Mathematisches Institut. Georg-August- Universitat by Yuri Tschinkel

By Yuri Tschinkel

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Additional info for Mathematisches Institut. Georg-August- Universitat Gottingen. Seminars Summer Term 2004

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RÓ✡Û➭③⑨⑧❯❒❙④✡③⑨⑥✬✉❢❶➐③✬❫❛❷✝✉❢Ñ ⑤ ✇rÓ✡⑤ Ò✕✉❢Ñ Ñ❞⑧❯Ó❊⑧❯Ñ ③⑨⑥✬⑤ ❶ ⑤ ❶ ✕ ①❈⑤⑦⑧◆①ï⑧▲✉❙⑥✏③❙❸ ✕ ③Ú①✝✉➢➤❙③ H 1,1 (X) t3 (p) = 1 + p3 + (1 + p)p1,1 − #X(Fp ). ✉❢❺②③✍❮❊③❑✔✝❶❈③⑨❮➐❒❭➤❙③▲❺ Q ❸ ❉●❋ ➴❢❋➠➱✷➺✓❩❭❚ ❩❙ã❯❋€ä✕❆➭❩❭❇✷➼➹▼☞❉②❘◆❘➠➾❯➻❭❚ ➱➢■ ➄✁➅❉➆✝➇❿➈✝➆✔➉➬➷✹➌✜↕ ♥♥➥ ø ➼❨➻ ➱❙❩❭➼❨✄❘ ❜➪➼➹▼❈❘❯❉②❘ï❩❭❉②❘❃❩❭➼Ú❚ ❘◆❩➢■●➼ ➻❭➚❭❘❯❉ Q ➾❯➻❭❉➪➶➞▼☞❋➠❑◆▼✈➼➹▼❈❘➮❳❬➻⑨➱❭❇❊❚ ❩❭❉●❋€➼♠✂❬❋⑦■ù❘●■●➼❨❩❙ã❯❚ ❋⑦■②▼❈❘◆➱ ➐ 50 ➺ ➔❛➒✜➽➞➛➅➜➞➑ ♥ è♥①❈③▲❺②③✧✉❢❺②③✯⑥✏③▲➤❙③▲❺◆✉❢Ñ❛④✡③❯✇②①❈❒❊❮❈⑥➞Ò➹❒❙❺✴③⑨⑥r✇◆✉❢Û❈Ñ ⑤⑦⑥✏①❈⑤ ❶❈➄✍✇②①❈③♥④✡❒❊❮❊❷❈Ñ⑦✉❢❺②⑤ ✇rÓÚÒ➹❒❙❺❫❺②⑤ ➄❙⑤⑦❮ Ð✯✉❢Ñ⑦✉❢Û❈⑤ ê❛ë✬✉❢❷✂ì➠✉❢❶✝❮➸Ò➹❒❙❺✧❶❈❒❙❶❊❽î❺②⑤ ➄❙⑤⑦❮ïÐ✯✉❢Ñ⑦✉❢Û❈⑤ ê❛ë✬✉❢❷➭í✓✇②①❈❺②③▲③❯Ò➹❒❙Ñ⑦❮❈⑥✧❮❊③❑✔✝❶❈③⑨❮➘❒❭➤❙③▲❺ Q ❻ t❃③❯✇②①❈❒❊❮✂❐✗❋✴è♥①❈③✆❳☞③▲❺②❺②③●ê✑❉✝✉❢Ñ ✇②⑤ ❶❈➄❛⑥✯⑧❯❺②⑤ ✇②③▲❺②⑤ ❒❙❶➅❻ t❃③❯✇②①❈❒❊❮ Ï ❋➁❄ ⑤ Ñ ③⑨⑥✧④✡③❯✇②①❈❒❊❮❏❻ t❃③❯✇②①❈❒❊❮ï✳❼ ❋✴Ù☎Ñ ➄❙③▲Û❈❺◆✉❢⑤⑦⑧➃⑧❯❒❙❺②❺②③⑨⑥✏×➭❒❙❶✝❮❊③▲❶✝⑧❯③✡ì➠è➞✉❭✇②✗③ ✖ ⑥☎⑧❯❒❙✙❶ ✘r③⑨⑧●✇②❷❈❺②③➢í●❻ t❃③❯✇②①❈❒❊❮➘☛Õ ❋✬Ð✬❒❙❶❈⑤ Ò➹❒❙Ñ⑦❮❈⑥♥④✡③❯✇②①❈❒❊❮❏❻ t❃③❯✇②①❈❒❊❮❃✑❾ ❋❫⑩❨❶❛✇②③▲❺②④✡③⑨❮❊⑤⑦✉❭✇②✿③ ➍❛✉❙⑧❯❒❙Û❈⑤⑦✉❢❶✝⑥▲❻ ❻❍➦ ☞ r✰❽ ➂✹✇ tì ❄✂③✡⑥✏①❈❒❙❷❈Ñ⑦❮✈❺②③▲④❬✉❢❺❃❂➸✇②①✝✉❭✇➪t❃③❯✇②①❈❒❊❮❈⑥✟❐✡✉❢❶✝❮ Ï ✉❢❺②③ù③❯↔❈⑧❯Ñ ❷✝⑥✏⑤ ➤❙③▲Ñ Ó➸Ò➹❒❙❺Ú❺②⑤ ➄❙⑤⑦❮✂Ð✯✉❢Ñ⑦✉❢Û❈⑤ ê ë✬✉❢❷ï✇②①❈❺②③▲③❯Ò➹❒❙Ñ⑦❮❈⑥▲❸ ✕ ①❈⑤ Ñ ③➪✇②①❈③➮❒❢✇②①❈③▲❺➃t❃③❯✇②①❈❒❊❮❈⑥✍✉❢❺②③➮✉❢×❈×❈Ñ ⑤⑦⑧▲✉❢Û❈Ñ ③ÚÒ➹❒❙❺✍❺②⑤ ➄❙⑤⑦❮ï✉❙⑥ ✕ ③▲Ñ Ñ➞✉❙⑥♥Ò➹❒❙❺ ❶❈❒❙❶❊❽î❺②⑤ ➄❙⑤⑦❮ïÐ✯✉❢Ñ⑦✉❢Û❈⑤ ê❛ë✬✉❢❷➸✇②①❈❺②③▲③❯Ò➹❒❙Ñ⑦❮❈⑥▲❻ í ➉ ➽❧➔ ➜ ➔❛➓➢✭➓ ➔ ➓ ➭ ➔ ❺ ➒⑨➟ ➙ ➆➭➑✡→❢➓❭➟➹➒✜➔❛➓❭➟➠➛ ➙✪❋ • ❥➅③❯✇ Û➭③➮✉✡⑧❯❷✝⑥✏×➸Ò➹❒❙❺②④❰❒❢Ò ✕ ③▲⑤ ➄❙①❛✇ 4 ❒❙❶ï⑥✏❒❙④✡③ Γ (N ) ✉❢❶✝❮ ✕ ❺②⑤ ✇②③ f ⑤ ✇②① q = e ✉❢❶✝❮ a (1) = 1 f (q) = a (n)q ✕ ❳☞❷❈×❈×➭❒❛⑥✏③Ú✇②①✝✉❭✇ ♠ì ❳☞③▲❺②❺②③➢í Ò➹❒❙❺☎✉❢Ñ Ñ❏➄❙❒☞❒❊❮➸×❈❺②⑤ ④✡③⑨⑥ p ➧ t (p) = a (p) ❰ì ❉✝✉❢Ñ ✇②⑤ ❶❈➄❛⑥◆í Ò➹❒❙✐❺ ✔✝❶❈⑤ ✇②③▲Ñ Ó➸④❬✉❢❶☞Ó➐➄❙❒☞❒❊❮➐×❈❺②⑤ ④✡③⑨⑥ p ❻ è♥①❈③▲❶✩⑤ ❶☞➤❙✗❒ t❂☞⑤ (p) ❶❈➄➍=✇②①❈③➈a Ð✬(p)①❈③▲Û➭❒❢✇◆✉❢❺②③▲r ➤ ✲✍③▲❶✝⑥✏⑤ ✇rÓ❀è♥①❈③▲❒❙❺②③▲④ï❸✧✇②①❈③ ⑥✏③▲④✡⑤⑦⑥✏⑤ ④✡×❈Ñ ⑤ ✔➭⑧▲✉❭✇②⑤ ❒❙❶✝⑥ ❒❢Ò➪✇②①❈③✈✇ ✕ ❒ 2❽❨❮❊⑤ ④✡③▲❶✝⑥✏⑤ ❒❙❶✝✉❢Ñ➃➁Ú✉❢Ñ ❒❙⑤⑦⑥➐❺②③▲×❈❺②③⑨⑥✏③▲❶❛✇◆✉❭✇②⑤ ❒❙❶✝⑥➐✉❙⑥②⑥✏❒❊⑧❯⑤⑦✉❭✇②③⑨❮✷✇②❒ X ✉❢❶✝❮ f ✉❢❺②③ ③ ❫❛❷❈⑤ ➤❭✉❢Ñ ③▲❶❛✇⑨❸❈⑥✏❒✟✇②①✝✉❭✇ L(X, s) = L(f, s) ❸❈❷❈×➘✇②❒❬✆✉ ✔✝❶❈⑤ ✇②③➮❶☞❷❈④ùÛ➭③▲❺✧❒❢❡Ò ❀✓❷❈Ñ ③▲❺✧Ò➠✉❙⑧●✇②❒❙❺◆⑥▲❻ ✬ ➑ → ➔❛✜➒ ➽➞➛➅➜ ❋ ❼ ➟ ❺ ➔☞➦ ⑩❨❭❶ •❄ ⑤ Ñ ③⑨✱⑥ ✖✝×❈❺②❒☞❒❢Ò❫❒❢Ò❫✇②①❈➄③ ❳☞①❈⑤ ④ù❷❈❺◆✉➢ê❊è➞✉❢❶❈⑤ Ó❛✉❢④❬✉✟⑧❯❒❙✙❶ ✘r③⑨⑧●✇②❷❈❺②③ÚÒ➹❒❙❺✍③▲Ñ Ñ ⑤ ×❊✇②⑤⑦⑧➮⑧❯❷❈❺②➤❙③⑨⑥▲❸❈×❈❺②⑤ ④✡③ ×❈Ñ⑦✉➢Ó❙③⑨❮➮➤❙③▲❺②Ó➪⑧❯❺②❷✝⑧❯⑤⑦✉❢Ñ❛❺②❒❙Ñ ③➃ì➹Û✝✉❙❲⑧ ❂❙③⑨❮➮❷❈×✟Û☞Ó➪×❈❺②⑤ ④✡③ í●➁❻ ❉❈❒❙❺➞❺②⑤ ➄❙⑤⑦❮✡Ð✯✉❢Ñ⑦✉❢Û❈⑤ ê❛ë✬✉❢❷ ✇②①❈=❺②③▲3③❯Ò➹❒❙Ñ⑦❮❈⑥♥❒❭➤❙③▲❺ Q ❸❈✉✡⑥✏⑤ ④✡⑤ Ñ⑦✉❢❺☎⑧❯❺②⑤ ✇②③▲❺②⑤ ❒❙❶➘①✝✉❙⑥♥Û➭③▲③▲❶ï③⑨⑥r✇◆✉❢Û❈=Ñ ⑤⑦⑥✏5①❈③⑨❮❏❻ ➄✁➅❉➆✝➇❿➈✝➆✔➉➬➷✹➌☛➷ ♥ ❰ì ✲✍⑤ ③▲❷❈Ñ ③❯Ò➠✉❢⑤ ✇➞✉❢❶✝❮Út✈✉❢❶❈❒❙①✝✉❢❺②④❬✉➢Ó☞❷❈④ ③ ⑧ ➺ ⑩❜❶ ⑦ ❸▲ë✓❒❙❷❊❽✏Ð✬①❈⑤⑦✉❢❶❈➄♥ë➃✈⑤ ③ ➍ ✷➟ ⑦ í ✾✕❘❯➼ ã◆❘✡❩➸❉●❋ ➴❢❋➠➱✂➺✓❩❭❚ ❩❙ã❯❋€ä✕❆➭❩❭❇✂➼➹▼☞❉②❘◆❘➠➾❯➻❭❚ ➱ï➱❙❰❘ ☎✓❖❞❘◆➱ï➻❭➚❭❘❯❉ Q ➐ ✤➭❇⑨ø❙ø✝➻➢■▲❘ù➼➹▼❈❩❭➼ X ■▲❩❭➼➽❲ ❋⑦t■ ☎✯❘●■ù➻❭X❖❞❘✟➻r➾Ú➼➹▼❈❘✬➾❯➻❭❚€❚ ➻❭➶✴❋€❖☞➴➘➼➽➶✬➻➸❑◆➻❭❖❞➱❭❋€➼➽❋➠➻❭❖❈■▲❪ ❉②❘◆➱❭❇✝❑❯➼➽❋➠➻❭❖✂❩❭➼ 3 ❩❭❖❞➱ 7 ❜✬➻❭❉ ➍ ■✒➎➎ X ▼❈▼❈❩➢❩➢■✍■Ú➴❛➴❛➻⑨➻⑨➻⑨➻⑨➱➐ ï ➱ ❉②❘◆➱❭❇✝❑❯➼➽❋➠➻❭❖➈❩❭➼ 5 ❩❭❖❞➱➐■▲➻❭❳❬❘✍ø❞❉●❋€❳❬❘ p ≡ ±2 (mod 5) ➶✴❋€➼➹▼ t (p) ❖❞➻❭➼♥➍→➨ ➱❭❋€➚⑨X❋⑦■●❋➠ã❯❚ ❘➐❑ã ✂ 5 ➐ ❍❞▼❈❘❯❖ X ❋⑦■➪❳❬➻⑨➱❭❇❊❚ ❩❭❉ ➐ ➎ ❺ ➆❜➔❛➏➞➓❭➔❈➟➠→ ➐ ➛✝➓➢✭➓ ➔☞✦➑ ➼✬➛ ➙ ❧➜ ➔ ➙ ✔→ ➔▲➒ ➉ ➔❊✜➒ ➔➫➩ ➑❬→❙➛ ✰➙ ➠ ➔☞→❢➒⑨➝✕✭➓ ➔ ➔ • ➁➃⑤ ➤❙③▲❶✟✉➃❺②⑤ ➄❙⑤⑦❮✡Ð✯✉❢Ñ⑦✉❢Û❈⑤ ê❛ë✬✉❢❷➪✇②①❈❺②③▲③❯Ò➹❒❙Ñ⑦❮ù❒❭➤❙③▲❺ Q ❸❢⑤ ✇◆⑥❫④✡❒❊❮❊❷❈Ñ⑦✉❢❺②⑤ ✇rÓ➮⑧▲✉❢❶ùÛ➭③♥③⑨⑥r✇◆✉❢Û❈Ñ ⑤⑦⑥✏①❈③⑨❮ Û☞Ó➊⑧❯❒❙❶✝⑥r✇②❺②❷✝⑧●✇②⑤ ❶❈➄ ✉❢❶✷✉❢Ñ ➄❙③▲Û❈❺◆✉❢⑤⑦⑧➸⑧❯❒❙❺②❺②③⑨⑥✏×➭❒❙❶✝❮❊③▲❶✝⑧❯③➐✇②❒➍✉✂④✡❒❊❮❊❷❈Ñ⑦✉❢❺✡❺②⑤ ➄❙⑤⑦❮✮Ð✯✉❢Ñ⑦✉❢Û❈⑤ ê❛ë✬✉❢❷ ✇②①❈❺②③▲③❯Ò➹❒❙Ñ⑦❮➸❒❭➤❙③▲❺ Q ❻✓è♥①❈⑤⑦⑥✧✉❢×❈×❈❺②❒❛✉❙⑧◆①➸⑤⑦⑥♥Û✝✉❙⑥✏③⑨❮➸❒❙❶➘✇②①❈③➪⑧❯❒❙✙❶ ✘r③⑨⑧●✇②❷❈❺②③➪❒❢Ò❫è➞✉❭✇②③➃✇②①✝✉❭✇ ➼➹▼❈❘✡❋⑦■▲➻❭❳❬➻❭❉➽ø✝▼☞❋⑦■●❳✚ã◆❘❯➼➽➶✬❘◆❘❯❖ ➼➽➶✬➔➻ ✢☎❩❭❚ ➻❭❋⑦■✡❉②❘îø❞❉②❘●■▲❘❯❖✝➼❨❩❭➼➽❋➠➻❭❖❈■✟❋⑦■✡❋€❖❞➱❭❇✝❑◆❘◆➱❃❑ã ✂❃❩❭❖➊❩❭❚ ➴❛❘❯❲ ã❯❉②❩❭❋➠❑✡❑◆➻❭❉●❉②❘●■➽ø✝➻❭❖❞➱❙❘❯❖❞❑◆❘ ➐ ❉➞⑤ ❺◆⑥r✇ ③➸❶❈③▲③⑨❮ ✇②❒✂⑧❯❒❙❶✝⑥r✇②❺②❷✝⑧●✇✟❺②⑤ ➄❙⑤⑦❮➊Ð✯✉❢Ñ⑦✉❢Û❈⑤ ê❛ë✬✉❢❷ ✇②①❈❺②③▲③❯Ò➹❒❙Ñ⑦❮❈⑥ù❒❭➤❙③▲❺ ❸ ①❈⑤⑦⑧◆①➊✉❢❺②③ ✉❙⑥②⑥✏❒❊⑧❯⑤⑦✉❭✇②③⑨✕ ❮❬✇②❒❬④✡❒❊❮❊❷❈Ñ⑦✉❢❺✧➄❙❺②❒❙❷❈×✝⑥▲❻❫è♥①❈⑤⑦⑥♥①✝✉❙⑥✧Û➭③▲③▲❶ï❮❊❒❙❶❈③➪Û☞ÓïÐ✍✓❻ ❳❊⑧◆①❈❒☞③▲❶➅❻ Q ✕ ❶ ③ ➜ ✝→ ➽ ➯❜➯ ⑦ Ð✓Òr❻❃❹✯③⑨✉❢❷❈➤☞⑤ Ñ Ñ ➲③ ③ ➪ ➔☞➔ ➯ ➏ ⑦ ❧í ✾✕❘❯➼ ➄✁➅❉➆✝➇❿➈✝➆✔➉➬➷✹➌✘➟ ♥ ♠ì ❳❊⑧◆①❈❒☞③▲ñ ã◆❘✟❩➸❑◆➻❭❖☞➴❢❉●❇✝❘❯❖❞❑◆❘➪■●❇✝ãî➴❢❉②➻❭❇⑨ø➍➻r✇➾ ☎✓❖✝❋€➼❨❘➪➧ ❋€❖❞➱❙❃❘ ✰✟❜♥❩❭❖❞➱❬❚ ❘❯➼ C := (H/Γ) ã◆Γ❘➮⊂➼➹▼❈❘➮P❳❬SL➻⑨➱❭❇❊(Z) ❚ ❩❭❉ ❑❯❇❊❉●➚❭❘ ➐ ✾✕❘❯➼ S ã◆❘ï➼➹▼❈❘❃❇❊❖✝❋€➚❭❘❯❉◆■▲❩❭❚❏➾❯❩❭❳✟❋€❚ ✂➊➻r➾❃❘❯❚€❚ ❋ ø❞➼➽❋➠❑ ❑❯❇❊❉●➚❭❘●■❃➻❭➚❭❘❯❉ C ➐ ✾✕❘❯➼ Y := ã◆❘✟➼➹▼❈❘ù■▲❘❯❚ ➾◆❲ ☎✯ã◆❘❯❉✧ø❞❉②➻⑨➱❭❇✝❑❯➼✍➻r➾ S ❜☎❩❭❖❞➱➘❚ ❘❯➼ X ã◆❘❬❩➐■●❳❬➻⑨➻❭➼➹▼❄❉②❘●■▲➻❭❚ ❇❊➼➽❋➠➻❭❖➈➻r➾ S × S 0 ∞ f n 2πiz f n=1 3 f 3 f 3 Γ Γ Γ P1 Γ 2 ∗ Γ Γ ❜✇ ❿➁ q ✏✑✍❿✇✮✠✽✡✷➇❼✠✳②✹①✟②✎③✈✇ ➈ ❽ ❿② ➂❯➀➺②✎④❜✇❤✏✑✍❿✇❤✏✳✒ ②✎④ ❤✍ ❍❞▼❈❘❯❖ ➴❢❉②➻❭❇⑨ø➭■▲❪ Y➐ X ❋⑦■✡❩➐❉●❋ ➴❢❋➠➱✂➺✓❩❭❚ ❩❙ã❯❋€ä✕❆➭❩❭❇✂➼➹▼☞❉②❘◆❘➠➾❯➻❭❚ ➱➸➶➞▼❈❘❯❖ Γ ❻❍➭ ❋⑦■✡➻❭❖❞❘✡➻r➾➮➼➹▼❈❘✯➾❯➻❭❚€❚ ➻❭➶✴❋€❖☞➴ï■●❋ ✰ Γ(3), Γ1 (4) ∩ Γ(2), Γ1 (5), Γ1 (6), Γ0 (8) ∩ Γ1 (4), Γ0 (9) ∩ Γ1 (3).

