) can be written as a partial Laurent series
¢(z)
=
a- N ( z - a )N
a- 1 +1 + · · · + + (z a--aN)N(z1 - a-) + 1/J ( z )
with 1/J (z) E
K[z - a],
and we define the residue of¢ at a to be
Res(¢(z) dz) z=o:
=
a-1 ·
The residue of ¢ at oo is defined by using the substitution z
Res (¢(z) dz)
z=CXJ
=
=
1/w; thus
Res (-¢(w- 1 )w- 2 dw) .
w=O
(a) Compute all of the nonzero residues of each of the following functions. In each case, check that the sum of the residues is 0.
(b)
¢1 ( z)
=
=
=
(c)
:
z2 + 2 z2 2 2 . (ii) ¢2 (z) 2 z - 3z + 2 z - 4z 2 5z If ¢( z) has a simple pole at a, prove that the residue of ¢ at a is the value of the function ( z - a)¢( z) evaluated at z a. More generally, for any integer n ;::o: 0, let 1/Jn ( z) be the function 1 dn 1/Jn ( z ) n! dzn ( (z - a) n+1 ¢(z) ) . (i)
=
3
314
Exercises Here we are taking formal derivatives in K(z). Note that in characteristic p, one must be careful to "cancel" the n! before setting p = 0. (If you have not seen this kind of computation before, rewrite the expression for 'l/Jn ( z) to make it clear that it makes sense in any characteristic.) Suppose that ¢(z) has a pole of exact order n 2: 1 at a, i.e., writing ¢(z) = F(z)/G(z) as a ratio of polynomials with no common factors, we have n = ordz=a ( G(z) ) . Prove that
Res(¢(z) dz) = 'l/Jn -t(a). z=o: (d) Prove that
(¢(z) dz) # o } { P E lP'1 (K) : Res p
is a finite set. (e) Assume that K is algebraically closed and prove the Cauchy residue formula
L �s(¢(z) dz) = O.
PEIP'1 (K)
(This is a hard exercise. Try it first under the assumption that ¢(z) has simple poles and ¢( oo ) # oo, which suffices for (f).) (f) Use (e) to prove that Theorem 1 . 14 is true in arbitrary characteristic, and use it to deduce that Corollary 5 . 1 9 and Proposition 5.20(c) are true. Hence Fatou sets are nonempty for all algebraically closed nonarchimedean fields. 5.11. (a) Let p 3 be a prime and let c E Cp satisfy lei > 1. Prove that the function ¢(z) = z2 + c has exactly one nonrepelling periodic point, namely the totally ramified fixed point at infinity. (b) * Suppose that ¢(z) E Cp(z) has an indifferent periodic point, i.e., a periodic point P whose multiplier satisfies I Ap (¢) I = 1 . Prove that ¢( z) has infinitely many indifferent periodic points. (This result is due to Rivera-Letelier [372].) (c) ** Do there exist rational maps ¢(z) E Cp(z) that have at least two, but only finitely many, nonrepelling periodic points? (From (b), the nonrepelling periodic points would have to be attracting.)
2:
Section 5.5. The Dynamics of (z2
- z)/p
5.12. Let S be a finite set, let SN be the space of sequences on S with the symbolic dynamics metric, and let L : sN --> sN be the left shift map. (See Section 5.5. 1 .) (a) Let a, f3 E SN and suppose that limn � = p ( Ln (a) , L n (/3)) = 0. Prove that there exists an m such that L m (a) = Lm (/3 ). (b) Prove that the periodic points o f L are dense i n SN . (c) Prove that there is an element "( E sN with the property that the orbit
Ch ('Y) = {C('Y) : n E N} is dense in sN One says that L is topologically transitive on sN (Hint. Create "( by first listing all possible blocks of length 1 , then all possible blocks oflength 2, and so on.) (d) Let a E sN . Prove that the backward orbit 0£ (a) = Un2: 0 L - n (a) is dense in sN . 0
0
315
Exercises (e) More precisely, prove that the backward orbit of any following sense: For all (3 E SN and all k 2:: 0,
o:
E SN is equidistributed in the 1
(#S) k . (In fact, prove that the limit stabilizes as soon as n
>
k.)
5.13. Let rj;(z) = (z 2 - z)/p with p 2:: 3. Use Exercise 5.12 and the identification provided by Proposition 5.24 to prove that the backward orbit of a point a E :J( rjJ) is equidistributed in the following sense: For all b E :J(r/>) and all radii r = p-k,
5.14. Let p 2:: 3 and for any c E Cp, let r/>c(z) = (z2 - cz) jp. (a) If c E z;, prove that the statement of Corollary 5.25 is true for rPc · (b) More generally, prove an analogous statement if c is a unit in a finite extension of Zp. (c) What happens if c E pZp? 5.15. Let d 2:: 2 be an integer, let p
rj;(z)
=
>
d be a prime, and consider the dynamics of the map
)(_ z -z(-''-z_ --_1'-"
2)
+ 1) --'·_,_(z_-_ d --
---''·-·
p
over Cp. Prove that :J ( rjJ) can be described using symbolic dynamics and use this identification to prove the following generalization of Corollary 5.25: :J( rjJ) C IQip, the set Pern( rjJ) consists ofthe fixed point at oo and dn repelling points, the Julia set :J( rjJ) is the closure of the repelling periodic points, and rjJ is topologically transitive on :J ( rjJ) . Section 5.6. A Nonarchimedean Version of Mantel's Theorem 5.16. Complete the proof of Theorem 5.26 (see page 265) by writing down the details in the case that o: = 0. 5.17. Let be a collection of power series that converge on D( a , r ), and suppose that there is a point o: E K such that o: � ¢( D(a, r) ) for all rjJ E <1>. We proved in Theorem 5.26 that there is a constant C = C ( o:, a, r) such that
p(rj;(z), rj;(w)) S Cp(z, w)
for all rjJ E
and all z, w E D(a, )
r .
Find an explicit value for the Lipschitz constant C in terms of o:, a, and r. 5.18. Let be a collection of rational, or more generally meromorphic, functions
D(a, r) -> IP'1 (K), and suppose that o: , (3 E IP'1 ( K) do not lie in rjJ (D( a, r) ) for all rjJ E . Our proof ofthe nonar chimedean Monte! theorem (Theorem 5.27) shows that there is a constant C C( o:, (3, a, r) such that =
p(rj;(z), rf;(w)) S Cp(z, w) Find an explicit value o f C i n terms o f
o: ,
E and all z, w E D(a, r). (3 , a, and r. for all rjJ
316
Exercises
Section 5.7. Periodic Points and the Julia Set 5.19. As in the statement of Lemma 5.34, let A, B C Cp be bounded sets for which there are constants 0 < 8 ::; 1 ::; � such that
sup I ::; �, sup 1 13 1 ::; �, and aEA,inf(3EB p(o:, !3) = 8 > 0. aEA Ia: (3EB For each ( o:, 13) E A x B, let La , f3 (z) (13 - o:)z + o:. During the proof of Lemma 5.34 we =
showed that
p ( La , f3(z), La',f3' (z ' ) ) ::;
�24
max {p (o: , o:'), p(l3, 13'), p(z, z') } for all o: , o:' E A, all 13 , 13' E B , and all z , z' E IP 1 (Cp)· Improve this result b y reducing the exponent o f � and/or 8 in the constant. Try to find the •
best-possible exponents. 5.20. Let A, B , C c IP1 (Cp) be three sets that are at a positive distance from each other, i.e., there is a constant 8 > 0 such that
aEA,inf(3EB p( o:, 13) 2: 8, aEA,inf-yEC p(o:, 'Y) 2: 8, For any triple of points (o:, 13, 'Y) E A x B x C, define
inf
(3EB, -yEC
p(l3, 'Y ) 2: 8.
to be the unique linear fractional transformation satisfying
La,(3, -y (0)
= 0: ,
Prove that the map
( o: , /3 , ')' , z)
f-->
La,(3, -y (z),
is Lipschitz and find an explicit Lipschitz constant depending only on 8. (This exercise gener alizes Lemma 5.34, which is the case that C = { oo} is the single point at infinity.) 5.21. This exercise can be used in place of Lemma 5.34 to prove Mantel 's theorem with moving targets (Theorem 5.36). (a) Let P, Q E IP 1 (K) be distinct points. Prove that there is a linear fractional transformation A = AP, Q E PGL2 (K) satisfying
A(O)
=
P,
A(oo)
=
Q,
IRes( A) I
=
p(P, Q).
(Hint. If A = ( � S) is normalized to satisfy max { l a l , l b l , l ei , ldl } = 1, then Exer cise 2.8 says that the resultant of A is equal to the determinant ad - be.) (b) Let A, B C Cp be bounded sets that are at a positive distance from each other. Prove that the map
(P, Q , R) i s Lipschitz.
f-->
A P, Q ( R) ,
317
Exercises Section 5.8. Nonarchimedean Wandering Domains
5.22. Let X be a topological space and let D be a collection of open subsets of X that form a base for the toplogy of X. We use D to define disk components as described on page 277. (Note that the sets in D need not be actual disks, since the space X is merely assumed to be a topological space.) (a) Let U c X be an open set, let P1 , P2 E U be points, and let V1 and V2 be the disk components of U containing H and P2 , respectively. Prove that either V1 = V2 or vl n v2 = 0. (b) Prove that U is a disjoint union of disk components. (c) Prove that the disk components of U are open. 5.23. Let X be a topological space and let D be a collection of open subsets (disks) of X that form a base for the topology of X. An open set U C X is defined to be disk-connected (relative to D) if for every pair of points P, Q E U there is a finite sequence of disks D 1 , D 2 , . . . , D n C U such that for all l ::; i <
D i n DH 1 -I 0
n.
(a) Let U1 and U2 be disk-connected subsets of X. Prove that either
(b) Let U C X be an open set and let P E U. Prove that there is a maximal disk-connected open subset of U containing P. (c) Prove that the maximal disk-connected open subset of U containing P described in (b) is in fact the disk component of U containing P. This gives an alternative definition of disk component. 5.24. Prove that the standard closed disks with rational radius (cf. Remark 5.7) in 1P' 1 (1Cp) are exactly the images via elements of PGL2 (ICp) ofthe unit disk D(O, 1). Similarly, the standard open disks of rational radius are the images of D(O, 1 ) . 5.25. Let K/Qp b e a finite extension and let ¢(z)
E K(z) be a rational function o f degree
d � 2 with oo E :F( ¢) . Prove that ¢ is hyperbolic, i.e., ¢ satisfies the conditions of Theo rem 5 .46, if and only if ¢ has the following property: For every finite extension L/ K there are a set U C constants b > a > 0, and a continuous function u :
L containing .:J( ¢) n L, positive U [a, b] such that ___,
for all a:
E U.
This shows that p-adic hyperbolicity and classical (complex) hyperbolicity can both be char acterized as saying that a map is everywhere expanding on the Julia set with respect to some reasonable metric. For further information about hyperbolic maps in the classical setting; see, for example [95] or [302]. 5.26. * Let ¢ : lP' 1 ___, lP' 1 be a rational map of degree at least 2 and let P E lP'1 (ICp) be a recurrent critical point that is in the Fatou set :F( ¢) . Prove that P is a periodic point.
