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ÉÁ;¾Á/ÇÃ{¿Á/Î Þ$Æ σ yà ¾Î ¿ÀÁÅÊDÒ;Ãy  Á/ΣÒ;ÃyÇ¿ÊÉÇÃyÑÀÐ Ò9ÉÇÊÏÑì!ÃÒ2¿Åj¿ÇÃy¾ ÅÐ¥¿Ð¥ÓªÁ;Â¥ÆZʾñ¿ÀÁhÅÁ;¿nÊÍÊÇÐ Á;¾$¿Á/Î 0 σ Á/ÎÉÁvÅ;Ü 1 è ptÀÁ<ÉÁ;ÊËaÁ2¿ÇÐÄÒUÁ/ÎÉÁvÅaÃyÇÁ<¿ÀÁUÊÇÞÐ¥¿ÅaÊÍ ÈtÃy¾Îß¿ÀÁ<ÓªÁ;Ç¿ÐÄÒ2ÁvŵÃyÇÁ<¿ÀʪÅÁUÊÍ σ1 (Γ) ÜUéåÁZÎÁ;¾Êy¿ÁñÞ$Æ Ãy¾ Î ¿ÀÁ/ÅÁ£ÅÁ2¿Å;È!Ãy¾ÎåÞ$Æ Ãy¾ Î ¿ÀÁ;Ð Ç σ0 (Γ) Ag (Γ) V (Γ) a(Γ) v(Γ) Ò;ÃyÇÎÐľÃyÂÄÅ;Ü hÊy¿Á¿ÀÃ{¿ Ð Å9ÃÌÉyÇÃ{ÑÀMÜqcÍ ÈÕnÁSÎÁ/¾Êy¿Á?ÞªÆ Ð¥¿ÅÊÇÐ ÉÐ ¾dÃy¾ Î Γ a ∈ A(Γ) a(0) ÞªÆ Ð ¿ÅhÁ;¾ÎÜtéåÁS¿ÃyÔÁS¿ÀÁÒ2ʾ$ÓªÁ;¾$¿Ð ʾd¿À Ã{¿ Ð Å¿ÀÁ *ÊÇÞÐ¥¿hÊÍ Ãy¾ Î a(1) a(1) σ0 a a(0) ÐÄÅn¿ÀÁ ÊÇÞÐ¥¿9ÊÍ Ü ÊÇ9Ò2ʾªÓªÁ;¾Ð Á;¾ Ò2ÁÈÕnÁ¢ÊÍ ¿Á;¾UÑÏ¿ g Ü σ0 σ1 (a) a ¯ = σ1 (a) éåÁGÎÁ;ã¾Á È,ÅÏ ÒÀ<¿À ÿ ÜnptÀÁ¢ÊÇÞÐ¥¿ÅÊÍ σ (Γ) = σ (Γ)σ (Γ)−1 σ σ σ2 = 1 σ (Γ) ÃyÇÁaÒ/Ãy  Á/ÎJ¿ÀÁ Z2:R<407µÊÍ 1 ÜÌéä0ÁµÎÁ/¾Êy¿Á¿ÀÁµÅÁ2¿SÊÍE0ÍÃÒ21Á/Å¢ ÞªÆ Ãy¾ ÎJÐ¥¿Å¢Ò/Ã{ÇÎ2Ð ¾ ÃyÂ Γ F (Γ) ÞªÆ Ü ptÀÁó Á;¾Éy¿À ÊyÍGÃßÒ'ÆÒ2 ÁJÊyÍ ÇÁ/ÅÑÜ ìµÐÄÅZ¿ÀÁmÓyÃy Á/¾Ò'ÆîÊÍ?¿ÀÁ f (Γ) σ0 (Γ) è σ2 (Γ) Ò2ÊÇÇÁ/ÅÑʾÎÐ ¾ÉÌÓªÁ;Ç¿Á;í ÇÁ/ÅÑÜÍÃÒ2Áì'Ü gÓªÁ;ÇÆ£ÍÃÒ2Á¢ÐÄÅ9Ãy¾<ÊÇÐ è Á;¾$¿Á/Î< ʨÊÑÈÍÊÇ/ÈÐÄÍ ¿ÀÁ;¾ Ü b = σ2 (a) b(0) = a(1) hÅMÃy¾?Á;íÃyËaÑ ÁÈ{ÃtÉÇÃyÑÀ Á/ËGÞ,Á/ÎÎÁ/Î?Ð ¾GÃÒ2ÊËa Ñ ÃÒ'¿MÊÇÐ Á;¾$¿Á/΢ÅÏÇÍÃyÒ;ÁÐÄÅMÃt¿ÊÑÊyÂÄÊ6É Γ ÐÄÒ;Ãy¨ÇÁvÃ{ÂÄÐ Ý/Ã{¿Ð ʾÌÊÍ ÃÍÃ{¿ *ÉÇÃyÑÀMÜ\ptÀÁEÑ,Á/ÇËGÏ¿Ã{¿ÐÄÊy¾ ÊÍ ÐÄÅOÉÐ¥ÓªÁ;¾ÌÞªÆS¿ÀÁEÑÇÊyëÁ/Ò2¿Ð ʾ σ Γ ÊÍM¿ÀÁ?¾Á;Ð ÉÀ¨Þ,ÊÇÀʨʨÎÅnÊÍ\¿ÀÁÓªÁ;Ç¿ÐÄÒ;Á/Åʾ£¿ÀÁ¿Ãy¾ÉÁ/0¾ª¿9ÑÂÄÃy¾Á/ÅÃ{¿¿ÀÁ/ÅÁ¢Ñ,ÊÐ ¾$¿Å;Ü ptÀÁ?ÉÁ;¾¨ÏÅ ÊÍM¿ÀÁ¢ÍÃ{¿ *ÉyÇÃyÑÀ ÐÄÅt¿ÀÁ;¾UÎÁ;ã¾Á/ΣÞ$ÆZ¿ÀÁ?gjÏÂÄÁ;ÇÍÊÇËGÏ Ãõ g(Γ)
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W (Γ) i\Á2¿ j Þ,Át¿ÀÁ ÊÇÞÐ¥¿ÊO(e) Í Ã{¾ Î j Ãy¾Î j j ÍðÊÇ (a1 , . . . , ak ) (b1 , . . . , bl ) j ∈ {1, . . . , |G|} G a(0) È$ÅÏÒÀÌ ¿À Ã{¿ Ç v Á Å Ñ Ü \ ì Ä Ð ! Å ¿ À Á Ê Ç Þ Ð ¿ Ê Í Ç / Á Å M Ñ Ü ' ì q Ü t p À ; Á a ¾ Ç 2 Á ¿ Ç Ã ' Ò ¿ Ä Ð ¾ ? É ¿ÀÁ a(1) {aj1 }j è {bj1 }j a ¯è a ÊÇÞÐ¥¿ ÉÐ¥ÓªÁE¿ÀÁj¾Á2ÕäÓªÁ;Ç¿ÐÄÒ;Á/Å ÜOptÀÁEÉÇÃyÑÀ O(e) sj = (aj , . . . , ajk , bj2 , . . . , bjl ) WO(e) (Γ) ÐÄÅn¿ÀÁ?ÍÃ{¿*ÉÇÃyÑÀZÕ9Ð ¿ÀdÅÃyËaÁ ÑÇÐ¥ÓÿÁ¢Ê2Í ÈÃy¾ Î<ÅÃyËaÁ Á2íÒ2Á;Ñ¿ j j σ1 O(e) = {(a , b )} σ0 ÍðÊÇt¿ÀÁ?ÊÇÞÐ¥¿ÅÊÍ Ãy¾ Î ÇÁ/ÑÂÄÃÒ2ÁvÎ<ÞªÆZ¿ÀÁ?¾Á2Õ 1ÊÇÞ1Ð ¿ j Ü a(0) a(1) {sj }j hÅtÕnÁSÒ;Ãy¾dÇÁ2¿ÇÃÒ'¿9¾Á;Ð¥¿ÀÁ;ÇÃÌ ʨÊÑMȾÊÇ9Ãy¾dÊÇÞÐ ¿Õ9ÀÐÄÒÀmÒ2ʾ$¿ÃyÐ ¾ Å9ÅÊËaÁ¢ÍÃÒ2Á/Å;ÈÕnÁ ÀÃ/ÓªÁ Ü m Ð ¾Ò;Á Ãy¾ Î
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pMÊUÐ ÂÄ ÏÅ¿ÇÃ{¿Áµ¿ÀÁµÍÃÒ'¿S¿À Ã{¿¢¿ÀÁµÒ2ÊËGÞÐ ¾ Ã{¿ÊÇÐÄÃyÂEÎÁ/ÅÒ2ÇÐÄѿРʾåÊÍnfhÏÇÕ9Ð¥¿ÝñÅÑ ÃyÒ;Á/Å¢Á;¾ Ò2ÊÎÁ/ÅO¿ÀÁ;ÐÄÇqÊÇÞÐ ÍÊÂÄÎÌÅ¿ÇÏÒ2¿ÏÇÁÈyÕnÁt Р¾ÔS¿ÀÁ;Ð ÇÊyÇÞÐ ÍÊÂÄÎgjÏ Á;ÇÒÀ ÃyÇÃÒ'¿Á;ÇÐÄÅ¿ÐÄÒyÈ¿Ê¿ÀʪÅÁ ÊÍ\ËaʨÎÏ ÐÅÑÃÒ2Á/Å/Ü é7ÜptÀ¨ÏÇſʾçÀÃÅ?Á2í¨¿Á/¾ÎÁ/Îó¿ÀÁa¾Êy¿ÐÄʾçÊÍEgjÏ Á/ÇSÒÀ ÃyÇÃÒ'¿Á;ÇÐÄÅ¿ÐÄÒÌ¿ÊUÊÇÞÐ ÍÊÂÄÎÅ O Õ9ÀÐÄÒÀîÑʪÅÅÁ/ÅÅ£ÅÊËaÁJÒ2Á;  ÏÂÄÃyÇUÎÁ/Ò2ÊËaÑ,ʪÅÐ¥¿Ð ʾ Ò2ÊËÌÑ Ã{¿ÐÄÞ ÁmÕ9Ð ¿ÀêÐ¥¿ÅZÊÇÞÐ ÍÊÂÄÎ (Ci )i Å¿ÇÏ Ò'¿ÏÇÁªÜptÀÁ;¾ χorb (O) =
X (−1)dim(Ci ) |G(Ci )|
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Õ9ÀÁ;ÇÁ ÐÄÅt¿ÀÁ¢ã¾Ð¥¿Á?ÉÇÊÏÑUÃÅÅʨÒ;ÐÄÃ{¿ÁvΣ¿ÊaÁvÃÒÀUÒ;Á; Â)Ü G(C ) fhÁ;ÇÁªÈÏ ÅiÐ ¾ÉµptÀÁ;ÊÇÁ/Ë ¨Ü ¨ÈÕnÁ?À Ã/ÓªÁ χorb (Hg (G, R) × P(Rb>0 )) =
X (−1)a(∆j )−1 × |Z(G)| |AutG (Γj , ϕj )|
[Γj ,ϕj ]
Õ9ÀÁ;ÇÁ ÜaptÀÁ/¾ÈÅÐ ¾Ò;Á ÈMÏÅÐ ¾É£¿ÀÁagjÏÂÄÁ;Ç?ÍðÊÇ ∆ = Γ /ϕ χorb (P(Rb>0 )) = (−1)b−1 ËGÏÂÄÃÈÕtÁ¢j ÎÁ/ÎÏjÒ2Áªõ j χorb (Hg (G, R)) =
X (−1)s(∆j ) × |Z(G)| . |AutG (Γj , ϕj )|
[Γj ,ϕj ]
éåÁÇÁvÅÏË Ê¾Ì¿ÀÁ9Ð ÅÊËÌÊÇÑÀÐÄÅË Ò2 ÃyÅÅÁ/Å Ð ¾ ÎÁ;í¨ÐľÉ?¿ÀÁhÒ2Á;  ÏÂÄÃyÇnÎÁvÒ2ÊËh [Γ /ϕ = ∆ ] Ñ,ʪÅÐ¥¿Ð ʾçÊÍ ÈMÃy¾Îm¿ÀÁ;¾JÕnÁÌÏ jÅÁG¿jÀÁaÇÁ;ÂÄjÃ{¿Ð ʾçÞ,Á;¿*ÕnÁ/Á;¾çÃyÏ¿ÊËaÊÇÑÀÐÄÅ˵ŠM 0 ÉÇÊÏÑ Å ptÀÁ;ÊÇgÁ/Ë ,(b1,...,b Übyìjt¿)ʵÊÞ¿ÃyÐ ¾ è χorb (Hg (G, R)) =
X (−1)s(∆) X |Aut(∆)| |Aut(∆)| |Aut(∆, ψ)| [∆]
ψ
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χorb (Hg (G, R)) = dorb × χorb (Mg0 ,(b1 ,...,bt ) ).
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V (761 X .A+8 bd7/1iRG46GP>+*B7 ,"GPO cMi*n1;)N46GPDF>:5B7[E>=.03.DF761;7 03, .A+8 0;7 ,2>d7 413*n`75nLc Y C − {0} x y .A+8o5B761 bU7aCP*B`P7 AHbL c+OTGP0,2GPDF7|*BA137CP70 V c"):*-,W*-,.8%7CP0;7 7 f f :Y →X x = yn n>1 Q?.5BGP*-,/46G `70i)+GP,;7IQ?.5BGP*-,CP0;GP<+>H*-,1;)+7X4L%45B*-4|CP03G<+> GPOG0@8%70 c+i):7 0;7a13):7 n Cn n CP7A:7 03.1;GP0IGPOW13):7NQ?.5BGP*-,9C03G<+>13. h 7 , fui*n1;) bd7*BA+C.J>+0;*BDF*n13*n`7 y 7→ ζn y ζn ∈ C ]¤1;)H0;GG1WGPO(<+A:*n1L=k@ V \OGPA+7?`%*B76i, .A+8 A+G1j2<+,K1.P,13GP>UGP5BGPCP*-4.5,;>+.P467 , c%b+<%1.P, n X Y 46GP>:*B7 ,/GPO1;)+79. FA:7X`.0;*B761L cU1;)+7A *-,1;)+7XDFGP0;>+):*-,;D 46GP0;0;7 ,2>UGPA=8%*BA+Ct1;G A1 − {0} f 1;)+7X*BA+45B<+,;*BGPAJGPO\OT<+A+461;*BGPAJl+75-8+C,c cUCP*n`7AmbL V c")+*-,*BA=465B<=,2*BGPA Y C(x) ,→ C(y) x 7→ y n i)+G,27/Q?.5-G*S,\CP0;GP<+>F*-, ci)+G,27WCP7 A:70@.13GP0 *-,$.XQ?.5BGP*S,\l:75S897[137A=,2*BGPAGPOz8%7CP0;7 7 n C .P413,|bL Vmfq130;*-4135nL,2>U7 . h *BA+C+c*BOR1;)+7EQ?.5BGP*S,|CP0;GP<+>vGPO"n46G `703,_.P41@,aGPAv13):7 y 7→ ζn y 5B7OT1 c%13)+7A13):7IQ?.5BGP*-,C03G<+>oGPO(l+75-8+,W.P4613,WGPA13):7|0;*BCP)1 Vxk \A13)+*-,I76[:.DF>:5B7c1;)+4 7 =<+A+8+.D97 A1@.b 5 c")+7GP0;7D OTGP094G `P7 0;*BA+Cm,;>+.P467 ,I*BDF>+5B*B7 ,I13):7 +<+A+8+.DF7A13.R 5 c")+7GP0;7D OuGP0913):7H7[137A=,2*BGPA GPOOu<+A+413*BGPA l:7 5-8:,c",2*BA=467H*BA1;7 0;DF7 8%*S.137 4GP0;0;7 ,2>UGPA=813Gm*BA1;7 0;D97 8%*-.137ol:7 5-876[M1;7 A+,2*BGPA=, fui):7037 46G `703, Z → X M ⊃ C(x) *-,W1;)+7|Ou<:A=413*BGPAHl+75-8HGPO kVsNGP0;7|CP7A+703.5B5BLc+GPA+7_4.AN46GPA=,2*-8:70?Q?.5BGP*-,/46G `703,WGPO M Z ,;43)+7DF7 ,1;)=.1/.0;7_A+G1A+7 467 ,;,;.0;*B5nLJ8%7l+A+7 8HG `70 cU.A=8H*BAN13)=.1?46GPA1376[1/)=. `7|bdG1;) C .5BCP7b:0@.*S4a.A=8CP7GPDF76130;*-4aOuGP0;D,"GPOQ?.5BGP*-,W1;)+7GP02LV \AGP038:7091;Gv76[M1;7 A+813):7o*S8%7 .GPO?46G `70;*BA+Cv,2>=.P46713G1;)+*-,F,271213*BA+C:cGPA+7oA:77 8:,I1;G 1;)=.1HCP7A+703.5B*Be 7 ,13)+7v465S.,3,oGPO 8%7l+A+7. 465-.P,;,HGPOXl:A+*n1;7D9GP0;>+):*S,2D, f : Y → X 46G `70;*BA+C,2>+.P467 ,Jfu*-A1;)+7J46GPDF>:5B7[DF76130;*-4H13GP>UGP5BGPCL=kIOuGP0t4GDF>+5B76[`P.03*B761;*-7 ,V c")+7 46GPA+8:*n13*BGPAGPObd7*BA+C.46G `703*BA:C,2>+.P467H*-A 1;)+7 U.0;*S, h *W13GP>UGP5BGPCL 8%GM7 ,FA+G1t8%Gv1;)+*-,c ,2*BA=467|.Ao*B0;037 8%<=46*Bb+5B7|,;4@):7 D97 i*B5B5A:G1W)+. `7|.ALE*B0;037 8%<=46*Bb+5B7|46G`P7 03,"*BA13)+*-,W,27 A+,27c X G1;)+70"1;)=.A13):7|*-8:7A1;*B1LoD.>ZVf ?.DF75nLc+*BO *-,76`7A:5BLo46G `70;7 8G `70.F8:7A+,;7 Y →X GP>U7AF,2761 c1;)+7A9*B1*-,\.?8:*-,uj2GP*BA1<+A+*BGPA9GPOU46GP>+*B7 ,GPO cCP5BGPb=.5B5nLVxI k \A=,K137 .P8IGPA:7"<=,27 ,13):7 X A:G13*BGPAGP O +u £ B£E@« £6¬@§cz*'V 7PVzl+A:*n1379DFGP0;>+):*-,2D, ,2<+4@)m1;)=.1a5BG%4.5B5nL f :Y →X .1W76`702LE>dGP*BA1WGPO cM13)+7|,;43)+7DF7 *-,"CP*n`7 AG`P7 0 bL >dGP5nL%A:GPDF*-.5-, Y Y X m f1 , . . . , f m *BA `.0;*S.b:5B7 , cU.A=8N,2<+4@)H1;)=.11;)+7 .P46GPb:*S.AJ8%761370;DF*BA=.A1 *S, m y1 , . . . , y m (∂fi /∂yj ) 5BG%4.5B5nLX*-A`7 021;*Bb+5B7P! V c")+7R>UGP*BA1*S,(1;)=.1OuGP0,2>+.P467 ,G `70 c13):*-,46GPA+8:*n13*BGPA9*-,7 <+*n`.5B7A1 C 1;G )+. `%*BA+Ct.F5BG%4.5,27 413*BGPAJA:7 .07`P7 02Lo>UGP*-A1IfgbL13):I 7 \A`70@,27 +<+A+461;*BGPA c")+7GP0;7DHc f i):7037 5BG%4.5,;7 413*BGPAMXDF7 .A+,/*BAN13):7946GPDF>:5B7[ND971;0;*S4 k .A=8HOuGP0?.l+A:*B1;7ID9GP0;>+):*S,2DHc 0 ,;.1;*-,;OSL%*BA+CE1;)+*-,/5-.1;1;7 0a46GPA=8%*B1;*BGPA*-,/7 <+*n`.5B7 A1?13GEbU7 *BA:Co.tl+A:*B1;7946G `7 0;*BA:Co,2>=.P467fg*BA 1;)+7|46GPDF>:5B7[DF761;03*-4_,27A+,;7k@V =GP0l+A+*n1; 7 61@.5B7W46G `703,GPO=.AI*B0;0;7 8%<=46*Bb+5B7",;4@):7DF7 cGA+71;)+7AI)=.P,. +<+A+8+.DF7A13.5 X c"):7GP0;7 D GPOQ?.5BGP*-C , c"):7GP02Lv.P,_.bdG `7P¨ V \O"GA+70;7 ,K130;*-41@,a13GJ46GPD9>+5B76[`.0;*-761;*-7 ,cGPA+7 GPb%1@.*BA+,_13):7CP7GPD971;0;*S4E,2*B1;<=.1;*BGPA8:*-,;46<=,;,27 8.bdG `7PV XA+8*BOGA+7E0;7 ,K130;*-41@,X13G GPO X 1;)+7tOuGP0;D fuOuGP0I,2GPDF7El+75-8 k@c1;)+7AGA+7E0;7 46G `703,FQ?.5BGP*-,_13)+7GP02LOuG0Xl+75-8 Spec K K 76[M1;7A=,2*BGPA=,|fui*n13) 4G030;7 ,2>UGPA=8%*-A:CI13G.9l+75-8o76[M137A+,2*-GPA k@V Spec L → Spec K L⊃K
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fu*'V 7PV13):7|Qa.5-GP*-,RC03G<+>=,$GPOzl+A:*B1;7 613.5B7a46G `703,RGPO k@V$qM*BDF*B5-.0;5nLc%GPAt13)+7CP7A+70;*S45B76`7 5 X Tf i)+70;7.0;b:*n1303.02Lb+03.A+4@):*-A:CJ*-,9.5B5BG iR7 8Uk@cGPA+7tiRGP<:5-85B* h 7E1;Gm)+. `7.A76[%>:5B*S46*n18%76] ,;460;*B>:1;*-GANGO13):7t § « 5:¤£41B«T§¬3«5 GPO1;)+7?Ou<+A+413*BGPAol+75-8 GK = Gal(K s /K) K GPO fui)+70;7 *-,/1;)+7F,27>=.0@.b+5B7945BG,2<:0;79GPO k@I V c")+*-,/*-Am1;<+0;AJiRGP<+5-8m>+0;G `%*-8%79.A X Ks K 76[%>:5B*S46*n1?8%7 ,3460;*B>:1;*BGPANGPO(13)+7|l:A+*n137_<+G1;*B7 A1@,WGPO c=*'V 7PV:1;)+7IQ?.5BGP*-,CP0;GP<+>+,WGPOl+A:*B1;7 GK l:75S876[M137A+,2*-GPA+,WGPO V K
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7 A @ 1 . $ 5 6 4 P G = A K , @ 1 . A 6 1 @ k V X C . B * Z A ( c 3 1 ):7 x = πy n π V c")+*-,?46G `70 c Y 46G `70/*-,aQa.5BGP*-,cUi*n1;)4L:465-*-4FQ?.5BGP*-,/CP0;GP<+>NCP7A+703.1;7 8NbL g : y 7→ ζn y .5BGPA:C?i*n13)F*n1@,Q?.5BGP*-,\.P413*BGPAcP*S,8:7l+A:7 8XbLI>UGP5nL%A+GPD9*S.5S,G `70 b+<%1$.OS1370\.a4@)+.A+CP7 C0 GPOd`.0;*-.b:5-7 , c13):7/46G `70*-,CP*n`7AtbLF>UGP5BLPGPDF*-.5-,G `70 ¯ fT`%*BePV n c%.A+8 z = π 1/n y z =x Q ke V \ANOg.P41 c=*BO *-,.ALH46<+02`7|1;)=.1/4.ANbd7_8:7l+A:7 8HG `70 ¯ fg7PV C+V=*BO *S, Q g : z 7→ ζn z X X 1;)+7I46GPDF>:5B7 D97 A1aGPOl:A+*n1375nLJD.AL ]¤>dGP*BA13,*BA 1 kc=13):7 Am.ALNl:A+*n137 13.5B7946G `70/GPO Q P 1;)+7I*BA=8%<=467 8m46GPDF>:5-76[J4<:02`7 4 .AJ*-AOu.P41a*n13,;75BO$bU7F8%7l+A:7 8NG `70 ¯ fg.5BGPA+CEi*n13) XC Q *n13,RQ?.5BGP*-,$.P41;*-GPAZc*BAF1;)+7 ]KQ?.5BGP*S,\4.P,27 ,27 7 `7D.0 h wMV-rPV ?bU75BGi?k@t V XA=8F,2*BA+47"1;)+70;7 G 0 .0;7GPA+5nLvl:A+*n1375nLDF.ALv>UGP5nL%A+GPD9*S.5S,_*BA`PGP5B`P7 8zc*n1I4.A76`7AbU7t8:7l+A:7 8vG `70X,2GPDF7 A<+DXbU70Wl+75-8 V K c")+7 h 76L<+7 ,K13*BGPA)+70;7*-,i)=.1$1;)+*-,\AM<+D_bd70$l:75S8F*-,c*BAF1370;D,$GOU13)+7"13GP>UGP5BGPCL9GPO 1;)+7I46G `70 V a$L `*B7D.A:AY Y , d\[%*-,K137A+4 7 c")+7GP0;7DHc=1;)+7FQ?.5BGP*-,?4G `703,GPO\.tCP*n`7Amb+.P,27 .0;7a465-.P,;,2*-l:7 8 7PV C:V13)+GP,;7/G `70 1 4GP0;0;7 ,2>dGA=813GX13):7?l+A+*n1;7a<+G1;*-7A13, 0 X P − {0, 1, ∞} GPO13):7HOu0;77oCP0;GP<+>GPA1i"GvCP7A:7 03.1;GP03,VpqGv*-A1;)=.1t4.P,27PcCP*B`P7 A .vl:A+*n1;7HCP0;GP<+> G 1;GPCP7613):70ai*n13).>=.*B0aGPOCP7A+703.13GP03,czi)+.1|*S,/13):7FAM<:DXbd70al+75-8 G `7 0?i):*-4@)13):7 K 46GP0;0;7 ,2>dGPA+8:*BA:C ]2Q?.5BGP*-,W46G `70"GPO 1 *S,"8:7l:A+7 8 >XW413<+.5B5nLcM1;)+*-,"l:75S8EGPO G P − {0, 1, ∞} 8%7l+A+*n1;*BGPA *-,/A:G1/<+A+*-<:75nLJ8%71;703D9*-A:7 8Zcd.5n13):GP<+CP)N1;)+70;7X*-,?.A *-8:7 .54.A=8%*S8:.137 K OTGP0 c(DFG13*n`.1;7 8bLQ?.5BGP*-,_1;)+7GP02LV ?.DF75nLc*BO c1;)+7A ¯ K ω ∈ G := Gal(Q/Q) ω .P413,IGPA13)+7o,2761FGPOXfg*-,2GPDFGP0;>+):*-,2D 45-.P,;,27 ,9GPO@k ]2Q?.5BGP*-,94QG `703,c\bL.P41;*-A:CGPA13):7 G 46GM7 46*B7A13, V \O_. ]2Q?.5BGP*-,4G `70E*-,8:7l+A:7 8 G `70o.AM<:DXbd70tl+75-8 cW1;)+7Ap.AL G K DX<+,K1R4 .0302LI13):*S, ]KQa.5BGP*-,46G `70$13G|*B13,275-OVqMG|iR7DF. L ¯ ¯ ω ∈ Gal(Q/K) ⊂ Gal(Q/Q) G 46GPA+,2*S8%70_1;)+7 £ m« F¥t« 5 OTGP0|13):7 ]KQa.5-G*S,X46G `70 c8%7 l:A+7 813GNbU7F13):7tl%[%7 8 M G l:75S8vGPOW.5B5\13):7 Y ,_*BA 13)+.1I4.0;0;L1;)+7 ]2Q?.5BGP*-,X46G `7 0|1;GJ*n13,27 5BOV c")+*-,|*S,|13):7A ω G G 46GPA1@.*BA+7 8v*BA76`702Ll:7 5-8vQGPOR8:7l:A+*n13*BGPAGPO1;)+7 ]2Q?.5BGP*-,_46G `70 VtsNGP0;7G `70_*n1_*-,a13):7 G T=£6¬@§ £@g'«JGPO13):7l+75-8+,|GPOW8%7 l:A+*n13*BGPA .A=8v*BA h 76Lv4 .,;7 ,tfg7PV C+V*BO *-,X.bU75B*S.AGP0 G 0 )+.P,\1;0;*B`M*-.5z467A1;7 0k*n1R*-,$1;)+7<+A:*-<+7DF*BA+*BD.5dl:7 5-8GPOZ8:7l+A:*n13*BGP A ^ V /A+74 .At13):7A *BA`P7 ,K1;*-CP.137"13)+7W0;75-.13*BGPA+,2)+*B>Fbd761iR77AF1;)+7|f'.0;*n13):DF7613*-4 k$Q?.5BGP*-,13)+7GP02LIGPO .A=8X13):7 M fuCP7GPDF76130;*-4 kQ?.5BGP*-,"13)+7GP02LEGPO(1;)+7|C*B`P7 AN46G `70 V \A>+.0213*-46<+5-.0 c*BO *-,X. U.0;*-, h *GP>U7A,2<:b=,2761XGPO13):¨ 7 `*B7D.A+A,2>+):70;7 1 c.A+8 X P *BO *-, h A:GiAo1;GtbU7a13):7_DF*BA+*BD.5l+75-8HGPO8:7l:A+*n13*BGPAHGPO1;)+7|CP*n`7A ]2Q?.5BGP*-,/C46G `70 c M G 1;)+7A *-,.EQ?.5BGP*S,CP0;GP<:>HG `70 .A+8N):7A=467|G `7 0W1;)+7_l:75S8 c=b> L ^/*B5Bbd7021 Y , G M (x) M
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B* A .ICP7GPDF76130;*-4aOuGP0;DHc+46GPD9*-A:COT0;GPD 1;)+7|>+.03.5B5B7 5ZbU71iR7 7AH4G `P7 0;*BA+CFCP0;GP<:>=,".A=8NQ?.5-G*S, CP0;GP<:>=,GOXl:7 5-876[M137A+,;*BGPA+,c?.A=8p*BA .A¡.0;*n13):DF7613*-4vOTGP0;DHc?4GDF*BA+COu0;GPD l:7 5-8:,oGPO 8%7l+A+*n1;*BGPAVfc")+*-,,2*n13<+.13*BGPAJ4.AHbU7|76[%>:0;7 ,;,27 8o*BAJ.A:G13):70"iW. Lc%`%*-.913):7 5% ¥£6U¤ £@©P\§ £ 5+£6d@£ ¯ → π1 (X) → Gk → 1. 1 → π1 (X) -* ,X.HCP7GPD971;0;*S4.5B5nL46GPA+A:7 41;7 8`.0;*B761LvG `70X.Hl+75-8 *-,|1;)+7t.b=,2GP5B<:1;7 X k 0 Gk Q?.5BGP*-,CP0;GP<:> .A=8 ¯ c1;)+7W,2>+.P467"GPb%1@.*BA:7 8XbL_0;7 CP.038:*BA+C G`P7 0 ¯ 0 Gal(k/k) X X = X ×k k¯ h R V " c + ) 7
