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Mathematisches Institut. Georg-August- Universitat by Yuri Tschinkel

By Yuri Tschinkel

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Extra resources for Mathematisches Institut. Georg-August- Universitat Gottingen. Seminars Summer Term 2004

Example text

RÓ➐❒❢Ò ❷✝⑥✏⑤ ❶❈➄✡✇②①❈➄③ ❳☞③▲❺②❺②③●✑ê ❉✝✉❢Ñ ✇②⑤ ❶❈➄❛⑥ ❯⑧ ❺②⑤ ✇②③▲❺②⑤ ❒❙❶➅❻➞Ù☎Ñ⑦⑥✏❒✆✲✍⑤ ③▲❷❈Ñ ③❯Ò➠✉❢⑤ ✇♥✉❢❶✝❮❬t✈✉❢❶❈❒❙①✝✉❢❺②④❬✉➢Ó☞❷❈④ V③ ⑧ ➺ ⑩❜❶ ⑦ ✉❢❶✝❮✟ë➃⑤❧③ ➍ ✷➟ ⑦ ①✝✉➢➤❙③ ③⑨⑥r✇◆✉❢Û❈Ñ ⑤⑦⑥✏①❈③⑨❮✟✇②①❈③➃④✡❒❊❮❊❷❈Ñ⑦✉❢❺②⑤ ✇rÓ✟❒❢Ò V ❷✝⑥✏⑤ ❶❈➄❏❄ ⑤ Ñ ③⑨⑥✯④✡③❯✇②①❈❒❊❮❏❻✴è♥①❈③➃✉❢Û➭❒❭➤❙③☎×❈❺②❒☞❒❢Ò➅❒❢Ò ❳❊✉❢⑤ ✇②❒❬✉❢❶✝❮➐ë✓❷❈⑤➅⑤⑦⑥✧④✡❒❙❺②③Ú➄❙③▲❒❙④✡③❯✇②❺②⑤⑦⑧➃✉❢Ñ ❒❙❶❈➄ù✇②①❈③➪Ñ ⑤ ❶❈③➪❒❢Ò✴è➞✉❭✇②③✗✖ ⑥✧⑧❯❒❙❶✙✘r③⑨⑧●✇②❷❈❺②③❙❻ ♣✤❼ ☞ r✰❽ ➂✹✇ Ï ❻✧Ý✧③⑨⑧❯③▲❶❛✇②Ñ Ó❙❸✙❳❊⑧◆①❩❺❊✇✏✇✚③ ➜ →✝➽ ⑩❜❶ ⑦ ①✝✉❙⑥➅✉❢×❈×❈Ñ ⑤ ③⑨❮✍✇②①❈③✬⑧❯❒❙❶✝⑥r✇②❺②❷✝⑧●✇②⑤ ❒❙❶Ú❒❢✳Ò ❳❊⑧◆①❈❒☞③▲❶➃✇②❒♥✇ ✕ ⑤⑦⑥r✇②③⑨❮ ⑥✏③▲Ñ Ò€♠❽ ✔✝Û➭③▲❺➮×❈❺②❒❊❮❊❷✝⑧●✇◆⑥ ①❈③▲❺②③ ⑤⑦⑥Ú✇②①❈③➐❶✝✉❭✇②❷❊❽ ❺◆✉❢Ñ✴×❈❺②✟❒ ✘r③⑨⑧●✇②⑤ ❒❙❶➍✉❢❶✝❮ (Sπ ⑤⑦⑥➮✉ï,❶❈pr) ❒❙❶❊❽➽✇②×❺②⑤ ➤☞⑤⑦✉❢(SÑ✬✉❢❷❊✇②❒❙,④✡π❒❙◦❺②×❈pr) ①❈⑤⑦⑥✏④ ✕ ❒❢Ò P pr⑤ ❶❛✇②③▲❺◆⑧◆①✝✉❢❶❈➄❙⑤ ❶❈➄ ❻➐Ð✬①❈❒☞❒❛⑥✏⑤ ❶❈➄ ✉❢×❈×❈❺②❒❙×❈❺②⑤⑦✉❭✇②③▲Ñ Ó❙❸❏①❈③❬❒❙Û❊✇◆✉❢⑤ ❶❈③⑨❮✈Ò➹❒❙❷❈❺➪④✡❒❙❺②③✟❺②⑤ ➄❙⑤⑦❮➍Ð✯✉❢Ñ⑦✉❢Û❈⑤ ê 0, 1, ∞ ë✬✉❢❷✮✇②①❈❺②③▲③❯Ò➹❒❙Ñ⑦❮❈⑥➐❒❭➤❙③▲π❺ Q ✉❙⑥②⑥✏❒❊⑧❯⑤⑦✉❭✇②③⑨❮➊✇②❒➊⑧❯❷✝⑥✏× Ò➹❒❙❺②④❬⑥➐❒❢Ò ✕ ③▲⑤ ➄❙①❛✇ 4 ✉❢❶✝❮✮Ñ ③▲➤❙③▲Ñ ✉❢❶✝❮ ❻ ➆☎⑤⑦⑥➸④✡③❯✇②①❈❒❊❮❀⑥✏①❈❒❙❷❈Ñ⑦❮ ✕ ❒❙❃❺ ❂➈Ò➹❒❙❺➸❒❢✇②①❈③▲❺➸➄❙❺②❒❙❷❈×✝⑥➐Ñ ⑤⑦⑥r✇②③⑨❮✮⑤ ❶ 10, 17, 21 è♥①❈③▲❒❙❺②③▲④ ❾❊❻ ❼❈❻ 73 ➐ ➛ ➙ ❃➟ ➒î➛ ❺ ➜➞➑ ➺ ➔❛✜➒ ➽➞➛➅➜ ❋ • è♥①❈⑤⑦⑥❃④✡③❯✇②①❈❒❊❮ ①✝✉❙⑥ïÛ➭③▲③▲❶➣❮❊③▲➤❙③▲Ñ ❒❙×➭③⑨❮✩Û☞Ó Ð✯✉❢❶✝❮❊③▲Ñ⑦✉❙⑥▲❸➪❮❊③➍Ñ⑦✉❀ð➃⑥②⑥②✉✮✉❢❶✝➌ ❮ ⑥✍⑤ Ñ Ñ ③▲➄❛✉❙⑥ ③ ➐ ➜ ❺→➼➻⑨❢❸ ⑩❜❶ ⑦ ❸❭✉❢❶✝❮ ✕ ❒❙❃❺ ❂❊⑥❞Ò➹❒❙❺➅❒❙❶❈③❯❽î×✝✉❢❺◆✉❢④✡③❯✇②③▲❺❞Ò➠✉❢④✡⑤ Ñ ⑤ ③⑨⑥❏❒❢Ò➭Ð✯✉❢Ñ⑦✉❢Û❈⑤ ê❛ë✬✉❢❷➪⑧❯❒❙④✡×❈Ñ ③❯✇②③✓⑤ ❶❊❽ ✇②③▲❺◆⑥✏③⑨⑧●✇②⑤ ❒❙❶➐✇②①❈❺②③▲③❯Ò➹❒❙Ñ⑦❮❈⑥▲❻❫è♥①❈③Úñ✓⑤⑦⑧▲✉❢❺◆❮❛✑ê ❉❈❷✝⑧◆①✝⑥✯❮❊⑤ ❁❞③▲❺②③▲❶❛✇②⑤⑦✉❢Ñ❞✬③ ❫❛❷✝✉❭✇②⑤ ❒❙❶✝⑥✬❒❢Ò✕✇②①❈③⑨⑥✏③✍Ò➠✉❢④✡⑤ Ñ ⑤ ③⑨⑥ ✇②❷❈❺②❶ï❒❙❷❊✇✧✇②❒❬Û➭③ù➁Ú✉❢❷✝⑥②⑥✧①☞Ó☞×➭③▲❺②➄❙③▲❒❙④✡③❯✇②❺②⑤⑦⑧➃⑥✏③▲❺②⑤ ③⑨⑥▲❻✓Ù✯✇✍⑧❯❒❙❶❈⑤ Ò➹❒❙Ñ⑦❮➘×➭❒❙⑤ ❶❛✇◆⑥➮ì➹⑤➽❻ ③❙❻ ❸✝✉❙⑥②⑥✏⑤ ➄❙❶❈⑤ ❶❈➄ ⑥✏×➭③⑨⑧❯⑤⑦✉❢Ñ♥➤❭✉❢Ñ ❷❈③⑨⑥✟✇②❒✂✇②①❈③ï×✝✉❢❺◆✉❢④✡③❯✇②③▲❺●í●❸✬⑧❯③▲❺✏✇◆✉❢⑤ ❶➊❒❙❶❈③❯❽î×✝✉❢❺◆✉❢④✡③❯✇②③▲❺➮Ò➠✉❢④✡⑤ Ñ ⑤ ③⑨⑥✡➄❙⑤ ➤❙③➘❺②⑤⑦⑥✏③➸✇②❒ ❺②⑤ ➄❙⑤⑦❮❃Ð✯✉❢Ñ⑦✉❢Û❈⑤ ê❛ë✬✉❢❷➐✇②①❈❺②③▲③❯Ò➹❒❙Ñ⑦❮❈⑥▲❸✝✉❢❶✝❮➐✇②①❈③▲⑤ ❺☎④✡❒❊❮❊❷❈Ñ⑦✉❢❺②⑤ ✇rÓ➐⑧▲✉❢❶ïÛ➭③➪③⑨⑥r✇◆✉❢Û❈Ñ ⑤⑦⑥✏①❈③⑨❮➘⑥r✇②❷✝❮❊Ó☞⑤ ❶❈➄ ×➭③▲❺②⑤ ❒❊❮❈⑥♥❒❢Ò➞✇②①❈③ÚÒ➠✉❢④✡⑤ Ñ ⑤ ③⑨⑥▲❻ ↕❯➙ ✜➒ ➔❛✭➓ → ➔☞➜➞➟➠➔❊✜➒ ➔➾➽➅➔❈→❙➛✝➏❫➟➠➔ ➙ ➑ ❋ • è♥①❈⑤⑦⑥✡④✡③❯✇②①❈❒❊❮✮①✝✉❙⑥✡❶❈❒❢✇➐Ó❙③❯✇❬×❈❺②❒❊❮❊❷✝⑧❯③⑨❮✷❺②③⑨⑥✏❷❈Ñ ✇◆⑥▲❻ ⑩ ✕ ❒❙❷❈Ñ⑦❮✷Ñ ⑤ ❂❙③➘✇②❒➍④✡③▲❶❛✇②⑤ ❒❙❶➊✇②①❈⑤⑦⑥ ✉❢×❈×❈❺②❒❛✉❙⑧◆①❃①❈❒❙×❈⑤ ❶❈➄➐✇②①✝✉❭✇➪⑥✏❒❙④✡③▲❒❙❶❈③ù④✡⑤ ➄❙①❛✇ ✕ ❒❙❃❺ ❂➘❒❙❶❄⑤ ✇⑨❻➮è♥①❈③ïì➽➁➃❺②⑤ ✉✡✇②①➭íù❋€❖✝➼❨❘❯❉●❳❬❘◆➱❭❋➠❩❭➼❨❘ ❦❊❩❙❑◆➻❙ã❯❋➠❩❭❖➐❒❢Ò❫✉➐Ð✯✉❢Ñ⑦✉❢Û❈⑤ ê❛ë✬✉❢❷➸✇②①❈❺②③▲③❯Ò➹❒❙Ñ⑦❮ ⑤⑦⑥☎❮❊❑③ ✔✝❶❈③⑨❮➸Û☞Ó X Γ1 (6) J 2 (X) = P1 Γ1 (6) 1 H 1,2 (X) ⊕ H 0,3 (X) .

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 ).