318
Exercises
Section 5.9. Green Functions and Local Heights 5.27. Let K be a field with a nonarchimedean absolute value v, let ¢ : lP'1 ---+ lP' 1 be a morphism of degree d 2:: 2, let = ( F, G) : A? ---+ A? be a lift of ¢ satisfying I I I v = 1, and let Y be the associated Green function. If I Res(F, G) I v =1- 1, or equivalently if ¢ has bad reduction at ¢, prove that there exists a point ( x, y ) E A; ( K) such that
This is the converse to Proposition 5.58(c). 5.28. Let K be a field with an absolute value gree d 2:: 2, and let : A2 ---+ A2 be a lift of ¢. (a) Prove that the map
g(x, y)
v,
let ¢ : lP' 1
---+
lP' 1 be a morphism of de
= logii (x , y) ll v - dlog ll (x , y) ll v
induces a well-defined function g : lP'1 ( K) ---+ R (b) Prove that g is Lipschitz, i.e., prove that there is a constant C = C( ¢) such that
l
g (P) - g(Q) I � Cpv(P, Q) for all P, Q E lP' 1 (K).
(c) If v is nonarchimedean, prove that g is locally constant. More precisely, if the lift is chosen to satisfy IIIIv = 1, prove that g(P)
=
g(Q) for all P, Q E 1P'1 (K) with
Pv(P, Q ) < IRes() I v·
(d) Define a modified Green function Q q, by Prove that Qq, is a Holder continuous function on lP' 1 ( K). In other words, prove that Qq, is well-defined on lP'1 ( K) and that there are positive constants C and <5 such that for all P, Q E lP' 1 (K). (Hint. Show that Qq, P) is given by the telescoping series
'L':=o d n g ( ¢n (P) . For n small n, say n � N, estimate the difference I g ( ¢ ( P) - g (¢n ( Q)) I using (b) and the fact that ¢ is Lipschitz, and for large n use an elementary bound. Then make an appropriate choice for N.) (
)
-
)
5.29. The definition of the canonical height h¢(P) as the limit of d n h ( ¢n (P) is not prac tical for numerical calculations, even for P E lP'1 (Q) , because one would need to compute the exact value of points ¢n ( P) whose coordinates have 0(dn ) digits. A better method to compute h¢(P) is as the sum ofthe Green functions Y
)
Y
This covers all but finitely many absolute values, so it remains to devise an efficient method to compute Y,v (x, y) in the remaining cases.
Exercises
319
Let P = [x, y] E IP'1 (K) and use the algorithm described in Figure 5 .6 to define a sequence of triples with u;, x;, y; E K.
This exercise asks you to prove that N iterations ofthe algorithm gives the value of 9
(c) As in Exercise 5.28, we let
g(X, Y ) log ii (X, Y ) ll v - d log llcx , Y ) t The homogeneity of shows that g is a well-defined function on IP' 1 (K), and Proposi =
tion 5.57 says that g is bounded. Prove that
9
N 9
(5.48)
where the big-0 constant depends only on . Deduce that the algorithm described in Figure 5.6 computes 9
I
5.30. Implement the algorithm described in Figure 5.6 to compute the Green function 9¢, oo for the archimedean absolute value on IP'1 (JR.). (a) Let ¢(z) = z + 1/z and compute 9¢,=(x, y) to (say) 8 decimal places for each of the points ( 1 , 1), (2, 1), and (5, 2). Check your program by verifying that your values satisfy
Compute the canonical height hq,(1) and compare it with the value that you obtained in Exercise 3.20. (Hint. The map ¢ has good reduction at all primes.) (b) Let ¢(z) = (3z 2 - 1)/(z 2 - 1) and note that ¢2 ( 1 ) = 3 and ¢3 (1) = ¥ · Com pute 9¢,=(x, y) for each of the points ( 1 , 1), (3, 1), and (13, 4). Why does 9¢,= ( 3 , 1) not equal 4 9¢,=( 1 , 1)? What is the difference between these two values?
5.31. Let ¢ E Q(z) be a rational map of degree at least 2 and let P E IP'1 (Q). Write an effi cient computer program to compute hq,(P) as a sum of Green functions. (For primes p not di viding Res( ), use Proposition 5.58 to compute 9¢,p (x, y). For primes dividing Res( ) and for the archimedean place, use the Green function algorithm in Figure 5 .6.) Compute hq, ( a ) to 8 decimal places for each of the following maps and points. (a) ¢(z) = z 2 - 1, a = �·
320
Exercises
INITIALIZATION
q, as q, = (F, G) with F, G E K[X, Y] N = Desired number of iterations Set x0 = x and y0 = y and Green = 0 MAIN LOOP: i = 0, 1, . . . , N i I nc rement Green By d - iog max { l x; lv, IYi lv } I f l x; lv :S: IYi lv S e t u; = x;jy; Compute x i +l = F(u;, 1) and Yi +l = G(u;, 1) Els e IYi lv < l x; lv S e t u; = y;jx; Compute x i + I = F(1, u;) and Yi +l = G(1, u;) Wr i t e Set
2,
END MAIN LOOP Ret urn the Value
Green
Figure 5.6: An algorithm to compute the Green function Q.p(x, y). = z2 + 1, a = � · = 3z2 - 4 , a = 1. 1 (d) ¢(z) = z + - , a = 1 . (b) ¢(z) (c) ¢(z)
(e) ¢(z)
=
z 3z2 - 1 , z2 1 _
a = 1 and a = 3. Check that hq,(3)
�
4hq,(1). 1
1
5.32. Let K be an algebraically closed field with an absolute value v , let ¢ : IP' --> IP' be a morphism of degree d ::;,: 2, fix a lift of ¢, and for each homogeneous polyno mial E E K[x, y], let 5..q, , E be the associated local canonical height function. (See Theo rem 5.60.) (a) If E1 , E2 E K[x, y] are homogeneous polynomials, prove that
)..¢ , E 1 E2 ( P) = )..¢ ,E1 ( P) + 5..q, ,E2 ( P) at all points such that E1 ( P) =I 0 and E2 ( P) =I 0. (b) Let D = n1 (Q1 ) + n2 (Q 2 ) + · · · + nr(Qr) be a divisor on IP'1 , i.e., D is a formal sum of points with n1 , . . . , nr E Z. Use (a) to associate to D a local height function (c) Prove that there is a rational function fv E K(z) with the property that
at all points where 5.. q, ,v (¢(P)) and 5.. q, ,v( P) are defined. (d) Let g(z) E K(z) be a rational function and let D9 be the divisor ofg, which by definition is the formal sum
Exercises
321 D9
= L ordQ(g)( Q ).
Q EII'l(K) (Here ordQ (g) is the order of vanishing of g at the point Q ; see Example 2.2.) Prove that
the function
P
>----+
5-..q, ,v9
(P) + loglg(P) l v
extends to a bounded continuous function on all of lP'1 (K) . Section 5 . 1 0. Berkovich Space and Dynamics 5.33. Let ¢(z) in z - a, say
E
Cp[z] be a polynomial and let a
Prove that
E
D(O, 1). Write ¢(z)
as a polynomial
¢(D(a, r)) = D (¢(a), s) ,
where the radius s of the image disk is given by
s = r · max{ lcl l, lc2 l r, lc3 l r2 , . . . , l cd l rd } . (Hint. We already proved that ¢ (D (a, r)) is a disk; see Proposition 5. 16. Now use the maxi mum modulus principle to find the radius.) 5.34. This exercise develops some elementary properties of bounded seminorms as defined on page 296. (a) Let I · I be a nonconstant seminorm. Prove that I 0 I = 0 and I l l = 1. (b) Suppose that I · I is an I · ll wbounded seminorm. Let f(z) = c be a constant polyno mial. Prove that If I = l e i is the usual absolute value on Cp . (c) Suppose that we replace property (3) of bounded seminorm with the usual triangle in equality If + gl ::; If I + lg l. Prove that I · I satisfies (3). (d) Suppose that we replace property (4) of bounded seminorm with the weaker statement that there is a constant K such that If I ::; K I I f II R for all f E Cp [z ] . Prove that (4) is true, i.e., we can take K = 1. 5.35. Prove that the seminorms associated to points of Types II, III, and IV are actually norms. (Hint. For Type IV, use the fact that the limiting radius is positive. See Remark 5.65.)
5.36. Let d 2: 2. Prove that the map ¢( z) = z d has no Type-IV fixed points in A 8. Does cp(z) have any Type-IV periodic or preperiodic points? 5.37. Let D(a, r) be a disk with 0 �
D(a, r) and let f(z) = 1/ z . Prove that f (D(a, r)) = {z- 1 : l z - a l :S: r} = D (a - 1 , r/lal 2 ) .
5.38. Let
d ¢(z) = L cd E Cp[z] i=O
be a polynomial with lc; l ::; 1 and l ed I
The polynomial ¢ induces a map ¢ : D8 --+ D8 on the Berkovich disk. (a) Prove that the Gauss point �o, 1 is fixed by ¢.
= 1.
(5.49)
Exercises
322
(b) Let a E Cp be a fixed point of c/> : Cp --+ Cp. Prove that �a , o is fixed by ¢. (c) Let a E Cp be a neutral fixed point, i.e., ct>' (a) = 1, and let 0 � r � 1. Prove that �a,r is fixed by ¢. (In other words, if a is a neutral fixed point, then the entire line segment La,o C D 8 is fixed by ¢.) (d) Prove that every Berkovich fixed point of c/> of Type I, II, or III is one of the three types listed in (a), (b), (c). (Hint. Use the explicit description of c/>(�a,r) given in Exercise 5.33.) (e) Can c/> have fixed points of Type IV?
l l
5.39. Let c/>( z ) = 2:: c;zi E Cp[z] be a polynomial as in (5.49). Prove directly that the Julia set of c1> is .J8 (c/>) = { 6,0 } . (This is a special case of the result described in Example 5.79, since the conditions on c/> imply that it has good reduction.) 5.40. For any nested sequence of closed disks
regardless of whether the intersection is empty, we define a seminorm in the usual way,
(a) Let r = lim r; . If the intersection n D (a; , r;) is nonempty and a is a point in the intersection, prove that lfl a,r = lfla ,r · Thus every point in D 8 , regardless of whether it is of Type I, II, III, or IV, can be represented by a nested sequence of closed disks. (b) Two nested sequences of closed disks D (O, 1) D (O, l)
::::l D (a1 , r l ) ::::J
D (b 1 , s1 )
::::l
::::J
D (a2 , r2 ) ::::l . . , D (b2 , s2 ) ::::J . . . , •
are defined to be equivalent if the following two conditions are true: • For every i 2 1 there exists a j 2 1 such that D (bj , Sj ) c D (a; , r;). • For every i 2 1 there exists aj 2 1 such that D (aj , rj ) C D (b; , s;). Prove that two sequences of disks are equivalent if and only if their seminorms I · l a,r and I · l b,s are equal. 5.41. Let �a,r E D 8 be a Type-IV point. We defined the "line segment" running from �a, r to the Gauss point �o,1 to be the set
00
L a, r
=
{�a,r } U U {�ai , t : r; � t � r; - d . i= l
(Note that r0
=
1 by definition.) Let r =
lim;___, 00 r;. Prove that the map
h : [r, 1]
h(t)
---->
L a,r ,
=
{ �
�a,r f t = r, �ai ,t Ifr; � t � Ti- l ,
is a homeomorphism, where [r, 1] C R has the usual topology and D 8 has the Gel'fond topology. Thus La,r is indeed a line segment.
