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L (X, s) = P (p ) × ( ⑩❨❶❃×✝✉❢❺✏✇②⑤⑦⑧❯❷❈Ñ⑦✉❢❺⑨❸❊⑤ Ò i = d = ✇②①❈③ù❮❊⑤ ④✡③▲❶✝⑥✏⑤ ❒❙❶ï❒❢Ò X ❸ ✕ ③➮⑥✏⑤ ④✡×❈Ñ Ó ✕ ❺②⑤ ✇②③ L(X, s) ⑤ ❶❃×❈Ñ⑦✉❙⑧❯③ ❒❢Ò L (X, s) ✉❢❶✝❮➘⑤ ✇◆⑥✧Ñ ❒❊⑧▲✉❢Ñ❞Ò➠✉❙⑧●✇②❒❙❺♥Û☞Ó P (T ) ⑤ ❶✝⑥r✇②③⑨✉❙❮➸❒❢Ò P (T ) ❻ i et i X, i p −s −1 p= i p i −s −1 p= d p d p ❶ ♥➊➉ ➽❧➔➲→➈➛➅➜➞➝ ❺ ➔❊➓❭➟➹➒✒➓➈➛❜➒▼➔ ❺❃❺ ➟✲➼➞➒⑨➟➠→➸→❙➝✕➓ ➑ ➔☞➑➐➛ ➑ ➔❛➓ Q ☎❒ ③✍✉❙❮❈❮❊❺②③⑨⑥②⑥❫✇②①❈③✍④✡❒❊❮❊❷❈Ñ⑦✉❢❺②⑤ ✇rÓ✟❒❢Ò✕❮❊⑤ ④✡③▲❶✝⑥✏⑤ ❒❙❶ Ð✯✉❢Ñ⑦✉❢Û❈⑤ ê❛ë✬✉❢❷❬➤❭✉❢❺②⑤ ③❯✇②⑤ ③⑨⑥✓❮❊③❑✔✝❶❈③⑨❮ ❒❭➤❙③▲❺ Q✕ ❻✜✕ ❥➅③❯✇ E Û➭③✟✉❢❶✈③▲Ñ Ñ ⑤ ×❊✇②⑤⑦⑧ù⑧❯❷❈❺②➤❙③➮❒❭➤❙③▲❺ Q ❸❞✉❢❶✝❮❃1Ñ ③❯✇ ∆ ❮❊③▲❶❈❒❢✇②③ù⑤ ✇◆⑥➃❮❊⑤⑦⑥②⑧❯❺②⑤ ④✡⑤ ❶✝✉❢❶❛✇⑨❻ è♥①❈③▲❶❬⑤ ✇◆⑥ ❽❨⑥✏③▲❺②⑤ ③⑨⑥✴⑧▲✉❢❶❬Û➭③✍Û❈❷❈⑤ Ñ ✇✯❷❈×❬Û☞Ó✡⑧❯❒❙❷❈❶❛✇②⑤ ❶❈➄➪✇②①❈③☎❶☞❷❈④ùÛ➭③▲❺✬❒❢Ò F ê☞❺◆✉❭✇②⑤ ❒❙❶✝✉❢Ñ❈×➭❒❙⑤ ❶❛✇◆⑥ ❒❙❶ E ❻ L ❸ p ∞ L(E, s) = ①❈③▲❺②③ p ❺②❷❈❶✝⑥☎✉❢Ñ Ñ❏×❈❺②⑤ ④✡③⑨⑥▲❸❈✉❢❶✝❮ p ✕ 1 a(n) = −s 1−2s 1 − a(p)p + ε(p)p ns n=1 ⑤Ò p ∆ a(p) = ⑤Ò p|∆ ➁➃⑤ ➤❙③▲❶✮✉ ×❈❺②⑤ ④✡③ ❸♥③⑨✉❙⑧◆①✷Ñ ❒❊⑧▲✉❢Ñ❆❀✓❷❈Ñ ③▲❺ ❽➽Ò➠✉❙⑧●✇②❒❙❺✡❒❢Ò ⑧▲✉❢❶✮Û➭③❄❮❊③❯✇②③▲❺②④✡⑤ ❶❈③⑨❮ ③❯↔❊×❈Ñ ⑤⑦⑧❯⑤ ✇②Ñ Ó➮❷✝⑥✏⑤ ❶❈➄✍✇②①❈③✧p✉❢Û➭❒❭➤❙③✯❮❊③⑨⑥②⑧❯❺②⑤ ×❊✇②⑤ ❒❙❶➅❻❡➆☎p❒ ✕ ③▲➤❙③▲❺⑨❸⑨✇②①❈③▲L(E, ❺②③♥✉❢❺②③✯s)⑤ ❶✑✔✝❶❈⑤ ✇②③▲Ñ Ó➮④❬✉❢❶☞Ó✿❀✓❷❈Ñ ③▲❺ Ò➠✉❙⑧●✇②❒❙❺◆⑥▲❻✴Ù ❶✝✉❭✇②❷❈❺◆✉❢Ñ✌❫❛❷❈③⑨⑥r✇②⑤ ❒❙❶ï⑤⑦✱⑥ ❋ ➎◆■Ú➼➹▼❈❘❯❉②❘✟❩❭❖☛✂ù➴❢❚ ➻❙ã◆❩❭❚❛➾◆❇❊❖❞❑❯➼➽❋➠➻❭❖❄➼➹▼❈❩❭➼✧➱❙❘❯➼❨❘❯❉●❳✟❋€❖❞❘●■➪➼➹▼❈❘ ❲î■▲❘❯❉●❋➠❘●■ L L(E, s) ➏ p + 1 − #E(Fp ), ε(p) = 1 0, ±1, ε(p) = 0 ❜✇ ❿➁ q ✏✑✍❿✇✮✠✽✡✷➇❼✠✳②✹①✟②✎③✈✇ ➈ ❽ ❿② ➂❯➀➺②✎④❜✇❤✏✑✍❿✇❤✏✳✒ ❻ ②✎④ ❤✍ ➃ ♥è ①❈③✟✉❢❶✝⑥ ✕ ③▲❺✧✇②❒➐✇②①❈⑤⑦⑥✥❫❛❷❈③⑨⑥r✇②⑤ ❒❙❶✈⑧❯❒❙④✡③⑨⑥✧Ò➹❺②❒❙④ ✉❍❫❛❷❈⑤ ✇②③✟❮❊⑤ ❁❞③▲❺②③▲❶❛✇Ú⑥✏❒❙❷❈❺◆⑧❯③❙❻☎⑩❨❶✝❮❊③▲③⑨❮❏❸✝✇②①❈⑤⑦⑥ ⑤ Ñ Ñ➞✇◆✉✟❂❙③ù❷✝⑥✍✇②❒➘④✡❒❙❺②③✟✉❢❶✝✉❢Ñ Ó❛✇②⑤⑦⑧➮❒❙Û✑✘r③⑨⑧●✇◆⑥▲❻❏❳☞❒ ③ ⑤ Ñ Ñ✴❶❈❒ ❮❊③❑✔✝❶❈③✟④✡❒❊❮❊❷❈Ñ⑦✉❢❺➃➄❙❺②❒❙❷❈×✝⑥▲❸ ✕ ④✡❒❊❮❊❷❈Ñ⑦✉❢❺➮Ò➹❒❙❺②④❬⑥✟✉❢❶✝❮➈⑧❯❷✝⑥✏×➍Ò➹❒❙❺②④❬⑥▲❻✽❥➅③❯✇ H ❮❊③▲✕ ❶❈❒❢✇②✕ ③➐✇②①❈③➘❷❈✕ ×❈×➭③▲❺✏❽î①✝✉❢Ñ Ò☎⑧❯❒❙④✡×❈Ñ ③❯↔➍×❈Ñ⑦✉❢❶❈③ ✉❢❶✝❮❬Ñ ③❯✇ Û➭③✍✇②①❈③✍➄❙❺②❒❙❷❈×➐❒❢Ò ⑤ ❶❛✇②③▲➄❙❺◆✉❢Ñ✝④❬✉❭✇②❺②⑤⑦⑧❯③⑨⑥ ⑤ ✇②①➘❮❊③❯✇②③▲❺②④✡⑤ ❶✝✉❢❶❛✇ ✉❢❶✝❮ ×❈❷❊✇ P SLSL(Z)(Z)= SL (Z)/ ± I ✕ 2①❈×③▲❺②2③ I ❮❊③▲❶❈❒❢✇②③⑨⑥✯✇②①❈③➪⑤⑦❮❊③▲✕ ❶❛✇②⑤ ✇rÓ➸④❬✉❭✇②❺②⑤ ↔➸❒❢Ò❫❺◆✉❢1✳❶ ❂ 2 ❻ ↔ ➆✒↕❦➙✰➛✡➜✡➛✍➇❿➙➩➨✹➌✡➊ ♥ ❥➅③❯✇ N 1 Û➭③➮✉❢❶ï⑤ ❶❛✇②③▲➄❙③▲❺⑨❸❊✉❢❶✝❮➘Ñ ③❯✇ 2 