Ӊ஦ࡍλΠώϛϡʔϥʔཧ࿦΁ͷ༠ʢ͍͟ͳʣ͍ʬϨΫνϟʔϊʔτ൛ʭ๬݄৽Ұʢ‫ژ‬େ਺ཧ‫ݚ‬ʣ

2015 ೥ 04 ݄
http://www.kurims.kyoto-u.ac.jp/~motizukiʮա‫ࡏݱͱڈ‬ͷ‫ڀݚ‬ʯ

§1. Hodge-Arakelov ཧ࿦తಈ‫͚෇ػ‬§2. Teichmüller ཧ࿦తͳม‫ܗ‬§3. ର਺ɾςʔλ֨ࢠ§4. Ӊ஦ࡍੑͱԕΞʔϕϧ‫ز‬Կ๬݄৽Ұʢ‫ژ‬େ਺ཧ‫ݚ‬ʣ

2§1. Hodge-Arakelov ཧ࿦తಈ‫͚෇ػ‬·ͣ͸۩ମతͳ࿩͔Β࢝ΊΑ͏ɻ࣮਺ h ∈ R≥0 ͸ʮ্͔Β཈͍͑ͨ ྔʯͱʂʣ͠ɺྫ͑͹ɺ͋Δ੔਺ N ≥ 2 ʹରͯ͠౳ࣜʢʹ ‘a ∈ a’ ͷҰछͷʮఆྔ൛ʯ

N ·h





def

= h + h + ... + h= h͕੒Γཱͭ͜ͱ͕෼͔͍ͬͯΔͱ͢Δɻʢ͜͜Ͱɺొ৔͢Δʮhʯͨͪͷ ରশੑʢʹʮࠨลͷΈʯɺʮࠨล ∪ ӈลʯʣʹ஫ҙʂʣ͢Δͱɺ؆୯ͳࣜม‫ܗ‬Λࢪ͢ͱɺɹ
(N − 1) · h = 0,
ɹͭ·Γɺɹ h = 0
ͱͳΔ͜ͱ͕‫͖Ͱ݁ؼ‬Δɻ͜ͷٞ࿦ͷ؆୯ͳʮਐԽ‫ܗ‬ʯͱͯ͠ɺ
ʮൺֱతখ͞
͍ʯఆ਺ C ∈ R ʹରͯ͠ɺෆ౳ࣜ

N ·h ≤ h+C͕੒Γཱͭ͜ͱ͕෼͔͍ͬͯΔͱ͢Δͱɺ

(N − 1) · h ≤ C,

ɹͭ·Γɺɹ h ≤1
N −1 · Cଈͪɺʮh ͸౰ॳͷ‫ظ‬଴௨Γɺ্͔Β཈͑ΒΕΔʯ͜ͱ͕‫͖Ͱ݁ؼ‬Δɻ

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ͯ͞ɺ࣍͸ ପԁ‫ۂ‬ઢ ʹ͍ͭͯߟ࡯ͯ͠ΈΑ͏ɻʢp ਐ‫ॴہ‬ମ΍ C ্ͷʣTate ‫ۂ‬ઢdef

E = Gm /q Z ͷ৔߹ɺૉ਺ l ʹରͯ͠ l ౳෼఺ʹؔ͢Δࣗવͳ‫׬‬શ‫͕͋ྻܥ‬Δɿɹ
0 −→ μl −→ E[l] −→ Z/lZ −→ 0
ͭ·Γɺ࣍ͷΑ͏ͳ ඪ४త ͳର৅͕ͨͪͱΕΔɿ
ʮ৐๏త෦෼ۭؒ ɹ μl ⊆ E[l]ʯɹͱɹʮੜ੒‫ ݩ‬ɹ ±1 ∈ Z/lZʯҎԼͰ͸ɺE = ପԁ‫ۂ‬ઢ/਺ମ F , ɹ ૉ਺ l ≥ 5 Λ‫ݻ‬ఆ͢Δɻ·ͨ E ͸ F ͷ͢΂ͯͷ༗‫ݶ‬ૉ఺ʹ͓͍ͯ҆ఆؐ‫ݩ‬Λ࣋ͭͱԾఆ͢Δɻ͢ΔͱɺҰൠʹ͸ɺE[l] ʹରͯ͢͠΂ͯͷ bad multiplicative reduction ͷ༗‫ݶ‬ૉ఺ʹ্͓͍ͯ‫ه‬ͷඪ४తͳʮ৐๏త෦෼ۭؒʯͱʮੜ੒‫ݩ‬ʯͱҰக͢ΔେҬత ͳʮ৐๏త෦෼ۭؒʯͱʮੜ੒‫ݩ‬ʯ͸ ଘࡏ͠ͳ͍ʂ