Exercises
323
5.42. There is a natural metric on the Berkovich disk D8 coming from the tree structure. We first identify the interval [0, 1] with each of the line segments
La,o
=
{ �a,r
: 0 .,:; r .,:; 1} E D , -B
so each La,o has length 1. We then define the distance between two points �a, r and �b,s to be the length of the shortest path in the tree connecting them. Denote this distance by ��:(�a,r, �b,s). (For Type-IV points, take the limit.) (a) If �a,r and �b,s both lie on some line segment Lc,o, prove that ��:(�a,r, �b,s) = lr - s l is simply the distance between them on that line segment. (b) In general, prove that
��:(�a,r, �b,s)
=
max { lr - s l , 2lb - al
- r - }. s
(Hint. How far above �a,r and �b,s do the line segments La,r and Lb,s merge? Go up one
line segment and then down the other.) (c) Prove that the set
{ � E D8 : ��:(�, �o, ! ) < �}
contains no Type-I points. Prove that every neighborhood of �0, 1 in the Gel'fond topol ogy contains infinitely many Type-I points. Deduce that the path metric 11: does not define the Gel'fond topology, and indeed that ��: is not even continuous in the Gel'fond topol ogy! In Baker and Rumely's terminology, the set D8 with the metric ��: is called the small model. There is also a big model, in which the edges are reparameterized so that if D(a, r) c D(b, s ) , then the distance from �a,r to �b,s is J log(r / s ) J . In particular, points of Type I are at infinite distance from each other and from all points of Types II, III, and IV. It is the big model that is best adapted for doing potential theory on Berkovich space; see [26] for details. 5.43. For points of Type I, II, and III in D8, the Hsia kernel is defined to be
8 (,;a, r , ,;b, s )
=
max(r, Ia - bl) . s,
(For points of Type IV, one takes the appropriate limit.) The Hsia kernel is used in studying potential theory on D8; see [26, 209]. (a) Prove that 8(�a,r, �a,r ) = r. (b) Prove that 8(�a,o, �b,o) = Ia - bl. Thus 8 extends the usual norm on ICp, where we identify the unit disk in ICp with the Type-I points in D8. (c) Let �c,t be the point where the line segments La,r and Lb,s first meet. Prove that o(�a,r, �b,s) = t. (d) Prove that
8(�, (') .,:; max{o(�, (), o(( , (' ) } 5.44. There are natural inclusions
D(O, 1 ) -
'---->
D , -B
In each case, prove that the restriction of the Gel'fond topology on the Berkovich space yields the usual metric topology on the smaller space, where we use the usual p-adic met ric on D(O, 1) and A 1 (ICp) and the p-adic chordal metric on lP'1 (ICp) .
Exercises
324 5.45. Let ¢ be a rational map of degree at least 2, let P E hood of P with the property that
¢( U ) �
U
and
n¢
n ;:> O
n
lP'5, and let U c lP'5 be a neighbor
( U)
= { P} .
Prove that P is a Type-I point and that, considered as a point in 1P'1 (Cp), the point P is an attracting fixed point for ¢.
Chapter 6
Dynamics Associated to Algebraic Groups In the forest of untamed rational maps live a select few whose additional structure allows them to be more easily domesticated. They are the power maps, Chebyshev polynomials, and Lattes maps, whose complex dynamics were briefly discussed in Section 1 .6. The underlying structure that they possess comes from an algebraic group, namely the multiplicative group for the power maps and Chebyshev poly nomials and elliptic curves for the Lattes maps. Although such maps are special in many ways, they yet provide important examples, testing grounds, and boundary conditions for general results in dynamics. In this chapter we investigate some of the algebraic and arithmetic properties of the rational maps associated to algebraic groups. 6.1
Power Map s and the Multiplicative Group
The simplest rational maps are the power maps given by monic monomials,
where the integer d may be positive or negative. These maps obviously commute with one another under composition,
A more intrinsic description of the maps associated to the monic monomials Md is that they are endomorphisms of the multiplicative group,
More precisely, there is an isomorphism of rings, 325
326
6. Dynamics Associated to Algebraic Groups
The fact that the Md are endomorphisms of the multiplicative group Gm makes it quite easy to describe their preperiodic points. IP'1 be the power Proposition 6.1. Let d E Z with l d l .2': 2 and let Md : IP'1 map Md(z) z d. Then -+
=
PrePer(Md)
=
(Gm)tors
=
{ ( E Gm : C
=
lfor some n .2':
where recall that J.Ln denotes the group ofn1h roots ofunity.
1}
=
U J.Ln ,
Proof We proved this long ago for any abelian group G and homomorphism z zd with d .2': 2; see Proposition 0.3. The proof for d :::; -2 is similar and left to the reader. D t--+
The iterates of Md are given by
so the periodic points of Md are also easy to characterize, l d l .2': 2 and let ( E Per�* (Md) be a point of exact pe 2. Then the multiplier ofMd at ( is given by
Proposition 6.2. Let
riod
n
:2:
Proof Using M:J:(z)
=
z dn , we can directly compute
In particular, if we are working over C, then every periodic point of Md in C * is repelling. On the other hand, over a p-adic field with p f d, the multiplier dn is a unit, so the periodic points are indifferent. And if p I d, then all of the periodic points are attracting. Of course, there are also the two superattracting fixed points 0 and oo . It is not hard to find all rational maps that commute with the power maps. In particular, we can compute the automorphism group Aut(Md). Proposition 6.3. Let K be afield and let Md(z) z d be the d1h-power map for some l d l .2': 2. Further, if K hasfinite characteristic p, assume that p f d. (a) The set ofrational maps that commute with Md(z) is given by =
{
=
Md o ¢}
=
{ cze : c E ILd- l and e E Z}.
327
6.1. Power Maps and the Multiplicative Group
(b) The automorphism group ofMd is
1 Aut(Md ) = {az : a E 1-Ld- d U { bz- : b E 1-Ld- d , where we recall that the automorphism group Aut(¢) of any rational map ¢ is the set off E PGL2 (k) satisfying ¢! = ¢. In particular, Aut(Md ) is a dihe dral group oforder where is the number of ( d 1 }'1 roots ofunity in K * . Proof (a) It is clear by a direct computation that the indicated maps cze commute with Md (z), so it suffices to prove that they are the only commuting maps. Suppose that ¢(z) E K(z) commutes with Md (z), so ¢(zd ) = ¢(z) d .
2n,
n
-
Let ( E k be a primitive
n
qy(z) = zkn 'I/Jn (Z dn ). 1/Jn ( z) = zj An ( z ) for some ing An (0) -:f. 0 and An (0) -:f. oo. Then
We write
integer j and some
deg(¢(z)) = deg(zkn 'I/Jn (Zdn )) = deg(zkn +jdn An (Zdn ) )
2::
An ( z )
E
K ( z)
satisfy
deg( An (Zdn ) ) = dn deg(An (z)) .
n,
Hence deg(An ) = 0 for sufficiently large which proves that ¢(z) has the form ¢( z) = cze for some c E k* and some e E Z. With this information in hand, it remains only to observe that ¢(zd) = ¢(z)d if and only if c = cd, i.e., if and only if either c = 0 or c E 1-Ld-1 · (b) By definition, Aut(¢) is the set of rational maps of degree 1 that commute with ¢. It follows from (a) that Aut(¢) is the set of all cz± 1 with c E 1-Ld-1 . In particular, if we let fa(z) = az and g(z) = z - 1 , then Aut(Md ) = {fa : a E 1-Ld- d U {fa o g : a E 1-Ld- 1 } , and the dihedral nature of the group law is evident from the identities fa o g = g o fa -1 ·
0
328
6. Dynamics Associated to Algebraic Groups
Example 6.4. We can use the map Md ( z) = z d to illustrate the construction of dynamical units in Section 3. 1 1 . First we use Theorem 3.66, which says that if o: has exact order n and gcd(i - j, n) = 1, then
Taking j
= 0 and gcd(i , n)
=
1, this implies that
di - 1 - 1 is a unit for all primitive (dn - 1 ) 81 roots of unity o:. d 0: -1 - 1
o:
These are examples of classical cyclotomic units. Similarly, let m and n be positive integers with m f n and n f m. Then Theo rem 3.68 says that if o: is a primitive ( dm - 1)81 root of unity and if (3 is a primi tive ( dn - 1 )81 root of unity, then o: - (3 is a unit. These are again classical examples of cyclotomic units. Example 6.5. Recall that a nontrivial twist of a rational map ¢( z) E K ( z) is a rational map 7/J(z) E K(z) such that 7/J(z) is PGL2 ( K )-conjugate to ¢(z), but 7/J(z) is not PGL2 (K)-conjugate to ¢(z). See Section 4.9 for the general theory. Since the power maps Md(z) = z d have a large automorphism group, they tend to have many twists. For example, the twists associated to the subgroup { cz : c E J.ld- 1 } of Aut(¢) are given by (cf. Example 4.81) K* /(K*) d-1 Twist(Md), a azd .
f----+
-+
There are also some rather complicated-looking twists associated to the subgroup {z, z - 1 } C Aut(Md) · Each b E K * /(K*) 2 leads to a twist
o,t;:d/2 (2�)
b' z "
I
) b' z''+' . � ck 'f
o
(6. 1 )
I
See Example 4.82 for the derivation of this formula. 6.2
Chebyshev Polynomials
The multiplicative group Gm has a nontrivial automorphism given by inversion z z -1 , and the quotient of Gm by this automorphism is isomorphic to the affine line A 1 via the map f---7
The inversion automorphism commutes with the �h-power map Md(z) z d , so when we take the quotient of Gm, we find that Md(z) descends to give a map on the quotient space A 1 . This leads to the following definition (cf. Section 1 .6.2).
=
329
6.2. Chebyshev Polynomials
The cfh Chebyshev polynomial is the polynomial Td( w) E Z[w] satisfy ing the identity
Definition.
in the field Q(z).
(6.2)
Of course, we need to show that that Td exists. In the next proposition we prove the existence of the Chebyshev polynomials and describe some of their algebraic properties. Proposition 6.6. For each integer d ;:::
Q[w] satisfying
0 there exists a unique polynomial Td ( w) E in thejield Q(z).