K ❸☞✇②①❈③▲❶ K ❒❢Ò Q ✇②❒Ú③▲❶✝⑥✏❷❈❺②③✬✇②①✝✉❭✇ L(N S(XK ) ⊗ Q , s) = ζK (s − 1)ρ(XK ) = ζK (s − 1)20 ①❈③▲❺②③ ⑤⑦⑥✍✇②①❈❧③ ✲✍③⑨❮❊✱③ ❂☞⑤ ❶✝❮❄é▲③❯✇◆✉❭❽➽Ò➹❷❈❶✝⑧●✇②⑤ ❒❙❶❄❒❢Ò ✿❻ ❄✂③✡⑥✏①❈❒❙❷❈Ñ⑦❮✈❺②③▲④❬✉❢❃❺ ❂➘✇②①✝✉❭✇➃✇②①❈③ ✓ñ✕ ⑤⑦⑧▲✉❢❺◆❮➸ζ❶☞❷❈(s) ④ùÛ➭③▲❺♥⑤⑦⑥✧✉❢❶ï✉❢❺②⑤ ✇②①❈④✡③❯✇②⑤⑦⑧➃⑤ ❶☞➤❭✉❢❺②⑤⑦✉❢❶❛✇ ✕ ①❈⑤⑦⑧◆①➘K⑤⑦⑥♥➤❙③▲❺②Ó➐⑥✏③▲❶✝⑥✏⑤ ✇②⑤ ➤❙③✍✇②❒✟✇②①❈③✥✔✝③▲Ñ⑦❮➸❒❢Ò ❮❊③❑✔✝❶❈⑤ ✇②⑤ ❒❙❶➸❒❢Ò✕✉❢Ñ ➄❙③▲Û❈❺◆✉❢⑤⑦⑧✧⑧❯Ó❊⑧❯Ñ ③⑨⑥▲❻❫⑩îÒ➅❶❈❒❢✇✧✉❢Ñ Ñ❞✉❢Ñ ➄❙③▲Û❈❺◆✉❢⑤⑦⑧✧⑧❯Ó❊⑧❯Ñ ③⑨⑥✯✉❢❺②③✍❮❊③❑✔✝❶❈③⑨❮❬❒❭➤❙③▲❺✯✉✿✔❈↔❊③⑨❮ ✔✝③▲Ñ⑦❮❏❸✴③❙❻ ➄✝❻ ❸ ❸❫✇②①❈③➸×❈❺②❒❙Û❈Ñ ③▲④➡❒❢Ò✍❮❊③❯✇②③▲❺②④✡⑤ ❶❈⑤ ❶❈➄✈✇②①❈③ L❽❨⑥✏③▲❺②⑤ ③⑨⑥ L(N S(X) ⊗ Q ), s) ⑤⑦⑥ ⑥r✇②⑤ Ñ Ñ➅❒❙×➭③▲❶➅❻ Q K ❜✇ ❿➁ q ✏✑✍❿✇✮✠✽✡✷➇❼✠✳②✹①✟②✎③✈✇ ➈ ❽ ❿② ➂❯➀➺②✎④❜✇❤✏✑✍❿✇❤✏✳✒ ❻ ②✎④ ❤✍ ➳ ☎❒ ✕ ✕ ③➪⑧▲✉❢❶ï⑥r✇◆✉❭✇②③➃✇②①❈③➪④✡❒❊❮❊❷❈Ñ⑦✉❢❺②⑤ ✇rÓ➐❺②③⑨⑥✏❷❈Ñ ✇◆⑥✯Ò➹❒❙❺☎⑥✏⑤ ❶❈➄❙❷❈Ñ⑦✉❢❺ K3 ⑥✏❷❈❺✏Ò➠✉❙⑧❯③⑨⑥▲❻ ■●❇❊❉➹➾❯❩❙❑◆❘ X ▼❈❩➢■ ➄✁➅❉➆✝➇❿➈✝➆✔➉➛↕✰➌☛➨ ♥ ♠ì ❳☞①❈⑤ ❒❊❮❈✉❃✉❢❶✝❮ ⑩❨❶❈❒❛⑥✏③ ③ ➜➅✜↕ ➤❜➤ ⑦ í✟❱✯➚❭❘❯✄❉ ✂❃■●❋€❖☞➴❢❇❊❚ ❩❭❉ ❩❃❳❬➻⑨➱❙❘❯❚✯➱❙❰❘ ☎✓❖❞❘◆➱❄➻❭➚❭❘❯❉✡■▲➻❭❳❬❘➐❖✝❇❊❳❬ã◆❘❯❆❉ ☎✯❘❯❚ ➱ K ❜➃❩❭❖❞➱❃❋€➼➠✵■ ➑ù❩➢■◆■▲❘❯✹ä K3 ❪✡❘❯❋€❚✴➬⑨❘❯➼❨❩❭❲ ➾◆❇❊❖❞❑❯➼➽❋➠➻❭❖ ❋⑦■✍➴❢❋€➚❭❘❯✳❖ ❜✬❇⑨ø ➼❨➻➸❝❩ ☎✓❖✝❋€➼❨❘➪❖✝❇❊❳❬ã◆❘❯❉➮➻r➾✍❱✯❇❊❚ ❘❯❉✓➾❯❩❙❑❯➼❨➻❭❉◆❲■ ❜♥❑ã ✂ ζ(X , s) ❸ K ζ(XK , s) = ζK (s)ζK (s − 1)20 L(s − 1, χ2 )L(s − 1, χ ¯2 ) ➶➞▼❈❘❯❉②❘ ❋⑦■✡➼➹▼❈❘ùÿ✟❘◆➱❙❘◆◗❭❋€❖❞➱➸➬⑨❘❯➼❨❩❭❲ ➾◆❇❊❖❞❑❯➼➽❋➠➻❭❖➊➻r➾ ❩❭❖❞➱ ❋⑦■✟➼➹▼❈❘➝➑ù❘◆❑◆◗❙❘ L ❲ ■▲❘❯❉●❋➠❘●■➪ζ➶✴❋€➼➹(s) ▼❄❩✡■●❇❊❋€➼❨❩❙ã❯❚ ❘❈✢♥❉②➻➢■◆■▲❘❯❖❞❑◆▼❈❩❭❉②❩❙❑❯➼❨❘❯❉ χ ➐ K L(s, χ ) Ù ④✡❒❊❮❊❷❈Ñ⑦✉❢❺✟ì➠⑧❯❷✝⑥✏×➭í♥Ò➹❒❙❺②④ ⑤⑦⑥☎❶❈❒❢✇✍×❈❺②③⑨⑥✏③▲❶❛✇✍⑤ ❶ï✇②①❈⑤⑦⑥☎✇②①❈③▲❒❙❺②③▲④ ❒❢Ò✪❳☞①❈⑤ ❒❊❮❈✉➐✉❢❶✝❮ï⑩❨❶❈❒❛⑥✏③❙❻ è♥①❈③➐④✡❒❊❮❊❷❈Ñ⑦✉❢❺②⑤ ✇rÓ❄❒❢Ò☎⑥✏⑤ ❶❈➄❙❷❈Ñ⑦✉❢❺ K3 ⑥✏❷❈❺✏Ò➠✉❙⑧❯③⑨⑥➪❒❭➤❙③▲❺ Q ⑤ ❶➈❒❙❷❈❺ù⑥✏③▲❶✝⑥✏③❬①✝✉❙⑥➪Û➭③▲③▲❶➍③⑨⑥r✇◆✉❢Û❊❽ Ñ ⑤⑦⑥✏①❈③⑨❮❃Û☞➀Ó ❥➅⑤ ➤☞❶❈Ô❙❝❻ ❥➅⑤ ➤☞❶❈Ôù✉❢❶✝✉❢Ñ Ó☞é▲③⑨❮➘✇②①❈③➮✇ ✕ ❒ 2❽❨❮❊⑤ ④✡③▲❶✝⑥✏⑤ ❒❙❶✝✉❢Ñ➞➁Ú✉❢Ñ ❒❙⑤⑦⑥☎❺②③▲×❈❺②③⑨⑥✏③▲❶❛✇◆✉❭✇②⑤ ❒❙❶✝⑥ ✉❙⑥②⑥✏❒❊⑧❯⑤⑦✉❭✇②③⑨❮❬✇②❒✡✇②①❈③➪❺◆✉❢✳❶ ❂ 2 ④✡❒❢✇②⑤ ➤❙③ T (X) ❻ Ô ③ ➞ ➟ ➑ ⑤ ➶ ⑦ ❏í ✾✕❘❯➼ X ã◆❘➐❩➘■●❋€❖☞➴❢❇❊❚ ❩❭❉ K3 ■●❇❊❉➹➾❯❩❙❑◆❘➸➱❙❰❘ ☎✓❖❞❘◆➱❃➻❭➚❭❘❯❉ ➄✁➅❉➆✝➇❿➈✝➆✔➉➛↕✰➌✜↕ ♥ ❰ì ❥➅⑤ ➤☞❶❈✧ ⑦ ❋ ➪ ■ ❬ ❳ ⑨ ➻ ❭ ➱ ❊ ❇ ❚ ❭ ❩ ❉ ➐ ❍❞▼❈❩❭➼✯❋⑦❲■ ❜✓➼➹▼❈❘➪➼➽❉②❩❭❖❈■▲❑◆❘❯❖❞➱❙❘❯❖✝➼❨❩❭❚❛ø✝❩❭❉●➼ T (X) ❋⑦■Ú❳❬➻⑨➱❭❇❊❚ ❩❭❉ù❩❭❖❞➱ ❞ ❍ ❈ ▼ ❯ ❘ ❖ Q➐ X 2 K 2 L(T (Y ) ⊗ Q , s) = L(g, s) ➶➞▼❈❘❯❉②❘ g ❋⑦■➃❩✟❑❯❇☞■➽ø➮➾❯➻❭❉●❳❰➻r➾☎➶✬❘❯❋ ➴➢▼☞➼ 3 = 2 + 1 ➻❭❖➸■▲➻❭❳❬❘Ú❑◆➻❭❖☞➴❢❉●❇✝❘❯❖❞❑◆❘✧■●❇✝ãî➴❢❉②➻❭❇⑨ø Γ (N ) ➻❭❉ Γ (N ) ➶✴❋€➼➹▼❀❩➊❑◆▼❈❩❭❉②❩❙❑❯➼❨❘❯❉ ➐ ➑ù❘❯❉②❘ Γ (N ) = { a b ∈ Γ (N ) | a ≡ d ≡ 1 á✍❚ ■▲➻ù➼➹▼❈❘➃■➽ø✝❩❙❑◆❘➮➻r➾Ú❑❯❇☞■➽øù➾❯➻❭❉●❳ù■Ú➻r➾✍➶✬❘❯❋ ➴➢▼☞c➼ 3 d➻❭❖ Γ (N ) ❋⑦■Ú❘❯❳☎ø❞➼♠✂➬❜❫■▲➻ù➶✬❘ (mod N ) } ➐ ➻❭❇❙➴➢▼☞➼✯➼❨➻❬➼➽➶✴❋⑦■●➼➭➾❯➻❭❉●❳ù■ù❑ã ✂➸❩➘❑◆▼❈❩❭❉②❩❙❑❯➼❨❘❯❉ ➐ ⑩❨❶➸④✡❒❙❺②③➃❺②③▲×❈❺②③⑨⑥✏③▲❶❛✇◆✉❭✇②⑤ ❒❙❶❬✇②①❈③▲❒❙❺②③❯✇②⑤⑦⑧☎Ò➹❒❙❺②④ù❷❈Ñ⑦✉❭✇②⑤ ❒❙❶➅❸❛✇②①❈⑤⑦⑥✯✇②①❈③▲❒❙❺②③▲④❰⑧▲✉❢❶➸Û➭③➪⑥r✇◆✉❭✇②③⑨❮➸✉❙⑥ Ò➹❒❙Ñ Ñ ❒ ✕ ⑥▲❻ Ô ③ ➞ ➟ ➑ ⑤ ➶ ⑦ ✵í ✾✕❘❯➼ ã◆❘✬➼➹▼❈❘✧❑◆➻❭❳☎ø✝❩❭➼➽❋➠ã❯❚ ❘❞➾❯❩❭❳✟❋€❚ ✂➪➻r➾ ❲✏➱❭❋€❳❬❘❯❖❈■●❋➠➻❭❖❞❩❭❚ ➄✁➅❉➆✝➇❿➈✝➆✔➉➛↕✰➌☛➷ ♥ ❰ì ❥➅⑤ ➤☞❶❈❢ ❲✏❩❙➱❭❋➠✆❑ ✢☎❩❭❚ ➻❭❋⑦■✍❉②❘îø❞❉②❘●■▲❘❯❖✝➼❨❩❭➼➽❋➠➻❭❖❈■✍❩➢■◆■▲➻⑨❑❯❋➠❩❭π➼❨❘◆➱ù➼❨➻ T (X) ❩❭❖❞➱➮❚ ❘❯➼ L(π, s) ã◆2❘☎❋€➼➠■ L ❲î■▲❘❯❉●❋➠❘●■ ➐ ❍❞▼❈❘❯❖➊➼➹▼❈❘❯❉②❘➘❃❘ ✰❙❋⑦■●➼➠■➸❩❃❇❊❖✝❋ ✁ ❇✝❘➸❑❯❇☞■➽ø❃➾❯➻❭❉●❳ ➻r➾✡➶✬❘❯❋ ➴➢▼☞➼ ❜✍❚ ❘❯➚❭❘❯❚✬➼➹▼❈❘➘❑◆➻❭❖❞➱❭❇✝❑❯➼❨➻❭❉➸➻r➾ ❩❭❖❞➱➸❩ùÿ➮❋€❉●❋➠❑◆▼☞❚ ❘❯➼ ➍ ➻⑨➱❙✒➱ ➎➘❑◆▼❈❩❭❉②❩❙❑❯➼❨❘❯❉ ε(p) = g( ) ■●❇✝❑◆▼ï➼➹▼❈3❩❭➼ L(π, s) = L(g, s) ➐ π ➟ ➆✔➉➡➠❿➈✔➢♥↕✰➌✘➟ ♥ ⑩îÒ X ⑤⑦⑥✍❶❈❒➐Ñ ❒❙❶❈➄❙③▲❺➃⑥✏⑤ ❶❈➄❙❷❈Ñ⑦✉❢❺ùì➹③❯↔☞✇②❺②③▲④❬✉❢рí●❸➭✇②①❈③ù④✡❒❊❮❊❷❈Ñ⑦✉❢❺②⑤ ✇r➀ Ó ❫❛❷❈③⑨⑥r✇②⑤ ❒❙❶ ⑤⑦⑥❬⑥r✇②⑤ Ñ Ñ✧❒❙×➭③▲❶➅❾ ❻ ➆☎❒ ✕ ③▲➤❙③▲❺⑨❸✴⑤ Ò X ①✝✉❙⑥✟ñ✓⑤⑦⑧▲✉❢❺◆❮➈❶☞❷❈④ùÛ➭③▲❺ ρ(X) = 19 ❸ X ⑤⑦⑥✟✬③ ❫❛❷❈⑤ ×❈×➭③⑨❮ ⑤ ✇②①➘➄✉ ❳☞①❈⑤ ❒❊❮❈✉➢ê☞⑩❨❶❈❒❛⑥✏③✍⑥r✇②❺②❷✝⑧●✇②❷❈❺②③✍✉❢❶✝❮➐⑧❯❒❙❶✝⑥✏✬③ ❫❛❷❈③▲❶❛✇②Ñ Ó ⑤⑦⑥✬③▲⑤ ✇②①❈③▲❺♥✆✉ ✦Ú❷❈④✡④✡③▲❺♥⑥✏❷❈❺✏Ò➠✉❙⑧❯③ ❒❙✕ ❺➪✉➘❮❊❒❙❷❈Û❈Ñ ③❬⑧❯❒❭➤❙③▲❺➃❒❢Ò✯➀✉ ✦Ú❷❈④✡④✡③▲❺➪⑥✏❷❈❺✏Ò➠✉❙⑧❯③ï✡ì ③ ➜➅✜↕ ➤❜➤ ⑦ X✉❢❶✝❮✂✉❢Ñ⑦⑥✏➴❒ ③ ➺ ➛✝➓ ➯ ➸ ⑦ í●✣❻ ❳☞❒➸✇②①❈③▲❺②③ ⑤⑦⑥➪✇②①❈③➸❷❈❶✝❮❊③▲❺②Ñ Ó☞⑤ ❶❈➄❃③▲Ñ Ñ ⑤ ×❊✇②⑤⑦⑧➸⑧❯❷❈❺②➤❙③ ✕ ①❈⑤⑦⑧◆①➍➄❙⑤ ➤❙③⑨⑥➪❺②⑤⑦⑥✏③❬✇②✽❒ ✦Ú❷❈④✡④✡③▲❺ù⑥✏❷❈❺✏Ò➠✉❙⑧❯③❙❸❫✉❢❶✝❮✂✇②①❈③ ④✡❒❊❮❊❷❈Ñ⑦✉❢❺②⑤ ✇rÓ➮❒❢Ò✝✇②①❈③♥✇②❺◆✉❢❶✝⑥②⑧❯③▲❶✝❮❊③▲❶❛✇◆✉❢Ñ❊×✝✉❢❺✏✇✴④❬✉➢Ó➮Û➭③✧❺②③⑨✉❢Ñ ⑤ é▲③⑨❮ùÛ☞Ó➪✇◆✟✉ ❂☞⑤ ❶❈➄➃✇②①❈③☎⑥✏Ó☞④✡④✡③❯✇②❺②⑤⑦⑧ ❃⑥ ❫❛❷✝✉❢❺②③✟❒❢Ò✯✉ï⑧❯❷✝⑥✏×❄Ò➹❒❙❺②④ ❒❢Ò ③▲⑤ ➄❙①❛✇ ✉❙⑥②⑥✏❒❊⑧❯⑤⑦✉❭✇②③⑨❮ï✇②❒➘✇②①❈③✡❷❈❶✝❮❊③▲❺②Ñ Ó☞⑤ ❶❈➄➘③▲Ñ Ñ ⑤ ×❊✇②⑤⑦⑧❬⑧❯❷❈❺②➤❙③❙❻ ⑩❨❶❬Ò➠✉❙⑧●✇⑨❸❛✇②①❈⑤⑦⑥♥✉❢×❈×❈❺②❒❛✉❙⑧◆①✡①✝✉❙⑥✬✕ Û➭③▲③▲❶➘⑧▲✉❢2❺②❺②⑤ ③⑨❮✡❒❙❷❊✇✯Û☞✣Ó ❥➅⑤ ❶❈➄➄ ❥➅❒❙❶❈❷➄ ③ ➞ ➛ ✰➙ ⑩❜❶ ⑦ Ò➹❒❙❺✯✉ù⑧❯③▲❺✏✇◆✉❢⑤ ❶ Ò➠✉❢④✡⑤ Ñ Ó➸❒❢Ò K3 ⑥✏❷❈❺✏Ò➠✉❙⑧❯③⑨⑥ ✕ ⑤ ✇②①ïñ✓⑤⑦⑧▲✉❢❺◆❮➸❶☞❷❈④ùÛ➭③▲❺ 19 ❻ 1 0 1 0 0 −d p ❻✎➚ ☞ r✰❽ ➂✹✇ ➶❫♥➊➉ ➽❧➔➲→➈➛➅➜➞➝ ❺ ➔❊➓❭➟➹➒✒➓ ➒î➛✝➓➐➓❭➟✩➆➭➟➠➜ ➐ ➔ ❺ ➔❊➏❫➟ ➓✷➍ ❈➔ ➝✮➒✜➽✕➓✭➔●➔●➒î➛ ❺ ➜➞➑✡➛ ➑ ➔❛➓ Q ❸☎❒ ③ ⑤ Ñ Ñ❢✇②❺②Ó✍✇②❒✍➄❙③▲❶❈③▲❺◆✉❢Ñ ⑤ é▲③➞✇②①❈③✬④✡❒❊❮❊❷❈Ñ⑦✉❢❺②⑤ ✇rÓ➃❺②③⑨⑥✏❷❈Ñ ✇◆⑥❏✇②❒ÚÐ✯✉❢Ñ⑦✉❢Û❈⑤ ê❛ë✬✉❢❷➃✇②①❈❺②③▲③❯Ò➹❒❙Ñ⑦❮❈⑥ ❮❊③❑✔✝❶❈③⑨✕⑤❮❃✕ ❒❭➤❙③▲✕ ❺ Q ❻✜❉❈❒❙❺➪Ð✯✉❢Ñ⑦✉❢Û❈⑤ ê❛ë✬✉❢❷ï✇②①❈❺②③▲③❯Ò➹❒❙Ñ⑦❮❈⑥▲❸➭✇②①❈③✟❹✯③❯✇✏✇②⑤❫❶☞❷❈④ùÛ➭③▲❺◆⑥ B , B ✉❢❺②③➮❶❈❒❢✇ ✔❈↔❊③⑨❮ï⑧❯❒❙❶✝⑥r✇◆✉❢❶❛✇◆⑥▲❸☞❶❈❒❙❺♥✇②①❈③✿❀✓❷❈Ñ ③▲❺✍⑧◆①✝✉❢❺◆✉❙⑧●✇②③▲❺②⑤⑦⑥r✇②⑤⑦⑧❢❻➞Ù ❶✝✉❭✇②❷❈❺◆✉❢▼ Ñ ❫❛❷❈③⑨⑥r✇②⑤ ❒❙❶➘⑤⑦✱⑥ ❋ ➎◆■✡➼➹▼❈❘❯❉②❘➘❩❭❖✷❩❙ã●■▲➻❭❚ ❇❊➼❨❘➘❑◆➻❭❖❈■●➼❨❩❭❖✝➼✍➶➞▼☞❋➠❑◆▼➊ã◆➻❭❇❊❖❞➱➢■✡➼➹▼❈❘➘❩❙ã●■▲➻❭❚ ❇❊➼❨❘➐➚❭❩❭❚ ❇✝❘➸➻r➾❬➼➹▼❈❘✟❱✯❇❊❚ ❘❯❉ ❑◆▼❈❩❭❉②❩❙❑❯➼❨❘❯❉●❋⑦■●➼➽❋➠❑ E(X) ➏ è♥①❈③ï⑧❯❷❈❺②❺②③▲❶❛✇✡❺②③⑨⑧❯❒❙❺◆❮✂Ò➹❒❙❺✟✇②①❈③ï❷❈×❈×➭③▲❺✡Û➭❒❙❷❈❶✝❮➈Ò➹❒❙❺ù✇②①❈③❃✉❢Û✝⑥✏❒❙Ñ ❷❊✇②③➘➤❭✉❢Ñ ❷❈③➘❒❢Ò☎✇②①❈➀③ ❀✓❷❈Ñ ③▲❺ ⑧◆①✝✉❢❺◆✉❙⑧●✇②③▲❺②⑤⑦⑥r✇②⑤⑦⑧✍⑤⑦⑥ ❻✴è♥①❈⑤⑦⑥✧⑤⑦⑥✧Û✝✉❙⑥✏③⑨❮➸❒❙❶❃✉➢➤❭✉❢⑤ Ñ⑦✉❢Û❈Ñ ③➃③❯↔❈✉❢④✡×❈Ñ ③⑨⑥♥❒❢Ò✬Ð✯✉❢Ñ⑦✉❢Û❈⑤ ê ë✬✉❢❷➍✇②①❈❺②③▲③❯Ò➹❒❙Ñ⑦❮❈⑥✟⑧❯|E(X)| ❒❙❶✝⑥r✇②❺②❷✝⑧●✇②③⑨❮ 960Û☞Ó ×❈①☞Ó❊⑥✏⑤⑦⑧❯⑤⑦⑥r✇◆⑥➮❷✝⑥✏⑤ ❶❈➄✂❹♥✉❭✇rÓ☞❺②③▲✎➤ ✖ ⑥➪❺②❑③ ➉✝③❯↔❊⑤ ➤❙③➐×➭❒❙Ñ Ó❛✇②❒❙×➭③⑨⑥▲❻ ➆☎❒ ③▲➤❙③▲❺⑨❸❊t❃⑤ Ñ ③⑨⑥✧Ý✧③▲⑤⑦❮ï⑧❯Ñ⑦✉❢⑤ ④❬⑥✯✇②①❈③▲❺②③➮⑥✏①❈❒❙❷❈Ñ⑦❮➸❶❈❒❢✇☎Û➭③➮⑥✏❷✝⑧◆①ï✉✡⑧❯❒❙❶✝⑥r✇◆✉❢❶❛✇➮ì➹Û✝✉❙⑥✏③⑨❮➸❒❙❶ï①❈⑤⑦⑥ ③❯↔❊×➭✕ ③▲❺②⑤ ③▲❶✝⑧❯③ ✕ ⑤ ✇②①❃t✈t❃ñ ì➠t❃⑤ ❶❈⑤ ④❬✉❢Ñ✕t❃❒❊❮❊③▲Ñ➅ñ✓❺②❒❙➄❙❺◆✉❢④➐í✏í●❻ ❄✂✥ ③ ✔✝❺◆⑥r✇✧④❬✟✉ ❂❙③➪✉✡⑧❯❺②❷✝❮❊③➪⑧❯Ñ⑦✉❙⑥②⑥✏⑤ ✔➭⑧▲✉❭✇②⑤ ❒❙❶➸❒❢Ò✬Ð✯✉❢Ñ⑦✉❢Û❈⑤ ê❛ë✬✉❢❷➐✇②①❈❺②③▲③❯Ò➹❒❙Ñ⑦❮❈⑥▲❻ ↔ ➆✒↕❦➙✰➛✡➜✡➛✍➇❿➙➩➷✹➌✡➊ ♥ Ù✹Ð✯✉❢Ñ⑦✉❢Û❈⑤ ê❛ë✬✉❢❷➘✇②①❈❺②③▲③❯Ò➹❒❙Ñ⑦❮ ⑤⑦⑥➃⑥②✉❢⑤⑦❮➸✇②❒➐Û➭③➸❉●❋ ➴❢❋➠➱➮⑤ Ò h = 0 ì➠⑥✏❒ ✇②①✝✉❭✇ B = 2í●❻✬ð☎✇②①❈③▲❺ ✕ ⑤⑦⑥✏③❙❸ X ⑤⑦⑥☎⑥②✉❢⑤⑦❮➐✇②❒✡Û➭③❬❖❞X➻❭❖✝❲❨❉●❋ ➴❢❋➠➱❭❻ ➟ ➆✔➉➡➠❿➈✔➢❒➷✹➌❲➝ ♥ Ù ❺②⑤ ➄❙⑤⑦❮✂Ð✯✉❢Ñ⑦✉❢Û❈⑤ ê❛ë✬✉❢❷❃✇②①❈❺②③▲③❯Ò➹❒❙Ñ⑦❮✈⑤⑦⑥➃⑤ ❶✝❮❊③▲③⑨❮❃✇②①❈③✟❶✝✉❭✇②❷❈❺◆✉❢Ñ➞➄❙③▲❶❈③▲❺◆✉❢Ñ ⑤ é⑨✉❭❽ ✇②⑤ ❒❙❶➐❒❢Ò✕✉❢❶❬③▲Ñ Ñ ⑤ ×❊✇②⑤⑦⑧➃⑧❯❷❈❺②➤❙③❙❻❫è♥①❈③☎✇②①❈⑤ ❺◆❮➐❹✯③❯✇✏✇②⑤➭❶☞❷❈④ùÛ➭③▲❺ B = 2 ✉❢❶✝❮➐✉❙⑧▲⑧❯❒❙❺◆❮❊⑤ ❶❈➄❙Ñ Ó➮✇②①❈③▲❺②③ ⑤⑦⑥✧✉ 2❽❨❮❊⑤ ④✡③▲❶✝⑥✏⑤ ❒❙❶✝✉❢Ñ➅➁Ú✉❢Ñ ❒❙⑤⑦⑥✯❺②③▲×❈❺②③⑨⑥✏③▲❶❛✇◆✉❭✇②⑤ ❒❙❶➘✉❙⑥②⑥✏❒❊⑧❯⑤⑦✉❭✇②③⑨❮❬✇②❒❬⑤ ✇⑨❻ ➙ ➠ ➔☞→❢➒⑨➝✕✭➓ ➔➴➒î➛✝➓➐➓❭✩➟ ➆➭➟➠➜ ➐ ➔ ❺ ➔❊➏❫➟ ➓✷➍ ➔❈➝ ✜➒ ➽✕✭➓ ➔●➔●➒î➛ ❺ ➜➞➑❬➛ ➑ ➔❛➓ Q ♥ ➉ ➽❧➔❯→➈➛➅➜➞➝ ❺ ➔❊➓❭➟➹✒➒ ➓➊→❙➛ ✰ ✾✕❘❯➼ ◆ ã ✈ ❘ ✂ ❩ ● ❉ ❋ ❢ ➴ ➠ ❋ ✮ ➱ ✓ ➺ ❭ ❩ ❚ ❩❙ã❯❋€ä✕❆➭❩❭❇ ➼➹▼☞❉②❘◆❘➠➾❯➻❭❚ ➱➈➱❙❰❘ ☎✓❖❞❘◆➱➍➻❭➚❭❘❯❉ ❍❞▼❈❘❯❖ ❋⑦■ï❳❬➻⑨➱❭❇❊❚ ❩❭❉ ❍❞▼❈❩❭➼✍X❋⑦❲■ ❜✍➼➹▼❈❘❯❉②❘➐❃❘ ✰❙❋⑦■●➼➠■❬❩✈❑❯❇☞■➽øï➾❯➻❭❉●❳ f ➻r➾✡➶✬❘❯❋ ➴➢▼☞➼ 4 = 3 +Q1➐ ➻❭❖➍■▲➻❭❳❬X❘ Γ (N ) ■●❇✝❑◆▼ ➐ ➼➹▼❈❩❭➼ 2 3 2,1 3 3 0 ❇⑨ø➊➼❨➻✈❩✿☎✓❖✝❋€➼❨❘✡❖✝❇❊❳❬ã◆❘❯❉❬➻r➾➮❱✯❇❊❚ ❘❯❉✧L(X, ➾❯❩❙❑❯➼❨➻❭s)❉◆■ =➐ ➑ùL(f, ❘❯❉②❘ s),N ❋⑦■➐➱❭❋€➚⑨❋⑦■●❋➠ã❯❚ ❘ï➻❭❖✝❚ ✂✈ã❑✂ùø❞❉●❋€❳❬❘●■➸➻r➾ ã◆❩❙➱❬❉②❘◆➱❭❇✝❑❯➼➽❋➠➻❭❖ ➐ è♥①❈⑤⑦⑥✧⑧❯❒❙✙❶ ✘r③⑨⑧●✇②❷❈❺②③ ✕ ✉❙⑥✯Ò➹❒❙❺②④ù❷❈Ñ⑦✉❭✇②③⑨❮➸⑤ ❶ïtï❽✧➆➮❻✓❳❊✉❢⑤ ✇②❒❬✉❢❶✝❮➸ë✓❷❈⑤✚③ ➜ ➍ ⑩ ♠ ⑦ ❻ ➟ ➆✔➉➡➠❿➈✔➢❒➷✹➌☛➨ ♥ ❐❙❻✧è♥①❈⑤⑦⑥Ú⑧❯❒❙❶✙✘r③⑨⑧●✇②❷❈❺②③✟⑤⑦⑥Ú✉➘⑥✏×➭③⑨⑧❯⑤⑦✉❢Ñ✴⑧▲✉❙⑥✏③✟❒❢Ò✓✇②①❈③✣❉❈❒❙❶❛✇◆✉❢⑤ ❶❈③●ê❊t✈✉❢é▲❷❈❺Ú⑧❯❒❙❶✙✘r③⑨⑧●✇②❷❈❺②③➮✇②①✝✉❭✇ ❘❯➚❭❘❯✄❉ ✂✈❋€❉●❉②❘◆➱❭❇✝❑❯❋➠ã❯❚ ❘ï➻⑨➱❙➱ ❲✏➱❭❋€❳❬❘❯❖❈■●❋➠➻❭❖❞❩❭❴❚ ✢☎❩❭❚ ➻❭❋⑦■➸❉②❘îø❞❉②❘●■▲❘❯❖✝➼❨❩❭➼➽❋➠➻❭➅ ❖ ➡➽❑◆➻❭❳✟❋€❖☞➴✟➾◆❉②➻❭❳ ➴❛❘◆➻❭❳❬❘❯➼➽✄❉ ✂✫➢✧■②▼❈➻❭❇❊❚ ➱➘ã◆❘➮❳❬2➻⑨➱❭❇❊❚ ❩❭❲❉ ❜✓❇⑨ø ➼❨➻➸❩✂❍✝❩❭➼❨❘➮➼➽➶✴❋⑦■●➼ ➐ Ï ❻✧Ù☎Ñ⑦⑥✏❒➸✇②①❈⑤⑦⑥➪⑤⑦⑥Ú✉ï⑥✏×➭③⑨⑧❯⑤⑦✉❢Ñ✓⑧▲✉❙⑥✏③✟❒❢Ò✬✇②①❈❈③ ❳☞③▲❺②❺②③✡⑧❯❒❙✙❶ ✘r③⑨⑧●✇②❷❈❺②③✡✉❢Û➭❒❙❷❊✇Ú✇②①❈③❬④✡❒❊❮❊❷❈Ñ⑦✉❢❺②⑤ ✇rÓ ❒❢Ò♥✇②①❈③➐❺②③⑨⑥✏⑤⑦❮❊❷✝✉❢Ñ✓④✡❒❊❮ p 2❽❨❮❊⑤ ④✡③▲❶✝⑥✏⑤ ❒❙❶✝✉❢Ñ✯➁Ú✉❢Ñ ❒❙⑤⑦⑥➪❺②③▲×❈❺②③⑨⑥✏③▲❶❛✇◆✉❭✇②⑤ ❒❙❶✝⑥➪✉❭✇✏✇◆✉❙⑧◆①❈③⑨❮❄✇②❒ ⑧❯③▲❺✏✇◆✉❢⑤ ❶➘❒❊❮❈❮➘❮❊⑤ ④✡③▲❶✝⑥✏⑤ ❒❙❶✝✉❢Ñ❏×❈❺②✟❒ ✘r③⑨⑧●✇②⑤ ➤❙③Ú➤❭✉❢❺②⑤ ③❯✇②⑤ ③⑨⑥✯❒❭➤❙③▲❺ ❻ ❼❈❻✧è♥①❈⑤⑦⑥➞④❬✉➢ÓÚÛ➭③✯❺②③▲➄❛✉❢❺◆❮❊③⑨❮➪✉❙⑥➞✉➃⑧❯❒❙❶✝⑧❯❺②③❯✇②③✬❺②③⑨✉❢Ñ ⑤ é⑨✉❭✇②⑤ ❒❙❶➪❒❢Ò❈✇②Q①❈✐③ ❥✕✉❢❶❈➄❙Ñ⑦✉❢❶✝❮❈⑥➅ñ✓❺②❒❙➄❙❺◆✉❢④ï❻ ❜✇ ❿➁ q ✏✑✍❿✇✮✠✽✡✷➇❼✠✳②✹①✟②✎③✈✇ ➈ ❽ ❿② ➂❯➀➺②✎④❜✇❤✏✑✍❿✇❤✏✳✒ ❻ ②✎④ ❤✍ ❮ Û➭③✈✉❄❺②⑤ ➄❙⑤⑦❮✮Ð✯✉❢Ñ⑦✉❢Û❈⑤ ê❛ë✬✉❢❷➈✇②①❈❺②③▲③❯Ò➹❒❙Ñ⑦❮ ❺ ➟➠→❙➟➹➒➍❧➜ ➔☞➑⑨→❢➓❭➟✲➼➞➒⑨➟➠➛ ➙ ➛❜➒ ♥ ❥➅③❯✇ ❊❮ ③❑✔✝❶❈③⑨❮➸❒❭➤❙③▲❺ Q ❻✴è♥①❈③▲❶ X ✉❢Ñ ✕ L(X, ✉➢Ó❊⑥✬①✝s)✉❙⑥✯✉❢❶➸⑤ ❶❛✇②③▲X➄❙❺◆✉❢Ñ➭④✡❒❊❮❊③▲Ñ➽❻✇❥➅③❯✇ p Û➭③Ú✉ù➄❙❒☞❒❊❮➐×❈❺②⑤ ④✡③❙❸ ✉❢❶✝❮➐Ñ ③❯✇❝❉❈❺②❒❙Û Û➭③✍✇②①❈③✿❉❈❺②❒❙Û➭③▲❶❈⑤ ❷✝⑥♥④✡❒❙❺②×❈①❈⑤⑦⑥✏④ï❻✴è♥①❈③▲❶➐✇②①❈③➪⑧◆①✝✉❢❺◆✉❙⑧●✇②③▲❺②⑤⑦⑥r✇②⑤⑦⑧✧×➭❒❙Ñ Ó☞❶❈❒❙④✡⑤⑦✉❢Ñ ①✝✉❙⑥✯✇②①❈③ÚÒ➹❒❙❺②④ P (T ) ⑤ ✇②① |t (p)| 2p P (T ) = 1 − t (p)T + p T ∈ 1 + T Z[T ] ✕ è♥①❈③▲❶ ➤ ➓❫➼ p p p 3 3 2 3/2 3 1 1 − t3 (p)p−s + p3−2s ❙➄ ❒☞❒❊❮ ➄❙❒☞❒❊❮ ①❈③▲❺②③ (∗) ⑥r✇◆✉❢❶✝❮❈⑥✯Ò➹❒❙❺♥✇②①❈③ÚÒ➠✉❙⑧●✇②❒❙❺◆⑥♥⑧❯❒❙❺②❺②③⑨⑥✏×➭❒❙❶✝❮❊⑤ ❶❈➄ù✇②❒✡Û✝✉❙❮➘×❈❺②⑤ ④✡③⑨⑥▲❻ ✕ Ù✍⑥✟Ò➹❒❙❺✡③▲Ñ Ñ ⑤ ×❊✇②⑤⑦⑧ï⑧❯❷❈❺②➤❙③⑨⑥✟❒❭➤❙③▲❺ ❸✓✇②①❈③ïÑ ❒❊⑧▲✉❢✐Ñ ❀✓❷❈Ñ ③▲❺ ❽➽Ò➠✉❙⑧●✇②❒❙❺✡❒❢Ò ❸✓✇②①✝✉❭✇❬⑤⑦⑥▲❸ ③⑨⑥②⑥✏③▲❶❛✇②⑤⑦✉❢Ñ Ñ Ó t (p) ❸❊⑧▲✉❢❶➐Û➭③➃❮❊③⑨⑥②⑧❯❺②⑤ Û➭Q③⑨❮❬⑤ ❶❬✇②③▲❺②④❬⑥✬❒❢Ò❏✇②①❈③➃❶☞p❷❈④ùÛ➭③▲❺✬❒❢Ò FL(X, ❽î❺◆✉❭✇②⑤ s)❒❙❶✝✉❢Ñ❈×➭❒❙⑤ ❶❛✇◆⑥ ❒❙❶ X ❻ ❸❊Ñ ③❯✇ ❉❈❒❙❺✧③⑨✉❙⑧◆① i, 0 i 6 ✇②❺◆✉❙⑧❯③ (❉❈❺②❒❙Û | H (X,¯ Q )) t (p) := Û➭③➃✇②①❈③➃✇②❺◆✉❙⑧❯③➃❒❢Ò✕✇②①❈✿③ ❉❈❺②❒❙Û ❒❙❶➸✇②①❈③ÚÔ❯✇◆✉❢Ñ ③ ❽❨✉❙❮❊⑤⑦⑧ i❽➽✇②①ï⑧❯❒❙①❈❒❙④✡❒❙Ñ ❒❙➄❙Ó✟➄❙❺②❒❙❷❈×➅❻❫è♥①❈③▲❶➸✇②①❈③ ❥➅③❯Ò➠⑥②⑧◆①❈③❯✇②✜ é ✔❈↔❊③⑨❮➘×➭❒❙⑤ ❶❛✇✧Ò➹❒❙❺②④ù❷❈Ñ⑦✉✟✉❙⑥②⑥✏③▲❺✏✇◆⑥✬✇②①✝✉❭✇ Pp (p−s )−1 = (∗) L(X, s) = (∗) p: p: 3 p i ∗ p i et ∗ p 6 #X(Fp ) = (−1)i ti (p) = t0 (p) − t1 (p) + t2 (p) − t3 (p) + t4 (p) − t5 (p) + t6 (p).

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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