Ӊ஦ࡍλΠώϛϡʔϥʔཧ࿦΁ͷ༠ʢ͍͟ͳʣ͍ʬϨΫνϟʔϊʔτ൛ʭ 3͕ɺԾʹ ଘࡏ͢Δʢʂ
ʂʣͱԾఆ͠Α͏ɻ͢ΔͱɺE ΛʮେҬత৐๏త෦෼ۭؒʯ
Ͱ ׂΔ ͜ͱʹΑͬͯಘΒΕΔಉछࣸ૾Λ E → E ∗ ͱॻ͘ͱɺ֤ bad ͳ༗‫ݶ‬ૉ఺ʹ͓͍ͯͦΕͧΕͷ q-parameter ͸࣍ͷΑ͏ͳؔ܎ࣜΛຬͨ͢ɿlqE= qE ∗

q-parameter ͔ͨͪΒఆ·Δ ਺࿦త࣍਺ ∈ R Λɺlog(qE ), log(qE ∗ ) ͱॻ͘ͱɺ্ͷؔ܎ࣜ͸࣍ͷΑ͏ʹͳΔɿ
l · log(qE ) = log(qE ∗ ) ∈ RҰํɺͦΕͧΕͷପԁ‫ۂ‬ઢͷ ߴ͞ Λ htE , htE ∗ ∈ R ͱॻ͘ͱɺ

ht(−) ≈ 16 · log(q(−) )ʢͨͩ͠ɺʮ≈ʯ͸ʮ༗քͳఆ਺ࠩΛআ͖ʯͱ͍͏ҙຯʣͱͳΓɺ·ͨ Faltings(1983) ͷ༗໊ͳ ඍ෼ʹؔ͢Δ‫ʹ ࢉܭ‬ΑΓɺɹhtE ∗  htE + log(l)ͱͳΔ͜ͱ͕෼͔Δɻैͬͯɺପԁ‫ۂ‬ઢ͕ొ৔͢Δલͷٞ࿦ͱಉ༷ʹɺl · htE  htE + log(l),ɹͭ·Γɺɹ htE 1l−1 · log(l) constantଈͪɺପԁ‫ۂ‬ઢ E ͷߴ͞ htE Λ ্͔Β཈͑Δ ͜ͱ͕Ͱ͖ͨͷͰɺʢద੾ͳԾఆͷԼͰ͸ʣ͜ͷΑ͏ͳʮେҬత৐๏త෦෼ۭؒʯΛ࣋ͭପԁ‫ۂ‬ઢ E ͷಉ‫ܕ‬ྨ͸ ༗‫ͱ͍͜ͳ͔͠ ݸݶ‬Λ‫݁͢ؼ‬Δ͜ͱ͕Ͱ͖Δɻ্ͷΑ͏ͳٞ࿦͸͍ͣΕʮେҬత৐๏త෦෼ۭؒʯ౳Λ࣋ͭͱ͸‫ݶ‬Βͳ͍ʮ࣋ͭʯͱ͍͏ʢඇ‫࣮ݱ‬తҰൠͷପԁ‫ۂ‬ઢ ʹ ֦ு͍ͨ͠ ͷ͕ͩɺͦͷલʹɺͳʂʣԾఆͷԼͰɺ্ͷٞ࿦ͱ͸গ͠ҧ͏ΞϓϩʔνΛ‫ݕ‬౼͍ͨ͠ɻɹ

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ͦͷͨΊʹ΋ɺHodge-Arakelov ཧ࿦ ͷ ‫ج‬ຊఆཧ ʹ‫ݹ‬యతͳ Ψ΢εੵ෼ʮಈ‫ܥܘ‬ͷඪ४తͳࢁͷମੵʯʹ

∞
−∞

e

−x2

dx =

√


π ʹ ʮภ֯‫ܥ‬ͷप‫ظ‬ʯͷҰछͷʮ཭ࢄతଟ߲ࣜ൛ʯΛɺઌ΄Ͳͷʮ࣋ͭʯͱ͍͏ԾఆͷԼͰʮ෮शʯ
͢Δඞཁ͕͋Δɻ๬݄৽Ұʢ‫ژ‬େ਺ཧ‫ݚ‬ʣ

4ʮ཭ࢄ൛ʯͷϞσϧɿdf (x)dx

f (x+δ)−f (x)
δ
δ→0



= lim

f (X + 1) − f (X)΍ɺ࣍਺͕ d ະຬͷ Z ܎਺ͷଟ߲ࣜʹؔ͢Δ࣍ͷ ‘ಉ‫’ܕ‬ɿ

Z[X]