(6.3)
We call Td the d1h Chebyshev polynomial. (a) Td( w) is a monic polynomial ofdegree d in Z[w]. (b) Td(Te (w)) = Tde (w)for all d, e 2: 0. (c) Td( -w) = ( - 1 ) d Td( w ) . Thus Td is an oddfunction ifd is odd and it is an even function ifd is even. (d) The Chebyshev polynomials satisfy the recurrence relation
(e) For all d ;::: 1, the d1h Chebyshev polynomial is given by the explicitformula
Proof Suppose first that there do exist polynomials Td ( w) satisfying (6.3 ) . Then
To(z + z-1) = z0 + z- 0 = 2 , T1 (z + z - 1 ) = z + z- 1 , T2 (z + z-1 ) = z2 + z- 2 = (z + z- 1 ) 2 - 2 , so we see that T0 ( w) = 2 , T1 ( w) = w, and T2 ( w) = w2 - 2 are uniquely determined monic polynomials with integer coefficients. Still assuming that the set of Chebyshev polynomials exists, we next observe that they satisfy (z + z - 1 )Td+ 1 (z + z -1 ) - Td(z + z- 1 ) d d d = (z + z -l ) (z + I + z- - 1 ) - (z + z - d ) d+I + z - d - 2 = z = Td+ 2 (z + z -1 ) .
Putting z + z-1 = w, this means that Td+ 2 (w) = wTd+ I (w) - Td(w). Hence ifthe Chebyshev polynomials exist, they are unique, because they are completely deter mined by the recurrence
330
6. Dynamics Associated to Algebraic Groups
To(w) = 2, T1(w) = w, and Td+2 (w) = wTd+1 (w) - Td (w) for d �(6.6) O. And from this recurrence we see immediately by induction that Td (w) is a monic polynomial of degree d in Z[w]. We now tum this argument around and use the recurrence (6.6) to define a se quence ofpolynomials Td (w). We claim that Td (w) then satisfies (6.3). This is clear for T0 and T1 , so we assume that (6.3) is true for T0 , T1 , . . . , Td+ l and use (6.6) with w = z + z - 1 to compute Td+2 (z + z- 1 ) = (z + z - 1 )Td+ l (z + z- 1 ) - Td (z + z - 1 ) = (z + z- 1 )(zd+1 + z-d- 1 ) (zd + z -d ) = zd+2 z -d-2 . Hence (6.3) is true for Td+2 , so by induction we have for all d � 0. _
+
This proves that the recurrence defines polynomials satisfying (6.3), so Chebyshev polynomials of every degree exist. We have now shown that Chebyshev polynomials exist, are unique, and have the properties in (a) and (d). Next we make repeated use of (6.3) to compute
Td (Te (z+z- 1 )) = Td (ze +z-e ) = (ze )d +(ze ) -d = zde +z-de = Tde (z+z - 1 ). Hence Td (Te (w)) = Tde (w), which proves (b). To prove (c), we replace z by -z in (6.3) to obtain Td ( - (z + z- 1 )) = Td ( -d z + ( -z)d- 1 ) d d d d = (-z) + (-z) - = (-1) (z + z - ) = (-1) Td (z z - 1 ). Therefore Td ( -w) = ( -1) dTd (w), which is (c). Next we prove the explicit summation formula (6.5) given in (e). Substitut ing d = 1 and d = 2 into the formula yields the correct values T1 ( w) = w and T2 (w) = w2 - 2. We now assume that the formula is correct up to Td+ 1 and use the recurrence (6.4) to check it for d + 2. Thus Td+2 (w) = wTd+ l (w) - Td (w) = w "'""' l (-1) k d+d +1 -1 k (d + k1 - k) wd+l -2k O�k � (d+ ) /2 d_ (d - k) wd-2k "'""' ( -1)k _ O�k �d/2 d - k k "'""' (-1) k d+ 1 (d+ k1 - k) wd+2-2k O�k� (d+ 1) /2 d+ 1 - k "'""' (-1)k- 1 d - dk + 1 (d -k k +1 1) wd-2k+2 1 � k �d/2+ 1 +
�
�
�
_
�
6.2. Chebyshev Polynomials
331
T2 = w2 - 2 T3 = w 3 - 3w T4 = w4 - 4w2 + 2 T5 = w5 - 5w3 + 5w T6 = w6 - 6w4 + 9w2 - 2 T7 = w7 - 7w5 + 14w3 - 7w Ts = w8 - 8w6 + 20w4 - 16w2 + 2 T9 = w90- 9w7 + 27w5 - 30w3 + 9w Tw = w1 - 10w8 + 35w6 - 50w4 + 25w2 - 2 Tn w11 - l lw90+ 44w7 - 77w5 + 55w3 - l lw T12 = w 12 - 12w 1 + 54w8 - 112w6 + 105w4 - 36w2 + 2 =
Table 6. 1 : The first few Chebyshev polynomials.
(d + k1 - k) + d (d +k 1 - k) ( d + 1) - 1 wd+2-2k ' "" (- l ) k l d+ k O::::;k:S:d/2+1 where we use the standard convention that (;:,) 0 if or if 0. A simple algebraic calculation that we leave for the reader (Exercise 6.4) shows that (d + 1) (d + k1 - k) + d (d +k -1 -1 k) d + 2 (d + 2 - k) = d+ l - k d+2 - k k . Hence Td+2 (w) = "" (-l) k d +d +2 -2 k (d + k2 - k) wd+2-2k ' O:s; k :S: (d+2)/2 �
=
n<m
m <
�
which completes the proof of (e).
D
Remark 6.7. As mentioned in Section 1 .6.2, the classical normalization for the
Chebyshev polynomials is
d ( z +2z- 1 ) = zd +2 z-d '
T.
or equivalently,
332
6. Dynamics Associated to Algebraic Groups
Td(cos B)
=
cos(dB)
for all B E R
The two normalizations are related by the simple formula Td ( w) = Td ( 2w) . We have chosen the alternative normalization because it has better arithmetic properties. In particular, the map Td : lP'1 lP'1 has good reduction at all primes. The classical normalization Td has bad reduction at 2. As with the power maps, it is not difficult to describe the periodic points of the Chebyshev polynomials and to compute their multipliers. We state the result and leave the computation as an exercise.
�
->
Proposition 6.8. Let Td ( w) be the d1h Chebyshev polynomialfor some d 2: 2.
(a) Thefixedpoints ofTd in A 1 (C) are
{ 2 ( 27rj ) 0 cos
d
+
1
:
::;
d+ 1
j ::; -2
} { 2 ( 27rj1 ) 0 < j < -2- } . u
cos
d-
d- 1
:
(b) The multipliers ofTd at itsfixedpoints are given by
( (}:j1 )) ( d2=j1 ) ) >.rd ( 2 Ard 2 cos
=
-d
d+1 for O < j < -, 2
cos
=
d
=
d2 .
for O < j < --, 2
Ard ( ± 2)
d- 1
(Note that - 2 E Fix(Td) ifand only ifd is odd.) In general, the periodic points and multipliers ofTd can be derivedfrom the above formulas using T:J: = Tdn and Pern (Td) = Fix(T:J:). Proof See Exercise 6.5.
D
We now prove an analogue of Proposition 6.3 for Chebyshev polynomials. Theorem 6.9. Let K be afield and let Td(w) be the d1h Chebyshev polynomialfor some d 2: 2. Further, if K hasfinite characteristic p, assume that p f d. (a) The automorphism group ofTd is given by
Aut(Td) =
{1
ifd is even, J.L2 ifd is odd.
(b) Assume that K does not have characteristic 2. Let ¢( w) E K (w) be a rational map that commutes with Td(w), i.e., ¢ (Td(w) ) ¢(w) = ±Te (w)
=
Td ( ¢(w) ) . Then
for some e 2: 1 .
The minus sign is allowed if and only if d is odd. (See Theorem 6.79 for a stronger result.)
6.2. Chebyshev Polynomials
333
Proof (a) The assertion that Aut(Td) c J.L2 is an immediate consequence of (b), since (b) implies that any f E Aut(Td) satisfies f ( w) = ±T1 ( w) = ±w . However,
since the proof of (b) is somewhat intricate, we give a direct and elementary proof of(a). Suppose that f E PGL 2 satisfies T1 = Td . The polynomial Td has a unique totally ramified fixed point at oo (cf. Exercise 6.8), and T1 similarly has a unique totally ramified fixed point at f - 1 (oo), so the equality T1 = Td tells us that f - 1 (oo) = oo. Hence f(w) = aw + b is an affine transformation. (The same ar gument applies to any polynomial not of the form awd .) Proposition 6.6(c) says that Td( w) satisfies Td( -w) = ( - 1 ) dTd(w ), so in par ticular, (6.7) Td(w) = wd + (terms of degree at most d - 2).
(K)
The identity T1 (w) = Td (w) with f(w) = aw + b can be written as We evaluate both sides using (6.7) and look at the top degree terms. This gives ad wd + dad -1 bwd- 1 + (terms of degree at most d - 2) = aw d + (terms of degree at most d - 2). Hence and By assumption, d i=- 0 in the field K, so we conclude that ad-1 = 1 and b = 0. In order to pin down the value of a, we use the explicit formula for Td(w) given in Proposition 6.6(e). In fact, we need only the top two terms, Td( w) = wd - dwd-2 + (terms of degree at most d - 4). By assumption we have Td(aw) = aTd(w), so Hence ad = a and a d- 2 = a, where we again use the assumption that d i=- 0 in the field K. It follows that a2 = 1 . Further, a = - 1 is possible only if ( - 1 ) d = - 1, so when d is odd. This completes the proof that Aut(Td) is trivial if d is even and is equal to J.L2 if d is odd. (b) It is easy to verify that the Chebyshev polynomial Td (w ) cannot be conjugated to a polynomial of the form cwd (Exercise 6.8). It follows that any rational map commuting with Td( w) is necessarily a polynomial, a fact whose proof we defer until later in this chapter; see Theorem 6.80. We now describe a proof due to Bertram [69] that the only polynomials commuting with Td are ±Te. We begin with two lemmas. The first characterizes the Chebyshev polynomials as the solutions of a nonlinear differential equation, and the second explains how to exploit such equations.
334
6. Dynamics Associated to Algebraic Groups
Lemma 6.10. Assume that K does not have characteristic 2. Let d :2:: 1 and let F( w ) be a polynomial solution to the differential equation
(6.8) Then F(w)
= ±Td(w ).
±Td (w ) are solutions. We differentiate the functional equation (6.2) defining the Chebyshev polynomials to obtain the identity
Proof We first check that
T�(z + z - 1 ) ( 1 - z - 2 ) = dz d- 1 - dz - d- 1 , and then solve for T� , Putting w = z + z- 1 as usual and noting that w2 - 4 = ( z - z- 1 ?, we compute
(
)
zd -d 2 (4 - w2 )T�(w) 2 = ( 4 - (z + z - 1 ) 2 )d2 z �:- 1 = -d2 (zd _ z - d ) 2 = d2 (4 - (zd + z - d ) 2 ) = d2 (4 - Td(w ) 2 ) . This proves that ±Td (w ) are solutions to (6.8). Next suppose that F(w) is any polynomial solution to (6.8). If is identi cally 0, then (6.8) implies that F ( w ) = ±2 = ±To (w ), so we are done. We may thus assume that F' ( w) -=f. 0. We differentiate both sides of (6.8) and divide by ( w ) to obtain (4 - w 2 )F" (w) - wF'(w) d2 F(w) = 0. (6.9) In particular, Td ( w ) is a solution to (6.9). Suppose now that F is any polynomial of degree that is a solution to (6.9). We write F( w) = awk + bw k- 1 + · · · with a -=f. 0 and substitute into (6.9). The leading term is
F'(w) 2F'
+
k
so we must have = d. In other words, we have shown that every nonzero poly nomial solution of (6.9) has degree d. But F(w) - aTd (w ) is a polynomial of degree strictly less than d that satisfies (6.9); hence F( w) = aTd ( w ). Finally, substituting w = 2 into (6.8) yields F(2) = ±2, while we know that Td (2) = Td(1 1 - 1 ) = 1 d + 1 d = 2. Hence F(w) = ±Td (w ), which completes the proof D of Lemma 6.1 0.
k
+
Lemma 6.1 1. Let A( w ) be a polynomial ofdegree a polynomial ofdegree d :2:: 2 satisfYing
r
:2::
A(w)F'(w r = dr A(F(w)) .