2 Γ0 (N ) = { 2 a c 2 b d 2 ∈ P SL2 (Z) | c ≡ 0 (mod N ) } ⊂ P SL2 (Z) Û➭③➮✉✡⑧❯❒❙❶❈➄❙❺②❷❈③▲❶✝⑧❯③Ú⑥✏❷❈Û❈➄❙❺②❒❙❷❈×➘❒❢Ò P SL (Z) ❻ ❥➅③❯✇ Û➭③✈✉✂❶❈❒❙❶❊❽î❶❈③▲➄❛✉❭✇②⑤ ➤❙③➸⑤ ❶❛✇②③▲➄❙③▲❺⑨❻✮Ù ❳❬➻⑨➱❭❇❊❚ ❩❭❉➪➾❯➻❭❉●❳ f ➻r➾➘➶✬❘❯❋ ➴➢▼☞➼ k 1 ➻❭❖ k ⑦ ⑤ ☎ ⑥ ✉✡⑧❯❒❙④✡×❈Ñ ③❯↔☞❽î➤❭✉❢Ñ ❷❈③⑨❮➸①❈❒❙Ñ ❒❙④✡❒❙❺②×❈①❈⑤⑦⑧☎Ò➹❷❈❶✝⑧●✇②⑤ ❒❙❶ï❒❙❶ H ⑥②✉❭✇②⑤⑦⑥rÒ➹Ó☞⑤ ❶❈➄✟✇②①❈③ÚÒ➹❒❙Ñ Ñ ❒ ✕ ⑤ ❶❈➄ Γ (N ) ✇②❺◆✉❢❶✝⑥rÒ➹❒❙❺②④❬✉❭✇②⑤ ❒❙❶➸❺②❷❈Ñ ✗③ ❋ Ò➹❒❙❺ z ∈ H ✉❢❶✝❮ a b ∈ Γ (N ).

R✰❽ ➂✹✇ ❮❊③▲➄ P = B ì€✇②①❈③ i❽➽✇②①➸❹✯③❯✇✏✇②⑤❞❶☞❷❈④ùÛ➭③▲❺♥❒❢Ò X í✬✉❢❶✝❮❬⑤ ✇◆⑥✯❺②③⑨⑧❯⑤ ×❈❺②❒❊⑧▲✉❢Ñ✝❺②❒☞❒❢✇◆⑥✯✉❢❺②③➃✉❢Ñ ➄❙③▲Û❈❺◆✉❢⑤⑦⑧ ⑤ ❶❛✇②③▲➄❙③▲❺◆⑥ ✕ ⑤ ✇②①❃✉❢Û✝⑥✏❒❙Ñ ❷❊✇②③Ú➤❭✉❢Ñ ❷❈③ p ì❰✲✍③▲Ñ ⑤ ➄❙❶❈③➢í●❻ ❥➅③❯✇ ➁Ú✉❢Ñ ¯ Û➭③✟✇②①❈③➐✉❢Û✝⑥✏❒❙Ñ ❷❊✇②③❬➁Ú✉❢Ñ ❒❙⑤⑦⑥➃➄❙❺②❒❙❷❈×➅❻✡è♥①❈③▲❶❄✇②①❈③▲❺②③✡⑤⑦⑥➪✉❢❶ ❽❨✉❙❮❊⑤⑦⑧ Ô❯✇◆✉❢Ñ ③➮➁ÚG✉❢Ñ ❒❙=⑤⑦⑥♥❺②③▲×❈❺②(③⑨Q/Q) ⑥✏③▲❶❛✇◆✉❭✇②⑤ ❒❙❶ i p i i/2 ⑩❨❶➘×✝✉❢❺✏✇②⑤⑦⑧❯❷❈Ñ⑦✉❢❺⑨❸☞❷✝⑥✏⑤ ❶❈➄ ❻ X i ¯ ρiX, : G → GL(Het (X, Q )). ρiX, ( ❈❺②❒❙Û ) ❸ ✕ ③➪❮❊③❑✔✝❶❈③➃✇②①❈③ i❽➽✇②①✂ì➠⑧❯❒❙①❈❒❙④✡❒❙Ñ ❒❙➄❙⑤⑦⑧▲✉❢рí L❽❨⑥✏③▲❺②⑤ ③⑨⑥✬❒❢Ò ❉ p ↔ ➆✒↕❦➙✰➛✡➜✡➛✍➇❿➙➞➝❉➌✡➊ ♥ è♥①❈③ ❽➽✇②①❬ì➠⑧❯❒❙①❈❒❙④✡❒❙Ñ ❒❙➄❙⑤⑦⑧▲✉❢рí