<d

d−1


∼

‘→’

Z

0

ʢ[Hartshorne], I, §7 ΛࢀরʂʣHodge-Arakelov ཧ࿦ʢ≈ 1990 ೥୅‫ޙ‬൒ʣͰ͸ɺ͜ΕΒͷϞσϧͷԆ௕ઢ্Ͱɺ‫ݹ‬యతͳ Arakelov ཧ࿦Ͱ͸ ແ‫ݶ‬ૉ఺ ʹ͓͍ͯ༻͍ΒΕΔ ղੳ ʹ ‘∂, ∂, Green ؔ਺౳’ ͷ ཭ࢄ൛ Λߏங͢ΔͷͰ͋Δɻ

Hodge-Arakelov ཧ࿦ ͷ ‫ج‬ຊఆཧ ʹ۩ମతʹ͸ɺςʔλؔ਺͓Αͼͦͷඍ෼Λɺl ౳෼఺ʹ੍‫͢ݶ‬Δ
͜ͱʹΑͬͯಘΒΕΔಉ‫ܕ‬͸࣍ͷΑ͏ʹఆࣜԽͰ͖Δɿ
†Γ(E , L)

<l

∼

→



l

j=−l

2

q j · OFͨͩ͠ɺɹɾE † → E ͸ E ͷʮීวϕΫτϧ֦େʯɺdefɹɾ
ʮ< lʯ͸ͦͷ֦େʹؔ͢Δʮ૬ରత࣍਺ʯɺl = (l − 1)/2ɺ
ɹɾL ͸ʢඇࣗ໌ͳʣ2 ౳෼఺ʹ෇ਵ͢Δઢଋɺdefɹɾ
ʮq ʯ͸ bad ͳ༗‫ݶ‬ૉ఺ʹ͓͚Δ q-parameterɺq = q 1/2l ɺɹɾࠨล ʹ͸ɺHodge ﬁltration F−i ͕ೖΓɺF−i /F−i+1 ͸ɺʢେମʣ
∼

⊗(−i)

→ ωE

ɹʢi = 0, 1, . . . , l − 1; ωE (≈ ‘d log(U )’) ͸‫Ͱ఺ݪ‬ͷ༨઀ଋʣɺ

ɹɾӈล ʹ͸ɺ௚࿨෼ղͱཱ྆తͳʮΨϩΞͷ࡞༻ʯ͕ೖΔɻ͜ͷಉ‫ܕ‬͸‫ݩ‬ʑF ্Ͱఆٛ͞ΕΔ΋ͷ͕ͩɺF ͷ͢΂ͯͷૉ఺ʹ͓͚Δࣗવͳ ੔ߏ଄ɾ‫ͱ ྔܭ‬ʢ΄΅ʣཱ྆త Ͱ͋Δɻಉ༷ͳಉ‫ܕ‬͸ ପԁ‫ۂ‬ઢͷϞδϡϥΠɾελοΫ ্Ͱ΋ߟ࡯͢Δ͜ͱ͕Ͱ͖ɺ
ಉ‫ܕ‬ͷূ໌͸ਖ਼ʹಉ‫ܕ‬ͷ྆ลʹ‫ݱ‬ΕΔϕΫτϧଋͷ ࣍਺ [−] ͕Ұக͢Δ ͱ͍͏ ‫ʹ ࢉܭ‬ΑΔ΋ͷͰ͋Δʢ஫ɿ
ʮlog(q)ʯ͸ q-param. ʹΑͬͯఆ·ΔҼࢠʣɿ1l · LHS1l · RHS≈ − l12 ·≈ − 1l ·