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Shξ = HomOX,ξ (Fξ , Gξ ) ⊗OX,ξ HX,ξ ,
Tξ = HomHX,ξ (Fξ ⊗OX,ξ Hξ , Gξ ⊗OX,ξ Hξ ).
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q*BDF*B5S.035nLc0;7>+5-.P46*BA+C bL czi"79GPb%1@.*BAv.Av.A+.5BGPCPGP<+,?5BGPA+C76[:.P41_,27 <:7A=467 O H (∗)h 0 . A+8H13):7 0;7I.0;7ofg4GDFDX<:1;*BA+CMkD.>+, OT03GD 7 .P43)J1;7 0;D *-A 1;Gt13):7946GP0;0;7 ,;>UGPA+8:*BA+C ε (∗) 1;703D *BA V!a$L913):7*-A+8:<+413*n`7)L%>UG13):7 ,;7 ,cP13):7D.> *S,$.At*-,2GPDFGP0;>:)+*-,2D GPA7 .P4@) (∗)h ε GPO13)+7XGP<%1370?OuGP<:01370;D,?.bdG `7PV|qMGEbLH1;)+7 *n`8 7 7DFD.:c *S,?.AJ*-,;GDFGP0;>+):*S,2D GPA ε V H q (X, O(n)) =GP<:0213)cMiR7a)=.A=8:5B7/13):7aCP7 A:70@.54.P,27P V aL.I`.A+*-,2)+*BA:C913):7GP0;7 D GPO\Q/0;G13):7A=8%*-7 4 h f Q/0 r ,277.5-,2G ^/021wMc)+.> V \\\@V c c"):7 G037D wMV ukcz13)+7 ]¤1;)46GP):GPDFGP5BGPCLm`.A:*-,2)+7 , q 0 OTGP0 . /GM761;)+70;*S.AN1;GP>UGP5-GCP*S4.5(,2>=.P467_GPO8:*BDF7A+,;*BGPA *-O VafOV ^/0216wMc:> V ! (OuGP0 n q>n 1;)+798%7 l:A+*n1;*-GPAmGPO T¥t£6:§¤«=VkqMGEiR794.Am>:03GM477 8JbLJ8%7 ,;47A+8:*BA+C*BA+8:<+461;*BGPAGPA V q L q*BA=467 *S,I46GP):7 0;7A1 c*n1I*-,X.J<:G13*B7A1IGPOW.J,2)+7 .O ^0216wMc\)=.>Z[ V \\c F E = O(n ) GP0 V !MV-r cZ,;. LJi*B1;) h 70;A+75 V c")+7F.P,;,2G%46*S.137 85-GA+Co7[%.P4i 1_,27 i <+7A+479*BA=465B<=8%7 ,c(*BA N >+.0219 H q (X, N) → H q (X, E) → H q (X, F) → H q+1 (X, N) → H q+1 (X, E)
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OTGP5B5BG i, 0 r=0 .P,"iR7 5B5'Vq70;037PY ,W5-.1370/.0;CP<+DF7A1*BA q7 y *S,.9`.0;*-.A1GPAH1;)+*-,*-A+8:<+413*n`7_>+0;GMGPO(1;)=.1 ,2*BDX<:5B13.A+7GP<+,25BLI>+0;G `7 ,$] Q X] Q X.A= 8 c"):7 G037Dt , X .A+ 8 apfg*'V 7P¦V c")+7GP0;7D,$wMVwMV-rPcwMVxwV y:c .A+8NwMVxwV !I.bUG `7kV c")+7GP0;7D , Xp.A=8 aiR70;7>:037 467 8:7 89bLI.aA+GPA%]>:0;GjK7 41;*B`7R`703,2*-GAGPO=13):G,27W0;7 ,;<:5n1@,c `M*-ePVOTGP0I>UGP5BL%8:*-,;4,I*BA c\.A=8DFGP0;7ECP7 A:703.5-5nLOuGP0Fq1;7*-A,2>+.P467 ,fu76[%>V YZ]\\\_.A+8 Cr Y\ GPO R.w 46OV.5-,2G Q/ < `_c>+>Vw Mc$w Py gkV c")+70;713GMG+c(13)+71iRGN13):7 GP0;7D,X.0;7 0 7 ,;,27A13*-.5B5nLJ7 <+*n`.5B7A1 V X5-,2G+cdA+G1i*-,K13*BA:Co*-,A+77 8:7 8NOuGP] 0 c"):7GP0;7 D a *BAJ1;)+7X7 .0;5B*B7 0 `703,2*BGPAHbU7 4.<+,27a1;)+7|,2>+.P47 ,Ri"70;7|A:G1W>+0;GjK7 461;*n`7?13)+70;7PV c")+7>+0;GMGOG O c"):7GP0;7 D X^*BA1;)+*-,97 .0;5B*B70,2761;13*BA:Cv<=,27 ,9.A .A=.5nLM1;*S4E>=.134@):*-A:C GPAv.Hq1;7*-Av,2>=.P467 V \A T .0;CP<:DF7A1 cZ.>+>:5B*-7 8J13GG `70;5S.>:>+*BA:CH46GPDF>+.P41|,2713, K 0 , K 00 X 1;)=.1,2*n13<=.13*BGPAcGPA:7R46GPA+,2*S8%70@,(D971;0;*S4$A+7*BCP)MbUGP0;)+GMGM8+, 0 00 GPO 0 00 037 ,2>U7 41;*n`7 5nLc U ,U K ,K .A+89GPA:74@):GMG,27 ,\CP7A:7 03.1;*BA+C_,27 413*BGPA+, 0 .A+8 00 f1 , . . . , fk0 ∈ M(U 0 ) f1 , . . . , fk00 ∈ M(U 00 ) OTGP0t1;)+7JCP*n`7Ap,2)+7 .O GPA 0;7 ,2>d7 413*n`75nLV +0;GPD13):*-,8+.13.:cWGPA:7J>:03GM8:<+47 , M U 0 , U 00 CP7A:7 03.1;*BA+C|,27 413*BGPA=, cPi):7 0;7 *-,\.AFGP>U7AFA+7*BCP)MbUGP0;)+GMGM8IGPO g1 , . . . , gk ∈ M(U ) U K= V " c : ) * , * , % 8 P G + A / 7 % ` * E . R . 2 0 3 1 . A Y ,
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7 ; 0 7 U 0 := X0 ∩ X1 U U = U 0 ∪ U1 U = Xj − {j} OTGP0 V a$LJ1;)+7.bUG `7F8%*-,;4<+,;,2*-GPAZc −1 czi)+70;j7 *S, j = 0, 1 f (X0 ) = IndG Y Y →X .m4L:465B*S44G `709b:03.A=43)+7 8GPA+5nL.1 c$.A=8CP*n`7AbL13):77 A0<=.01;*BGPA m 0 fTi)+070;7 0 y = x *S,13):79GP038%7 0?GO k@VIqM*BDF*B5-.0;5nL −1 cdi):70;7I1;)+7Ib:003.A=43)+7 8m46G `70 G m a f (X1 ) = IndA1 Y1 *-,/CP*B`P7 AmbL n fui):7 0;7 *-,1;)+7IGP038:70/GPO kV_qM*BA=467X1;)+7IG `70;5S.> Y1 → X 1 y1 = x − 1 n b 8:G7 ,?A:G1|DF7761a13):7Fb+03.A+43)5-GM4<+, cziR7F)=. `P791;)=.1 −1 0 *-,?13):7 U 0 = X0 ∩ X1 S f (U ) 1;0;*n`%*-.5 ]2Q?.5BGP*-,46G `70 V c")+7 ,27R*BA=8%<=467 8I46G `7 03,)+. `7"46GPA+A:7 41;7 8I46GPD9>dGPA:7 A1@, 0; G IndG U 1;)=.1E.037H0;7 ,2>U7 41;*B`P7 5nL *BA+8:176[%7 8bL1;)+7H5-7OS1E46G,2713,GPO .A=813):7N*-8:7A13*n1L A , A , 10 46G,2761F46GP0;0;7 ,2>UGPA=8:,_1;Gm1;)+746GPDF>UGPA+7A1I0;7 ,;>U7 413*n`75BL4GA1@0.*BA:*-1A:C V /b+,27 02`7 η0 , η 1 , η 1;)=.1"13):7a*-8:7A13*n1LH46GPDF>UGPA+7A1WGPO * , 6 4 P G A @ 1 . B * + A
7 8 B * o A 3 1 + ) a 7 * % 8
7 A 3 1 n * 1 H L 6 4 P G D9>dGPA:7 A1@, 0 IndG 1 U GPO13):7aG13):70W1iRGF*BA=8:<+467 84G `703,c:bd7 4.<=,27 >=.P,;,27 ,"1;)+0;GP<:CP) V ˜ι η ,η ,η c(<:0;A+*BA+C_13):*S,R.0;GP<+A+8Zci"7/GPb%1@.*BAE1;)+7?8:7 ,2*B037 8 >=.1@43)+*BA+CX0;7 46*B0>U76/1 OTGP0"46GPA=,K1;0;<=413*BA:C 1;)+7 ]2Q?.5BGP*-,|46G `70|GPO i*n13)vCP*n`7Avb:03.A=43)v4L:465B78:7 ,;460;*->%13*BGPA 9 `70/13)+7 G U (a, b, c) .bUG `7$GP>d7AI,;7613, .A+8 c13. h 7$13)+7*BA=8:<+467 8X46G `70@, . + A 8 ci):7 0;7 U0 U1 IndG IndG A 0 V0 A 1 V1 .A+8 .0;7F0;7 ,2>U7 461;*n`75BLmCP*n`7AvbL m .A+8 n c.A+8 V0 → U 0 V1 → U 1 y0 = x y = x−1 i):7037 c V $*-4 h .a>dG*-A1 G `70 GPA913):7*-8:7A1;*n1L1 46GPDF>UGPA:7 A1@,GPO A = hai A = hbi η 1/2 7 .P43)XGPOM130):7 ,27$1iRG*BA=18%<=467 8X46G `703, 13)M<+, *-,.Wi"75B5n]8%7 l:A+7 8_>UGP*BA1GPA_7 .P4@)XGPO%1;)+7 ,27 g(η) 0 *BA+8:<=467 846G `703,c(OTGP0X.AL V c")+7F*BA=8:<+467 8v46G`P7 03,|7 .P43)0;7 ,K130;*-41X1;Go13):7F130;*n`%*-.5 Q g ∈G ]KQa.5-G*S,46G `70GPA1;)+7G `70;5S.> A:G i >=.P,K1;71;GPCP7613):7 0\13):7?46GPD9>dGPA:7 A1@, G U0 = U ∩ U 0 GPOZ13)+7 ,27/130;*n`%*-.54G `703,RbLE*-8:7A1;*-OSL%*BA+C+c%Ou0GP0"7 .P14@) cM1;)+7a46GPDF>dGA+7A1WGPO g∈G IndG V0 0 46GPA1@.*BA+*BA:C i*B1;)p1;)+746GPDF>UGPA+7A1GPO 6 4 P G A 3 1 . B * + A B * : A C ; 1 = ) . 1 U > P G B * A 1 O V Ac" )+7 V g(η) IndG 1 A1 0;7 ,2<+5n1*-,"13):7_8:7 ,2*B0;7 8H46G `70 V V →U
c"):7.bUG`P7$7[%.DF>+5B7bU7 CP*BA+,Zi*n13)I.CP0;GP<:> .A=8_.b+03.A+4@)X4L:465B7"8%7 ,;40;*B>:1;*BGPA c G (a, b, c) .A+8v46GPA=,K130;<+41@,a1;)+7t46G `70 i n * 3 1 v ) 3 1 + ) . _ 1 : b 3 0 . = A 3 4 ) 4 : L 6 4 B 5 t 7 : 8 6 7 ] V → U = P1 − {0, 1, ∞} ,;460;*B>:1;*-GAT V \Av8%GP*BA+Co,2G+cd*B1aC*B`7 ,/1;)+7946G `C70?5-GM4 .5-5nLm*BAm1;7 0;DF,?GPO\7 <=.1;*BGPA=,/G`P7 0/1iRG 1;GP>UGP5-GCP*S4.5(G>d7AJ8%*-,34, .A+8 cU.A=8H*BA=,K1;0;<=413*BGPA+,OuGP0>=.1@43)+*BA+CtGPAH1;)+7_G `70;5-.>V U0 U1 c")<=,t*n1ECP*n`7 ,1;)+7m46G `70.A=.5nLM1;*S4.5B5nL fgA:G1.5BCP7b+03.*-4.5B5BLc,2*BA+47N13):7 Y ,E.0;7NA+G1 Ui U.0;*-, h *G>d7AN,2<:b=,2761@,@k@V
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GPORGP038:70 c.A=8b+03.A=43)4L:465B7E8%7 ,;40;*B>:1;*BGPA V `70 1;)+7t46G `70 C3 = hgi 3 (g, g, g) U0 *-,/CP*n`7 AmbLfgGPA:794G>LJGPO@k 3 .A=8NG`P7 0 *B1a*-,/CP*n`7 AmbL 3 VT^7 0;7Pc y0 = x 0 U1 y1 = x − 1 G `70 c(13)+7tCP7A:7 03.1;GP0 .P413,XbL ci):7 0;7 V a$LQ]XQ]XXc U g g(y ) = ζ y ζ = e2πi/3 1;)+7t46G `i 7 0_4 .AbU7E8%7 ,3460;*BbU7 8.5BCP7b+03.*-4.i 5B5nLc*'V 7P3 VibLv>UGP5nL%A+GP3D9*S.5S,_G `70 d.0;*-, h *$GP>U7A ,2761@,V XA+8*BA13):*S,I>=.021;*S46<:5S.097[%.DF>+5B7Pc\13):*S,F4.A76`7AbU7o8:GPA:7oCP5BGPb+.5B5nLG `7 0 c U bLH13):7F,2*BA:CP5-797 <+.13*BGPA 3 fui):7 0;7 k] V ^/70;7 GPA z = x(x − 1) g(z) = ζ3 z z = y0 f0 (x) c(i)+70;7 *S,|13):7t):GP5BGPDFGP0;>+):*-4Ou<+A+413*BGPAGPA ,2<=43)13)+.1 .A=8 U0 f0 (x) U0 f0 (0) = −1 * N A I . + A 7 B * P C M ) U b P G ; 0 + ) M G M G 8 P G O V 6 7 % [ : > 5 * 6 4 n * 3 1 n 5 L c f03 = x − 1 0 f0 (x) = −1 + 31 x + 19 x2 + · · · x=0 q*BDF*B5S.035nLc GPA c\i):7037 *S,X1;)+7)+GP5BGPDFGP0;>:)+*-4EOu<:A=413*BGPAGPA z = y1 f1 (x) U f1 (x) U ,2<+4@)t13)=.1 .A=8 3 1 ):7 0;7 *BAo.1 f1 (1) = 1 f1 = x 0 f1 (x) = 1 + 13 (x − 1) − 19 (x − 1)2 + · · · A:7*-C)MbdGP0;):GMG%8_GPO V"f /G1;7R1;)=.1OTGP013):*-,`702LI,2*BDF>+5B7W4G `70 c1;)+7RCP5BGPb=.5%7 <+.13*BGPA x=1 4.ANbU7ai03*n12137Am8%G iAJbLo*BA=,2>U7 41;*-GAR V a<:1*BAJCP7A+703.5'c=OuGP0A:GPA:].bU7 5B*-.ANCP0;GP<+>+,c:13):7 CP5BGPb+.5>dGP5nLMA+GPDF*-.57 <+.13*BGPA=,a.0;7IA+G1|.1|.5B5GPb`%*BGP<+,?Ou0;GPD 1;)+795BG%4.5GPA+7 ,cd1;)+GP<:CP) bLN] Q X] Q X 13):76LDX<=,K176[%*-,K1 Vk d\[:.DF>:5B7wMV y:V-rF0;7 <:*-0;7 ,I] Q X] Q X *BAG0@8%70|1;GN>=.P,;,|OT0;GPD 13):7t.A=.5BL13*-47 <+.13*BGPA+, fu5BG%4.5B5BLcGPAXDF76130;*-4GP>U7AI,2<:b=,2761@,@kz13G/.5-C7 b:03.*-47 <+.13*BGPA+,13)+.1.0;7$`.5B*S8_GPAI. d.03*-, h * GP>U7Ao8:7A=,27a,2<+b+,2761 ! V \AH.8+8:*n1;*-GAc%*n1<+,27 ,"*-8%7 .P,RGPO1;GP>dG5-GPCL^*BAo>=.0;1;*-4<:5-.0 c h A:G i5n] 7 8%CP7|GPO(1;)+7|Ou<:A=8+.DF7A1@.5ZCP0;GP<:>c:.A=81;)+7|7[M*-,21;7A=467_GPOGP>U7AN,;7613,W13)+.1G `70;5-.>J.A+8 V \Av5-.1370_,27 413*BGPA+,aGPO\13)+*-,?>+.>d70 cdi"7Ii*B5B5$8%*-,;4<+,;,?13):7 C 1;GPCP7613):70_46G `70?1;)+7F,2>=.47 U >:0;GPb+5B7D
GPOU>U7 0;OTGP0;DF*BA+C_.A=.5BGPCPGP<+,\46GPA=,K130;<+413*BGPA=,G `70\l:75S8:,\G1;)+701;)=.A c*BAGP038:70 C 1;GE<:A=8%7 03,K13.A=8N46G `70@,GPO\.5BCP7b+03.*-4X46<+02`7 ,G `7013):G,27Xl+75-8+,V +GP01;)=.1 c=iR7_i*-5B5,277 1;)=.1GPOS137AF.AF.A+.5BGPCaGPOd9 Q X] Q X 7[M*-,213,M .A+8I1;)=.1\.A=.5BGPC/i*B5B5+>U70;DF*n1\>=.P,;,;.CP7ROu0;GPD .A=.5BL13*-4@|46G `703,R1;GF.5BCP7b+03.*-4?GPA:7 , ! V X 8: * 46<:5B1L1;)=.1")=.P,RA:G1"L761WbU7 7AG `7034GPD97c ):G iR7`70 cz*-,?)+G i 1;Gl+A+8.A=.5BGPC,/GPO\13):79A+G13*BGPA=,?OT0;GPD 1;GP>UGP5-GCL$ bUG13)m037C.038%*BA+C 76[%>:5B*S46*n198:7 ,;40;*B>%13*BGPA=,_GPO"Ou<:A=8:.DF7A13.5$CP0;GP<:>=,_.A=8v037C.038%*BA+CH13):7tA:77 8OuGP0_)=. `M*BA+C G `70;5-.>+>:*BA+C|GP>U7 At,27613,/fui):*-4@)t*BAA+GPA%].0343)+*BDF7 8:7 .A46GPA1;7[1@,$8:G|A+G1$76[%*-,K1R*BAt.|A+GPA%] 1;0;*n`%*-.5i". L+kV A+7Xi". LJ.0;GP<+A+8N1;)+*-,?>+0;GPb:5B7 D *-,/13G46GPA=,2*-8:70?GPA:5BLJ467 0213.*BAm1L%>U7 ,/GPO 46G `703,cOuGP0\i):*-4@)o] Q X] Q X¡.5-GPA:7/,2< F47 ,/fu*¤V 7VPi)+70;71;)+7*-A:OuGP0;DF.13*BGPAEOT0;GPD
1;GP>dGP5BGPCL *-,"A+G10;7 <:*-0;7 8Uk@ V c")+7aA:7[176[:.D9>+5B7|*B5B5B<=,K1303.1;7 ,1;)+*-,V
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³ 761 bd7N.l:A+*n137NCP0;GP<:>c$i*n13)CP7A+703.13GP03, fui):G,27 G g1 , . . . , g r >:0;G%8:<+41?A+77 8mA:G1abU7 kV (761 bd79.,2761aGPO 8%-* ,K13*BA=41|>UGP*-A1@,/*-A 1 S = {ξ , . . . , ξ } 2r c\.A+846GPA+,2*S8%70X13)+7 ]2Q?.5BGP*-,946G `70;1*-A:Cm,;>+.P2r467 i*B1;)b+03.A+4@) P1C G V → U = P1C − S 4L:465B7_8%7 ,3460;*B>:1;*BGPA (g1 , g1−1 , g2 , g2−1 , . . . , gr , gr−1 ), (∗) i*n1;)p0;7 ,2>d7 41t13G.bUGP<+<+761tGPOa5-GGP>=, .1.b+.P,27J>UGP*BA1 V 71 σ ,...,σ ξ ∈ U bU7X13):7946GP0;037 ,2>UGPA=8%*BA+Cb:03.A=43)+7 18m46G `70 2rVIc"):*S,?46G `70?*S,/iR75B5\8%0 7l+A:7 8J,;*BA+47 Y → P1C 1;)+7>+0;GM8:<=41IGPO1;)+77 A130;*B7 ,9GPO *S, c$.A=8*n1F*S,94GA+A+7 4137 8,2*BA=4671;)+77A130;*B7 ,9GPO (∗) 1 CP7A+703.137 V c")+7E46G `7094.AbU7EGPb%1@.*BA+7 8bL. 4<%1;].A=8M]>+.P,K137t46GPA+,K130;<=41;*-GPA (∗) G .P,OuGP5B5BG i,:9?)+GG,2798%*S,ujKGP*BA1|,2*BDF>+5B7Hfg*'V 7PVdA:GPA:],275-OS]*BA1;703,;7 413*BA:CMk>=.1;)=, *BA s1 , . . . , s r c(i)+70;7 bd7CP*BA+,X.1 .A=8v7A=8:,_.1 V c. h 7 > 8:*-,K1;*-A+4194GP>:*B7 ,|GPO 1 c P1C s ξ ξ2j |G| P *BA+8:76[%7 8bL1;j)+775-7DF7A13,X2j−1 GPO > V `7 8%7l+A+7F13):7 1;GP>UGP5-GCLGPA1;)+7t8:*-,uj2GP*BA1I<+A:*-GPACGPO G
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GPA1;)+7 ]¤1;) 1;)+7 ,2746GP>+*B7 ,abLm*-8:7A1;*-OSL%*BA+Co1;)+7F0;*-C)1_)+.A=8m7 8%CP79GPOR. ,25-*n1Kt.5BGPA:C s g 46GP>LHGPO 1 1;G13):7X5B7OT1/)+.A=8H7 8%CP7_GPO13):7 ,25-*n1KI.5BGPA+C GPAH13):7 ]¤1;j)J4GP>LHGO 1 PC sj ggj P fTi*n13)E13):7?GP0;*-7A1@.1;*-GPAo.P,RGPA+7/>+0;GM477 8+,R.5BGPA+CX13):7a,25B*n1@,@k@Vc"):7?0;7 ,;<:5n13*BA+C9,2>=.P467?D.>+C, 1;G 1 *-AH1;)+7_GPb`%*BGP<=,"i". LcU.A+8N. i". LoOT0;GPD *B1*S,"13):7 ]2Q?.5BGP*-,/46G `70;*BA+Ct,2>+.P47|GPO PC S G i n * 3 1 N ) + b 3 0 . + A @ 4 N ) 6 4 % L 4 B 5 X 7 : 8
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. L @ 1 . 7 3 1 N G U b 7 ; 1 + ) 7 + l : b 3 0 F 7 + > ; 0 % G % 8 = < 4 _ 1 P G R O ; 1 + ) 7 Y _ , G ` 70 pri i Y Yi x V k P1R c")+797 .P,2*B7 ,K1X4.P,27F*-,a*-O$1;)+7Fl:75S8 46GPA1@.*-A+,|.>+0;*BDF*n13*n`7 ]¤1;)v0;GMG1|GOR<:A+*n1L V K n ζ c"):7AiR7DF. Lp1@. h 7 1;G bU713):7vl+75-8¡GPb%1@.*BA:7 8pbL¡.P8j2GP*BA:*-A:C .A ]'13)¡0;GMG1HnGPO L n 8%GM7 ,|A:G1a`.A+*-,2).1_.ALm>dGP*BA1|GPO c(.A=8 n−1 cZi)+70;7 f (x)(f (x) − α) f (x) ∈ R[x] S i):7037 V9f =GP0/76[:.DF>:5-7PcU*-O *-,/*BA+l:A+*n137PcUiR7XD. LJ43)+GMGP,;7 α ∈ m − {0} k f (x) = x − c OTGP0,2GPDF7 46GPD9>=.0;9 7 d\[:.D9>+5B7IwMV y%VxwMVxk /76[M1 c,2c<:∈>+>URG,20 7"13)+.1 8:G7 ,A:G1$4GA1@.*BA.a>:03*BDF*n1;*B`P7 ]'13)F0;GMG1GPOd<:A+*n1LIb+<%1\1;)=.1 K n *S,XA+G1I7 <+.5$13GJ1;)+743)=.03.P41;7 0;*-,K13*-4EGPO V c")+7AiR74 .A4GPA+,2*-8:70 c p K K 0 = K[ζn ] .A+8oi*B5B546GPA+,K130;<=41/.A ]4L:465B*-4 |<+D9DF7076[M137A+,2*-GPANGPO 0 i):*S43)J8:7 ,;467A=8+,"13Gt . n K (x) 8%7 ,;*B0;7 8o76[M137A+,2*-GPAHGO V c")+*-,"i*-5B5bU7|8:GPA:7a<=,2*BA+C46GPA=,K1;0;<=413*BGPA+,*BA q5n1 Z13GFl+A+8 K(x) .AF7 5B7DF7A1 ,2<+4@)F1;)=.1\13):776[M1;7 A+,2*BGPA n GPO 8:7 ,;467 A+8+, g(x) ∈ R[ζ , x] y − g(x) R[ζ , x] 1;Gt.A ]4L:465B*-4|76[M1;7A=,2*BGPnAHGPO i):G,27_465BG,27 8Hl:b+0;7a*-,.9D9G%4 h 46G `70 nV n R[x] qM>U7 4*Bl+4 .5B5nLcl+03,K1H,2<:>+>UG,27J13)+.1 *-,G%8:8ZV (761 bd7J13):7GP038%70GPOa13):7v4L:465B*-4 p s CP0;GP<:> ( c i n * 3 1 ) P C 7 : A