L❽❨⑥✏③▲❺②⑤ ③⑨⑥ L(X, s) = L(H (X, ¯ Q ), s) ⑤⑦⑥✧❮❊③❑✔✝❶❈③⑨❮➘Û☞Ó➐✇②①❈③✿❀✓❷❈Ñ ③▲❺✧i×❈❺②❒❊❮❊❷✝⑧●✇ ❮❊③❯✇ (1 − ρ (❉❈❺②❒❙Û )p ) × (⑥✏⑤ ④✡⑤ Ñ⑦✉❢❺♥Ò➠✉❙⑧●✇②❒❙❺☎✉❭✇ p = ) L (X, s) := ①❈③▲❺②③➃✇②①❈③➪×❈❺②❒❊❮❊❷✝⑧●✇✧❺②❷❈❶✝⑥✧❒❭➤❙③▲❺✯➄❙❒☞❒❊❮➸×❈❺②⑤ ④✡③⑨⑥▲✇❻ ✲✍⑤ ➄❙❺②③⑨⑥②⑥✏⑤ ❶❈➄✝❸ ✕ ③Ú①✝✉➢➤❙③ ✕ ⑥✏⑤ ④✡⑤ Ñ⑦✉❢❺♥Ò➠✉❙⑧●✇②❒❙❺☎✉❭✇ p = ).

Q Ò➹❒❙❺➸✇②①❈③ ❺◆✉❭✇②⑤ ❒❙❶✝✉❢Ñ❞③▲Ñ Ñ ⑤ ×❊✇②⑤⑦⑧➪⑥✏❷❈❺✏Ò➠✉❙⑧❯③⑨⑥ S Ò➹❒❙❺♥✇②①❈③➮✉❢Û➭❒❭➤❙③Ú④✡❒❊❮❊❷❈Ñ⑦✉❢❺✧➄❙❺②❒❙❷❈×✝⑥▲❻ ➄✁➅❉➆✝➇❿➈✝➆✔➉➬➷✹➌✘➯ ♥ tì ⑥✴③▲❺②⑤ Ñ ❿Ñ ③ ➍ ➝➞➟ ⑩❜❶ ⑦ íÚ❍❞▼❈❘✯■●❋ ✰➃❉●❋ ➴❢❋➠➱ù➺✓❩❭❚ ❩❙ã❯❋€ä✕❆➭❩❭❇✟➼➹▼☞❉②❘◆❘➠➾❯➻❭❚ ➱➢■✧❑◆➻❭❖❈■●➼➽❉●❇✝❑❯➼❨❘◆➱ ❋€❖✷❍❞▼❈❘◆➻❭❉②❘❯➳ ❳ ➲ ▼❈❩❭➚❭❘✟❃❘ ✰●ø❞❚ ❋➠❑❯❋€➼♥➱❙❰❘ ☎✓❖✝❋€❖☞➴➸❘ ✁ ❇✝❩❭➼➽❋➠➻❭❖❈■➪➻❭➚❭❘❯❉ ◗❏❇❊❉●➼➹▼❈❘❯❉●❳❬➻❭❉②✄❘ ❜✴➼➹▼❈❑❘ ✂❬❩❭❉②❘ ❩❭❚€❚✕❳❬➻⑨➱❭❇❊❚ ❩❭❉ ➐ ❍❞➐➸➵ ▼❈❩❭➼♥❋⑦❲■ ❜✓➼➹▼❈❘❯❉②❘➮❋⑦■ù❩➸❑❯❇☞■➽ø❬➾❯➻❭❉●❳ f ➻r➾Ú➶✬❘❯❋ ➴➢▼☞Q➼ 4➐ ➻❭❖ Γ ■●❇✝❑◆▼ï➼➹▼❈❩❭➼ Γ L(X, s) = L(f, s).

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Mathematisches Institut. Georg-August- Universitat Gottingen. Seminars Summer Term 2004 by Yuri Tschinkel


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