1 and suppose that F( w ) is (6. 10)
335
6.2. Chebyshev Polynomials
Suppose further that G ( w) is a polynomial of degree e i.e. , F(G(w)) = G(F(w)). Then
2:
0 that commutes with F,
A(w)G'(wr = e" A (G (w)) . Proof Consider the polynomial
B(w) = A(w)G'(wr - e" A(G(w)) . We assume that B(w ) -1- 0 and derive a contradiction, which will prove the desired result. First we observe that the leading coefficients of A( w)G' (w ) " and e" A ( G( w)) cancel, so deg B < re (strict inequality). Next we use the various definitions and given relations to compute
d" B (F(w)) = d" A(F(w))G' ( F(w) ) " - d" e" A( G( F(w))) = d" A( F(w)) G' ( F(w)) " - d" e" A(F(G(w))) = A(w)F' (wrG' (F(w)) " - e" A(G ( w))F' (G (w)) " = A(w)(G o F)'(wr - e" A ( G (w))F' ( G(w) ) " = A(w) (F o G)'(wr - e" A(G (w))F' (G (w)) " = A(w)F' ( G(w) ) " G'(wr - e" A( G(w) ) F' ( G(w ) ) " = F' (G (w)) " [A(w)G'(wr - e" A(G(w))] = F' (G(w)) " B(w)
definition of B, using F o G = G o F, using (6. 10) twice, chain rule, using F o G = G o F, chain rule, definition of B.
Taking degrees of both sides gives
(deg B)(deg F) = r(deg F - l)(deg G) + (deg B), and the assumption that deg F 2: 2 means that we can solve for deg B = r(deg G) = re. This contradicts the earlier strict inequality deg B < re. Hence B must be the zero
polynomial, which completes the proof of Lemma 6. 1 1 .
D
We now resume the proof of Theorem 6.9(b). Let ¢( w) be as in the statement (c) with ¢( w) a polynomial and let e = deg( ¢). Lemma 6. 10 tells us that
(4 - w2 )T�(w) 2 = d2 (4 - Td (w) 2 ) . Hence we can apply Lemma 6. 1 1 with A( w) = 4 - w2 and the commuting polyno mials ¢ and Td to deduce that (4 - w2 )¢(w) 2 = e2 (4 - ¢(w) 2 ) . Then another application of Lemma 6. 10 implies that ¢( w) = ±Te ( w). D
336
6. Dynamics Associated to Algebraic Groups
Using Theorem 6.9, it is easy to describe all of the twists of the Chebyshev poly nomials. Corollary 6.12. Continuing with the notation and assumptions from Theorem 6.9, ifd is even, then Td has no nontrivial k / K-twists, and ifd is odd, then each a E K*
yields a twist
1 Td ( /(i w) . Va Two such twists Td, a and Td,b are K-conjugate ifand only if ajb is a square in K*. Proof We use the description of Aut(¢) from (a). If d is even, then the automor phism group Aut(¢) is trivial, so Proposition 4.73 says that ¢ has no nontrivial twists. For odd d we have Aut(¢) = { ±z }, so the desired result follows from Ex D ample 4.81 (see also Example 4.75). Td ,a ( w)
=
Remark 6. 13. Over C, there is a short proof that Aut(Td) c J.t2 using the fact (Exercise 1 .31) that the Julia set of Td is J(Td) = [ - 2 , 2]. Then the assumption that TJ = Td implies that f maps the interval [ - 2 , 2] to itself. Since f is bijec tive on JP 1 (C), it follows in particular that f permutes the endpoints of the inter val [-2, 2]. Hence f( 2 ) = ±2 and f( -2) = =t=2. Writing f as f( w ) = aw + b, this gives two equations to solve for a and b, yielding b = 0 and a = ±1. Note that this
proof does not carry over to characteristic p, since, for example, working over P we have see Exercise 6. 10.
f
Aut(Tp) PGL2 (fp); =
6.3
A Primer on Elliptic Curves
The remainder of this chapter is devoted to rational maps associated to elliptic curves. In this section we give some basic definitions and review, without proof, some of the properties of elliptic curves that will be needed later. The reader should also review the summary of elliptic curves over C given in Section 1 .6.3. For further reading on elliptic curves and for the proofs omitted in this section, see for example [96, 198, 248, 250, 254, 257, 410, 412, 420]. 6.3.1
Elliptic Curves and Weierstrass Equations
An elliptic curve E over a field K (of characteristic different from 2 3) is described by a Weierstrass equation, which is an equation of the form
Definition.
and
(6. 1 1)
with a, b E K and 4a3 + 27b2 =!= 0. Of course, we really mean that E is the projec tive curve obtained by homogenizing equation (6. 1 1 ), so E has one extra point "at infinity," which we denote by 0. If K has characteristic 2 or then equations of the form (6. 1 1) are insufficient, and indeed they are always singular in characteristic 2, so one uses the generalized Weierstrass equation (6. 12)
3,
337
6.3. A Primer on Elliptic Curves
EI K E. E(K), L K. E E(K).K
Remark 6. 14. Let
E(K)K,K.
E,
be an elliptic curve defined over a field When we write we mean the geometric points of i.e., the points in for some chosen alge braic closure of If we want to refer to points defined over we always explicitly write and similarly we write for the points defined over some extension field of More intrinsically, an elliptic curve is a pair consisting of a smooth alge braic curve of genus 1 and a point E For convenience we often call an elliptic curve, with the understanding that there is a specified point We say that is defined over afield if the curve is given by equations with -coefficients and the point 0 is in Using the Riemann-Roch theorem, one can prove that every elliptic curve can be embedded in lP'2 by a cubic equation of the form (6. 12) with mapping to the point at infinity. (See [410, III §3].) Then, if the characteristic of is neither 2 nor we can complete the square on the left and the cube on the right to obtain the simpler Weierstrass equation (6. 1 1 ). In order to simplify our discussion, we will generally make this assumption. The discriminant and ]-invariant of the elliptic curve given by (6. 1 1) are defined by the formulas
t::. (E)
E, E(L) E
0 E. (E, 0)
j(E)
EE
K0. 0 K
EI K 3,
E
a, bis Enonsingular, K and let Eandbethus the curve given by the Weierstrass E is an elliptic curve, if and only t::. ( E) =/: 0. if (b) Two elliptic curves E and E' are isomorphic over K ifand only ifj(E) = j(E'). More precisely, E and E' are isomorphic ifand only if there is a u E K* such that a' = u4 a and b' = u6 b (a) Let equation (6. 1 1 ) Then
Proposition 6.15.
.
Proof See [410, III §1]. 6.3.2
D
Geometry and the Group Law
E
There is a natural group structure on the points of that may be described as follows. Let be any line in lP'2 • Then counted with appropriate multiplicities, the cubic curve and the line intersect at three points, say
LE
L
E n L = {P,Q,R}, where P, Q, and R need not be distinct. The group law on E is determined by the requirement that the sum of the points P, Q, R be equal to 0, P + Q + R = 0. The point 0 serves as the identity element of the group. The inverse of a point P, which we denote by P, is the third point on the intersection of E with the line through P and 0. -
338
6. Dynamics Associated to Algebraic Groups
Theorem 6.16. Let Ej K be an elliptic curve defined over afield K.
(a) The addition law described above gives E
=
E(K) the structure ofan abelian group. (b) The group law is algebraic, in the sense that the addition and inversion maps, E X E ( P,Q)>-+ P+Q E,
E P >-+ -P E
'
are morphisms, i.e., are given by everywhere defined rationalfunctions. (c) The subset E ( K) consisting ofpoints ofE that are defined over K is a subgroup ofE(K). D
Proof See [410, III §§2,3].
It is not hard to give explicit formulas for the group law on an elliptic curve, as in the following algorithm. Proposition 6.17.
(Elliptic Curve Group Law Algorithm) Let E be an elliptic curve
given by a Weierstrass equation
y2 = x3 + ax + b, and let P1 = (x 1 , Yl ) and Pz = (xz, yz) be points on E. IfX 1 = xz and Yl = -yz, then P1 + Pz = 0. Otherwise, define quantities A = XYzz -- XY11 ' xz - xl i l + 2b ' ifx1 = xz. A = 3x2yl+ a ' V = --xr--"---+-2axy--l Then y = AX + is the line through P1 and P2 , or tangent to E if P1 = P2 , and the sum of P1 and Pz is given by E
:
v
P = (x, y) is 2 - 8bx + a2 x([2]P) = x4 -4x2ax 3 + 4ax + 4b
As a special case, the duplication formula/or
Proof See [410, III.2.3] 6.3.3
Divisors and Divisor Classes
Definition.
A divisor on E is a formal sum of points D
= LEE np(P), P
D
339
6.3. A Primer on Elliptic Curves
with np E Z and all but finitely many np = 0. The set of divisors under addition forms the divisor group Div( E). The degree of a divisor d is deg(D) =
L np.
PEE
The degree map deg : Div(E) Z is a group homomorphism. There is a natural summation map from Div(E) to E defined by ----+
sum : Div(E) ----> E,
L np(P)
PE E
f------t
L [np] (P).
PE E
(N.B. The two summation signs mean very different things. The first is a formal sum of points in Div(E). The second is a sum using the complicated addition law on E.) The zeros and poles of a rational function f on E define a divisor div(f) =
L ordp(f)(P), PEE
where ordp(f) is the order of zero of f at P if f(P) = 0, and ordp(f) is negative the order of the pole off at P if f(P) = oo. A divisor of the form div(f) is called a principal divisor. The principal divisors form a subgroup ofDiv(E), and the quotient group is the Picardgroup Pic( E). Within Pic( E) is the important subgroup Pic0 (E) generated by divisors of degree 0. The next proposition describes the basic properties of divisors on E. Proposition 6.18. Let E be an elliptic curve.
(a) Every principal divisor on E has degree 0. (b) A divisor D E Div(E) is principal if and only if both deg(D) sum(D) = 0.