7 3 0 . ; 1 P G 0 N V : ) GMG,27 .A+8v5-761 Gal(K 0 /K) τ : ζn 7→ ζnm α ∈ m − {0} c"OuGP0,;GDF7 i):*-4@) 8%GM7 ,A:G1`.A:*S,2)GPA V 761 b = f (x)n − ζn p2 α f (x) ∈ R[x] S d b 7 3 1 : ) 7 ] 4 : L 6 4 B 5 * E 4 + l 7 5 8 6 7 M [ ; 1
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d£J§ 5 §£ 13Gbd7.,2<+b+,271OuG0Ei)+*-43) 13)+70;7 Y ⊂ Max A *-,o.A. 9A+GP*-8.5BCP7b+03. 1;)=.1o0;7>+0;7 ,27A13,E1;)+7mOu<:A=413GP0 A hY : B 7→ {φ : A → *'V 7PYV,2<+4@)13)+.1 V)f c")+*-,_*-,I4.5B5-7 8.A B | φ◦ (Max B) ⊂ Y } 0 hY (B) = Hom(AY , B) d« F§ 5 «¥tTm*BA a Q ` Vxb k X §+£@¤ ( d£|§ 5 § £ *-,.F,;<:b=,2761GPO Y ⊂ Max A 1;)+7aOTGP0;D
Y = {ξ ∈ Max A : |fi (ξ)| ≤ 1 (∀i), |gj (ξ)| ≥ 1 (∀j)}, i):7037 .0;7vl+A:*B1;7Ou.DF*B5-*B7 ,NGPOI75B7DF7A1@,HGPO V f)c")+7 ,27.0374.5B5B7 8 5 (f ), (gj ) A ¬;£6U «¥tiT+§ *BA aW Q ` Vxk c(.1;7,;):G iR7 8 c.:c $0;GP>dG,2*n13*BGPA MVxw _13)+.1J7`702L ,2>U76] 46*-.5.FA:7J,2<+b+,2761t*S,. FA:7Pc`%*BeV1;)=.1t*BO *-,CP*n`7 A bL .P,.bUG `7c$13):7A Y (fi ), (gj ) V N s P G ; 0 7 G ` 7 9 0 B * O *-,F.A. 9A+7 AY = A{fi ; gj−1 } := A{xi ; yj }/(fi − xi , 1 − gj yj ) Y ,2<:b=,2761\GPO c13)+7AI13)+7W4.A+GPA:*-4 .5:D.> *-,\.ab:* j27 413*BGPA c.:c $0;GP>UG] Max A Max AY → Y ,2*n1;*-GPA V y ] V \AmOg.P41 cz*n1a*-,?.E):GPDF7GPDFGP0;>:)+*-,2D c(.:c GP0 VZw91;( G $0;GP>V :V-r cU*BO\iR7ICP*n`7 1;)+713GP>UGP5BGPCL*-AEi):*S43)o.XOT<+A=8:.DF7A13.5z,KL%,21;7D GPOZA+7*BCP)MbUGP0;)+GG%8+,$GPO.X>dGP*BA1 Max A *-,XCP*n`7AbL,2761@,XGPOR13)+7tOuGP0;D OuG0 ξ0 Uε (g1 , . . . , gn ) = {ξ ∈ Max A : |gi (ξ)| < ε c:i):7037 .A=8Ei)+70;7 ,;.1;*S,2OSL V 1 ≤ i ≤ n} ε>0 g ,...,g ∈ A g (ξ ) = 0 c.1;7$8:7l:A+7 8 7 43)_46GP):GPDFGP5BGPCL?OTGP041G `70;*BA+Cn,zGPO%. FA:GP*-8?`.0;i*B76130*B7 , V =(Max A, A) bLIl+A:*B1;75BLID.AL9. FA+7,2<:b=,2761@,c.A+89>+0;G `7 8I)+*- , XW4L:465B*S46*n1L c")+7GP0;7D c.%$c c"):7GP0;7 D OTGP0 )+70;7 *-,1;)+7X>:037 ,2):7 .O1;)=.1?.P,;,2G%46*-.137 ,13G.AL Vw c=13)+.1 i H (V, O) = 0 i > 00 O . 9A+7?,2<:b=,2761"*n1@,". FA+GP*-8.5BCP7b:0@.%c%.A=8 *S,R.Xl:A+*n137?46G `7 0;*BA:C9GO bLE,2>U7 46*-.5z. FA:7 V V ,2<:b=,2761@,V"~f \AOg.P41 c13):*S,)+GP5-8:,\76`7A9i*B1;)t.al:A+*n13746G `7 0;*BA:C_GPO bLF. FA+7,2<:b=,2761@, ,277 V 0 a Q `_c VxwM$c c"):7GP0;7 D r VxI k X/,\._46GPA=,27 <:7A=467PcOuGP0$,2<=43)t.|46G`P7 0;*BA+C GPO .A=8F.AL § V V ]DFGM8:<:5-7 GPO$l:A+*n137X1L%>U7Pc 0 ˜ ) *-,?*-,2GPDFGP0;>+):*-4X13G M c.A+8 H i (V, M ˜) = 0 A M H (V, M OTGP0 c(.:V c c"):7 GP0;7D V ):7 0;7 ˜ *-,a1;)+7F>+0;7 ,2)+7 .O OTGP0 .A Y →M⊗ A Y M 0 . 9A+i7>,2<:0b=,2761_GPO © V c"):7 ,27.0;7.A+.5BGPC,aGPO$13)+79<=,2<=.5\Ou.P4613,aOuGP0?13)+7FA4G)+YGPDFGP5BGPCLmGPO . 9A+7_`.0;*B7613*B7 ,V?VsNGP0;7G `7 0 c+13):76LN*BDF>:5BLH1;)=.1 .A+8 ˜ .037X,;):7 . `7 , V \AJ>=.0;1;*-4<:5-.0 O M a Q `_c VxwMczGP0;GP5B5-.02Lmw ¤cU*BO .CP0;7 7_GPAN7 .P4@)NDF7DXbd70 GPO\.l:A+*n137X. FA:7 § f, g ∈ A U 46G `70;*BA+CGPO c:1;)+7AH1;)+76LH.037|7 <+.5 .A=8o*BOOuGP076`702L iR7X.0;7|CP*ni`7Am.FOT<+A+413*BGPA V i f 0 GPA ci*B1;)o.CP0;77DF7A1@,$GPA13):7/G `70;5S.>+,c1;)+7At1;)+76LD. LbU7 >+.1@43)+7 8M=^*'V 7PV1;)+70;7 i Ui *-,W.FOT<+A+461;*BGPA i):*S43)H0;7 ,K130;*-4613,"13GF7 .P4@) V f ∈A f X/,DF*BCP)1/bd7X76[%>U7 4137 8ZcU*BO *S,/.Am. FA:7XGP>di 7Am,2<:b=,2761/GPO$.A. 9A+GP*-8H`.0;*-761L c U V 1;)+7A1;)+7aD.> *-,W*BAjK7 461;*n`7PV /A:OuGP0213<:A=.1;75BLc%*B1*-,"A+G1,2<:0'jK7 461;*n`7Pc+7PV C+V A → Γ(U, O) bU7 4.<=,27FGPO"43)=.03.PU41370;*-,K13*-4FOu<+A+413*BGPA=,a5-* h 7 c(DF7A1;*-GPA:7 8.1|13):7bd7CP*BA:A+*BA+CHGPO$13):*S, fD ,27 413*BGPAZV\sJGP0;7G `70 c13):7?Ou<:A=413GP0 *-,"Ou.*B1;)+OT<+5'c%b:<:1RA+G1"Ou<:5-5nLtOg.*n13):Ou<:5 c(.:c A 7→ Max A
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7 , cW13):7 W PC[[x]] W◦0 Γ P1C[[x]] Z A:GP0;D.5B*Be .13*BGPAtGO 1 *BA V c"):7b+03.A+4@)95-GM46<=,\GPO t 46GPA=,2*-,K1@,\GPOdl:A+*n1375nL P Z◦ W → P1 DF.ALo*-0;0;7 8:<+46*-b:5B7I46C[[x]] GPDF>dGA+7A13, R V XOT1;70/.t4@)+.A+C7_GPO`.0;*S.b:5B7 ,C[[x]]0 GPA 1 c y = xm y PC((x)) iR7DF. Lv.P,;,;<:DF7F13)=.1X76`702Lvb:03.A=43)46GPDF>UGPA:7 A1X>+.P,;,27 ,a1;)+0;GP<+C)v13):7t465-GP,;7 8>UGP*BA1 cZ.A+8H13)+.1?A+Gtb+03.A=43)m46GPDF>UGPA+7A1/G13):7013)+.A >+.P,;,;7 ,13):0;GP<+CP)m.ALHG13):70 (x, y) (x) >UGP*BA1GPAo13):7X465BG,27 8Hl:b+0;7|GPO 1 V XC.*BAH<+,;*BA:¨ C Xb+)L.A h .0 Y , 7DFD..A=8 $<+0;*n1L PC[[x]] GPI O a03.A=43) (GM46<=,ciR746GPA+465-<+8:7J13)+.11;)+7m0;7 ,K130;*-413*BGPA¡GPO G `70 *-,. W C[y −1 ][[x]] 8%*-,gjKGP*BA1a<:A+*BGPAmGPO$46GPDF>UGPA:7 A1@,/CP*n`7AbL u O P G ? 0 ; , G F D I 7 X D + < n 5 3 1 B * : > 5 7 P G O d c i *n13) wN = x N n 7 .P43)0;7 8:<+47 846GPDF>UGPA+7A1_GPOR1;)+7t45BG,27 8l+b:037FGPO bd7*BA+CJ.N46GPDF>:5B7[v5B*-A:7PVmq*BA=467 W *-,9,2>+5B*n1G `70 c*B19OuGP5B5BG i,I1;)=.1 V c")M<=,_1;)+7>:<+5B5Bb=.P4 h GPO W ◦ → Z◦ y = ∞ N = n G `70 *S,.I1;0;*n`%*-.5(46G `70 V W →Z C[y −1 ][[x]] qM*BA=467_13):7ICP7A+703.5(l+b:037_GPO *-,/CP7GPDF76130;*-4.5B5nLm*B0;0;7 8:<+4*Bb:5-7PcU13):79465BG,27 8 W → P1C[[x]] l:b+0;7E*-,F46GPA+A:7 461;7 8Zc\bL U.0;*-, h *'Y ,tGPA:A+ 7 4137 8:A:7 ,; , c")+7GP0;7D ^/021wM c \\\c$GP0 VRrPrPV y V qG9bLt1;)+7?>+0;76`%*BGP<=,R>+.0@.CP03.>+)Zc13):7|46GPDF>UGPA+7A13,"GPOZ13)+7a465-GP,;7 8l+b:0;7?GPO .5B5zDF7761 W .1F.m,2*BA+CP5B7>dGP*BA19G `70 VqMGJ1;)+7>:<+5B5Bb=.P4 h GPO G `70 (x = y = 0) W∗ W → P1C[[x]]
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7 : 8 + < 4 B * b:5-7 ] W ∗ → Spec C[[x, y]] Γ Q?.5BGP*-,a46G `7 0 V_sNGP0;7 G `70 * ? , * 2 , P G F D P G ; 0 + > : ) * I 4 ; 1 G G ` 7 0 c 2 , B * = A 6 4 I 7 7 .P43) W ∗ /N Spec S Spec R *-,"CP*n`7 AHbL n VqGF*n1*-,.9>:03G>d70,2GP5B<%13*BGPAH1;G91;)+7|CP*n`7AH7DXbU7 8:8:*BA+C9>+0;GPb:5-7DHV =x /G137a13)+.z1/*BAN 1;)+*-,/4.P,27Pc+13)+7|>+0;GMGPO,2)+G i,WDFGP0;7 91;)=.1 *-,W13):7I4L:465B*-4_CP0;GP<+> c G Cn .A+8H13)+.1?G `70 cU13)+7X>+<:5B5-b+.P4 h GPO * , 3 1 ; 0 n * % ` * . " 5 ' f 2 , B * = A 6 4 I 7 ; 1 + ) I 7 ; , . F D X 7 S * , C((y))[[x]] W ∗ → Z∗ 1;0;<+7aG `70 kV −1 C[y ][[x]] \§ £ 1£6d£6¬3 Z@§ £ 71 bU7/1;)+7?b:0@.A=43)5BG%46<=,RGPO ∗ c:.A+8E5B761 bd713):7 B Z → X∗ C 13.A+CP7A146GPA:7/13G .1"13):7|465BG,27 8>dGP*BA1 V c")M<+, *-,W.I<:A+*BGPAGPO(l:A+*n1;7 5nLED.AL B (x, y) C 5B*BA+7 , 1;)+0;GP<:CP) *BA V XOT1;7 0|.43)=.A:CP79GPO\`P.03*-.b:5B7 ,?GPO\13):79OuG03D C (ax + by) (x, y) X∗ c:iR7aD. Lo.P,;,2<+D97a13)+.1 8%GM7 ,"A+G146GPA13.*-Ao1;)+7a5-GM4<+,WGPO V y 0 = y − cx C (y = 0) (761 V 761 bd713):7\76[:467 >%13*BGPA+.5 ˜ bU713):7\b+5BG iW]¤<+>|GO X ∗ .11;)+7$465BG,27 8|>UGP*BA1 (x, y)
E X 8%*n`%*-,2GP0 1;)+*-,H*-,H. 46GP>LpGPO 1 ci*n13) >=.0@.DF761370 V 761 bU71;)+7 PC t = y/x τ ∈ T 0 465BG,27 8 >dGP*BA1 1;)+*-,E*S,i)+70;7 DF7713,t13):7m>:03G>d70t1;03.A=,2OuG03D (x = y = t = 0) 0 E GPO V 71 ˜ ˜ bU7H13)+7NA+GP0;D.5-*Be7 8 >+<:5B5Bb=.P4 h GO Z ∗ → X ∗ V a$L13):7 (y = 0) Z → X >:0;7`M*BGP<=,|>=.03.CP03.>:)c1;)+*-,X*-,_<+A+03.D9*-l:7 8*BA.NA+7*BCP)MbUGP0;)+GG%8vGPO 76[:467>:1X>UG,;,2*Bb+5nL τ .5BGPA:C V$qMGIG `70"1;)+7|46GPD9>+5B76137a5BG%4.5z0;*-A:C ˆ GO *BA ˜ c%1;)+7?>+<:5-5Bb+.P4 h X = C[[x, t]] τ OX,τ E ˜ ˆ ˜ GPO Z˜ → X ˜ ∗ := Spec O ˜ *S,t03.DF*Bl+7 8GPA+5nL G `70 (x = 0) V 7Ni*B5-5 Z˜ ∗ → X 46GPA+,K130;<=41. ]2Q?.5BGP*-,X,τ 46G `70 ˜ ˜ 8%GPDF*BA=.1;*-A:C Z˜ V f¤qM77 *BCP<+0;7 , !MV y:V-r .A+8 W → X Γ !MV y:V-rPrPVxk (761 bU7J.46GPA:A+7 4137 846GPDF>UGPA+7A1tGPO ˜ ∗ V c")M<+, ˜ ∗ ˜ ∗ *-,Qa.5BGP*-,i*n13) Z˜0∗ Z Z0 → X CP0;GP<:> c$.A+8 ˜ ∗ V 6 7 1 U b E 7 ; 1 + ) o 7 2 , + < b+C03G<+>C7 A:70@.137 8 G ⊂ G Γ0 ⊂ Γ Z˜ ∗ Z = IndG bL .A+08 fg*-8:7A1;*BOTL%*BA:C iG*B1;0) 0 c.A+8 i*B1;) k@Q V c")M<+, N G0 N N o1 ⊂ Γ G 1oG ⊂ Γ V a$LmR.P,27trPc=1;)+70;7X*-,/.0;7 CP<:5-.0*-0;0;7 8:<+46*-b:5B7XA+GP0;D.5 ]2Q?.5BGP*-,/46G `70 Γ0 = N o G 0 Γ0 ˜∗ → X ˜ ∗ → Z˜ ∗ G `70 ˜ ∗ 13)+.198:GPDF*BA+.137 , Z˜ ∗ c.A+8,2<=43)13)+.1I1;)+7E>+<:5B5-b+.P4 h GPO W W 0 0 * F , ; 1 ; 0 B * M ` * . ¤ 5 V " c + ) . 1 * , c * 9 , 130)+7130;*n`%0*-.5 ] ˜ 0 = Spec C((t))[[x]] ˜ 0 := W ˜ ∗ × ˜∗ X ˜0 N X W 0 0 X Q?.5BGP*-,46G `70GPO ˜ 0 c . = A 8 3 1 : ) 7 2 ] ? Q . B 5 P G * , 6 4 G ` 7 0 S * ? , K j = < K , 1 ˜0 ˜0 → X ˜0 Γ0 Z := Z˜ ∗ × ˜ ∗ X W 0 V c")M<+,1;)+07 ]KQ?.5B0GP*S,$X 46G `70 ˜ ∗ = ) P . , ; 1 + ) W 7 + > ; 0 P G U > 70;1LI1;)=.1 Γ0 ˜ 0 t Γ ˜ ∗ ∗ ˜ IndG0 Z0 Γ W := IndΓ0 W0 → X *n13,R>+<:5-5Bb+.P4 h ˜ 0 * ( , K j = < K , 1 c i : ) 7 3 0 7 ˜ ∗ × ˜∗ X ˜0 ˜0 IndΓG0 Z˜00 = IndΓG Z˜ 0 W := W Z˜ 0 = IndG G0 Z 0 X *-,R13)+7a>+<:5-5Bb+.P4 h ˜ ∗ 0V ˜ Z ×X˜ ∗ X (761 c.A=85B761 bU713):7H46GPDF>+5B7613*BGPAGPO ˜ .5BGPA+C *'V 7PV 0 X U = E − {τ } X0 U0 X = cUi):7037 V (761 0 z c . = A N 8 B 5 6 7 1 V 0 0 ˜ Spec C[s][[y]] s = x/y = 1/t Z = Z ×X˜ X W = IndΓG Z 0 c")<=,W1;)+7_>+<:5B5-b+.P4 h 0
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Galois Groups and Fundamental Groups MSRI Publications Volume 41, 2003
Constructive Dierential Galois Theory B. HEINRICH MATZAT AND MARIUS VAN DER PUT
Abstract.
We survey some constructive aspects of dierential Galois theory and indicate some analogies between ordinary Galois theory and dierential Galois theory in characteristic zero and nonzero.
Contents
Introduction Classical Theory 1. 2. 3. 4.
Linear Dierential Equations PicardVessiot Extensions Monodromy and the RiemannHilbert Problem The Constructive Inverse Problem
Modular Theory 5. 6. 7. 8.
Iterative Dierential Modules and Equations Iterative PicardVessiot Theory Local Iterative Dierential Modules Global Iterative Dierential Modules
427 431 436 440 446 451 455 461
References
INTRODUCTION The aim of this article is to survey some constructive aspects of dierential Galois theory and to indicate some analogies between ordinary Galois theory and dierential Galois theory in characteristic zero and nonzero. We hope it may serve as an appetizer for people who work in ordinary Galois theory but are not familiar with the dierential analogue. In the rst part we start with a constructive foundation of the PicardVessiot theory in characteristic zero mimicking Kronecker's construction of root elds. 425
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B. HEINRICH MATZAT AND MARIUS VAN DER PUT