l 
j=1

l−1
i=0

i · [ωE ] ≈ − 2l · [ωE ]lj 2 · [log(q)] ≈ − 24
· [log(q)] = − 2l · [ωE ]Ӊ஦ࡍλΠώϛϡʔϥʔཧ࿦΁ͷ༠ʢ͍͟ͳʣ͍ʬϨΫνϟʔϊʔτ൛ʭ 5Ұํɺ਺ମ ্ͷઃఆʹ໭ΔͱɺFi ͸্‫ه‬ͷ ௚࿨෼ղ ͱ ཱ྆͠ͳ͍ ͨΊɺͦͷ௚࿨෼ղͷ௚࿨Ҽࢠ΁ͷࣹӨΛ༻͍ΔͱɺʢຆͲͷʣj ʹରͯ͠ʢ͍ΘΏΔʮ਺࿦త Kodaira-Spencer ࣹʯͷʮ਌ྨʯͱ΋‫͑ݴ‬Δʣઢଋͷʢඇྵͳʣࣹ
2(OF ≈) F0 → q j · OFΛ࡞Δ͜ͱ͕Ͱ͖ɺ਺࿦త࣍਺ degarith (F0 )≈ 0 ͱͳΔͨΊɺପԁ‫ۂ‬ઢdefͷϞδϡϥΠͷ ର਺ඍ෼ Ωlog
M |E ʹΑΔߴ͞Λ htE = 2 · degarith (ωE ) =log
degarith (ΩM |E ) ͱॻ͘ͱɺɹ
1
6 · degarith (log(q))= htE < constantʢͨͩ͠ɺʮlog(q)ʯ͸ q-parameter ʹΑͬͯఆ·Δ਺࿦తҼࢠʣͷΑ͏ͳ ෆ౳͕ࣜ‫݁͞ؼ‬ΕΔʂͨͩ͠ɺ؊৺ͳ େҬత ͳ ৐๏త෦෼ۭؒ ͱ ੜ੒‫ ݩ‬͸࣮ࡍʹ͸ ଘࡏ͠ͳ͍ ͨΊɺ͜ͷٞ࿦͸௚ͪʹద༻͢Δ͜ͱ͸Ͱ͖ͳ͍ɻɹ
͔͠͠ࠓ౓͸࣍ͷΑ͏ͳ ಥഥࢠ΋ͳ͍ʢʂʣ͜ͱΛߟ͑ͨ͘ͳΔɻ΋͠ྫ͑͹ɺ
 2
→ q
qj
j=1,... ,lͱ͍͏ରԠʹΑͬͯɺ਺ମ F ͷ ࣗ‫ݾ‬ಉ‫ ܕ‬Λఆٛ͢Δ͜ͱ͕Ͱ͖ͨΒͲ͏ͳΔ͔ɻʮࣗ‫ݾ‬ಉ‫ܕ‬ʯ͸ ਺࿦తઢଋͷ࣍਺ Λඞͣ อͭ Θ͚͔ͩΒɺ্‫ه‬ରԠͷӈล ͷ࣍਺ͷઈର஋͸ ࠨล ͷʢฏ‫ۉ‬తʂʣ࣍਺ͷઈର஋ͱൺ΂ͯʮখ͍͞ʯ
ͨΊɺಉ͡Α͏ʹɺ
16 · degarith (log(q))= htE < constantͷΑ͏ͳ ෆ౳ࣜ ͕‫݁͞ؼ‬ΕΔʂ
ʂ͔͠͠ࠨลͷ΋ͪΖΜɺͦͷΑ͏ͳ਺ମͷࣗ‫ݾ‬ಉ‫ܕ‬͸࣮ࡍʹ͸ ଘࡏ͠ͳ͍ʂ
2
ʮ{q j }ʯͱӈลͷʮq ʯΛɺͦΕͧΕ ผʑͷʮʢ௨ৗ‫ܕ‬ͷʣ‫؀‬ɾεΩʔϜ࿦ʯʹʮ਺࿦తਖ਼ଇߏ଄ʯʹॴଐ͢Δ΋ͷͱ‫ݟ‬၏͠ɺॴ๬ͷରԠʹ
ʮHA ཧ࿦ΛσΟΦϑΝϯτε‫ز‬ԿʹԠ༻͢Δ্Ͱͷো֐ʯʹର͢ΔҰछͷʮಉٛ൓෮తղܾʯɹ
 2
qj