0 and
(c) The summation map induces a group isomorphism
sum : Pic0 (E) ----> E. Proof See [410, 111.3.4 and 111.3.5]. 6.3.4
D
Isogenies, Endomorphisms, and Automorphisms
Definition. An isogeny between two elliptic curves E1 and E2 is a surjective morphism 'ljJ : E1 E2 satisfying 'lj;( 0) = 0. (Note that any nonconstant mor phism E1 E2 is automatically finite and surjective.) The curves E1 and E2 are said to be isogenous if there is an isogeny between them. Remark 6. 19. Every nonconstant morphism 'ljJ E1 E2 is the composition of an isogeny and a translation (cf. [410, III.4.7]). To see this, let ¢(P) = 'lj;(P) - 'lj;(O). Then the map ¢ : E E is a morphism, and ¢( = 0, so ¢ is an isogeny. Hence ----+
----+
:
----+
----+
0)
'lj;(P) = ¢(P) + 'lj;(O) is the composition of an isogeny and a translation.
6. Dynamics Associated to Algebraic Groups
340
Remark 6.20. We observe that an isogeny is unramified at all points. This fol
lows from the general Riemann-Hurwitz formula (Theorem 1 .5) applied to the map
7/J : E1 E2 , 2g(E1 ) - 2 = (deg 7j;)(2g (E2 ) - 2) + 2::: (ep('lj;) - 1 ) . The elliptic curves E1 and E2 both have genus 1, and the ramification indices sat isfy ep('lj;) :2: 1, so it follows that every ep('lj;) is equal to 1, so 7/J is unramified. Theorem 6.21 . An isogeny 7/J : E1 E2 is a homomorphism ofgroups, i.e., 7/J(P + Q) = 7/J(P) + 7/J( Q) for all P, Q E E1 (K). -+
P E E1
-+
0
Proof See [410, 111.4.8]
7/J : EE1 . E2 Q Q,7/J 2 7/J deg(7/J) 7/J. Theorem 6.22. Let 7/J : E1 -+ E2 be an isogeny ofdegree d. Then there is a unique isogeny � : E2 E1 , called the dual isogeny of 7/J, with the property that �(7/J(P)) = [d]P and 7/J(�(Q)) = [d]Q for all P E E1 and Q E E2 .
The degree of an isogeny is the number of points in the inverse image for any point E This number is independent of the point since, as noted earlier, is an unramified map. It is clear that if > 1, then is not invertible, since it is not one-to-one. However, there does exist a dual isogeny that provides a kind of"inverse" for -+
'lj;- 1 (Q)
-+
0
Proof See [410, III §6].
End( E),E
E,
Definition. Let be an elliptic curve. The endomorphism ring of which is de noted by is the set of isogenies from to itself with addition and multipli cation given by the rules
E
E d(E) End(E).
(In order to make n into a ring, we also include the constant map that sends every point to 0.) The automorphism group of denoted by is the set of endomorphisms that have inverses. Equivalently, is the group of units in the ring
E,Aut(E) = End( Aut(E)*E), Every integer m gives a multiplication-by-m morphism in End( E). For m this is defined in the natural way as m terms
>
0
[m] : E -+ E, [m](P) = P + P + · · · + P . For m < 0 we set [m](P) = -[ -m](P), and of course [O](P) = 0. This gives an embedding of Z into End( E), and for most elliptic curves (in characteristic 0), there are no other endomorphisms.
341
6.3. A Primer on Elliptic Curves
E d( )
An elliptic curve E is said to have complex multiplication if n E is strictly larger than Z. The phrase "complex multiplication" is often abbreviated by CM. Example 6.23. The elliptic curve E y2 x3 + x has CM, since the endomorphism Definition.
:
'ljJ : E
-----+
=
'1/J(x, y) ( - x , iy ),
E,
=
i s not in Z . An easy way to verify this assertion i s to note that
'ljJ2 (x , y ) (x, - y ) = -(x , y ),
=
so 'ljJ2 [ - 1]. This gives an embedding of the Gaussian integers Z[i] into via the association m + ni [m] + [n] '1/J, and in fact it is not hard to show that is isomorphic to Z[i]. Example 6.24. More generally, there are two special families of elliptic curves that have CM, namely those with a 0 and those with b 0. These are the curves
= End(E)
f---)
End(E)
o
= = j(E�) = 1728, End(E�) = Z[i], Aut(E�) = J-L4 , j(E� ) = 0, End(E� ) = Z[p], Aut(E� ) = J-L6 .
E� : y2 = x3 + ax, E� y2 = x3 + b, Here p = ( -1 +A)I2 denotes a cube root ofunity and of unity. :
1-Ln
is the group ofn1h roots
Of course, all of the E� are isomorphic over an algebraically closed field, since they have the same j-invariant, and similarly for all of the E�. However, the curves in each family may not be isomorphic over a field K that is not algebraically closed. This is an example of the phenomenon of twisting as described in Section 4.8 (see also [410, X §5]). Proposition 6.25. Let E I K be an elliptic curve. Then the endomorphism ring of
E (a) End(E) = Z. (b) End( E) is an order in a quadratic imaginaryfield F. This means that End(E) is a subring ofF and satisfies End( E) Q = F. In particular, End( E) is a subring offinite index in the ring of integers of F. (c) End( E) is a maximal order in a quaternion algebra. (This case can occur only ifE is defined over afinitefield.)
is one of thefollowing three kinds ofrings:
0
0
Proof See [410, III §9].
The automorphisms of an elliptic curve are very easy to describe. Proposition 6.26. Let K be afield whose characteristic is not equal to or and let EI K be an elliptic curve. Then
2 3
{ Aut(E) =
J-L2 J-L4
J-L6
ifj (E) =1- 0 and j (E) =1- 1728, ifj(E) = 1728,
ifj( E)
=
0.
342
6. Dynamics Associated to Algebraic Groups D
Proof See [410, III § 1 0].
Aut(E)
Remark 6.27. It is easy to make the description of in Proposition 6.26 com pletely explicit. Assuming that E is given by a Weierstrass equation (6. 1 1) as usual,
for an appropriate choice of n there is an isomorphism [ · ] : J.Ln
_____. Aut(E), (6. 13) Here we take n 4 if j ( E ) = 1728, we take n 6 if j (E) 0, and we take n 2 otherwise. Of course, for n 2 and n 4 the formula simplifies somewhat to (x, � ) and (x, �- 1 ), respectively. =
=
y
=
=
y
=
=
Minimal Equations and Reduction Modulo p
6.3.5
Let K be a local field with ring of integers R, maximal ideal p, and residue field Rip . As in Section 2.3, we write x for the reduction of x modulo p .
k
=
Let EI K be an elliptic curve defined over a local field K. A minimal Weierstrass equation for E is a Weierstrass equation whose discriminant Ll(E) has
Definition.
minimal valuation subject to the condition that the coefficients of the Weierstrass equation are all in R.
k
2 3,
Example 6.28. If does not have characteristic or then a Weierstrass equation (6. 14)
for E is minimal if and only if a,
bER
and
min{ 3 ordp ( ) 2 ordp (b) } < 12. a ,
k 2 3,
2 3,
In general, if the residue field does not have characteristic or then any Weier strass equation (6. 14) can be transformed into a minimal equation by a substitution of the form (x, y) (u2 x, u3 y ) for an appropriate u E K*. If has characteristic or then a minimal Weierstrass equation may require the general form (6.12). There is an algorithm of Tate [412, IV §9] that transforms a given Weierstrass equation into a minimal one. �--'
k
Definition. Fix a minimal Weierstrass equation for EI K. Then we can reduce the coefficients of E to obtain a (possibly singular) curve I We say that E has good reduction if is nonsingular, which is equivalent to the condition that Ll(E) E R* . In any case, we obtain a reduction modulo p map on points,
E k.
E
E(K)
_____.
E(k),
P f-------+ P.
Proposition 6.29. If E has good reduction, then the reduction modulo p map
E(K) --+
E(k) is a homomorphism.
Proof See [410, VII.2. 1].
D
343
6.3. A Primer on Elliptic Curves
Remark 6.30. For elliptic curves defined over a number field K, we say that E has good reduction at a prime p of K if it has a Weierstrass equation whose coefficients
are p-adic integers and whose discriminant is a p-adic unit. Note that one is allowed to use different Weierstrass equations for different primes. If there is a single Weier strass equation that is simultaneously minimal for all primes, then we say that E K has a global minimal Weierstrass equation. Global minimal equations exist for el liptic curves over Q, and more generally for elliptic curves over any number field of class number 1, but in general the existence of global minimal equations is somewhat subtle; see [410, VIII §8] and [48]. We discussed a related notion of global minimal models of rational maps in Section 4.1 1 .
I
6.3.6
Torsion Points and Reduction Modulo p
The kernels of endomorphisms help to determine the arithmetic properties of elliptic curves. Definition. Let E be an elliptic curve. For any endomorphism 'ljJ E End(E) we write E['lj;] = Ker('lj;) = { P E E : 'lj;(P) = 0 } . Of particular importance is the kernel of the multiplication-by-m map, E[m] = { P E E : [m]P = 0 } . The group E[m] is called the m-torsion subgroup ofE . The union of all E[m] is the torsion subgroup of E,
Etors =
I
U E[m].
m� l
Theorem 6.31. Let E K be an elliptic curve and assume that either K has char acteristic 0 or else that K has characteristic p > 0 and p f m. Then as an abstract group, E[m] = ZlmZ x ZlmZ. In other words, E [m] is the product oftwo cyclic groups oforder m. Proof See [410, III.6.4]. D
The next result gives conditions that ensure that the reduction modulo p map respects the m-torsion points. It may be compared with Theorem 2.2 1 , which tells us what reduction modulo p does to periodic points of a good-reduction rational map. Theorem 6.32. Let K be a local field whose residue field has characteristic p,
let E K be an elliptic curve with good reduction, and let m ;::: 1 be an integer with p f m. Let E(K) [m] denote the subgroup of E[m] consisting ofpoints defined over K, i.e., E(K) [m] = E[m] n E(K). Then the reduction map
I
E(K) [m] --+
E(k)
is injective. In other words, distinct m-torsion points have distinct reductions mod ulo p.
344
6. Dynamics Associated to Algebraic Groups
D
Proof See [410, VII.3. 1].
EIK K,
K.
K. E[m]
Let be an elliptic curve defined over the field Then the points in are algebraic over so their coordinates generate algebraic extensions of An im mediate corollary of the preceding theorem limits the possible ramification of these extensions. Corollary 6.33. Let be a local field whose residue field has characteristic p, let be an elliptic curve with good reduction, and let m 2: 1 be an integer with p f m. Then the field obtained by acijoining to the coordinates of the m-torsion points of is unramified over
K
EI K
K EK (E[ml) K. ProofSketch. Let K' K(E[ml), let p ' be the maximal ideal of the ring of integers of K' , and let k' be the residue field. Suppose that CY E Gal( K' I K) is in the inertia group. Then CY fixes everything modulo p', so in particular, for all P E E[m]. (6. 1 5) CY(P) P ( mod p') But from Theorem 6.32, the reduction map E[m] ----+ E(k') is injective, so (6. 1 5) implies that CY(P) = P for all P E E[m]. The points in E[m] generate K' IK, CY fixes K' . Hence Gal(K' I K) has trivial inertia group, so K' IK is unramified. (For further details, see [41 0, VII.4. 1].) D Remark 6.34. The coordinates of the points in E[m] are algebraic over K, so the absolute Galois group Gx Ga!(KIK) acts on E[m] compatibly with the group =
=
so
=
structure. In this way we obtain a representation p :
G x --+ Aut(E[m]) <:::< GL2 (ZimZ).