This leads to a smallest dierential eld extension (with no new constants) containing a full system of solutions of a (system of) linear dierential equation(s) with a linear algebraic group as dierential Galois group. Then we explain the Galois correspondence between the intermediate dierential elds of a Picard Vessiot extension and the Zariski closed subgroups of the dierential Galois group. On the way we deal with the question of solvability by elementary functions, comparable to the question of solvability by radicals in ordinary Galois theory. In Chapter 3 we describe the link between the dierential Galois group and the monodromy group over the complex numbers generalizing the eective version of Riemann's existence theorem used in (ordinary) inverse Galois theory [MM]. Further we recall the solution of the inverse dierential Galois problem over C in the case of monodromy groups (RiemannHilbert problem) given by Plemelj (1908) and its completion by Tretko and Tretko [TT] for dierential Galois groups. Finally in Chapter 4 we outline the constructive solution of the inverse problem for connected groups over general algebraically closed elds of characteristic 0 recently given by Mitschi and Singer [MS]. In the second part we develop a PicardVessiot theory in positive characteristic. For this purpose ordinary derivations these cause new constants in any nonalgebraic extension are replaced by a family of higher derivations, called iterative derivations in the original paper of Hasse and Schmidt [HS]. They have already been used earlier by Okugawa [Oku] to outline a PicardVessiot theory in characteristic p > 0. Here we follow a new approach developed in [MP] based on the study of iterative dierential modules (ID-modules) and corresponding projective systems. This allows us to construct (iterative) PicardVessiot extensions in the same formal way as in characteristic 0. We again obtain as IDGalois groups reduced linear algebraic groups dened over the eld of constants and we establish a Galois correspondence between the intermediate ID-elds of a PicardVessiot extension and the reduced closed subgroups of the corresponding ID-Galois group. In Chapter 7 we determine the structure of ID-modules and ID-Galois groups over local elds these are trigonalizable extensions of connected solvable groups by nite local Galois groups and solve the inverse problem for these groups. Finally in Chapter 8 we solve the inverse problem of dierential Galois theory over global elds of positive characteristic and prove an analogue of the Abhyankar conjecture for dierential Galois extensions. The main sources (sometimes used without a reference) are the introductory texts of Magid [Mag] and the second author [Put2] for the classical part, for the modular part there are the research paper [MP] combined with the notes [Mat]. Dierent approaches for dierential equations in positive characteristic have been developed, for example, by Katz [Kat2] and André [And].
CONSTRUCTIVE DIFFERENTIAL GALOIS THEORY
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Acknowledgement. The authors thank the Mathematical Science Research Institute in Berkeley (MSRI) for its hospitality and support during the research program Galois Groups and Fundamental Groups, and for giving us the opportunity to present most of the results given in this article in a series of lectures at the MSRI. Among other things, the solution of the inverse problem of ID-Galois theory for connected groups over global elds (Theorem 8.4) and the proof of the connected dierential Abhyankar conjecture (Corollary 8.5) have been achieved during our stay in Berkeley.
CLASSICAL THEORY 1. Linear Dierential Equations 1.1. Derivations. In this rst section we collect some well-known facts on derivations and dierential rings. The proofs can be found, for example, in [Jac], Chapter 8.15. Let R be a commutative ring (always with unit element). A map ∂ : R → R is called a derivation of R if
∂(a + b) = ∂(a) + ∂(b) and ∂(a · b) = ∂(a)b + a∂(b) for all a, b ∈ R. An element c ∈ R with ∂(c) = 0 is a dierential constant. The set of dierential constants forms a ring denoted here by C(R). Further a ring R together with a derivation ∂ of R is called a dierential ring (D-ring) (R, ∂). From the denition we immediately obtain the formulas ³a´ 1 ∂ = 2 (∂(a)b − a∂(b)) in case b ∈ R× , (11) b b X µk ¶ ∂ k (ab) = ∂ i (a)∂ j (b) (12) i i+j=k
for a, b ∈ R and i, j, k ∈ N . Now let (R, ∂R ) and (S, ∂S ) be two D-rings. Then a ring homomorphism ϕ ∈ Hom(R, S) is called a dierential homomorphism (D-homomorphism) if ϕ ◦ ∂R = ∂S ◦ ϕ. The set of all D-homomorphisms is denoted by HomD (R, S). An ideal A of R with ∂R (A) ⊆ A is called a dierential ideal (D-ideal). It can be shown that in case R is a Ritt algebra, i.e., Q ≤ R, the nil radical of any D-ideal again is a D-ideal. A corresponding statement does not hold anymore in positive characteristic (see [Kap], I.4). If (R, ∂R ) is a D-ring and S ⊆ R a multiplicatively closed subset with 0 ∈ /S −1 −1 we have a canonical map λS : R → S R from R into the quotient ring S R. Then by (11) there exists a uniquely determined derivation ∂S −1 R of S −1 R such that ∂S −1 R ◦ λS = λS ◦ ∂R . In particular, if R is an integral domain, ∂R can be extended uniquely to its quotient eld F = Quot(R). A eld F with derivation ∂F is called a dierential eld (D-eld).
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Finally, let E/F be a nitely generated separable eld extension of a D-eld (F, ∂F ) with separating transcendence basis x1 , . . . , xr . Then for all y1 , . . . , yr ∈ E there exists exactly one extension ∂E of ∂F on E with ∂E (xi ) = yi for all i. In particular, an extension of ∂F to a separably algebraic eld extension E/F always exists and is unique.
1.2. Linear dierential operators. From now on, (F, ∂F ) denotes a D-eld P n
of characteristic 0. Then ` := k=0 ak ∂ k with ak ∈ F and an 6= 0 is called a linear dierential operator of degree deg(`) = n over F (D-operator) and F [∂] is the (noncommutative) ring of linear dierential operators over F . Now let (E, ∂E ) be a D-eld extension of F . Then an element y ∈ E is called a solution of ` if y is a solution of the homogeneous linear dierential equation
`(y) =
n X
ak ∂ k (y) = 0.
(13)
k=0
The set of all solutions of ` in E forms a vector space over the eld of constants C(E) of E and is named the solution space VE (`) of ` in E .
Let (F, ∂F ) be a D-eld of characteristic 0 and ` ∈ F [∂] a D-operator . Then for all D-eld extensions (E, ∂E ) ≥ (F, ∂F ) the solution space VE (`) of ` is a vector space over C(E) with dimC(E) (VE (`)) ≤ deg(`). Proposition 1.1.
The proof of Proposition 1.1 relies on the fact that the Wronskian determinant
wr(y1 , . . . , yn ) := det(∂ i−1 (yj ))ni,j=1
(14)
of linearly independent elements yj ∈ E over C(E) is dierent from zero (see [Mag], Theorem 2.9). In the special case of equality in Proposition 1.1, VE (`) is called a complete solution space. The rst fundamental question now concerns the existence of a D-eld extension E/F such that VE (`) is a complete solution space. However, before answering this question we want to study some preliminary examples and to introduce a slightly more general setting. For the examples let F = C(t) be the eld of rational functions over the complex numbers C with derivation ∂ = ∂t := d/dt and E ≥ F the eld of analytic functions. Take ` = ∂ 1 − a ∈ F [∂] with a ∈ C × . Then `(y) = 0 if ∂(y) = ay . Therefore the solution space is given by VE (`) = C · exp(at) and every nontrivial solution is transcendental over E .
Example 1.2.1.
1 In the case ` = ∂ 1 − nt with n ∈ N any solution of ` in E √ n belongs to VE (`) = C t and therefore is algebraic over F .
Example 1.2.2.
Example 1.2.3. A solution of the inhomogeneous dierential equation ∂(y) = f ∈ F × is also a solution of the degree 2 homogeneous dierential equation `(y) = ∂ 2 (y) − f −1 ∂(f )∂(y) = 0. The solution space of the latter consists of
CONSTRUCTIVE DIFFERENTIAL GALOIS THEORY
429
R VE (`) = C ⊕ Cg where g = f dt denotes a solution of the inhomogeneous equation. This may be an element of F as for f = 1 or transcendental over F as for f = 1t . These examples show that solutions and solution spaces of linear dierential equations may algebraically behave very dierently.
1.3. Systems of linear dierential equations. Any solution y ∈ E of ` ∈ F [∂] leads to a solution y = (y, ∂ 1 (y), . . . , ∂ n−1 (y))tr ∈ E n of the matrix dierential equation
∂(y) = A` y,
0 ... where A` = ... 0 −a0
1 0 .. .. . . .. .. . . ··· ··· ··· ···
··· .. . .. . 0 −an−2
0 .. . 0 1 −an−1
∈ F n×n ,
and vice versa. Now we start with an arbitrary A ∈ F n×n and dene the solution space of A to be
VE (A) := {y ∈ E n | ∂(y) = Ay}. This again is a vector space over the constant eld of E of dimension less than or equal to n. Two matrices A and B ∈ F n×n are called dierentially equivalent, or D-equivalent, if every solution z ∈ VE (B) can be transformed into a solution y ∈ VE (A) by a matrix C ∈ GLn (F ), i.e., if VE (A) = CVE (B). The latter is equivalent to the matrix identity B = C −1 AC − C −1 ∂(C). Assume for a moment that A ∈ F n×n admits a complete solution space over some D-eld extension E ≥ F , i.e., there exists a matrix Y ∈ GLn (E) with ∂E (Y ) = AY . Such a matrix is called a fundamental solution matrix of the system of dierential equations ∂(y) = Ay over E . If Y, Y˜ ∈ GLn (E) are two fundamental solution matrices for the same A, then it is easy to verify that these can only dier by a matrix C ∈ GLn (C(E)), i.e., Y˜ = Y C . Using this information, one obtains the following partial converse of the statement above.
Let (F, ∂) be a nontrivial D-eld of characteristic 0 and A ∈ F n×n . Assume that there exists a D-eld extension E/F such that the matrix dierential equation dened by A has a complete solution space over E . Then A is D-equivalent to a matrix A` ∈ F n×n dened by a linear dierential operator ` ∈ F [∂].
Proposition 1.2.
A proof of Proposition 1.2 is presented in [Kat1]. In Section 2.1 we will see that the assumption on the existence of a fundamental solution matrix over some extension eld is superuous.
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B. HEINRICH MATZAT AND MARIUS VAN DER PUT
1.4. Dierential modules. Another very common way to describe linear dierential equations are dierential modules. A dierential module or D-module for short is a module M over a D-ring (R, ∂R ) together with a map ∂M : M → M with the properties
∂M (x + y) = ∂M (x) + ∂M (y) and
∂M (ax) = ∂R (a)x + a∂M (x)
(15)
for x, y ∈ M and a ∈ R. The solution space of M is dened by
V (M ) = {x ∈ M | ∂M (x) = 0}. ∼ V (M ) ⊗C(R) R. In case (M, ∂M ) and M is called a trivial D-module if M = (N, ∂N ) are two D-modules over R, an element ϕ ∈ HomR (M, N ) is called a dierential homomorphism (D-homomorphism) if ϕ ◦ ∂M = ∂N ◦ ϕ. Obviously the D-modules over R together with the D-homomorphisms form an abelian category denoted by DModR . Now assume that R is a D-eld F with eld of constants K . Then it is easy to verify that DModF with the tensor product over F becomes a tensor category over K . Here the tensor product M ⊗F N is provided with the derivation
∂M ⊗N (x ⊗ y) = ∂M (x) ⊗ y + x ⊗ ∂N (y)
(16)
and the dual vector space M ∗ = Hom(M, F ) with
(∂M ∗ (f ))(x) = ∂F (f (x)) − f (∂M (x))
(17)
for x ∈ M, y ∈ N and f ∈ M ∗ . Then (F, ∂F ) is the unit element of DModF with EndDModF (F, ∂F ) = K . If in addition K is algebraically closed, DModF even forms a Tannakian category using the forgetful functor
Ω : DModF → VectF ,
(M, ∂M ) 7→ M
from the category DModF into the category of vector spaces over F (see [Del]). However, this will not be used in the sequel. The link between D-modules and systems of linear dierential equations is L given in the following way. Let M = i=1 bi F be a nite-dimensional Dmodule over F with basis {b1 , . . . , bn }. Then by (15) the action of ∂ is uniquely determined by n X ∂M (bj ) = bi aij with aij ∈ F. (18) i=1
Pn Thus for i=1 bi yi = By ∈ M with B = (b1 , . . . , bn ) and y = (y1 , . . . , yn )tr ∈ F n the two statements By ∈ V (M ) and ∂F (y) = −Ay where A = (aij ) ∈ F n×n are equivalent because of
∂M (By) = ∂M (B)y + B∂F (y) = B(Ay + ∂F (y)).
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Therefore a D-module M with representing matrix A ∈ F n×n of ∂M leads to a system of linear dierential equations over F with matrix −A. In particular, the solution space V (M ) of M coincides with V (A) and thus is a vector space over K with dimK (V (M )) ≤ dimF (M ).
2. PicardVessiot Extensions 2.1. PicardVessiot rings and elds. Now we are coming back to the
questions raised in Section 1.2: For a linear dierential equation ∂(y) = Ay over a D-eld F of characteristic 0 with (algebraically closed) eld of constants K , does there always exist a D-eld E with dimK (V (M ⊗F E)) = dimE (M ⊗F E)? (The latter number equals dimF (M ).) For this purpose we dene a Picard Vessiot ring (PV-ring) R for A to be a dierential ring (R, ∂R ) ≥ (F, ∂F ) with the following properties: (21) R is a simple D-ring, i.e., R only contains trivial D-ideals. (22) There exists a fundamental solution matrix over R, i.e., there exists a Y ∈ GLn (R) such that ∂R (Y ) = A · Y . (23) R is generated over F by the coecients yij of Y = (yij )ni,j=1 and det(Y )−1 . It is easy to verify that a nitely generated simple D-ring is always an integral domain and that R and even Quot(R) do not contain new constants. The next proposition is basic for all that follows.