j=1,... ,l→qΛɺ૬ҟͳΔਖ਼ଇߏ଄ Λ࣋ͭϦʔϚϯ໘ͷؒͷ ٖ౳֯ࣸ૾ ͷΑ͏ͳ΋ͷͱࢥ͏ͱͲ͏ͳΔ͔ɻ๬݄৽Ұʢ‫ژ‬େ਺ཧ‫ݚ‬ʣ6ʮಉٛ൓෮తͳղܾʯ͕੒ཱ͢ΔΑ͏ͳʮิॿͷ෣୆ʯΛߏ੒্ͨ͠Ͱɺ
஫ɿͦͷิॿͷ෣୆ͱʮ‫ݩ‬ͷ෣୆ʯͷ ൺֱʢʹಉ‫ܕ‬ʁຆͲಉ‫ܕ‬ʁʣΛߦͳ͏ͱ͍͏
ͷ͸ɺ਺࿦‫ز‬Կɺ͍΍਺ֶͷʮৗ౟खஈʯͰ͋Δʂผͷ‫ํ͍ݴ‬Λ͢Δͱɺ
ʮಉٛ൓෮తͳղܾʯΛҰ୴ ͨͩͰ ͍͍͔ͨͩͯΒɺͦΕʹΑͬͯੜ͡Δ ͓௼ΓΛ‫צ‬ఆ͢Δͱ͍͏ख๏Ͱ͋Δɻ
ʮಉٛ൓෮తͳղܾʯͷ‫ݹ‬యతͳྫɿҰ୴ आΓͨࡒ࢈ Λ༻͍ͯ঎ച౳ͷࣄ‫ۀ‬ʹΑΓ໥͚ͨ ৽ͨͳࡒ࢈ Λར༻ͯ͠आΓͨࡒ࢈Λ རࢠ෇͖ Ͱฦࡁ͢Δͱ͍͏ʢ‫ݹ‬୅จ໌ʹ૎Δʂʣ࢓૊Έɻ༗ཧ਺ͷՃ‫ݮ‬৐আ͔͠஌ΒΕ͍ͯͳ͍ઃఆʹ
ର͢Δɺෆఆ‫͓ ݩ‬ΑͼͦΕΛ༻͍ͨ ୅਺తͳࣜม‫ ܗ‬ͷಋೖɻ2ʙ4 ࣍ࣜͷղͷࠜ߸౳ʹΑΔ໌ࣔతͳද͔ࣔ͠஌ΒΕ͍ͯͳ͍ઃఆʹର͢Δɺந৅తͳʮମʯ
΍ʮΨϩΞ‫܈‬ʯͷಋೖɻͭ·Γɺʢઌ΄Ͳͷ਺ମ্ͷପԁ‫ۂ‬ઢͷ࿩ʹ໭Δͱʣ௨ৗͷ‫؀‬ɾεΩʔϜ࿦Λɺ෦෼తʹղମ ͢Δ͜ͱʹΑͬͯॴ๬ͷରԠΛ࣮‫͢ݱ‬Δ͜ͱ͕Ͱ͖Δͱ͍͏͜ͱͰ͋Δɻͨͩ͠ɺ௨ৗͷ‫؀‬ɾεΩʔϜ࿦Λɺ෦෼తʹղମͯ͠ʹ ࿪Ίͯ ͠·ͬͨͱ͖ɺ
ͲͷҐͷ࿪Έ͕ੜ͡Δ͔Λ‫͢ࢉܭ‬Δ
ඞཁ͕͋Δɻ͜ͷ ૖େͳ‫͕ ࢉܭ‬ɺIUTeich ͷ಺༰ Ͱ͋Δɻ݁࿦͔Β͍͏ͱɺ۩ମతͳϨϕϧͰ͸ɺఴ͑ࣈ j ͷͱ͜ΖͰ͸ɺʢେମʣ

≤ j · (log-diﬀF + log-condE )Ґͷʮ࿪Έʯ͕ੜ͡ΔͷͰ͋Δɻ͢Δͱɺ্‫ه‬ͷପԁ‫ۂ‬ઢͷϞδϡϥΠɾελοΫ্ͷ‫ ͱࢉܭ‬શ͘ಉ͡ʢʹ j ʹؔ͢Δ ฏ‫ۉ‬஋ʂͷʮओཁ߲ʯͷ) ‫ʹࢉܭ‬ΑΓɺɹ
16 · degarith (log(q))= htE ≤ (1 + )(log-diﬀF + log-condE ) + constantͱ͍͏ ෆ౳ࣜ ͕‫݁͞ؼ‬ΕΔɻ͜Ε͕͍ΘΏΔʮSzpiro ༧૝ʢͷ‫ܗ͍ڧ‬ʣʯʢ⇐⇒ ʮABC ༧૝ʯʣͰ͋ΔɻӉ஦ࡍλΠώϛϡʔϥʔཧ࿦΁ͷ༠ʢ͍͟ͳʣ͍ʬϨΫνϟʔϊʔτ൛ʭ 7஫ɿઌ΄ͲͷʮN · h = hʯʹରԠ͢Δ΋ͷ͸ɺࠨӈ ͷʮhʯΛ ۠ผ ͢Δɹʮಉٛ൓෮తղܾʯ