In order to create a characteristic-0 representation, we fix a prime £ and combine all of the £-power torsion to form the Tate module
Tg(E) = lim E[£n] Zg /Zg. +-
/Zg
<:::'
x
E
(Here denotes the ring of £-adic integers.) Then the £-adic representation of is the homomorphism PE ,£ :
Gx --+
Aut(Tg(E)) � GL2 (Zg).
These representations are of fundamental importance in the study of the arithmetic properties of elliptic curves. 6.3.7
The Invariant Differential
E : y2 x3 + ax + bE
Let be an elliptic curve given by a Weierstrass = equation. The invariant differential on (associated to the given Weierstrass equa tion) is the differential l-form Definition.
dx 2dy . 2y 3x + a
WE = - =
345
6.3. A Primer on Elliptic Curves
The next result explains why the invariant differential is so named and shows that it linearizes the group law in a useful way. Theorem 6.35. Let E be an elliptic curve given by a Weierstrass equation and let wE be the associated invariant differential on E. (a) For any given point Q E E, let TQ : E E be the translation-by-Q map defined by TQ P) = P + Q. The differentialform wE is translation-invariant in the sense that for every Q E E.
-----+
(
(b) The differentialform wE is holomorphic at every point of E. (c) Up to multiplication by a nonzero constant, wE is the only holomorphic trans lation-invariant 1-form on E. (d) For every m E the differentialform wE satisfies
Z
[m] * wE = mwE. D
Proof See [410, III §5].
The invariant differential can also be used to fix an embedding of the endomor phism ring of E into C. Of course, this is of interest only when E has CM, since there is only one way to embed into
Z
Proposition 6.36. Let E /
R = {a E
Then for each a E R there is a unique endomorphism [a] E End(E ) satisfYing
[a] * (w)
=
(6. 16)
aw,
and this association defines a unique ring isomorphism
[ · ] R � End( E). :
( Without the normalization (6. 16), the isomorphism R � End( E) is unique only up to complex conjugation ofR.) Proof We fix the isomorphism E(
dz
to define an isomorphism F
-----7 E(
see Section 1 .6.3 and [410, VI §3]. Then F * (w)
= F* ( dx2y ) db:Jg:J' ((zz)) dz. =
=
346
6. Dynamics Associated to Algebraic Groups
Cl Cl Cl1f; E(C) C C C 1f; CCI C.
Let 'lj; E End(E), so using the identification � L, the endomorphism 'lj; defines a holomorphic map 'lj; L L satisfying 'lj;(O) 0. We claim that the analyticity implies that 'lj; lifts to a map of the form 'lj; = for a unique E and hence in particular that 'lj; is a homomorphism. (Compare with the algebraic statement of this fact given in Theorem 6.2 1 .) To prove this, we first observe that the covering map L is the universal cover of L, so we can lift 'lj; to some holomorphic map Further, the fact that lifts 'lj; means that satisfies :
---+
:
a C,
=
---+
---+
:
1f;
---+
1f;(z + w) - 1f;(z) E L
( z) az C1f;I
z C and all w E L. Fixing w E L, we find that the map z f--+ 1f;(z + w) - 1f;(z) is a holomorphic map from C to the discrete set L, so it must be constant. Thus for each w E L there is a number (w) E C such that 1{;(z + w) 1f;(z) + c(w) for all z E C. (6. 17) Writing 1{;( z) 2::: ai z i as a convergent power series, one easily checks that the relation (6. 1 7) forces 1f; to be linear, say 1f;(z) az+ (3. Then the assumption 'lj;(O) 0 tells us that (3 E L, so az and az + (3 descend to the same map on Cl L. Hence 'lj; lifts to a map of the form 1f; ( z) az. Further, a is unique, since if 'lj; ( z) also lifts to a' z, then the map z (a-a')z sends C to L, hence must be constant, so a a'. Finally, we observe that in order for az to descend to ClL, the complex number a must satisfy aL L. Thus the association 'lj; a gives a map End( E) R, and it is clear that the map 'lj;(z) az satisfies 'lj;* ( dz) adz, so with our identifications, we have 'lj;* (w) Next we check that the resulting map End(E) R is a ring homomorphism. Let 'lj; 1 , 't/J2 E End(E). Then on Cl L we have 't/J 1 (z) a 1 z and 't/J2 (z) a2 z, so for all E
c
=
=
=
=
=
=
f--+
�
f--+
=
---+
=
= aw.
---+
=
=
It remains to check that every E R comes from some 'lj; E End( By def inition any E R induces a map on that descends to a holomorphic homomorphism L L. Using the theory of elliptic functions (see, e.g., [410, Theorem VI.4. 1 ]), one can show that every such holomorphic map D is given by rational functions, which shows that End(R) R is surjective.
a Cl Cl ---+
az az C
E) . E(C) E(C)
f--+
---+
---+
6.3.8
Maps from E to JP>1
E
The quotient of by a finite group of automorphisms gives a map from These quotient maps play an important role in dynamics.
Aut(E). E EThen If theIID1 quotient is given
Proposition 6.37. Let r be a nontrivial subgroup of curve is isomorphic to IID 1 and the projection map 1r explicitly by
EIf
E to IID1 .
:
---+
�
r = M2 (j (E) if { ( y) = x2 if r = f-L4 (j(E) = 1728
347
6.3. A Primer on Elliptic Curves
x
1f
X
'
arbitrary), only), only), only).
y
if r = M3 (j(E) = 0 x3 if r = f-L6 (j(E) = 0 Proof By definition, the quotient curve E /f is the curve whose function field is the subfield of K(E) = K(x, y) fixed by r . Using the explicit description (6. 13) of the action of Aut(E) on the coordinates of E, it is easy to find this subfield. For example, iff = JL2 , it consists of those functions that are invariant under (x, y ) (x, -y ), so the fixed field K(Et is K(x, y2 ) = K(x). As a second example, if r = JL6 , then we need functions invariant under (x, y) (px, -y ), where p is a primitive cube root of 1 . This fixed field is K ( El = K (x3, y2 ) = K (x3), since in this case the elliptic curve is given by an equation of the form y2 = x3 + b. The other cases are 0 similar. For later use, we prove that the isomorphism class of an elliptic curve E is deter mined by the critical values of any double cover E JID1 . Lemma 6.38. Let E be an elliptic curve defined over a field of characteristic not equal to 2 and let 1r : E ---+ JID 1 be a rational map of degree 2. Then 1r has exactly four critical values and they determine the isomorphism class ofE. Proof The Riemann-Hurwitz formula (Theorem 1 .5) for the map 1r : E JID 1 says that 2g (E) - 2 = (deg 7r)(2g(JID1 ) - 2) + L (ep(¢) - 1). The map 1r has degree 2, the elliptic curve E has genus 1, and JID 1 has genus 0, so we find that L (ep (¢) - 1) = 4. The ramification indices satisfy 1 e p ( ¢) deg 1r = 2, so we conclude that there are exactly four critical points, i.e., four points with ep ( ¢) = 2 and all other points satisfy e p ( ¢) 1. Further, these four critical points must have distinct images in JID1 , since for any point E JID1 we have L ep (¢) deg(1r). Let fi, t2 , t3 , t4 E JID1 be the four critical values of1r, i.e., the images ofthe critical points, and let f E PGL2 be the unique linear fractional transformation satisfying f---*
f---*
---+
---+
PEE
PEE
:S
=
t
PE7r- 1 (t)
Explicitly,
:S
=
348
6. Dynamics Associated to Algebraic Groups
(If any of t 1 , t2 , t3 equals oo, take the appropriate limit.) The quantity
is called the cross-ratio oftr, t2 , t3 , t4; cf. Section 2.7, page 7 1 . We let x f so x is a rational function of degree 2 on E with critical values 0, 1, oo, and "' To ease notation, we let =
o 1r, ·
Taking 0 to be the identity element for the group law on E, we see that
div (x)
=
2 (To ) - 2 ( 0 ) ,
div(x - 1)
=
div (x- K, ) = 2 (T,. ) - 2 ( 0 ) ,
2 (TI ) - 2 ( 0 ) ,
so To , T1 , T,. are in E[2] , i.e., they are points of order 2. The sum of the three non trivial 2-torsion points on any elliptic curve is equal to 0, so Proposition 6. 1 8 tells us that there is a rational function y on E with divisor
div (y) = (To ) + (TI ) + (T,.) - 3 ( 0 ) . After multiplying y by an appropriate constant, it follows that x and y are Weierstrass coordinates for E (cf. [198, IV.4.6] or [410, 111.3. 1 ]). More precisely, the rational functions x and y map E isomorphically to the curve with Weierstrass equation E : y2 = X (X 1) (X ) -
0
- K,
It is easy to compute the j-invariant of E in terms of"' (cf. [410, III. l . 7(b)]). We find that 2 " s ("' - "' + 1) 3 J (E) = 2 /'1,2 ( /'1, - 1) 2 so in particular j (E) is uniquely determined by t 1 , t2 , t3 , t4. Then we apply Propo sition 6. 1 5 (or [410, III.l .4(b )]), which says that the isomorphism class of E is de D termined by its j-invariant. This completes the proof of Lemma 6.38. '
6.3.9
Complex Multiplication
Let E/ K be an elliptic curve with complex multiplication defined over a number field. As described in Proposition 6.25, the endomorphism ring of E is isomorphic to a subring of the ring of integers of a quadratic imaginary field. We briefly recall an analytic proof of this important fact and then discuss the relationship between com plex multiplication and the ideal class group of the associated quadratic imaginary field. This material is used only in Section 6.6, so may be omitted at first reading. Proposition 6.39. Let E /C be an elliptic curve with complex multiplication, i.e., the endomorphism ring End( E) is strictly larger than Z. Choose a lattice L C C such that E(C) � Cj L, let
349
6.3. A Primer on Elliptic Curves
R = {a E C aL <:;;; L }, :
and let
[ · ] : R � End(E)
be the isomorphism described in Proposition 6.36. Then R is a subring of the ring of integers ofa quadratic imaginaryfield F, andfor any a E R, the degree of [a] is given by
deg[o:] = mY = NF;Q(a) ,
where a denotes the complex conjugate of a.
R, we choose a basis for L, say L = + have R # by assumption, so there exists an a E R with a � Write
Z. Zw1 Zw2 . We (6. 1 8) a, b, e, d E Z.
Proof To describe the ring 1
Z
w aw + bw2 and aw2 cw1 + dw2 with The numbers w1 and w2 are JR-linearly independent, so the relation (a-e- a a-b- d) (ww21 ) (00) a
=
=
1
=
implies that the matrix has determinant 0,
(6. 19)
a2 - (a + d)a + (ad - be) = 0.