Let (F, ∂F ) be a D-eld with algebraically closed eld of constants K of characteristic 0 and A ∈ F n×n . Then for the dierential equation ∂(y) = Ay there exists a PicardVessiot ring (R, ∂R ) over F and it is unique up to D-isomorphism . Proposition 2.1.
The construction of R is similar to Kronecker's construction of root elds in the case of polynomial equations. Let X = (xij )ni,j=1 be a matrix with over F algebraically independent elements xij . Then by Section 1.1 we can extend P ∂F uniquely to F [xij ]ni,j=1 by ∂U (X) = A · X , i.e., ∂U (xij ) = k=1 aik xkj , and to U := F [GLn ] = F [xij , det(xij )−1 ]ni,j=1 . Then (U, ∂U ) is a D-ring over F . By Zorn's Lemma there exists a maximal D-ideal P £U . The quotient R := U/P is a simple D-ring containing a fundamental solution matrix Y := κP (X), where κP denotes the canonical map κP : U → R = U/P . Obviously, R is generated over F by the coecients yij of Y and by det(Y )−1 such that by denition R is a Picard Vessiot ring. It nally remains to be checked that two PV-rings belonging to the same matrix A are D-isomorphic. This can be done by elementary computations (see [Put2], Proposition 3.4). The quotient eld E := Quot(R) of a PV-ring is called a PicardVessiot eld for A. It can be characterized without using R.
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B. HEINRICH MATZAT AND MARIUS VAN DER PUT
Let F and A ∈ F n×n be as in Proposition 2.1 and let (E, ∂E ) ≥ (F, ∂F ) be a D-eld extension . Then E/F is a PicardVessiot extension for A if and only if Proposition 2.2.
(a) the constant elds of E and F coincide , (b) there exists a Y ∈ GLn (E) with ∂E (Y ) = A · Y , (c) E is generated over F by the coecients yij of Y . A proof is given in [Put2], Proposition 3.5. These characterizing properties correspond to the classical denition of PV-elds (compare [Kap], III.11 and [Mag], Denition 3.2).
2.2. The dierential Galois group. As before, let R be a PV-ring and
E = Quot(R) a PV-eld over a D-eld F of characteristic 0 with algebraically closed eld of constants. Then an automorphism γ of R/F or E/F , respectively, is called a dierential automorphism (D-automorphism) if ∂◦γ = γ◦∂ . The group of all D-automorphisms is called the dierential Galois group (D-Galois group) of R/F or E/F , respectively, and is denoted by GalD (R/F ) = GalD (E/F ). Since GalD (E/F ) acts faithfully on the solution space VE (A), it is a subgroup of GLn (K). It can be characterized in the following way.
Let F be a D-eld of characteristic 0 with algebraically closed eld of constants and let R/F be a PV-ring for A ∈ F n×n with fundamental solution matrix Y = (yij ) ∈ GLn (R). Then
Proposition 2.3.
GalD (R/F ) = {C ∈ GLn (K) | q(Y · C) = 0
for all q ∈ P }
where P denotes the annulator ideal P = {q ∈ F [GLn ] | q(yij ) = 0}. A proof can be found for example in [Mag], Corollary 4.10. Since P is nitely generated, GalD (R/F ) consists of the K -rational points of a Zariski closed subgroup of GLn (K) ([Eis], Section 15.10.1) and therefore of a reduced linear algebraic group G over K . This already proves the rst part of the next proposition.
Let F be a D-eld of characteristic 0 with algebraically closed eld of constants K and E/F a PV-extension . Then there exists a reduced linear algebraic group G over K with GalD (E/F ) ∼ = G(K). In addition the xed eld E G(K) coincides with F . Proposition 2.4.
The last statement follows from the fact that for each z ∈ E \F a γ ∈ GalD (E/F ) can be constructed that moves z (see [Put2], Proposition 3.6). Now we return to our examples in Section 1.2. Again (F, ∂) denotes the D-eld (C(t), ∂t ). Let ` = ∂ − a ∈ F [∂] with a ∈ C× . Then by Example 1.2.1 the PV-eld for ` is given by E = F (y) and VE (`) = Cy for y = exp(at). The DGalois group GalD (E/F ) equals Gm (C) = GL1 (C) since any c ∈ GL1 (C) = C × denes a D-automorphism because of ∂(cy) = c∂(y). Example 2.2.1.
CONSTRUCTIVE DIFFERENTIAL GALOIS THEORY 1 In the case ` = ∂ − nt we obtain E = F (y) for y = GalD (E/F ) = Cn is the cyclic group of order n.
Example 2.2.2.
433
√ n
t and
For ` = ∂ 2 + 1t ∂ the PV-eld E is F (y) with y = log(t) and VE (`) = C ⊕ Cy . Because of (γ ◦ ∂)(y) = 1t = ∂(y), for any γ ∈ GalD (E/F ) there exists a c ∈ C with γ(y) = y + c. This proves GalD (E/F ) = Ga (C) = C . Example 2.2.3.
2.3. Torsors and Kolchin's Theorem. In order to prove a Galois corre-
spondence between the intermediate D-elds of a PV-extension E/F and the Zariski closed subgroups of GalD (E/F ) = G(K) we need a structural theorem due to Kolchin which shows that after a nite eld extension F˜ /F a dening PV-ring R inside E becomes isomorphic to the coordinate ring of GF˜ = G ×K F˜ , i.e., R ⊗F F˜ ∼ = F˜ [GF ]. This is a consequence of the fact that the ane scheme X = Spec(R) over F is a GF -torsor or a principal homogeneous space for GF , respectively. This means that GF acts on X via
Γ : X ×F G F → X ,
(x, g) 7→ x · g
(24)
and in addition
Id ×Γ : X ×F GF → X ×F X ,
(x, g) 7→ (x, x · g)
(25)
is an isomorphism of ane schemes over F (see [Put2], Section 6.2). Such a torsor X is called a trivial GF -torsor if X ∼ = GF where the action is given by the multiplication. The latter is equivalent to X (F ) 6= ∅ where as usual X (F ) denotes the set of F -rational points of X .
Let F be a D-eld of characteristic 0 with algebraically closed eld of constants , A ∈ F n×n and R a PV-ring for A over F . Further let G denote the reduced linear algebraic group over K with G(K) = GalD (R/F ) and GF := G ×K F . Then Spec(R) is a GF -torsor .
Theorem 2.5 (D-Torsor Theorem).
For the proof see for example [Put2], Section 6.2. Since the GF -torsor Spec(R) becomes trivial after a nite eld extension F˜ /F , the following version of Kolchin's theorem is an immediate consequence of the D-Torsor Theorem. Corollary 2.6 (Kolchin).
and setting X := Spec(R):
With the same assumptions as in Theorem 2.5,
(a) There exists a nite eld extension F˜ /F with X ×F F˜ ∼ = GF ×F F˜ . (b) X is smooth and connected over F . (c) The degree of transcendence of Quot(R)/F equals dim(G) (over K ).
2.4. The dierential Galois correspondence. Now we are ready to explain the dierential Galois correspondence. This can be stated as follows:
Let F be a D-eld of characteristic 0 with algebraically closed eld of constants K , A ∈ F n×n and E a PV-extension for A. Denote by G the reduced linear algebraic group over K with G(K) = GalD (E/F ). Then :
Theorem 2.7 (D-Galois Correspondence).
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(a) There exists an anti-isomorphism between the lattices
H := {H(K) | H(K) ≤ G(K) closed} and L := {L | F ≤ L ≤ E D-eld}
given by Ψ : H → L, H(K) 7→ E H(K) and Ψ−1 : L → H, L 7→ GalD (E/L). (b) If thereby H(K) is a normal subgroup , then L := E H(K) is a PV-extension of F with GalD (L/F ) ∼ = G(K)/H(K). 0 (c) Denote by G 0 the identity component of G and F 0 := E G (K) . Then F 0 /F is a nite Galois extension with Galois group GalD (F 0 /F ) ∼ = G(K)/G 0 (K). Besides Proposition 2.4, for (a) we have to use that for all Zariski closed subgroups H G the xed eld E H(K) is dierent from F . For the proof of this fact as well as for the proof of (b) Kolchin's theorem has to be used (compare [Put2], Section 6.3). As an application, we obtain a result comparable to the classical solution of polynomial equations by radicals. To this end we dene a PV-extension E/F to be a Liouvillean extension if it contains a tower of intermediate D-elds
F = F0 ≤ F1 ≤ . . . ≤ Fn = E
with Fi = Fi−1 (yi )
i) and ∂(y ∈ Fi−1 or ∂(yi ) ∈ Fi−1 or yi is algebraic over Fi−1 . Further a linear yi algebraic group G is called virtually solvable or solvable-by-nite if the connected component G 0 is a solvable group. Since in this case the composition factors of G 0 are isomorphic either to Gm or to Ga and D-Galois extensions of this type can be generated by solutions of ∂(y) = f y or ∂(y) = f with f ∈ F we nd from Theorem 2.7:
Corollary 2.8. A PV-extension E/F is Liouvillean if and only if its D-Galois group is virtually solvable .
For a more complete proof and further applications concerning integration in nite terms see for example [Mag], Chapter 6. As in the polynomial case, linear dierential equations with non (virtually) solvable Galois groups exist. We want to verify this statement with the Airy equation. For this purpose we rst explain an analogue of the square-discriminant criterion in ordinary Galois theory which is useful to reduce D-Galois group considerations to unimodular groups.
Let F be a D-eld of characteristic 0 with algebraically closed P eld of constants K , ` = nk=0 ak ∂ k ∈ F [∂] a monic dierential operator and E/F a PV-extension dened by ` or A` , respectively . Then the linear dierential equation over F ∂(w) + an−1 w = 0 (26) Proposition 2.9.
has a solution w in E with the properties (a) F (w)/F is a PV-extension with GalD (F (w)/F ) ≤ G m (K),
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(b) GalD (E/F (w)) ∼ = GalD (E/F ) ∩ SLn (K). For the proof let {y1 , . . . , yn } denote a K -basis of VE (`). Then any y ∈ VE (`) satisies `(y) = wr(y1 , . . . , yn , y) · wr(y1 , . . . , yn )−1 . (27) In particular, for the rst derivative of the Wronskian determinant w := wr(y1 , . . . , yn ) we obtain equation (26). Now any γ ∈ GalD (E/F ) acts on the fundamental solution matrix Y = (∂ i−1 (yj ))ni,j=1 of E/F via γ(Y ) = Y Cγ with Cγ ∈ GLn (K) and on w via γ(w) = w det(Cγ ). Hence w is left invariant by γ if and only if det(Cγ ) = 1. With the help of Proposition 2.9 we are able to compute the Galois group of the Airy equation. By Corollary 3.2 below the Airy equation ∂ 2 (y) = ty has no algebraic solution over the D-eld (F, ∂F ) = (C(t), ∂t ). Hence by Proposition 2.9 its Galois group G = G(C) is a connected closed subgroup of SL2 (C). In case G 6= SL2 (C) the linear algebraic group G would be reducible and VE (`) would contain a G-invariant line Cy . But then z := ∂(y)y −1 would be invariant under G and therefore belong to F . Obviously no element z of F = C(t) satisies Example 2.4.1.
∂(z) = ∂ 2 (y)y −1 − ∂(y)2 y −2 = t − z 2 . as can be seen from the reduced expression of z as a quotient of polynomials.
2.5. Characterization of PV-rings and PV-elds. The theorem of Kolchin
allows us to characterize the PV-ring R inside Quot(R).
Let F be a D-eld of characteristic 0 with algebraically closed eld of constants K and R a PV-ring over F with quotient eld E and Galois group G := GalD (R/F ) = G(K). Then for z ∈ E are equivalent :
Proposition 2.10.
(a )
z ∈ R,
(b )
dimK (KhGzi) < ∞,
(c )
dimF (F h∂ k (z)ik∈N ) < ∞.
Here KhGzi denotes the K -vector space generated by the G-orbit of z and F h∂ k (z)ik∈N is the F -vector space generated by all derivatives ∂ k (z) of z . The critical step is the one from (a) to (b). By the D-Torsor Theorem we may, after a nite extension, assume R = F [G]. Then the result follows from the fact that the action of G(F ) on F [G] is locally nite, i.e., F [G] is a union of nite-dimensional G-stable subspaces ([Spr], Proposition 2.3.6). It is quite natural to call an element z ∈ E with property (c) in Proposition 2.10 dierentially nite (D-nite). For such an element there exists, by denition, a nonconstant linear dierential operator `z ∈ F [∂] monic of minimum degree with `z (z) = 0. We call `z a minimal dierential operator of z . Given a basis z1 , . . . , zn of KhGzi, it can be constructed by
`z (y) =
wr(z1 , . . . , zn , y) , wr(z1 , . . . , zn )
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where wr denotes the Wronskian determinant dened in (14). In this notation, Proposition 2.10 tells us that the PV-ring R is characterized inside a PV-eld E = Quot(R) as the ring of D-nite elements. In the particular case of nite D-extensions E/F the PV-ring R coincides with E . Another implication of Proposition 2.10 is the following characterization of PV-elds.
Let F ≤ E be D-elds of characteristic 0 with algebraically closed eld of constants . Then E is a PV-extension of F if and only if
Theorem 2.11.
(a) E/F is nitely generated by D-nite elements , (b) E and F share the same eld of constants K , (c) for all D-nite z ∈ E yields dimK (VE (`z )) = deg(`z ), where `z ∈ F [∂] is the minimal D-operator of z . An elementary proof is presented for example in [Put2], Proposition 6.11.
3. Monodromy and the RiemannHilbert Problem 3.1. Regular and singular points. Let F = K(C) be the function eld of a
smooth projective curve C over an algebraically closed eld K of characteristic zero with a nontrivial derivation ∂F . Then C(F ) = K . Further for x ∈ C the completion of F with respect to the valuation dened by x is denoted by Fx . It is isomorphic to the eld of Laurent series K((t)) where t ∈ F denotes a local parameter at x. Now let E/F be a PV-extension dened by A ∈ F n×n . Then a point x ∈ C is called a regular point for E/F if A is D-equivalent to a matrix over Fx without poles, i.e., there exists a matrix B ∈ GLn (Fx ) such that
B −1 AB − B −1 ∂(B) ∈ K[[t]]n×n .
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This property can also be characterized by having a fundamental solution matrix over Fx = K((t)) :
Let F = K(C) as above and A ∈ F n×n . Then x ∈ C is a regular point for the PV-extension E/F dened by A if and only if the D-equation ∂(y) = Ay possesses a fundamental solution matrix Y ∈ GLn (Fx ). Proposition 3.1.
This result immediately implies
Let E/F be as in Proposition 3.1 with GalD (E/F ) = G(K) and let L be the xed eld of G 0 (K). Then the nite Galois extension L/F is unramied in all regular points x ∈ C for E/F .
Corollary 3.2.
In the particular case C = P 1 (projective line), the Galois group of a PV-extension E/F with at most one non regular point is connected. This applies, for example, to the Airy equation ∂ 2 (y) = ty in Example 2.4.1 since all nite points are regular.
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Non regular points x ∈ C for E/F are called singular points and the set of all singular points is called the singular locus SE/F of E/F . A point x ∈ SE/F is called tamely (weakly, regular) singular if there exists a B ∈ GLn (Fx ) such that
B −1 AB − B −1 ∂(B) ∈
1 K[[t]]n×n , t
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otherwise it is a wild (strong, singular) singularity. For tame singularities, an even stronger characterization can be given.
Let F = K(C) as above , A ∈ F n×n and E/F a PV-extension dened by A. Then x ∈ C is tamely singular if and only if there exists a B ∈ GLn (Fx ) and a constant matrix D ∈ K n×n such that B −1 AB − B −1 ∂(B) = 1t D.
Proposition 3.3.
For a sketch of proof see for example [Put2], Exercise 7. For later use we add a characterization of regular and tamely singular points in the language of D-modules which immediately follows from the denitions (31) and (32) above.
Let (M, ∂) be a D-module over F = K(C), x ∈ C , Mx := M ⊗F Fx and let t ∈ F be a local parameter for x such that Fx = K((t)).
Corollary 3.4.
(a) A point x ∈ C is regular if and only if Mx contains a ∂ -invariant K[[t]]-lattice . (b) x ∈ C is tamely singular if and only if Mx contains a δ -invariant K[[t]]-lattice where δ := t∂ .
3.2. The monodromy group. In the case of K = C the matrix B in Proposition 3.3 can be chosen to have coecients in the subeld Fxconv ≤ Fx = K((t)) of convergent Laurent series (see [Put2], Exercise 7 or [For], § 11.12). This allows us to analyze the local behaviour. Let F = C(C), A ∈ F n×n and E/F a PV-extension for A. Assume x ∈ C is a tame singularity and denote by t a local parameter at x. Theorem 3.5.
(a) Then ∂(y) = Ay possesses a local fundamental solution matrix of the form Y = B exp(C log(t)) with B ∈ GLn (Fxconv ) and C ∈ C n×n . (b) Via analytic continuation along a loop σ around x we obtain σ(Y ) = Y · Mσ with Mσ = exp(2πiC). For a proof see for example [For], § 11. The matrix Mσ ∈ GLn (C) is called a local monodromy matrix and is determined inside GLn (C) only up to conjugation. In order to simplify the notation we now restrict ourselves to the projective line C = P 1 (C). Then F = C(P 1 ) = C(t) is the eld of rational functions over C . Let S ⊆ P 1 (C) be a nonempty set of cardinality ]S = s < ∞ and let U := P 1 (C) \ S . Then the fundamental group of U with respect to a base point x0 ∈ U is known to be
π1 (U; x0 ) = hσ1 , . . . , σs | σ1 · · · σs = 1i
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where the σi are loops starting from x0 counterclockwise around the points xi ∈ S (compare [Ful], Chapter 19). Applying Theorem 3.5 and analytic continuation we obtain a homomorphism (the monodromy map)
µ : π1 (U ; x0 ) → GLn (C),
σ 7→ Mσ
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where the image is called the monodromy group Mon(E/F ) of E/F . Again Mon(E/F ) is only determined up to conjugacy inside GLn (C). Since Mσ ∈ GLn (C) acts on the solution space VE (A) spanned by the columns of Y it induces an automorphism γσ of E/F compatible with the dierentiation on E . Consequently
γ : Mon(E/F ) → GalD (E/F ), Mσ 7→ γσ
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denes a homomorphism from Mon(E/F ) to the D-Galois group of E/F , which in fact is a monomorphism. This already gives the rst part of the next theorem. (a) Let F = C(t) and E/F be a PV-extension . Then Mon(E/F ) is (isomorphic to ) a subgroup of GalD (E/F ). (b) If in addition the singular locus SE/F is tame , then Mon(E/F ) is Zariski dense in GalD (E/F ). Theorem 3.6.
The proof of (b) relies on the fact that systems of linear dierential equations with only tame singularities by Propositions 3.1 and 3.3 only admit locally meromorphic solutions and that meromorphic functions on P 1 (C) (xed by Mon(E/F )) are rational ([For], Corollary 2.9). In Example 2.4.1 of the Airy equation ∂ 2 (y) = ty we have SE/F = {∞}. Therefore π1 (P 1 (C) \ S) = π1 (A1 (C)) = 1 and Mon(E/F ) is trivial. But GalD (E/F ) = SL2 (C), hence ∞ is a wild singularity.
3.3. The RiemannHilbert Problem. We have seen that in the case of a
tame singular locus the D-Galois group coincides with the Zariski closure of the monodromy group. Therefore it is a fundamental question if every homomorphic image of π1 (U; x0 ) already appears as the monodromy group of a linear system of dierential equations possibly even with only tame singularities. This problem is named the RiemannHilbert problem for tame (regular) systems and is number 21 among the famous Hilbert problems. A positive solution has already been presented by Plemelj (1908) in the following form.
For any nite set S = {x1 , . . . , xs } ⊆ P 1 (C) and any Q set of matrices Mi ∈ GLn (C) with si=1 Mi = 1 there exists a tamely singular system of linear D-equations ∂(y) = Ay over C(t) with monodromy matrices Mi = Mσi around xi .
Theorem 3.7 (Plemelj).
This theorem can be seen as a dierential analogue and generalization of the algebraic version of Riemann's existence theorem (see for example [Voe], Theorem 2.13). A modern proof is given in [AB], Theorem 3.2.1. It relies on the
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theorem of Birkho and Grothendieck on the triviality of complex holomorphic vector bundles. A simplied version for noncompact Riemann surfaces, for example A1 (C), can be found in [For], § 30 and § 31. Here is an easy consequence of Theorem 3.7:
Every nitely generated subgroup G ≤ GLn (C) can be realized as the monodromy group of a system of homogeneous linear dierential equations over C(t) with tame singular locus .
Corollary 3.8.
3.4. The inverse problem over the complex numbers. The solution
of the RiemannHilbert problem is also the main ingredient for the solution of the inverse D-Galois problem over C(t). Namely by Theorem 3.6 it is enough to observe that all linear algebraic groups over C have nitely generated dense subgroups. This nal step of the solution of the inverse problem was settled by Tretko and Tretko only in 1979.
Any Zariski closed subgroup of GLn (C) possesses nitely generated dense subgroups .
Proposition 3.9.
For the proof see [TT], Proposition 1. Together with Theorem 3.7, Proposition 3.9 solves the inverse D-Galois problem over C(t) even with tame singularities.
Every linear algebraic group over C can be realized as a dierential Galois group over C(t) with tame singular locus . Theorem 3.10.
Unfortunately the above general solution of the inverse D-Galois problem over C relies on nonconstructive topological and cohomological considerations. In contrast to the case of nite groups it does not even carry over to algebraically closed elds of constants dierent from C due to the lack of a D-analogue of Grothendieck's Specialization Theorem. For connected groups the situation looks more pleasant. There is a new constructive solution of the inverse D-Galois problem due to Mitschi and Singer which is valid for all D-elds with algebraically closed eld of constants of characteristic 0. This will be outlined in the next section. Before that, however, we want to indicate a theorem of Ramis concerning realizations with restricted singular locus.
A linear algebraic group over C can be realized as a dierential Galois group over C(t) with at most one singular point if and only if it is generated by its maximal tori .
Theorem 3.11 (Ramis).
More generally a linear algebraic group G(C) over C can be realized as a D-Galois group over C(t) with singular locus inside S if and only if the same is true for the quotient by its maximal closed normal subgroup generated by tori. A proof is elaborated in [Ram].