⇐⇒ ʮN · hLHS = hRHS ʯͰ͋Γɺʮ࿪Έ ΛೝΊΔͱɺ۠ผ͠ͳ͍ͰࡁΉʯͨΊɺʮN · h ≤ h + C ʯɺଈͪʮʢ‫ܗ͍ڧ‬ͷʣSzpiro ༧૝ʯ͕ै͏ͷͰ͋Δɻ

§2. Teichmüller ཧ࿦తͳม‫ܗ‬C ্ͷ‫ݹ‬యతͳ Teichmüller ཧ࿦:ϦʔϚϯ໘্ͷ ೋ࣍ඍ෼ ʹ෇ਵ͢Δඪ४తͳ࠲ඪ z = x + iy Λ༻͍Δͱɺ
๲ு཰ 1 < K < ∞ ʹରͯ͠ɺTeichmüller ม‫ ܗ‬͸࣍ͷΑ͏ͳ‫هͰܗ‬ड़͢
Δ͜ͱ͕Ͱ͖Δɿz → ζ = ξ + iη = Kx + iyϙΠϯτ: ਖ਼ଇͳ࣍‫ ݩ‬1 ຊ ʹରͯ͠ɺԼ෦ͷ࣮࣍‫ ͕ݩ‬2 ຊ ͋Γɺɹͦͷ͏ͪͷ 1 ຊ ͕ ม‫ ܗ‬ͷର৅ͱͳΔͷʹରͯ͠ɺɹɹɹ΋͏ 1 ຊ ͸ ‫ݻ‬ఆ ͞ΕΔʢʹม‫ܗ‬ͷର৅ͱͳΒͳ͍ʣʂp ਐ Teichmüller ཧ࿦:ɾਖ਼ඪ਺ͷ૒‫ۂ‬త‫ۂ‬ઢʴϕΩྵͳ‫ݻ‬༗ଋʹ෇ਵ͢Δ p ਐతͳඪ४త্࣋ͪ͛
ɾϞδϡϥΠɾελοΫͷʢ૒‫ۂ‬తʹʣordinary ͳ locus ͱಉٛ൓෮త‫ۂ‬ઢɹ্ͷ ඪ४త Frobenius ্࣋ͪ͛ɹ ←→ C ্ͷཧ࿦ʹ͓͚Δ Poincaré ্൒ฏ໘ͷ‫ྔܭ‬ʴ Weil-Petersson ‫ྔܭ‬๬݄৽Ұʢ‫ژ‬େ਺ཧ‫ݚ‬ʣ

8

IUTeich ͱ pTeichʢ·ͨ͸ ීวతପԁ‫ۂ‬ઢ/P1 \ {0, 1, ∞}ʂʣ ͷؒͷྨࣅ:Z ্ͷ௨ৗͷεΩʔϜ࿦ ←→ Fp ্ͷεΩʔϜ࿦਺ମʢʴ༗‫ݸݶ‬ͷૉ఺ʣ ←→ ਖ਼ඪ਺ͷ૒‫ۂ‬త‫ۂ‬ઢ਺ମ্ͷҰ఺ൈ͖ପԁ‫ۂ‬ઢ ←→ ϕΩྵͳ‫ݻ‬༗ଋର਺ɾςʔλ֨ࢠ ←→ p ਐతඪ४త্࣋ͪ͛ ʴ Frob. ͷඪ४త্࣋ͪ͛
p

p

p

p

p

p

p

p

p

p

p

p

.......

p

p

p

p

p

p

p

p
p

p

਺࿦తͳ৔߹ɿ଍͠ࢉͱֻ͚ࢉɻίϗϞϩδʔ࣍‫ݩ‬:Z ͷΑ͏ͳ‫؀‬ͷ ‫ߏ؀‬଄ Λ ਺࿦తͳਖ਼ଇ࣍‫ ݩ‬1 ຊ ͱ‫ݟ‬၏͠ɺ
ʮ଍͠ࢉʯ

ͱ

ʮֻ͚ࢉʯ

(Z, +)

(Z, ×)