Hence a is an algebraic integer in a quadratic field. Further, we must have a � JR, since if a were real, then the relation (a - a = (and a � would contradict the JR-linear independence of and This proves that every element of R is an algebraic integer in a quadratic imaginary field, and hence R is a subring of the ring of integers of such a field. Finally, in order to compute the degree of the endomorphism [a] E E corresponding to a E R, we observe that
w1 w2 . )w1 bw2
Z)
:
----+
deg(a) = # Ker ( C/L � C/L) = (L : aL).
2
Z2
If a E then it is clear that ( L : aL) = a , since L = as an abstract group. Suppose now that a � Then continuing with the earlier notation, the transforma tion formulas (6. 1 8) imply that the index of aL in L is ( L : o:L) = ad - be. On the other hand, the product aa is the constant term in the minimal equation (6.19) for a over Q, hence also equal to ad
Z,
Z.
- be.
0
The theory of complex multiplication uses elliptic curves to describe the abelian extensions of a quadratic imaginary field F in a manner analogous to the descrip tion of abelian extensions of Q using torsion points in Gm, i.e., using roots of unity. For a complete introduction to the theory of complex multiplication, see, for exam ple, [257, Part II], [399, Chapter 5], or [412, Chapter II]. We now describe the tiny piece of the theory that will be needed in Section 6.6 in order to prove Theorem 6.62.
350
6. Dynamics Associated to Algebraic Groups
Let F be a quadratic imaginary field, let RF be the ring of integers of F, and let IF be the group of fractional ideals of F. If we fix an embedding F C then each fractional ideal E IF is a lattice c hence it determines an elliptic curve Ea whose complex points are
a
C,
a C;
(6.20) We observe that Ea has complex multiplication by RF, since any property that c hence it induces a holomorphic map
aa a,
a E RF has the
[a] : C/a �-------+ C/ a, z az, which in turn yields an isogeny [a] Ea Ea. In fact, since RF is the maximal order in F, we have End(Ea) = RF. We denote by E.eC(RF) the set �-------+
:
____,
E.eC(RF) = { isomorphism classes of elliptic curves E with End(E) � RF} . We thus have a natural map
a �-------+ (isomorphism class of Ea) · (6.21) We also observe that if we multiply a by a principal ideal, then the isomorphism class of Ea does not change, since for any c E K* there is an obvious isomorphism Cja � Cjca, z cz. Hence the map (6.21) induces a natural map from the ideal class group CF = IF jF* I----+
to elliptic curves with complex multiplication by RF,
Proposition 6.40. Let F be a quadratic imaginary field with ring of integers RF and ideal class group CF, and let hF = #CF be the class number of F. Then with notation as above, the natural map
is a bijection. In particular, there are exactly curves whose endomorphism ring is RF.
hF
isomorphism classes of elliptic D
Proof See [412, 11. 1 .2]. 6.4
General Properties of Lattes Maps
zn
Chebyshev polynomials arise by restricting the power map to the quotient of JID 1 by the finite group of automorphisms { z, z- 1 } . As already briefly described in Sec tion 1 .6.3, quotients of elliptic curves lead similarly to rational maps called Lata.�s maps. In this section we define and discuss general properties of these Lattes maps. For an excellent introduction to Lattes maps over including historical remarks and proofs of their basic geometric and analytic properties, see [300].
C,
351
6.4. General Properties of Latti�s Maps
A 1r : E
¢>E,:
7/J : E d 2:E,2
lP'1 lP'1 of degree rational map is called a Lattes map if there are an elliptic curve a morphism and a finite separable' covering lP' 1 such that the following diagram is commutative:
Definition.
�
�
�
(6.22)
lP' ' � lP'' . be an elliptic curve. Then the classical Example 6.41 . Let E = lP'1 yield formula for (Proposition 6. 1 7) and the isomorphism the Lattes map
2 x3 + ax + b : y x(2P) x : E/ { ±1} 2 - 8bx + a2 ¢>(x) = x(2P) = x4 -4x2ax 3 + 4ax + 4b Here 7/J is the duplication map 7/J ( P) [2]P, and the projection 1r is given by 1r(P) 1r(x, y) = x. Example 6.42. Let E be the elliptic curve E : y2 x3 + ax with j (E) 1728 and again let 7/J(P) = [2]P be the doubling map. If we take 1r(x, y) = x, then we are in the b 0 case of Example 6.4 1 , and we obtain the Lattes map ¢>(x) = x(2P) = 4(xX (2X-2 +a)a2) However, for this curve we may instead take (x, y) = x 2 . This gives a new Lattes map ¢ 1 . We find a formula for ¢ 1 using the relation (x - a)2 ) 2 (x - a)4 2 · q;,(x) (JX) 2 ( 4fo(x + a) 16x(x + a) Note that the map 1r(x, y) = x2 corresponds to taking the quotient of E by its automorphism group Aut(E) � 114 via the association described in Remark 6.27. Example 6.43. In a similar manner, the doubling map on the elliptic curve E : y2 = x 3 + 1 with j(E) 0 and Aut(E) 116 gives various Lattes maps corresponding to taking the quotient of E by the different subgroups of 116 . Explicitly, the Lattes maps corresponding, respectively, to 112 , 113 , and 116 are �
=
=
=
=
=
'
1r
=
=
=
=
We leave the verification of these formulas to the reader; see Exercise 6. 12. 1 The assumption that 1r i s separable i s relevant only when one i s working over a field o f characteris tic p, in which case it is equivalent to the assumption that 1r does not factor through the p-power Frobenius map.
6. Dynamics Associated to Algebraic Groups
352
We begin with an elementary, but useful, characterization of the preperiodic points of a Lattt��s map (cf. Proposition 1 .42). Proposition 6.44. Let ¢ be a Lattes map associated to an elliptic curve E. Then
PrePer(¢) = ;r (Etors ) · Proof Let ( E lP' 1 and let P E E be any point satisfying ;r(P) = (. We consider the orbits of ( and P. Thus ;r (O,p (P)) = 1r ({�n (P) : n � 0 }) = {;r�n (P) : n � 0} = {¢n ;r(P) : n � 0} = {¢n (() : n � 0 } = Oq, ((). The map 1r is finite, so this shows that O,p( P) is finite if and only if Oq, ( ( ) is finite. Hence
PrePer(¢) = ;r(PrePer(�)), and it i s left to prove that PrePer(�) = E1ors · We observe that the map � : E E has the form �(P) = �o (P) + T for some �o E End(E) and some point T E E. (See [410, III.4.7].) We are going to prove Proposition 6.44 in the case that � = �0 E End(E), i.e., assuming that = 0. �
T For the general case, which requires knowing that the point T is a point of finite or
der, see Exercise 6. 14. Suppose first that P E E1ors . say [n]P = 0 for some n � 1. Consider the images of the iterates �, �2 , �3 , . . . in the quotient ring End (E)/ n End (E). It follows from the description of End( E) in Proposition 6.25 that this quotient ring is finite, so we can find iterates i > j � 1 such that
�i �j (mod n End(E)). In other words, there i s an endomorphism E End(E) such that �i = �j + f3n. Evaluating both sides at P and using the fact that [n]P = 0 allows us to conclude that �i (P) = �j (P). Hence P E PrePer( �), which proves that E1ors C PrePer(� ). Next suppose that P E PrePer(�), say �i (P) = �J (P) for some i > j. We ----i i rewrite this as ( � - �j ) ( P) = 0 and apply the dual isogeny � - �j described in =
f3
Theorem 6.22 to obtain
[deg(�i - �j )](P) = 0. We know that �i -1- �j ' since deg(�) = deg(¢) � 2 and i > j, so �i - �j has positive degree. This proves that P E E1ors . which gives the other inclu sion PrePer(�) C Etors · 0 Many dynamical properties of a rational map can be analyzed by studying the behavior of the critical points under iteration of the map. This is certainly true for Lattes maps, whose postcritical orbits have a simple characterization, which we give after setting some notation.
353
6.4. General Properties of Lattt�s Maps
Let ¢ C1 C2 be a nonconstant rational map between smooth projective curves. The set of critical points (also called ramification points) of ¢ is denoted by Definition.
:
CritPt,p
=
----+
{P E C1 : ¢ is ramified at P}
{P E C1 ep(¢) 2 2}.
=
:
The set ofcritical values of ¢ is the image of the set of critical points and is denoted by CritVal¢
=
¢(CritPt¢) ·
If ¢ C C is a map from a curve to itself, the postcritical set is the full forward orbit of the critical values and is denoted by :
----+
00
PostCrit¢
00
U=O ¢n (CritVa1¢) U=l CritValn .
=
=
n
n
(See Exercise 6. 1 5.) Proposition 6.45. Let ¢ diagram (6.22). Then
:
IP'1
----+
IP'1 be a Lattes map thatfits into a commutative
CritVal7r
=
PostCrit .
In particular, a Lattes map is postcritically finite. Proof The key to the proof of this proposition is the fact that the map 'lj!
:
E E ----+
is unramified, i.e., it has no critical points, see Remark 6.20. (In the language of modem algebraic geometry, the map 'lj! is etale.) More precisely, the map 'lj! is the composition of an endomorphism of and a translation (Remark 6.19), both of which are unramified. For any n 2 1 we compute
E
CritVal7r
CritVal1r,p n CritVal¢n7r = CritValn U ¢n ( CritVal7r) :2 CritValn .
= =
because 'lj! is unramified, from the commutativity of (6.22), from the definition of critical value,
This holds for all n 2 1, which gives the inclusion CritVal7r :2
00
U=O ¢n (CritVa1¢)
=
PostCrit¢ .
n
In order to prove the opposite inclusion, suppose that there exists a point Po E satisfying and 1r(P0 ) � PostCrit¢ . (6.23) Po E CritPt7r Consider any point Q E 'lj! - I (Po). Then Q is a critical point of 1r'lj!, since 'lj! is unramified and 1r is ramified at 'lj! ( Q) by assumption. But 1r'lj! ¢1r, so we see that Q is a critical point for ¢1r.
E
=
6. Dynamics Associated to Algebraic Groups
354
On the other hand, ¢ (1r( Q ) )
=
1r(Po) tJ_ CritVal
=
cj>(CritPt¢) ,
so 1r(Q ) i s not a critical point for ¢. It follows that Q i s a critical point of 1r. Further, we claim that no iterate of 4> is ramified at 1r ( Q). To see this, we use the given fact that 1r(P0) is not in the postcritical set of 4> to compute for all n � 1, 1r(Po) tJ_ PostCrit
1r( Q) tJ_ PostCrit,p .
and
In other words, every point Q E 1/J-1 (Po) satisfies the same two conditions (6.23) that are satisfied by P0. Hence by induction we find that if there is any point Po satisfying (6.23), then the full backward orbit of 1/J is contained in the set of critical points of 1r, i.e., CritPt71'
:::>
00
U 'lj;- n (P0 ) .
n= 1
But 1/J is unramified and has degree at least 2 (note that deg 1/J
=
deg 4>), so
This is a contradiction, since has only finitely many critical points, so we conclude that there are no points Po satisfying (6.23). Hence 1r
P0 E CritPt1r ===} 1r(P0) E PostCrit