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4. The Constructive Inverse Problem 4.1. The logarithmic derivative. As before, let (F, ∂F ) be an arbitrary D-
eld of characteristic 0 with algebraically closed eld of constants K . Then the F -algebra D := F [X]/(X)2 = F + F e, where e2 = 0, is called the algebra of dual numbers over F . It has the advantage that the map
δ : F → D,
a 7→ a + ∂F (a)e
dened by the non-multiplicative derivation ∂F is a K -homomorphism. For a linear algebraic group G ≤ GLn,F over F the Lie algebra of G can be dened to be the F -vector space
LieF (G) := {A ∈ F n×n | 1 + eA ∈ G(F [e])} provided with the Lie bracket
[·, ·] : LieF (G) × LieF (G) → LieF (G),
(A, B) 7→ [A, B] := AB − BA.
It can be shown that in fact the Lie algebra as dened above is isomorphic to the tangent space of G at the unit point and therefore only depends on G and not on the chosen embedding G ≤ GLn,F .
Let G ≤ GLn,F be a linear algebraic group dened over a Deld F of characteristic 0 with derivation ∂F and with algebraically closed eld of constants . Then Proposition 4.1.
λ : G(F ) → LieF (G),
A 7→ ∂F (A)A−1
is a map from G(F ) to the Lie algebra of G over F . It has the property λ(A · B) = λ(A) + Aλ(B)A−1 . The proof of Proposition 4.1 is immediate (compare [Kov], Section 1). The map λ is usually called the logarithmic derivative. One of its nice features also stated in [Kov] is that it gives an upper bound for the D-Galois group.
Let (F, ∂F ) be a D-eld as above with eld of constants K , G a linear algebraic group over K and A ∈ LieF (G). Then the D-Galois group of a PV-extension E/F dened by ∂(y) = Ay is isomorphic to a subgroup of G(K).
Proposition 4.2.
For the proof we only have to observe that A ∈ LieF (G) implies that the dening ideal I £ F [GLn ] of GF is a D-ideal. Hence the maximal D-ideal P £ F [GLn ] dening the PV-ring R ≤ E contains a conjugate of I . By Proposition 2.3 this already entails the assertion. In the case where the eld F in question has cohomological dimension cd(F ) ≤ 1 there is a partial converse of Proposition 4.2. This relies on the famous Theorem of Springer and Steinberg ([Ser], III, § 2.3). Among the elds with this property
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are, for example, all elds of transcendence degree 1 over an algebraically closed eld (Theorem of Tsen, [Ser], II, § 3.3).
Let F be a perfect eld with cd(F ) ≤ 1. Then for every connected linear algebraic group G over F
Theorem 4.3 (Springer and Steinberg).
H 1 (GF , G(F alg )) = 0
where
GF = Gal(F alg /F ).
Here F alg denotes the algebraic closure of F and hence GF the absolute Galois group of F . Now let F be a D-eld with cd(F ) ≤ 1 and with algebraically closed eld of constants K . Since H 1 (GF , G(F alg )) classies the GF -torsors, with the assumptions of Theorem 4.3 all GF -torsors are trivial. Hence by the D-Torsor Theorem 2.5 then any PV-ring R over F with connected D-Galois group G(K) is isomorphic to the coordinate ring F [G] of G . Another consequence of Theorem 4.3 of Springer and Steinberg is the following converse of Proposition 4.2 (see for example [Put2], Theorem 4.4).
Let (F, ∂F ) be a D-eld of characteristic 0 with algebraically closed eld of constants K and cd(F ) ≤ 1, H ≤ GLn,K a connected closed subgroup , A ∈ LieF (H) ⊆ F n×n and E/F a PV-extension dened by A with connected Galois group Gal(E/F ) = G(K). Then there exists a B ∈ H(F ) such that Corollary 4.4.
B −1 AB − B −1 ∂F (B) ∈ LieF (G). In this case E/F can be generated by a dierential equation ∂(y) = Ay with A ∈ LieF (G). D-Galois extensions of this specic type are called eective PVextensions in this article. Obviously the existence of eective PV-extensions is restricted to connected groups.
4.2. Chevalley modules. Before tackling the inverse problem for connected
groups, we have to recall some basic notions and general structure theorems for linear algebraic groups G . The maximal connected solvable normal subgroup of G is called the radical of G and its maximal connected unipotent normal subgroup the unipotent radical of G . These are denoted by R(G) and U (G), respectively. Further G is called semisimple if R(G) = 1 and reductive if U(G) = 1. For a connected linear algebraic group we have the following structure theorem (see [Bor], IV, 11.22 and [Spr], Proposition 7.3.1 and 8.1.6).
Let G be a connected linear algebraic group over an algebraically closed eld K of characteristic 0.
Theorem 4.5.
(a) Then G is isomorphic to a semidirect product U o P of its unipotent radical U = U(G) and a maximal reductive subgroup P ≤ G (Levi complement ). (b) The group P is the product T · H of a torus T = R(P) ∼ = Grm and the connected semisimple group H = (P, P). More precisely , there exists a nite subgroup H = H ∩ T such that P ∼ = (T × H)/H .
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This already suggests a strategy for solving the inverse problem for connected groups. The rst step would be to realize tori and semisimple groups and the second to solve embedding problems with unipotent kernel. For the realization of connected semisimple groups we need some strengthening of the following theorem of Chevalley.
Let G be a linear algebraic group over K . Then for all closed subgroups H ≤ G there exist a K -vector space V , a linear representation %H : G → GL(V ) and a one-dimensional subspace W ≤ V such that
Theorem 4.6 (Chevalley).
H(K) = {h ∈ %H (G) | h(W ) ⊆ W }. For the proof see [Spr], Theorem 5.5.3. From this theorem it is fairly easy to deduce the following statement ([MS], Lemma 3.1).
Let G be a connected semisimple linear algebraic group over an algebraically closed eld K of characteristic 0. Then there exist a K -vector space V and a faithful linear representation % : G → GL(V ) with the following properties : Corollary 4.7.
(a) V contains no one-dimensional %(G)-submodule . (b) Any connected closed subgroup H of G leaves a one-dimensional subspace of V invariant . Such a module is called a Chevalley module for G in [MS]. Obviously the natural 2-dimensional representation of SL2 (K) already denes a Chevalley module for this group. In general, Chevalley modules are obtained by composing representations of the type of Theorem 4.6 and therefore are not of this simple structure.
4.3. Realization of connected reductive groups. The key lemma for the realization of semi-simple groups as dierential Galois groups over F = K(t) is the following.
Let F = K(t) be a eld of rational functions over an algebraically closed eld K of characteristic 0, G be a semisimple linear algebraic group over K with Chevalley module V and without loss of generality G ≤ GL(V ). Let A := A0 +tA1 ∈ LieF (G) with constant matrices A0 , A1 ∈ LieK (G), and E/F a PV-extension for A. Then GalD (E/F ) is a proper subgroup of G(K) if and only if there exists a vector w ∈ V ⊗K K[t] and a polynomial f ∈ K[t] of degree at most 1 with (A − ∂)w = f w. Proposition 4.8.
Obviously by Proposition 4.2 the group GalD (E/F ) is isomorphic to a subgroup H(K) of G(K). In case H(K) 6= G(K) by Corollary 4.4 there exists a B ∈ G(F ) such that A˜ := B −1 AB − B −1 ∂(B) ∈ LieF (H). Since V is a Chevalley module ˜ ∈ F v . But then for w := Bv there exists in addition a v ∈ V, v 6= 0, such that Av one obtains (A − ∂)w = f w ∈ F w with deg(f ) ≤ 1.
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Hence, one only has to nd constant matrices A0 and A1 such that (A−∂)w = f w has no solution. For the construction of such matrices we need the root space decomposition of L := LieK (G). This is given by
L = L0 ⊕
³M
´ Lα
α
where L0 denotes the Cartan subalgebra and the one-dimensional spaces Lα = KXα are the eigenspaces for the adjoint action of L0 on G corresponding to the non-zero roots α ∈ L∗0 , i.e., α : L0 → K . More precisely the adjoint action of L0 on L0 is trivial, and for any root α 6= 0 one has [C, Xα ] = α(C)Xα for all C ∈ L0 . The action of L0 on the Chevalley module V produces a similar decomposition L V = β Vβ into eigenspaces for a collection of linear maps β ∈ L∗0 . These are called the weights of V . Now we choose X A0 := Xα . (40) α6=0
In order to fulll the assumptions of Proposition 4.8, for A1 we choose an element in L0 satisfying the following conditions: (41) The α(A1 ) are non-zero and distinct for the non-zero roots α of L. (42) The β(A1 ) are non-zero and distinct for the non-zero weights of V . (43) The linear operator X −1 X−α Xα α(A1 ) α6=0
does not have positive integers as eigenvalues. Obviously the set of A1 ∈ L0 satisfying (41) and (42) is Zariski dense. Condition (43) can be fullled using a suitable multiple of A1 . Now Mitschi and Singer have proved the following result in [MS]:
With matrices A0 and A1 satisfying (40) to (43), the PVextension E/F in Proposition 4.8 generated by A = A0 + tA1 has the dierential Galois group G(K). Proposition 4.9.
In particular, any connected semi-simple linear algebraic group can be realized eectively as a dierential Galois group over F = K(t). The next step is the realization of tori T = G m (K)r , r ∈ N , as dierential Galois groups over F . This follows from the next result:
Let F = K(t) as in Proposition 4.8 and c1 , . . . , cr ∈ K linearly independent over Q . Then the PV-extension E/F generated by A = diag(c1 , . . . , cr ) ∈ LieK (G rm ) has the dierential Galois group G rm (K). Proposition 4.10.
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Obviously by Proposition 4.2 and Corollary 3.2 GalD (E/F ) is a connected subgroup of G rm (K). Hence the result follows from the fact that the solutions yj = exp(cj t) of ∂(y) = cj y are algebraically independent over F for j = 1, . . . , r. Since any connected reductive group is a quotient of a direct product of a connected semi-simple group and a torus by a nite group, from Proposition 4.9 and 4.10 we immediately obtain
Every connected reductive linear algebraic group over an algebraically closed eld K of characteristic 0 can be realized eectively as dierential Galois group over F = K(t). Theorem 4.11.
4.4. Embedding problems with unipotent kernel. In order to solve the
inverse problem of dierential Galois theory for arbitrary connected groups over F = K(t) by Theorem 4.11 it remains to solve dierential embedding problems with unipotent kernel. Here a dierential embedding problem is dened in the following way. Let L/F be a PV-extension with D-Galois group GalD (L/F ) ∼ = H(K) and let β
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1 → A(K) → G(K) → H(K) → 1
be an exact sequence of linear algebraic groups (in characteristic zero). Then the corresponding dierential embedding problem (D-embedding problem), denoted by E(α, β), asks for the existence of a PV-extension E/F with E ≥ L and a monomorphism γ which maps GalD (E/F ) onto a closed subgroup of G(K) such that the diagram res
1
- A(K)
- Gal(L/F ) Gal(E/F ) .... .. .. .. . ∼ γ ... = α .. .. ..? ? β - G(K) - H(K)
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- 1
commutes. The kernel A(K) is also called the kernel of E(α, β) and the monomorphism γ a solution of E(α, β). We say γ is a proper solution if γ is an epimorphism. Further the D-embedding problem is called a split embedding problem if the exact sequence splits (i.e., G(K) as an algebraic group is a semidirect product of A(K) with H(K)) and a Frattini embedding problem if G is the only closed supplement of A in G (i.e., any U ≤ G which satises AU = G already equals G ). Finally we say the embedding problem is an eective embedding problem, if L/F is an eective PV-extension (according to Section 4.1). The unipotent radical U of a linear algebraic group G possesses a closed complement H which is a reductive linear algebraic group (Levi complement). Thus (G/U)(K) ∼ = H(K) already can be realized eectively as D-Galois group over F . Hence to realize G(K) as D-Galois group it suces to solve an eective split embedding problem with unipotent kernel U(K). Dividing by the commutator
CONSTRUCTIVE DIFFERENTIAL GALOIS THEORY
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subgroup U 0 (K) of U (K) this embedding problem decomposes into an eective split embedding problem with abelian unipotent kernel
1 → U (K)/U 0 (K) → G(K)/U 0 (K) → H(K) → 1
(46)
and a Frattini embedding problem belonging to
1 → U 0 (K) → G(K) → G(K)/U 0 (K) → 1.
(47)
For the rst of these embedding problems we can use a recent result of Oberlies ([Obe], Proposition 2.4) based on a theorem of Ostrowski.
Every eective split D-embedding problem with (minimal ) unipotent abelian kernel has an eective proper solution over K(t), where K is algebraically closed of characteristic 0. Proposition 4.12.
Here the assumption of minimality can be neglected by direct decomposition of the kernel (compare [Obe], Reduction). The solvability of the second embedding problem already goes back to Kovacic ([Kov], Proposition 11). In our terminology it can be stated in the following way.
Every eective Frattini D-embedding problem has an eective proper solution over K(t), where K is algebraically closed of characteristic 0. Proposition 4.13.
For a sketch of the proof, denote dβ : LieF (G) → LieF (H) the surjective Lie algebra map induced by β : G → H and A ∈ LieF (H) a matrix dening an eective PV-extension L/F with isomorphism α : Gal(L/F ) → H(K). Then any inverse image B ∈ LieF (G) of A by dβ , i.e., dβ(B) = A, denes a PVextension E/F with GalD (E/F ) ≤ G(K) by Proposition 4.2 and E ≥ L. Hence by the Frattini property there exists an isomorphism γ : GalD (E/F ) → G(K) with in addition α ◦ res = β ◦ γ , i.e., γ is an eective proper solution of E(α, β). Combining Proposition 4.12 and 4.13 above with Theorem 4.11 we get a constructive solution of the inverse problem for connected groups (see [MS]).
Every connected linear algebraic group over an algebraically closed eld K of characteristic 0 can be realized eectively as dierential Galois group over F = K(t). Theorem 4.14 (MitschiSinger).
A nonconstructive variant of proof had already been presented in [Sin].
Added in Proof. A solution of the inverse problem in dierential Galois theory over K(t) for nonconnected groups has recently been obtained by J. Hartmann in her thesis [Har].
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B. HEINRICH MATZAT AND MARIUS VAN DER PUT
MODULAR THEORY 5. Iterative Dierential Modules and Equations 5.1. Iterative derivations. When trying to set up a dierential Galois theory in positive characteristic, one is confronted with the problem that the usual dierentiation, extended to transcendental extensions of a dierential eld, automatically causes new constants. This problem can be overcome using iterative derivations (also called higher derivations of innite rank in [Jac], 8.15). These were introduced for the rst time by H. Hasse and F. K. Schmidt [HS]. As before, let R be a commutative ring. A family ∂ ∗ = (∂ (k) )k∈N of maps ∂ (k) : R → R with ∂ (0) = idR is called an iterative derivation of R if X ∂ (k) (a + b) = ∂ (k) (a) + ∂ (k) (b), ∂ (k) (a · b) = ∂ (i) (a)∂ (j) (b),
∂ (i) ◦ ∂ (j)
µ ¶ i + j (i+j) = ∂ j
i+j=k
(51)
for all a, b ∈ R and i, j, k ∈ N . (Observe the modied product rule!) The pair (R, ∂ ∗ ) is then called an iterative dierential ring or ID-ring for short. An element c ∈ R is a dierential constant if ∂ (k) (c) = 0 for all k > 0. Again the set of all dierential constants forms a ring denoted by C(R). In case (R, ∂) is a dierential ring containing Q , i.e., a Ritt algebra, the maps 1 k (k) ∂ = k! ∂ dene an iterative derivation on R. (This observation has also led to the name divided powers.) In the case of positive characteristic p, the last condition in (51) implies (∂ (1) )p = 0, i.e., iterative derivations always have trivial p-curvature. The following example shows that in positive characteristic extensions of iterative derivations to transcendental extensions may maintain the constant rings in contrast to ordinary derivations. For ¡ ¢this purpose let F = K(t) be a eld of rational functions. Then ∂ (k) (tn ) = nk tn−k denes an iterative derivation on F denoted by ∂t∗ . Thus with the iterative derivation ∂t∗ , the ring of dierential constants remains K in any characteristic. Iterative derivations can also be characterized by the behaviour of their Taylor series. An iterative Taylor series of a ∈ R is dened by X Ta (T ) := ∂ (k) (a)T k (52) k∈N
with the higher derivations ∂ (k) instead of ∂ k . The following result was found by F. K. Schmidt ([HS], Satz 3):
A commutative ring R together with a family of maps ∂ (k) : R → R for k ∈ N is an ID-ring if and only if Proposition 5.1.
CONSTRUCTIVE DIFFERENTIAL GALOIS THEORY
447
(a) the Taylor map T : R → R[[T ]], a 7→ Ta (T ) is a ring homomorphism with I ◦ T = idR for I : R[[T ]] → R, Θ(T ) 7→ Θ(0), (b) the extended map X XX ˜ : R[[T ]rightarrowR[[T ]], T ai T i 7→ ∂ (j) (ai )T i+j i∈N
i∈N j∈N
(k) ˜ =T ˜ ◦ ∂ (k) . is a ring homomorphism with ∂T ◦ T ∗ of R Using iterative Taylor series it is easy to extend an iterative derivation ∂R −1 to quotient rings S R by expanding Ta/b (T ) := Ta (T )/Tb (T ). Obviously this extension is unique. In particular, an iterative derivation of an integral domain R uniquely extends to its quotient eld F = Quot(R) ([HS], Satz 5). For separable eld extensions, the following result is given in [HS], Satz 6 and Satz 7.
Let (F, ∂F∗ ) be an ID-eld and E/F a nitely generated ∗ separable eld extension . Then ∂F∗ extends to an iterative derivation ∂E of E . In case E/F is nite this extension is unique . Proposition 5.2.
The ring of dierential constants K of an ID-eld (F, ∂ ∗ ) is a eld which is separably algebraically closed in F . Corollary 5.3.
5.2. The Wronskian determinant. In positive characteristic the Wronskian
determinant as dened in the classical case may vanish even if the functions involved are linearly independent. Fortunately the iterative Taylor series preserve linear independency.
Let (F, ∂F∗ ) be an ID-eld with eld of constants K . Then for elements x1 , . . . , xn ∈ F linearly independent over K the iterative Taylor series Tx1 , . . . , Txn are linearly independent over F . Proposition 5.4.
The proof can be found in [Sch]. From this result one obtains the existence of elements di ∈ N with det(∂ (di ) (xj ))ni,j=1 6= 0. The set D = {d1 , . . . , dn } of natural numbers, which are the smallest (in lexicographical order) with this property is called the set of derivation orders of x1 , . . . , xn . The corresponding determinant (53) wrD (x1 , . . . , xn ) := det(∂ (di ) (xj ))ni,j=1 is called the Wronskian determinant of x1 , . . . , xn . Obviously the set of derivation orders only depends on the K -module spanned by the xj . With this modied Wronskian determinant we now obtain the following result familiar from characteristic zero.
Let (F, ∂F∗ ) be an ID-eld with eld of constants K . Then elements x1 , . . . , xn ∈ F with set of derivation orders D are linearly independent over K if and only if wrD (x1 , . . . , xn ) 6= 0. Corollary 5.5.
In characteristic 0 the set of derivation orders always coincides with {0, . . . , n−1} which is closed by ≤. On the contrary, in characteristic p > 0 each subset D ⊆ N
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B. HEINRICH MATZAT AND MARIUS VAN DER PUT
which is closed by the relation ≤p may appear as a set of derivation orders. Here k≤p l stands for the property that all coecients of the p-expansion of k are less than or equal to the corresponding coecients of l. This can be veried for example with (F, ∂F∗ ) = (K(t), ∂t∗ ) and {x1 , . . . , xn } = {td1 , . . . , tdn } for D = {d1 , . . . , dn }. In particular, in characteristic p ≥ n the set of derivation orders is always the same as in the classical case.
5.3. Iterative dierential modules. In positive characteristic it is more
suitable to dene dierential equations by introducing dierential modules rst ∗ (compare Section 1.4). For this purpose let (R, ∂R ) be an ID-ring with ring (k) ∗ = (∂M )k∈N of maps of constants S and M be an R-module. A family ∂M (k) (0) ∂M : M → M with ∂M = idM satisfying X (i) (k) (k) (k) (k) (j) ∂M (x + y) = ∂M (x) + ∂M (y), ∂M (a · x) = ∂R (a)∂M (x),
µ (i)
(j)
and ∂M ◦ ∂M =
i+j=k
¶ i + j (i+j) ∂M i
for all a ∈ R, x, y ∈ M and i, j, k ∈ N is called an iterative derivation on M , and ∗ (M, ∂M ) is called an iterative dierential module or ID-module for short. The S -module \ (k) V (M ) = Ker(∂M ) k>0
is called the solution space of M . Further M is called a trivial ID-module if M∼ = V (M ) ⊗S R. ∗ ∗ Given ID-modules (M, ∂M ) and (N, ∂N ) over R, an element ϕ ∈ HomR (M, N ) (k) is called an iterative dierential homomorphism (ID-homomorphism) if ϕ◦∂M = (k) ∂N ◦ϕ for all k ∈ N . The category of ID-modules over R with ID-homomorphisms as morphisms is denoted by IDModR . It is easy to check that in case R is a eld F , i.e., (F, ∂F∗ ) is an ID-eld, IDModF is an abelian category. It becomes a tensor category over the eld of constants K using the tensor product M ⊗F N with the iterative derivation X (i) (k) (j) ∂M ⊗N (x ⊗ y) = ∂M (x) ⊗ ∂N (y) (54) i+j=k
and the dual M ∗ = HomF (M, F ) with X (k) (i) (j) ∂M ∗ (f ) = (−1)j ∂F ◦ f ◦ ∂M
(55)
i+j=k
for all x ∈ M , y ∈ N , f ∈ M ∗ and i, j, k ∈ N . Then (F, ∂F∗ ) is the unit element of IDModF with EndIDModF (F, ∂F∗ ) = K . If in addition K is algebraically closed then IDModF together with the forgetful functor
Ω : IDModF → VectF ,
∗ (M, ∂M ) 7→ M
CONSTRUCTIVE DIFFERENTIAL GALOIS THEORY
449
is even a Tannakian category. As in the classical case we will not make use of this property in the sequel. From Corollary 5.5 we immediately obtain the following formal analogue of Proposition 1.1.
Let (F, ∂F∗ ) be an ID-eld with constant eld K and M ∈ an ID-module over F . Then for the solution space V (M ) of M we
Proposition 5.6.