૊߹ͤ࿦త࣍‫ ݩ‬1 ຊ૊߹ͤ࿦త࣍‫ ݩ‬1 ຊΛɺͦͷਖ਼ଇͳ 1 ࣍‫෇ʹݩ‬ਵ͢Δ Լ෦ͷ૊߹ͤ࿦త࣍‫ ݩ‬2 ຊ ͱ‫ݟ‬၏͢ʂ
ɹ... ɹ͜Ε͸ɺɹɹɹɹɾʢ૯‫ͳڏ‬ʣ਺ମ F/Q < ∞ ΍ɹɹɹɹɾp ਐ‫ॴہ‬ମ k/Qp < ∞ɹɹɹɹɹɹͷઈରΨϩΞ‫܈‬ͷ ίϗϞϩδʔ࣍‫ ݩ‬2 ຊɺɹɹɹɹʢ஫ɿGal(Qp /Qp ) ͷʮpro-l తͳ෦෼ʯ≈ Zl  Z×ɺ͋Δ͍͸ɺl ʂʣɹɹɹɹɾC× ͷ Լ෦ͷ࣮࣍‫ ݩ‬2 ຊɹɹɹɹɹɹɹɹʹରԠ͍ͯ͠Δ΋ͷͱ‫ݟ‬Δ͜ͱ͕Ͱ͖ΔɻӉ஦ࡍλΠώϛϡʔϥʔཧ࿦΁ͷ༠ʢ͍͟ͳʣ͍ʬϨΫνϟʔϊʔτ൛ʭ 9୯਺‫ͱ܈‬஋‫܈‬:p ਐ‫ॴہ‬ମ k/Qp < ∞ ͷ৔߹ɺԼ෦ͷ૊߹ͤ࿦త࣍‫ ݩ‬2 ຊ Λ࣍ͷΑ͏ʹଊ͑Δ͜ͱ΋ՄೳͰ͋Δɿ
Ok×

⊆

k×



૊߹ͤ࿦త࣍‫ ݩ‬1 ຊ

k × /Ok× (∼= Z)૊߹ͤ࿦త࣍‫ ݩ‬1 ຊ

ɹɹ... ɹෳૉ਺ͷ৔߹ͷ࣍ͷ௚ੵ෼ղʹରԠɿɹ C× = S1 × R>0

IUTeich ʹ͓͚Δ ਺ମͷʮਖ਼ଇߏ଄ͷม‫ܗ‬ʯͰ͸ɺςʔλؔ਺ Λ༻͍ͯ ஋‫ ܈‬͸ ม‫͢ ܗ‬Δ͕ɺ୯਺‫ ܈‬ͷํ͸ ม‫͍ͳ͠ܗ‬ʂ§3. ର਺ɾςʔλ֨ࢠ
Hodge theater ʮ•ʯͨͪͷඇՄ‫׵‬ʢʂʣͳʮ2 ࣍‫ݩ‬తͳʯਤࣜ:2 ࣍‫ݩ‬͸ p ਐ‫ॴہ‬ମ΍ʮ‫؀‬ʯͷԼ෦ͷ ૊߹ͤ࿦త࣍‫ ݩ‬2 ຊ ʹରԠ͍ͯ͠Δʂ
..
.
⏐
⏐

..
.
⏐
⏐

...

−→

•
⏐
⏐

−→

•
⏐
⏐

−→ . . .

...

−→

•
⏐
⏐

−→

•
⏐
⏐

−→ . . .

..
.

..
.

IUTeich ͱ pTeichʢ·ͨ͸ Witt ‫؀‬ʂʣ ͷؒͷྨࣅ:ɹɹɹ • = Z ্ͷεΩʔϜ࿦ʮҰࣜʯ ←→ Fp ্ͷεΩʔϜ࿦ʮҰࣜʯ⏐⏐ = log-link = ʮҰࣜʯಉ࢜ͷ షΓ߹Θͤ ←→ ਖ਼ඪ਺ʹ͓͚Δ Frobenius ࣹ