Az + b f( ) = (cz + d) f (z) c d cz + d Ù ❑❯❇☞■➽ø➪➾❯➻❭❉●❳ f ➻r➾♥➶✬❘❯❋ ➴➢▼☞➼ k ➻❭❖ Γ (N ) ⑤⑦⑥✓✉✍④✡❒❊❮❊❷❈Ñ⑦✉❢❺❫Ò➹❒❙❺②④➣➤❭✉❢❶❈⑤⑦⑥✏①❈⑤ ❶❈➄Ú✉❭✇✓✉❢Ñ Ñ❈⑧❯❷✝⑥✏×✝⑥ ❒❢Ò Γ (N ) ❻✴⑩❨❶ï×✝✉❢❺✏✇②⑤⑦⑧❯❷❈Ñ⑦✉❢❺⑨❸ f ①✝✉❙⑥✧❧✉ ❉❈❒❙❷❈❺②⑤ ③▲❺✧③❯↔❊×✝✉❢❶✝⑥✏⑤ ❒❙❶✂ì➠✉❭✇ ∞í●❸✝✉❢❶✝❮ ✕ ③➪⑧▲✉❢❶ ✕ ❺②⑤ ✇②③ ⑤ ✇②① q = e . f (q) = n = 1 a (n) q ✕ è♥①❈③ L❽❨⑥✏③▲❺②⑤ ③⑨⑥✬❒❢Ò✴✉✡⑧❯❷✝⑥✏×➸Ò➹❒❙❺②④ f ⑤⑦⑥✧❮❊❑③ ✔✝❶❈③⑨❮➘Û☞Ó 2 0 k 0 0 0 ∞ n f L(f, s) := 2πiz ∞ af (n) ns n=1 ❸☎❒ ③✂⑥r✇◆✉❭✇②③❄✇②①❈③✂❺②③⑨⑥✏❷❈Ñ ✇ï❒❢❏Ò ❄ ⑤ Ñ ③⑨⑥➘③❯✇❃✉❢Ñ➽❻ ①❈⑤⑦⑧◆①❀×❈❺②❒❭➤❙③⑨⑥❬✇②①❈③ ⑧❯❒❙❶✙✘r③⑨⑧●✇②❷❈❺②③✂❒❢Ò ☞①❈⑤ ④ù❷❈✕ê❺◆✉✡✕ ✉❢❶✝❮➘è➞✉❢❶❈⑤ Ó❛✉❢④❬✉ù⑤ ❶➘✇②①❈③➮✉✙✉❬❺②④❬✉❭✇②⑤ ➤❙③❙❻ ✕ ❸ ③ ➉ ❼ ⑤ ➶ ⑦ ❸✑③ ➪ ➐ ⑧ ➉ ⑩ ♠ ⑦ í❭✾✕❘❯➼ E ã◆❘➍❩❭❖ ➄✁➅❉➆✝➇❿➈✝➆✔➉➬➨✹➌❲➝ ♥ tì ❄ ⑤ Ñ ③⑨⑥➘③❯✇ï✉❢Ñ➽❻ ③ ❼ ➟ ❺ ⑤ ➶ ⑦ ✑ ❘❯❚€❚ ❋ ø❞➼➽❋➠❑➐❑❯❇❊❉●➚❭❘❬➻❭➚❭❘❯❉ Q ➐ ❍❞▼❈❘❯❖ E ❋⑦■ù❳❬➻⑨➱❭❇❊❚ ❩❭❲❉ ❜♥➼➹▼❈❩❭➼♥❋⑦■❲❜✯➼➹▼❈❘❯❉②❘✟❋⑦■✡❩➘❑❯❇☞■➽ø ➍ ❖❞❘❯➶✹➎✍➾❯➻❭❉●❳ ➻r➾➪➶✬❘❯❋ ➴➢▼☞➼ 2 = 1 + 1 ➻❭❖ Γ (N ) ■●❇✝❑◆▼ï➼➹▼❈❩❭➼ f ❋❘❜ L(E, s) = L(f, s) ➐ ➐ a(n) = a (n) ∀n ➐ ➑ù❘❯❉②❘ ❋⑦■➪➼➹▼❈❘✟❑◆➻❭❖❞➱❭❇✝❑❯➼❨➻❭❉ù➻r➾ E ➐ N ➟ ➆✔➉➡➠❿➈✔➢❒➨✹➌☛➨ ♥ è♥①❈③ù×❈❺②❒☞❒❢Ò✴❒❢✵Ò ❄ ⑤ Ñ ③⑨⑥✍③❯✇Ú✉❢Ñ➽❻✧⑤⑦⑥☎✇②❒➸⑧❯❒❙④✡×✝✉❢❺②③Ú✇②①❈③➮✇ ❒ ❽❨❮❊⑤ ④✡③▲❶✝⑥✏⑤ ❒❙❶✝✉❢Ñ ➁Ú✉❢Ñ ❒❙⑤⑦⑥✓❺②③▲×❈❺②③⑨⑥✏③▲❶❛✇◆✉❭✇②⑤ ❒❙❶✝⑥✴✉❢❺②⑤⑦⑥✏⑤ ❶❈➄➪Ò➹❺②❒❙④ E ✉❢❶✝❮ f ❻❫⑩îÒ❏✇②①❈③⑨⑥✏③☎✇ ✕ ❒ 2❽❨❮❊⑤ ✕ ④✡③▲❶✝2⑥✏⑤ ❒❙❶✝✉❢Ñ❞➁Ú✉❢Ñ ❒❙⑤⑦⑥ ❺②③▲×❈❺②③⑨⑥✏③▲❶❛✇◆✉❭✇②⑤ ❒❙❶✝⑥✴✉❢❺②③✧✬③ ❫❛❷❈⑤ ➤❭✉❢Ñ ③▲❶❛✇✬④✡❒❊❮ ì➹❒❙❺✓④✡❒❊❮ í●❸❢✇②①❈③▲❶❬✇②①❈③▲Ó✡✉❢❺②③✧✬③ ❫❛❷❈⑤ ➤❭✉❢Ñ ③▲❶❛✇⑨❸☞✉❢❶✝❮ ③⑨⑥r✇◆✉❢Û❈Ñ ⑤⑦⑥✏①➸✇②①❈✆③ ❳☞①❈⑤ ④ù❷❈❺◆✉➢ê❊è➞✉❢❶❈⑤ Ó❛✉❢④❬✉ù⑧❯❒❙3✙❶ ✘r③⑨⑧●✇②❷❈❺②③Ú⑤ ❶➘5 ✇②①❈③➮✙✉ ✉❬❺②④❬✉❭✇②⑤ ➤❙③❙❻ ❳ 0 f ❻❈➣ r✰❽ ➂✹✇ ☞ ➸❬♥➊➉ ➽❧➔➲→➈➛➅➜➞➝ ❺ ➔❊➓❭➟➹➒✒➓➈➛❜➒☎➑⑨➟ ➙ ➭➝ ❺ ➔❊➓♥➒✡➔✑➓❏➒▲➓✭➔●→➈➔ ❺→➔ K3 ➑⑨➝✕➓✭➒î➔❈→✔➔☞➑ ❸☎❒ ③ ⑤ Ñ Ñ✧✉❙❮❈❮❊❺②③⑨⑥②⑥Ú✇②①❈③➘④✡❒❊❮❊❷❈Ñ⑦✉❢❺②⑤ ✇rÓ✂❒❢Ò➃❮❊⑤ ④✡③▲❶✝⑥✏⑤ ❒❙❶ 2 Ð✯✉❢Ñ⑦✉❢Û❈⑤ ê❛ë✬✉❢❷ ➤❭✉❢❺②⑤ ③❯✇②⑤ ③⑨⑥▲❸ ❶✝✉❢④✡③▲Ñ ✕ Ó❙❸ ✕ K3 ✕ ⑥✏❷❈❺✏Ò➠✉❙⑧❯③⑨⑥▲❸❊❮❊③❑✔✝❶❈③⑨❮➘❒❭➤❙③▲❺ Q ❻ ❥➅③❯✇ Û➭③✟✉❢❶✈✉❢Ñ ➄❙③▲Û❈❺◆✉❢⑤⑦⑧ ⑥✏❷❈❺✏Ò➠✉❙⑧❯③❙❝❻ ❥➅③❯✇ N S(X) Û➭③➮✇②①❈➄③ ❸☎Ô▲❺②❒❙❶☞ê✳❳☞③▲➤❙③▲❺②⑤➅➄❙❺②❒❙❷❈× ❒❢Ò X ➄❙③▲X❶❈③▲❺◆✉❭✇②③⑨❮ïÛ☞Ó✈✉❢Ñ ➄❙③▲Û❈❺◆✉❢K3 ⑤⑦⑧➪⑧❯Ó❊⑧❯Ñ ③⑨⑥✍❒❙❶ ❻Úè♥①❈③▲❶ ⑤⑦⑥➃✉❬Ò➹❺②③▲✆③ ✔✝❶❈⑤ ✇②③▲Ñ Óï➄❙③▲❶❊❽ ③▲❺◆✉❭✇②③⑨❮➍✉❢Û➭③▲Ñ ⑤⑦✉❢❶➈➄❙❺②❒❙❷❈×➅❻❄è♥①❈③ Z❽î❺◆✉❢✳❶ ❂✂❒❢Ò NXS(X) ⑤⑦⑥✟N⑧▲✉❢S(X) Ñ Ñ ③⑨❮✂✇②①❈↔③ ➣✯❋➠❑◆❩❭❉②➱✂❖✝❇❊❳❬ã◆❘❯❉Ú❒❢Ò ✉❢❶✝❮✈❮❊③▲❶❈❒❢✇②③⑨❮❃Û☞Ó ❻ ❳☞⑤ ❶✝⑧❯③ S(X) ⊆ H (X, Z) ∩ H (X, R) ❸❞✇②①❈③ùñ✓⑤⑦⑧▲✉❢❺◆❮ ✥ X ❶☞❷❈④ùÛ➭③▲❺ ρ(X) ⑤⑦⑥❬✉❭✇❬ρ(X) ④✡❒❛⑥r✇ 20 ❻✷Ù NK3 ⑥✏❷❈❺✏Ò➠✉❙⑧❯③ ⑤⑦⑥✡✬③ ❫❛❷❈⑤ ×❈×➭③⑨❮ ⑤ ✇②①✷✇②①❈③ï×➭③▲❺✏Ò➹③⑨⑧●✇ Û➭③ ×✝✉❢⑤ ❺②⑤ ❶❈➄✈⑤ ❶✝❮❊❷✝⑧❯③⑨❮➍Û☞Ó✂✇②①❈③➸⑤ ❶❛✇②③▲❺◆⑥✏③⑨⑧●✇②⑤ ❒❙❶➍×✝✉❢⑤ ❺②⑤ ❶❈➄✝✽❻ X❥➅③❯✇ T (X) := N✕ S(X) ✇②①❈③✍❒❙❺✏✇②①❈❒❙➄❙❒❙❶✝✉❢Ñ❈⑧❯❒❙④✡×❈Ñ ③▲④✡③▲❶❛✇✯❒❢Ò N S(X) ⑤ ❶ H (X, Z) ✕ ⑤ ✇②①➐❺②③⑨⑥✏×➭③⑨⑧●✇✬✇②❒➪✇②①❈⑤⑦⑥✬×➭③▲❺✏Ò➹③⑨⑧●✇ ×✝✉❢⑤ ❺②⑤ ❶❈➄✝❻☎è♥①❈③▲❶ ⑤⑦⑥✍✉❬Ñ⑦✉❭✇✏✇②⑤⑦⑧❯③ù❒❢Ò✴❺◆✉❢✳❶ ❂ ❸❞✉❢❶✝❮ï⑤⑦⑥✍⑧▲✉❢Ñ Ñ ③⑨❮➘✇②①❈③ù➄❙❺②❒❙❷❈×❃❒❙❺ ✇②①❈③➪Ñ⑦✉❭✇✏✇②⑤⑦⑧❯③➪❒❢Ò✕✇②❺◆T✉❢(X) ❶✝⑥②⑧❯③▲❶✝❮❊③▲❶❛✇◆✉❢Ñ❏⑧❯Ó❊⑧❯Ñ ③⑨⑥♥❒❙❶ X22❻ − ρ(X) ❸☎❒ ③ ✕ ⑤ Ñ Ñ✕⑥✏⑤ ❶❈➄❙Ñ ③Ú❒❙❷❊✇☎✉✡⑥✏×➭③⑨⑧❯⑤⑦✉❢Ñ✕⑧❯Ñ⑦✉❙⑥②⑥✯❒❢Ò K3 ⑥✏❷❈❺✏Ò➠✉❙⑧❯③⑨⑥▲❻ ✕ ✕ ↔ ➆✒↕❦➙✰➛✡➜✡➛✍➇❿➙➙↕✰➌✡➊ ♥ Ù K3 ⑥✏❷❈❺✏Ò➠✉❙⑧❯③ X ⑤⑦⑥ï⑥②✉❢⑤⑦❮✮✇②❒➊Û➭③➍✉ ■●❋€❖☞➴❢❇❊❚ ❩❭❉❄ì➹❒❙❺➍❃❘ ✰❙➼➽❉②❘❯❳❬❩❭❚ í➐⑤ Ò ❻ ρ(X) = 20 ❸☎❒ ③✡⑧❯❒❙❶✝⑥✏⑤⑦❮❊③▲❺➪✉➘⑥✏⑤ ❶❈➄❙❷❈Ñ⑦✉❢❺ K3 ⑥✏❷❈❺✏Ò➠✉❙⑧❯③ X ❮❊❑③ ✔✝❶❈③⑨❮❄❒❭➤❙③▲❺ Q ❻ùè♥①❈③ L❽❨⑥✏③▲❺②⑤ ③⑨⑥✍❒❢Ò ✕✈✕ ⑦ ⑤ ✧ ⑥ ❊ ❮ ③ ❑ ✔✝❶❈③⑨❮➘Û☞Ó X ➆ 2 1,1 ⊥ H 2 (X,Z) 2 ❙➄ ❒☞❒❊❮ P (p ) ①❈③▲❺②③ï✇②①❈③❄×❈❺②❒❊❮❊❷✝⑧●✇➸❺②❷❈❶✝⑥❬❒❭➤❙③▲❺➐✉❢Ñ Ñ✍➄❙❒☞❒❊❮✷×❈❺②⑤ ④✡③⑨⑥❄ì ✕ ⑤ ✇②① (∗) ⑤ ❶✝❮❊⑤⑦⑧▲✉❭✇②③⑨⑥✡✇②①❈③✈Ò➠✉❙⑧●✇②❒❙❺ ✕ ⑧❯❒❙❺②❺②③⑨⑥✏×➭❒❙❶✝❮❊⑤ ❶❈➄ù✇②❒✡Û✝✉❙❮➸×❈❺②⑤ ④✡③⑨⑥◆í●❸✝✉❢❶✝❮ ❉❈❺②❒❙Û ¯ Q )) P (T ) = det(1 − T | H (X, ⑤⑦⑥✯✉❢❶➐⑤ ❶❛✇②③▲➄❙❺◆✉❢Ñ✝×➭❒❙Ñ Ó☞❶❈❒❙④✡⑤⑦✉❢Ñ✝❒❢Ò✕❮❊③▲➄❙❺②③▲③ 22 ✕ ①❈❒❛⑥✏③✍❺②③⑨⑧❯⑤ ×❈❺②❒❊⑧▲✉❢Ñ❈❺②❒☞❒❢✇◆⑥✓①✝✉➢➤❙③✧✇②①❈③➃✉❢Û✝⑥✏❒❙Ñ ❷❊✇②③ ➤❭✉❢Ñ ❷❈③ ❻❫è♥①❈③✍❮❊③⑨⑧❯❒❙④✡×➭❒❛⑥✏⑤ ✇②⑤ ❒❙❶✡❒❢Ò❞✇②①❈③☎Ñ⑦✉❭✇✏✇②⑤⑦⑧❯③⑨⑥ ⑤ ❶✝❮❊❷✝⑧❯③⑨⑥ ✇②①❈③➮❮❊③⑨p⑧❯❒❙④✡×➭❒❛⑥✏⑤ ✇②⑤ ❒❙❶➘❒❢Ò✕✇②①❈③ L❽❨⑥✏③▲❺②⑤ ③⑨⑥ L(X, s) ❋ H (X, Z) = N S(X) ⊕ T (X) L(X, s) = (∗) p −s −1 p: p ∗ p 2 et 2 L(X, s) = L(N S(X) ⊗ Q , s) · L(T (X) ⊗ Q , s).

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