IDModF have
dimK (V (M )) ≤ dimF (M ).
5.4. Projective systems. ID-modules over elds of positive characteristic can
be described by projective systems of vector spaces. To explain this connection, ∗ let (M, ∂M ) be an ID-module over an ID-eld (F, ∂F∗ ) of characteristic p > 0. Then \ (pj ) Ml := Ker(∂M ) (56) j
T
is a vector space over the eld Fl :=
(pj )
j
Ker(∂F
). Indeed, Ml is even an (kpl )
ID-module over Fl with respect to the iterative derivations (∂M (kpl ) (∂F )k∈N ,
)k∈N and
respectively. Further the embedding ϕl : Ml+1 → Ml is an Fl+1 linear map and denes a projective system (Ml , ϕl )l∈N . Moreover each ϕl can be extended uniquely to an isomorphism ϕ˜l : Ml+1 ⊗Fl+1 Fl → Ml . In order to prove dimFl+1 (Ml+1 ) = dimFl (Ml ) for the last statement one has to use the (pl )
triviality of the p-curvature (∂M )p = 0 on Ml (compare [Mat], Proposition 2.7). In fact ID-modules are characterized by the above properties.
Let (F, ∂F∗ ) be an ID-eld of characteristic p > 0. Then the category IDProjF of projective systems (Nl , ψl )l∈N over F with the properties
Theorem 5.7.
(a) Nl is an Fl -vector space of nite dimension and ψl is Fl+1 -linear , (b) each ψl extends to an isomorphism ψ˜l : Nl+1 ⊗Fl+1 Fl → Nl
is equivalent to the category IDModF . This equivalence is even compatible with the structure of Tannakian categories. The critical point in the proof is the denition of an iterative derivation on M := N0 . Dening Ml := (ψ0 ◦ · · · ◦ ψl−1 )(Nl ) we get Ml ⊆ Ml−1 ⊆ . . . ⊆ M . By property (b) an Fl -basis Bl = {b1 , . . . , bn } of Ml also is an F -basis of M . So Pn for all x ∈ M we can nd coecients ai ∈ F such that x = i=1 bi ai = Bl · a T (k) for a = (a1 , . . . , an )tr . Since by induction Bl ⊆ Ml ⊆ k
∂M (x) =
n X i=1
(k)
(k)
bi ∂F (ai ) = Bl ∂F (a).
(57)
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B. HEINRICH MATZAT AND MARIUS VAN DER PUT
Obviously this denition is independent of the choice of the bases Bl of Ml . The above step in the proof leads to the following formula for the iterative derivation which is basic for the introduction of iterative dierential equations. ∗ Let (M, ∂M ) be an ID-module over an ID-eld (F, ∂F∗ ) of characteristic p > 0 with corresponding projective system (Ml , ϕl )l∈N . Then
Corollary 5.8. (k)
(k)
∂M = ϕ˜0 ◦ · · · ◦ ϕ˜l ◦ ∂F ◦ ϕ˜l −1 ◦ · · · ◦ ϕ˜0 −1
for all k < pl+1 .
5.5. Iterative dierential equations. As before, M denotes an ID-module over an ID-eld F of characteristic p > 0 with projective system (Ml , ϕl )l∈N . Let Bl = {bl1 , . . . , bln } be a basis of Ml and Dl the representing matrix of ϕl with respect to Bl+1 and Bl , i.e., Bl+1 = Bl Dl for Bl = (bl1 , . . . , bln ) etc. Then Corollary 5.8 leads to the formula (k)
(k)
∂M (B0 ) = B0 D0 · · · Dl ∂F (Dl−1 · · · D0−1 ) for k < pl+1 (k)
(58)
(k)
because of B0 = Bl+1 Dl−1 · · · D0−1 and ∂M (B0 ) = Bl+1 ∂M (Dl−1 · · · D0−1 ). From (58) we get the following characterization of the solution space of an ID-module. ∗ Assume the characteristic is p > 0. Let (M, ∂M ) be an ID∗ module over an ID-eld (F, ∂F ) with corresponding projective system (Ml , ϕl )l∈N , basis {b1 , . . . , bn } of M , and B = (b1 , . . . , bn ). Then for y = (y1 , . . . , yn )tr ∈ F n , the following statements are equivalent : T Pn (a) By = i=1 bi yi ∈ V (M ) = l∈N Ml , −1 (b) yl := Dl−1 · · · D0−1 y ∈ Fln for all l ∈ N ,
Proposition 5.9.
(pl )
(pl )
(c) ∂F (yl ) = A◦l yl for all l ∈ N where A◦l = ∂F (Dl )Dl−1 , l
l
(p )
(p )
(d) ∂F (y) = Al y for all l ∈ N where Al = ∂F (D0 · · · Dl )(D0 · · · Dl )−1 . Here the equivalence of (a) and (b) directly follows from the denition of Ml and (58). The equivalence with (c) and (d) is derived from (pl )
(pl )
(pl )
(pl )
∂F (yl ) = ∂F (Dl yl+1 ) = ∂F (Dl )yl+1 = ∂F (Dl )Dl−1 yl and the corresponding equation for y = D0 · · · Dl yl+1 . The families of higher dierential equations in Proposition 5.9, (c) and (d) associated to the ID-module M are called an iterative dierential equation (IDE) (in its relative and its absolute version, respectively). In terms of the logarithmic (pl )
derivative associated to ∂F
λl : GLn (F ) → F n×n = Lie(GLn (F )),
(pl )
D 7→ ∂F (D)D−1
(59)
these read as (pl )
∂F (yl ) = λl (Dl )yl (pl )
with
∂F (y) = λl (D0 · · · Dl )y
λl (Dl ) ∈ Fln×n , with
λl (D0 · · · Dl ) ∈ F n×n .
We close the section with two typical examples:
(510)
CONSTRUCTIVE DIFFERENTIAL GALOIS THEORY
451
Let (F, ∂F∗ ) = (K(t), ∂t∗ ) be an ID-eld of characteristic p > 0 l and M = F b a one-dimensional vector space over F . Suppose Dl = (tal p ) ∈ Example 5.5.1.
(pl )
l
GL1 (Fl ). Then Al = ∂F (D0 · · · Dl )(D0 · · · Dl )−1 = (al t−p ) and the corresponding IDE is given by l
l
∂ (p ) (y) = al t−p y
for
l ∈ N.
∂t∗ ) with char(F ) = p > 0 and Let again (F, ∂F∗ )µ= (K(t), ¶ pl 1 al t ∈ GL2 (Fl ) we obtain Al = let M = F b1 ⊕ F b2 . For Dl = 0 1 µ ¶ 0 al λl (D0 · · · Dl ) = . Therefore the corresponding IDE simply is 0 0 µ ¶ µ ¶ l 0 al y1 ∂ (p ) (y) = y where y = . 0 0 y2 Example 5.5.2.
6. Iterative PicardVessiot Theory 6.1. Iterative PV-rings and elds. Surprisingly PicardVessiot rings and
elds in positive characteristic can formally be dened in the same way as in characteristic zero. Let (F, ∂F∗ ) be an ID-eld of characteristic p > 0 with algebraically closed eld of constants K and l
∂ (p ) (y) = Al y
with Al ∈ F n×n for l ∈ N
(61)
∗ an IDE over F as dened in the second line of (510). Let (R, ∂R ) be an ID-ring ∗ ∗ with R ≥ F and ∂R extending ∂F . Then Y ∈ GLn (R) is called a fundamental (pl )
solution matrix for the IDE (61) if ∂R (Y ) = Al Y for all l ∈ N . The ring R is called an iterative PicardVessiot ring (IPV-ring) if it satises the following conditions: (62) R is a simple ID-ring, i.e., R contains no nontrivial ID-ideals, (pl )
(63) there exists a Y ∈ GLn (R) with ∂R (Y ) = Al Y for all l ∈ N , (64) R over F is generated by the coecients of Y and det(Y )−1 . Again it is easy to verify that a nitely generated simple ID-ring is an integral domain with no new constants. The quotient eld of R is called an iterative PicardVessiot eld (IPV-eld).
Let (F, ∂F∗ ) be an ID-eld of characteristic p > 0 with algel braically closed eld of constants K . Then for every IDE ∂ (p ) (y) = Al y over F there exists an iterative PicardVessiot ring which is unique up to an iterative dierential isomorphism . Proposition 6.1.
By Section 5.5 the matrices Al have the form Al = λl (D0 · · · Dl ) with Dl = D(ϕl ). Then U := F [GLn ] = F [xij , det(xij )−1 ]ni,j=1 can be given the structure
452
B. HEINRICH MATZAT AND MARIUS VAN DER PUT
of an ID-ring in the following way: First we dene ∂U∗ on the vector space F hxij ini,j=1 simply by (pl )
∂U (xj ) = Al xj
for xj = (x1j , . . . , xnj )tr .
(65)
This corresponds to the projective system (Nl , ψl ) where Nl = Fl (Xl ) denotes −1 the Fl -vector space generated by the coecients of Xl = Dl−1 · · · D0−1 X and ψl the Fl+1 -linear map dened by ψl : Nl+1 → Nl , Xl+1 7→ Dl Xl+1 = Xl . Then by the product rule ∂U∗ uniquely extends to an iterative derivation on the polynomial ring F [xij ]ni,j=1 and nally on F [GLn ]. Now we can proceed as in the classical case: Factoring U by a maximal ID-ideal P we obtain an IPV-ring R with fundamental solution matrix Y = κP (X) which turns out to be uniquely determined by Al up to ID-isomorphism ([MP], Lemma 3.4). Again the IPV-eld E = Quot(R) can be described without referring to R (see [Mat], Proposition 4.8).
Let (F, ∂F∗ ) be an ID-eld of characteristic p > 0 with algebraically closed eld of constants and Al = λl (D0 · · · Dl ) ∈ F n×n . Then an ∗ ID-eld (E, ∂E ) ≥ (F, ∂F∗ ) is an IPV-eld for (Al )l∈N if and only if
Proposition 6.2.
(a) E does not contain new constants ,
(pl )
(b) there exists an Y ∈ GLn (E) with ∂E (Y ) = Al Y for all l ∈ N , (c) E is generated over F by the coecients of Y . Obviously Proposition 6.2 immediately implies the following minimality property for the solution space of the underlying ID-module M .
The IPV-extension E/F in Proposition 6.2 is a minimal eld extension of F such that dimK (VE (M )) = dimF M where VE (M ) = V (M ⊗F E).
Corollary 6.3.
6.2. The ID-Galois group. An automorphism of an IPV-extension R/F
or E/F is called an iterative dierential automorphism (ID-automorphism) if it commutes with ∂ (k) for all k ∈ N . Correspondingly the group of all IDautomorphisms of R/F (or E/F ) is called the iterative dierential Galois group (ID-Galois group) of R/F or E/F and is denoted by GalID (R/F ) = GalID (E/F ). This again is a maximal subgroup of GLn (K) respecting the maximal ID-ideal P of F [GLn ] used for the construction of R (compare Proposition 2.3). With similar arguments as in the classical case we can deduce ([Mat], Theorem 3.10):
Let F be an ID-eld of characteristic p > 0 with algebraically closed eld of constants K and E/F an IPV-extension . Then there exists a reduced linear algebraic group G dened over K such that GalID (E/F ) ∼ = G(K). Moreover the xed eld of G(K) equals F . Proposition 6.4.
From the preceding proposition it follows immediately that an IPV-extension E/F with nite ID-Galois group is an ordinary nite Galois extension. On the
CONSTRUCTIVE DIFFERENTIAL GALOIS THEORY
453
other hand a nite Galois extension E/F of an ID-eld (F, ∂F∗ ) is even an IPVextension since ∂F∗ uniquely extends to E and since every γ ∈ Gal(E/F ) is an IDautomorphism. To complete the proof we can use the following characterization of IPV-extensions ([Mat], Proposition 3.11).
Let E ≥ F be ID-elds of characteristic p > 0 over an algebraically closed eld of constants . Then E/F is an IPV-extension if and only if Proposition 6.5.
(a) there exists a nite-dimensional F -vector space V ⊆ E with E = F (V ) and (b) a group G of ID-automorphisms of E acting on V with E G = F .
Finite Galois extensions of ID-elds of characteristic p > 0 with algebraically closed eld of constants are IPV-extensions and vice versa . Corollary 6.6.
We now return to our examples in Section 5.5 where (F, ∂F∗ ) = (K(t), ∂t∗ ). l
Let Dl = (tal p ) as in Example 5.5.1 with corresponding IDE (pl ) −pl ∂ (y) = al t y and IPV-extension E/F . Then for all y ∈ VE (M ) and γ ∈ GalID (E/F ) µ ¶ µ ¶ γ(y) γ(yl+1 ) (pl ) (pl ) ∂E = ∂E = 0 for yl+1 = Dl−1 · · · D0−1 y y yl+1 Example 6.2.1.
such that γ(y) = cy with c ∈ K × , i.e., GalID (E/F ) is a subgroup of G m (K). P Q l l A formal solution of the IDE is given by y = l∈N tal p = t l∈N al p . This P represents an algebraic function if and only if the p-adic integer α := l∈N al pl belongs to Q , i.e., α = na with a, n coprime. Then GalID (E/F ) is cyclic of order n, otherwise Gal(E/F ) = G m (K) = K × . Example 6.2.2.
From Example 5.5.2 we know that the IDE for µ ¶ 1 al pl Dl = ∈ GL2 (F ) 0 1
is given by
µ l
∂ (p ) (y) = Al y,
where Al =
0 al 0 0
¶
µ and y =
y1 y2
¶ .
Obviously y2 ∈ K . Then the IPV-extension is generated by y1 , i.e., E = F (y1 ). (pl )
For γ ∈ GalID (E/F ) and y1 ∈ VE (M ) we have ∂E (γ(y1 ) − y1 ) = 0 such that γ(y1 ) = y1 + c with c ∈ K and Gal´ID (E/F ) ≤ G a (K). A formal solution of the ³P pl IDE is given by y1 = y2 with y2 ∈ K . This function is separably l∈N al t algebraic over F if and only if the sequence (al )l∈N becomes periodic. Then the ID-Galois group is a nite elementary abelian p-group, otherwise GalID (E/F ) ∼ = G a (K).
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B. HEINRICH MATZAT AND MARIUS VAN DER PUT
6.3. Kolchin's Theorem and the Galois correspondence. Now we are
ready to explain the ID-Galois correspondence. Again it relies substantially on Kolchin's theorem based on the following ID-torsor theorem. Theorem 6.7 (ID-Torsor Theorem). Let F be an ID-eld of characteristic p > 0 with algebraically closed eld of constants K , R an IPV-ring over F for some IDE with GalID (R/F ) ∼ = G(K) and GF := G ×K F . Then Spec(R) is a GF -torsor .
Here the proof given in [Put2], Section 6.2, in the classical case completely carries over by replacing all statements used for D-structures by the corresponding statements for ID-structures ([Mat], Theorem 4.4). Then Kolchin's theorem as stated in Corollary 2.6 is a formal consequence of it. As another consequence we get the ID-Galois correspondence in the following form ([MP], Theorem 3.5). Theorem 6.8 (ID-Galois Correspondence). Let F be an ID-eld of characteristic p > 0 with algebraically closed eld of constants K and E/F an IPVextension of some IDE with GalID (E/F ) ∼ = G(K). Then :
(a) There exists an anti-isomorphism between the lattices
H = {H(K) | H(K) ≤ G(K) reduced closed} and L = {L | F ≤ L ≤ E ID-eld}
given by Ψ : H → L, H 7→ E H(K) and Ψ−1 : L → H, L 7→ GalID (E/L). (b) If thereby H(K) is a normal subgroup then L := E H(K) is an IPV-extension of F with GalID (L/F ) ∼ = G(K)/H(K). The statement on nite ID-Galois extensions corresponding to Theorem 2.7(c) is already contained in Corollary 6.6.
6.4. Characterization of IPV-rings and elds. It remains to carry over the characterization theorems for PV-rings and PV-elds. Obviously the denition of a D-nite element has to be adjusted. Let E/F be an IPV-extension. Then z ∈ E is called iterative dierentially nite over F (ID-nite) if dimF (WE (z)) < ∞,
(k)
where WE (z) := F h∂E (z)ik∈N ,
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∗ with the iterative derivation ∂E of E . Then Proposition 2.10 translates into
Let F be an ID-eld of characteristic p > 0 with algebraically closed eld of constants , R/F an IPV-ring and E = Quot(R) with G := GalID (E/F ). Then for z ∈ E the following conditions are equivalent : Proposition 6.9.
(a ) z ∈ R,
(b ) dimK (KhGzi) < ∞,
(c ) dimF (WE (z)) < ∞.
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In the classical case the proof relies on the use of the minimal D-operator of z dened using the Wronskian wr(z1 , . . . , zr ) of a base of KhGzi. In positive characteristic this has to be replaced by a family of higher D-operators (k)
`(k) (y) :=
wrD (z1 , . . . , zr , y) , wrD (z1 , . . . , zr )
where the classical Wronskian determinant is replaced by the F. K. Schmidt Wronskian wrD dened in (53) with set of derivation orders D = {d1 , . . . , dr } (k) and where wrD denotes the Wronskian with derivation orders d1 , . . . , dr and k . Then KhGzi can be characterized as the K -vector space of solutions of (`(k) )k∈N in E , which is denoted by VE (z). Using this we nally get the following characterization of IPV-elds analogous to Theorem 2.11.
Let E ≥ F be ID-elds of characteristic p > 0 with algebraically closed eld of constants . Then E is an IPV-extension of F if and only if Theorem 6.10.
(a) E/F is nitely generated by ID-nite elements , (b) E and F share the same eld of constants K , (c) for any ID-nite element z ∈ E , dimF (WE (z)) = dimK (VE (z)). Complete proofs of Proposition 6.9 and Theorem 6.10 are presented in [Mat], Section 4.3.
7. Local Iterative Dierential Modules 7.1. Tamely singular ID-modules. For the denition of regular and tamely
singular ID-modules we use an ID-analogue of Corollary 3.4. Let F = K((t)) be the eld of power series over an algebraically closed eld K of characteristic p > 0 with ∂F∗ = ∂t∗ and M an ID-module over F with iterative (k) ∗ ∗ derivation ∂M . Then the members ∂M of the family ∂M generate a commutative K -algebra denoted by (k)
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DM := K[∂M |k ∈ N].
Corresponding to Corollary 3.4 (a) we call M a regular local ID-module if and only if M contains a DM -invariant K[[t]]-lattice (of full rank). In order to obtain an analogous denition for tamely singular local ID-modules as in Corollary 3.4 we have to replace ∂ (k) by δ (k) := tk ∂ (k) .
Let K be an algebraically closed eld of characteristic p > 0, F = K((t)) with ∂F∗ = ∂t∗ and M an ID-module over F . Then
Proposition 7.1.
(k)
0 DM := K[δM |k ∈ N]
(k)
is a commutative K -algebra with the additional property (δ (k) )p = δ (k)
(k)
with δM := tk ∂M
for k ∈ N.
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B. HEINRICH MATZAT AND MARIUS VAN DER PUT
Here the amazing second property immediately follows from µ ¶p µ ¶ n n n (k) p n n (δ ) (t ) = t = δ (k) (tn ). t = k k According to Corollary 3.4(b) a local ID-module M is called a tamely singular 0 ID-module if it contains a DM -invariant K[[t]]-lattice. Obviously any regular local ID-module is tamely singular. Moreover, all one-dimensional local ID-modules are tamely singular by Example 5.5.1. 0 acts on a nite-dimensional K -vector space V by Proposition 7.1 In case DM (k) the δ are commuting diagonalizable endomorphisms. Hence V possesses a 0 basis of common eigenvectors for DM . This already explains the rst part of
Let V be a K -vector space of dimension n ∈ N which is a 0 -algebra . Then the following hold : DM Ln 0 (a) There exists a direct sum decomposition V = i=1 Vi where each Vi is DM stable of dimension 1. (b) For each Vi = Kvi there exists an αi ∈ Z p such that Corollary 7.2.
(pl )
δM (vi ) = −
µ ¶ αi vi pl
where denotes the residue in F p . (pl )
Here the second statement follows from the fact that by the rule (δM )p = (pl )
(pl )
δM the elements ail ∈ K with δM (vi ) = −ail vi belong to F p . Hence αi := P ail pl ∈ Z p has the desired property. By abuse of language we call V = Ll∈N n i=1 Vi an eigenspace decomposition and αi ∈ Z p eigenvalues of the whole family (k) ∗ δM = (δM )k∈N . Using an induction process the eigenspace decomposition in Corollary 7.2 can be lifted to tamely singular ID-modules over F = K((t)). The result is the following ([MP], Proposition 6.1)
Let K be an algebraically closed eld of characteristic p > 0, F = K((t)) be an ID-eld with ∂F∗ = ∂t∗ and let M be a tamely singular local ID-module over F of dimension n. Ln (a) There exist αi ∈ Z p and a decomposition M = i=1 Mi of M into a direct sum of one-dimensional ID-submodules Mi = F bi with µ ¶ αi (pl ) δM (bi ) = − l bi . p Theorem 7.3.
(b) The ID-Galois group of the corresponding IPV-ring R/F is the maximal closed subgroup of G m (K)n preserving the Z -relations between the eigenvalues
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αi , i .e ., Gal(R/F ) = {(c1 , . . . , cn ) ∈ (K × )n |
n Y
cidi = 1 if
n X
i=1
di αi ∈ Z, di ∈ Z}.
i=1
In particular, if the αi are Z -linearly independent Gal(R/F ) is the full group G m (K)n . Here part (b) relies on the fact that algebraic relations over F between Qn solutions yi of Mi are of the simple form i=1 yidi = td0 with di ∈ Z . From Theorem 7.3 we further obtain the following characterization of regular and tamely singular local ID-modules by their ID-Galois groups. Corollary 7.4.
Let (F, ∂F∗ ), M and R be as in Theorem 7.3.
(a) M is tamely singular if and only if GalID (R/F ) is diagonalizable . (b) M is regular if and only if GalID (R/F ) is trivial . Part (a) follows directly from Theorem 7.3, thanks to the fact that all onedimensional local ID-modules are tamely singular. Then (b) follows from (a) by observing that in the regular case all eigenvalues equal zero.
7.2. The structure of local ID-modules. By Theorem 7.3 one-dimensional
ID-modules M over F = K((t)) are determined by their eigenvalues α ∈ Z p , and any α ∈ Z leads to the trivial ID-module. To be more precise, the isomorphism class of a one-dimensional ID-module is characterized by the congruence class α ¯ of its eigenvalue α modulo Z . Using tensor products, the set of isomorphism classes IDMod1F of ID-modules of dimension 1 becomes a group (IDMod1F , ⊗) where in the parameter space Z p /Z the group law translates into the addition. This proves
Let F = K((t)) be an ID-eld with ∂F∗ = ∂t∗ over an algebraically closed eld K of characteristic p > 0. Then Proposition 7.5.
(IDMod1F , ⊗) ∼ = (Z p /Z, +). If the dimension of a local ID-module M is greater than 1 then inside M we can always nd a nontrivial tamely singular ID-submodule and thus by Theorem 7.3 a nontrivial one-dimensional ID-submodule. Hence by induction on the dimension of M we obtain the rst half of the following Theorem 7.6. Let F = K((t)) be an ID-eld over an algebraically closed eld K of characteristic p > 0 with ∂F∗ = ∂t∗ , M an ID-module over F and R an IPV-ring for M . Then :
(a) M is a repeated extension of one-dimensional ID-modules . (b) Gal(R/F ) = G(K) is trigonalizable and there exists and exact sequence of nite groups 1 → P → G(K)/G 0 (K) → Z → 1
where P is a p-group and Z is a cyclic group of order prime to p.
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B. HEINRICH MATZAT AND MARIUS VAN DER PUT
The rst assertion in (b) is a direct consequence of (a) since G(K) can be embedded into the standard Borel subgroup Bn (K), and the exact sequence for G/G 0 follows from Hilbert theory. A complete proof can be found in [MP], Proposition 6.3 and Corollary 6.4.
7.3. The connected local inverse problem. The question remains if every
linear algebraic group with the two properties in Theorem 7.6 (b) appears as ID-Galois group over F . Before giving the solution in the connected case we have to explain the meaning of eectivity in the context of IPV-extensions. It is based on the following analogue of Proposition 4.2:
Let F be an ID-eld of characteristic p > 0 with algebraically closed eld of constants K and G a reduced connected linear algebraic group over K . Let M be an ID-module over F with associated projective system (Ml , ϕl )l∈N and representing matrices Dl (with respect to suitable bases of Ml ). Assume that Dl ∈ G(Fl ); then for the corresponding IPV-extension E/F we have GalID (E/F ) ≤ G(K).
Proposition 7.7.
As in the classical case the proof relies on the fact that the dening ideal I E F [GLn ] of GF is an ID-ideal with respect to the iterative derivation on F [GLn ] given by Al = λl (D0 · · · Dl ) according to Section 6.1 (see [MP], Proposition 5.3, or [Mat], Theorem 5.1). In the case of equality GalID (E/F ) = G(K) the eld extension E/F in Proposition 7.7 is called an eective IPV-extension. This further leads to the notion of an eective embedding problem as dened in Section 4.4 etc. In case the eld F has cohomological dimension cd(F ) ≤ 1 it follows from the Theorem 4.3 of Springer and Steinberg that all IPV-extensions E/F with connected Galois group are eective. More precisely in analogy to Corollary 4.4 we obtain ([Mat], Thm 5.9)
Let F be an ID-eld of characteristic p > 0 with cd(F ) ≤ 1 and with algebraically closed eld of constants K , H ≤ GLn,K a reduced connected closed subgroup and M an ID-module over F with projective system (Ml , ϕl )l∈N and Dl ∈ H(Fl ). Assume the ID-Galois group G(K) of M is connected . Then −1 there exist Cl ∈ H(Fl ) such that Cl Dl Cl+1 ∈ G(Fl ). Corollary 7.8.
Now we come back to the inverse problem. In the case of connected groups this problem restricts to the realization of reduced connected solvable linear algebraic groups over K . Such a group G is a semidirect product U o T of a unipotent normal subgroup U and a torus T . According to Proposition 7.5 T (K) can eectively be realized as ID-Galois group over F = K((t)) by a direct sum of Lr one-dimensional ID-modules M = i=1 Mi with eigenvalues αi ∈ Z p linearly independent over Z . Since any connected solvable group with nontrivial unipotent radical possesses a normal subgroup A isomorphic to G a ([Spr], Lemma 6.3.6) it remains to solve eective embedding problems with kernel G a . In analogy to Proposition 4.12 and 4.13 we obtain in positive characteristic:
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Every eective split ID-embedding problem with kernel G a has an eective proper solution over F , where F = K(t) or F = K((t)) and K is algebraically closed of characteristic p > 0.
Proposition 7.9.
Proposition 7.10. Every eective Frattini ID-embedding problem has an eective proper solution over F , where F = K(t) or F = K((t)) and K is algebraically closed of characteristic p > 0.
The next theorem is an immediate consequence of these two propositions.
Let K be an algebraically closed eld of characteristic p > 0. Then for every reduced connected solvable linear algebraic group G over K there exists an eective IPV-extension E/K((t)) with ID-Galois group G(K). Theorem 7.11.
7.4. The nonconnected local inverse problem. In order to solve the
general inverse problem we still have to solve embedding problems with connected kernel and nite cokernel. With the following theorem of BorelSerre [BoS] and Platonov (see [Weh], Lemma 10.10) this problem can be reduced to the solution of split embedding problems.
Let G be a linear algebraic group over an algebraically closed eld K . Then the connected component G 0 of G possesses a nite supplement . Theorem 7.12.
In the case of potential local ID-Galois groups we can prove in addition that the nite supplement H can be chosen to be of the form H = P o Z with P and Z as in Theorem 7.6(b) ([Mat], Proposition 8.4). From the inverse problem of ordinary Galois theory over K((t)) we know that nite groups of this type appear as Galois groups and hence as ID-Galois groups over F := K((t)) (compare [Bo+ ], 14.2). Therefore there exists an IPV-extension L/F with GalID (L/F ) ∼ = H. 0 Now we want to realize the semidirect product G (K) o H with the obvious action of H on G 0 (K) as an ID-Galois group over F . This leads to the following split embedding problem E(α, β) with homomorphic regular section σ . res
1
- G 0 (K)
- Gal(L/F ) Gal(E/F ) ........ .. ... .. ∼ γ ... = α .. .. .? ? - G 0 (K) o H β - H ←−
(72)
- 1
σ
In other words, we have to nd an IPV-extension E/L with connected Galois group GalID (E/L) ∼ = G 0 (K) such that E/F is an IPV-extension and in addition 0 GalID (E/F ) ∼ = G (K) o H (via an isomorphism γ with α ◦ res = β ◦ γ ). This problem can be attacked by the following criterion proved in [Mat], Theorem 8.2:
Let G ∼ = G 0 o H be a linear algebraic group dened over an algebraically closed eld K of characteristic p > 0 with regular homomorphic Proposition 7.13.
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B. HEINRICH MATZAT AND MARIUS VAN DER PUT
section σ : H → G(K). Further , let F be an ID-eld with eld of constants K and cd(F ) ≤ 1. (a) Let L/F be a nite Galois extension with Galois group isomorphic to H via α. Let
χ := σ ◦ α : Gal(L/F ) → σ(H) ≤ G(K),
η 7→ Cη
be the composite isomorphism . Then for all l ∈ N there exist elements Zl ∈ GLn (Ll ) satisfying η(Zl ) = Zl Cη for all η ∈ H ∼ = Gal(L/F ). Moreover , L = F (Z) with Z := Z0 . (b) Let E/L be an IPV-extension with Galois group isomorphic to G 0 (K) via an isomorphism γL : Gal(E/L) → G 0 (K) E G(K),
ε 7→ Cε .
Then there exist elements Yl ∈ G 0 (El ) satisfying ε(Yl ) = Yl Cε for all ε ∈ Gal(E/L) and Dl ∈ G 0 (Ll ) such that Yl+1 = Dl−1 Yl . Moreover , E = L(Y ) with Y := Y0 . (c) Suppose in addition that the following equivariance condition is satised : η(Dl ) = Cη−1 Dl Cη
for all l ∈ N, η ∈ H.
Then E/F is an IPV-extension with ID-Galois group isomorphic to G(K) and fundamental solution matrix ZY . Further , the isomorphism γL in (b ) can be extended to an isomorphism γ : GalID (E/F ) → G(K)
with res ◦α = β ◦ γ.
In order to solve the embedding problem E(α, β) above we thus have to construct an ID-module M over L having a system of representing matrices Dl ∈ G 0 (Ll ) as dened in Section 5.5 fullling the equivariance condition in (c). The latter can be transformed into Dl = Cη η(Dl )Cη−1 , i.e., Dl belongs to the group of F rational points of the L/F -form Gχ0 of G 0 with the twisted Galois action given by
η ∗ D = Cη η(D)Cη−1 = χ(η)η(D)χ(η −1 )
(73)
(compare [Spr], 12.3.7). This is the key observation for the proof of
For a potential local Galois group G(K) (as described in Theorem 7.6) the derived split ID-embedding problem E(α, β) given by (72) has a proper solution . Proposition 7.14.
For the proof we rst show that the L/F -form Gχ0 of G 0 is F -split ([Mat], proof of Proposition 8.3). Then the proof of Theorem 7.11 can be recycled to realize G 0 (K) as an ID-Galois group over L with matrices Dl ∈ Gχ0 (F ). Applying Proposition 7.13 yields the result. The next theorem now follows almost immediately from Proposition 7.14:
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461
Let K be an algebraically closed eld of characteristic p > 0. Then every trigonalizable reduced linear algebraic group G over K with G/G 0 ∼ = P oZ and P , Z as in Theorem 7.6 is the ID-Galois group of some IPV-extension E/K((t)). Theorem 7.15.
Let G be as in Theorem 7.15. Then G has a nite supplement H of type P o Z . As remarked above, H can be realized as ID-Galois group of a nite extension L/F . By Proposition 7.14 we can solve the split embedding problem E(α, β) ∼ = for G 0 (K) o H by γ : Gal(E/F ) −→ G 0 (K) o H . Using the regular (morphic) homomorphism
ψ : G 0 (K) o H → G(K),
(D, C) 7→ D · C
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˜ := E Ker(ψ◦γ) of ψ ◦ γ denes an IPV-extension E/F ˜ the xed eld E with ∼ ˜ GalID (E/F ) = G(K).
8. Global Iterative Dierential Modules 8.1. The singular locus. In this chapter let F/K be an algebraic function eld
of one variable over an algebraically closed eld K of characteristic p > 0, i.e., the function eld F = K(C) of a smooth projective curve C over K . Let M be an ID-module over F with projective system (Ml , ϕl ) and E/F a corresponding IPV-extension. Then a point x ∈ C is called a regular point of M (or of E/F respectively) if there exists a local parameter t for x, an open neighborhood ∗ V ⊆ C of x and a ∂M,t -stable O(V)-lattice Λ ⊆ M , where (pl )
(pl )
∂M,t = ϕ˜0 ◦ · · · ◦ ϕ˜l ◦ ∂t
◦ ϕ˜−1 ˜−1 0 l ◦ ··· ◦ ϕ
(81)
according to Corollary 5.8. The points which are not regular are called singular points and the set SM ⊆ C of singular points of M is referred to as the singular locus of M . The iterative chain rule guarantees that the notion of a regular point does not depend on the choice of the local parameter t. The following proposition is immediate and connects the regularity of points with the regularity of local ID-modules introduced in the last chapter.
Let F = K(C) be a function eld over an algebraically closed eld K of characteristic p > 0 and x ∈ C be a regular point of an ID-module M over F . Then Fx ⊗F M is a regular local ID-module over the completion Fx of F at x. Proposition 8.1.
Unfortunately this local property of regular points cannot be used for the definition as the following example shows. Let C = P 1 (K) be the projective line and F = K(t) its eld of rational functions with ∂F∗ = ∂t∗ . Further, let M be a l one-dimensional ID-module over F with Dl = (t − al )p ∈ G m (Fl ) for pairwise distinct al . Then we obtain an IDE by (pl )
l
∂F (yl ) = λl (Dl )yl = (t − al )−p yl
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B. HEINRICH MATZAT AND MARIUS VAN DER PUT
Q l which has the symbolic solution y = l∈N (t−al )p . The dierential Galois group lies inside G m (K) and is in fact the full multiplicative group by the considerations made in Section 6.2. Obviously every point x ∈ P 1 (K)\S with S = {al | l ∈ N} is regular. For x ∈ S , we can assume without loss of generality that x = a0 . Then Fx = K((t − a0 )) and thus y again denes an element in Mx . Consequently, Mx is regular for all x ∈ P 1 (K). In particular, all local ID-Galois groups are trivial, and GalID (E/F ) is not generated by the Galois groups of the localized modules.
8.2. Realization of connected groups. As explained in Section 7.3, a solvable connected group G = U o T can be realized over F = K(t) starting from an eective realization of T (K) over F by solving eective embedding problems with kernel G a (K). As in the local case T (K) can be realized eectively by a direct sum of one-dimensional ID-modules over F with p-adic eigenvalues linearly independent over Z . Hence from Propositions 7.9 and 7.10 we obtain also in the global case:
For every reduced connected solvable linear algebraic group G over an algebraically closed eld K of characteristic p > 0 the group of K rational points G(K) can be realized eectively as ID-Galois group over K(t). Proposition 8.2.
In the nonsolvable case rst we have to nd a substitute for Propositions 4.8 and 4.9 in the classical case. This is given by
Let G be a reduced connected linear algebraic group over an algebraically closed eld K of characteristic p > 0, let A be either G a or G m and l l l set Sl = K[tp ] or Sl = K[tp , t−p ], respectively . Suppose M is an ID-module over F = K(t) with projective system (Ml , ϕl )l∈N and representing matrices Dl of ϕl (with respect to a given basis of Ml ). Assume further the following properties are satised : Proposition 8.3.
l
(a) For all l ∈ N there exist γl ∈ Mor(A, G) such that Dl = γl (tp ) ∈ G(Sl ) and γl (1A(K) ) = 1G(K) . (b) For all m ∈ N the set {γl (A(K))| l ≥ m} generates G(K) as an algebraic group . (c) There exists a number d ∈ N such that the degree of γl is bounded by d for all l ∈ N . (d) If l0 < l1 < . . . is the sequence of natural numbers li for which γli 6= 1, then limi→∞ (li+1 − li ) = ∞.
Then the IPV-eld E for M is eective over F with GalID (E/F ) ∼ = G(K). Here in (c) the degree deg(γl ) is dened as the maximum of the degrees of the numerator and the denominator of the reduced expression of γl (with respect to l tp ). The proof of Proposition 8.3 is rather technical and can not be reproduced in this survey (compare [MP], Lemma 7.4, and [Mat], Theorem 7.14). But observe that the gap condition (d) mimicking the condition for Liouvillean transcendental numbers excludes all nonconnected subgroups.
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As a consequence of Proposition 8.3 we obtain the solution of the connected inverse problem. Theorem 8.4. Let F = K(t) be an ID-eld over an algebraically closed eld K of characteristic p > 0 with ∂F∗ = ∂t∗ and G be a reduced connected linear algebraic group over K . Then G(K) can eectively be realized as an ID-Galois group over F .
For the proof one observes rst that a maximal unipotent subgroup U(K) of G(K) can be realized via some M ∈ IDModF with projective system (Ml , ϕl ) satisfying conditions (a) to (c) of Proposition 8.3. A suitable choice of the sequences (al ) appearing in Example 6.2.2 for the chief factors Ai (K) of U(K) of type G a (K) also guarantees property (d). (Take for example ai,l ∈ F p and P αi = l∈N ai,l pl ∈ Z p algebraic independent over Q ). In the general case let T (K) be a maximal torus of G(K). Then G(K) is generated as an algebraic group by a nite number of conjugates of U(K) and T (K). By Proposition 8.2 T (K) has an eective realization via some N ∈ IDModF with projective system (Nl , ψl ) satisfying conditions (a) to (d) in Proposition 8.3. Combining dif˜ which again satisferent conjugates of ϕl and ψl we obtain an ID-module M es the four conditions. Hence the corresponding IPV-eld E is eective with l l l Gal(E/F ) ∼ = G(K). Because of D(ϕl ) ∈ G(K[tp ]) and D(ψl ) ∈ G(K[tp , t−p ]) from the proof we obtain in addition:
If G(K) in Theorem 8.4 above is generated by unipotent subgroups , it can be realized with at most one singular point at ∞. In the general case , G(K) can be realized with singular points at most in {0, ∞}. Corollary 8.5.
8.3. Realization of nonconnected groups. In order to solve the nonconnected inverse problem we need a version of Proposition 8.3 which not only works over F = K(t), but also over nite Galois extensions of F . Let K be an algebraically closed eld of characteristic p > 0 and let L = K(s, t) be a nite Galois extension of F = K(t) with ∂F∗ = ∂t∗ . Let C be an ane model of L/K dened by f (s, t) = 0 such that o = (0, 0) ∈ C is l l l a regular point . Then Ll = Lp = K(sp , tp ) has an ane model Cl dened by l l some fl (sp , tp ) = 0. Let G be a reduced connected linear algebraic group over K and let Gχ be an L/F -form of G dened by a regular homomorphic section χ : H := Gal(L/F ) → G o H as in (73) with Gχ (Fl ) ≤ G(Ll ). Let M be an ID-module over L with projective system (Ml , ϕl )l∈N and representing matrices Dl . Suppose the following properties are satised : Proposition 8.6.
(a) For all l ∈ N there exists a rational map γl : Cl → Gχ such that Dl = l l γl (sp , tp ) ∈ Gχ (Fl ) and γl (o) = 1G(K) . (b) For all m ∈ N the algebraic group over L generated by {γl (Cl ) | l ≥ m} contains G(K).
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B. HEINRICH MATZAT AND MARIUS VAN DER PUT
(c) There exists a number d ∈ N such that the degree of γl is bounded by d for all l ∈ N . (d) If l0 < l1 < . . . is the sequence of natural numbers li for which γli 6= 1, then limi→∞ (li+1 − li ) = ∞.
Then M denes an eective IPV-extension E/L with GalID (E/L) ∼ = G(K). Here in (c) the degree deg(γl ) denotes the maximum of the degrees of the numerator and the denominator of the divisors of the matrix entries of Dl in Ll (compare to Proposition 8.3). From Proposition 8.6 we can derive
Let K be an algebraically closed eld of characteristic p > 0. Then every ID-embedding problem over K(t) with connected kernel and nite cokernel has a proper solution . Proposition 8.7.
By the Theorem 7.12 of BorelSerre and Platonov the problem can be reduced to a split ID-embedding problem of the same type. Hence, thanks to Proposition 7.13, we only need to nd a sequence of matrices Dl ∈ Gχ0 (Fl ) which satisfy the conditions of Proposition 8.6. The group Gχ0 is generated as an algebraic group by nitely many F -split unipotent subgroups and F -tori (essentially from [Spr], Corollary 13.3.10). For any such unipotent subgroup the matrices needed may be found as in the proof of Theorem 8.4. By [Tit], III, Proposition 1.6.4 a single element suces to generate a dense subgroup of an F -torus, and such an element may be normed to satisfy condition (a) in Proposition 8.6. Finally, we splice these matrices together into a sequence satisfying the gap condition (d) in Proposition 8.6. Then we obtain an eective IPV-extension E/L with GalID (E/L) ∼ = G(K) by Proposi= G 0 (K) by Proposition 8.6 and GalID (E/F ) ∼ tion 7.13. Obviously Proposition 8.7 implies the solution of the nonconnected inverse problem.
Let G be a reduced linear algebraic group over an algebraically closed eld K of characteristic p > 0. Then G(K) appears as an ID-Galois group ∗ of an IPV-extension E/K(t) with ∂K(t) = ∂t∗ . Theorem 8.8.
8.4. The dierential Abhyankar conjecture. In Corollary 8.5 we have seen
that reduced connected linear algebraic groups which are generated by their closed unipotent subgroups can be realized as ID-Galois groups over F = K(t) with at most one singular point. This statement resembles the Abhyankar conjecture stated in [Abh] and proved by Raynaud [Ray]: Every nite group which is generated by its p-Sylow groups can be realized as a Galois group over F = K(t) unramied outside {∞}. Such groups are usually called quasi-p groups. In order to reduce an ID-embedding problem with connected unipotently generated kernel and nite quasi-p cokernel to split embedding problems of the same type we have to use the following variant of Theorem 7.12 ([Mat], Proposition 8.12).
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Let G be a unipotently generated linear algebraic group over an algebraically closed eld K of characteristic p > 0. Then G 0 (K) has a nite supplement which is a quasi-p group .
Proposition 8.9.
Next we have to adapt Proposition 8.6. Proposition 8.10. If the Galois extension L/F in Proposition 8.6 is unramied outside {∞} and Gχ is a connected unipotent F -split group , the IPV-extension E/L can be constructed unramied outside the places of L above {∞}.
With these preparations we are able to prove the following dierential analogue of the Abhyankar conjecture in the nonconnected case.
Let K be an algebraically closed eld of characteristic p > 0 and let F = K(t) an ID-eld with ∂F∗ = ∂t∗ . Let G be a unipotently generated reduced linear algebraic group dened over K . Then G(K) can be realized as an ID-Galois group over F with at most one singularity . Theorem 8.11.
By Proposition 8.9 the connected component G 0 (K) has a nite supplement H in G(K) which is a quasi-p group. Hence it suces to consider the corresponding split ID-embedding problem. By the classical Abhyankar conjecture proved by Raynaud [Ray] there exists a nite Galois extension L/F with Gal(L/F ) = H which is unramied outside {∞}. The composite χ : Gal(L/F )→H ˜ ,→ G(K) 0 0 denes a twisted form Gχ of G as used in Proposition 8.6. It can be shown that Gχ0 is F -quasi-split and contains a maximal closed F -split unipotent subgroup U ([Mat], proof of Theorem 8.14). Since Gχ0 (F ) is dense in Gχ0 (L) = G 0 (L), the group Gχ0 is generated by nitely many Gχ0 (F )-conjugates of U . Thanks to Proposition 8.10 these conjugates may be generated as algebraic groups over L by equivariant matrices with singular locus above {∞}. Using Proposition 7.13 (c), these matrices may be combined into a sequence which realizes G(K) as ID-Galois group over F with singular locus inside {∞}. At the end we want to call the reader's attention to the parallelism between the dierential Abhyankar conjecture in characteristic p > 0 as presented in Theorem 8.11 and the Theorem 3.11 of Ramis. It generalizes one of the Ramis Raynaud analogies between nite Galois extensions in characteristic p > 0 and PV-extensions in characteristic 0. More specic links, particularly those concerning tame and wild ramications and singularities respectively, are collected in the RamisRaynaud dictionary presented in the Bourbaki lecture notes [Put1].
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Marius van der Put Department of Mathematics University of Groningen P.O. Box 800 NL-9700 AV Groningen The Netherlands
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