n
n+1
n+1
n+2
−→ = Θ-link = ʮҰࣜʯಉ࢜ͷ షΓ߹Θͤ ←→ p /p
 p/p๬݄৽Ұʢ‫ژ‬େ਺ཧ‫ݚ‬ʣ10ʮ‫؀‬ʯͷԼ෦ͷ ૊߹ͤ࿦త࣍‫ ݩ‬2 ຊ ͱൺֱ͢Δͱɿ
 N−→ = Θ-link ←→ ‘×’ɿ N≥1 ∼p( =⇒ p1 ? p2 ? pλ1 (N  λ = 0) ?)=p⏐⏐ = log-link ←→ ‘×’  ‘+’ ( =⇒ p1 = 1 + . . . + 1 = p2 = 1 + . . . + 1 !!)[Θ±ell NF-]Hodge theater:ʮ[Θ±ell NF-]Hodge theaterʯ͸ɺ਺ମ F ্ͷପԁ‫ۂ‬ઢ E ʹؔ࿈ͨ͠ ௨ৗͷεΩʔϜ࿦త਺࿦‫ز‬Կ ͷϞσϧʮҰࣜʯͰ͋Δɻ΋ͬͱ۩ମతͳϨϕϧͰ͍͏ͱɺE/F ΍ͦͷ༷ʑͳ‫ॴہ‬Խ͔Βࣗવʹੜ͡Δ
ந৅తͳϞϊΠυ ΍ ΨϩΞ‫܈‬ɾ਺࿦త‫ج‬ຊ‫܈‬Λɺෳࡶͳ ‫ʹ ܥ‬ฤ੒ͨ͠΋ͷͰ͋Δɻฤ੒ͷഎ‫͋ʹޙ‬Δ ‫ݪ‬ཧɿɹ େҬతͳ ৐๏త෦෼ۭؒ ʴ ੜ੒‫ݩ‬ʢ§1 ΛࢀরʂʣɹɹɹͷΑ͏ͳ΋ͷΛ࣮‫͢ݱ‬ΔͨΊͷ l ౳෼఺ͷ؅ཧʂ± F
l -, Fl -ରশੑɹɹɹɹɹʢͨͩ͠ɺF= F×l /{±1},
l
def

{±1}



−l < . . . < −1 < 0< 1 < . . . < lF±= Fl  {±1}ʣ
l
def


⇒

1 < ...
< l

1 < ...

⇐

< l

⇓
±
↑
±

→
F±
l

⇓
±





↓

↑

←

±



. . . cf. ord. monodromy!

→
F
l





↓

←



. . . cf. s/sing.!

ฤ੒ͷ࢓૊Έͷ ώϯτɿp ਐϗοδཧ࿦ ʹ͓͚Δɺp-Hecke ରԠ ͷ special
ﬁber ͷߏ଄΍ Qp ্ͷପԁ‫ۂ‬ઢͷϞδϡϥΠ্ͷ େҬతͳ ৐๏త෦෼ۭؒ

(p-adic Tate module) ⊗ (p-adic ring of fns.)ɹɹɹɽɽɽ  ෆࢥٞʢʂʣͳ ‘⊗’ ʹΑΔ ‫ ఺ج‬ͷʮ૊߹ͤ࿦త࠶ฤ੒ʯ
ʂ←→ Hodge theater ʹ͓͚ΔɺઈରԕΞʔϕϧ‫ز‬Կʂʹ ‘ring of fns.’ ͷ ೋछྨ ͷ໾ׂʹର͢Δʮෆมྔʯ!

Ӊ஦ࡍλΠώϛϡʔϥʔཧ࿦΁ͷ༠ʢ͍͟ͳʣ͍ʬϨΫνϟʔϊʔτ൛ʭ11

log-Link:
਺ମ F ͷ nonarch. ͳ v ʹ͓͍ͯ log-link ͷ྆ଆʢʹఆٛҬͱ஋ҬʣͷͦΕͧΕͷ ‫ߏ؀‬଄ ͸ɺ‫؀‬४ಉ‫ͳͱܕ‬Βͳ͍ʢʂʣ‫ؔͰܗ‬࿈෇͚ΒΕΔɿ

logv : O× → k
k

def

def

ʢͨͩ͠ɺk ͸ k = Fv ͷ୅਺ดแɻ·ͨ Gv = Gal(k/k) ͱஔ͘ɻʣ
ϙΠϯτ: log-link ͸ɺ྆ଆͷ਺࿦త‫ج‬ຊ‫ ܈‬Πv ͷؒͷಉ‫ܕ‬∼

Πv → Πv
ͱɺΠv  Gv Λ‫ܦ‬༝ͨ͠ ࣗવͳ࡞༻ ʹؔͯ͠ ཱ྆త Ͱ͋Δɻ෇஋ v Λಈ͔͢ͱɺେҬతͳઈରΨϩΞ‫ ܈‬ͷ࡞༻ͱ΋ཱ྆తͰ͋Δɻ·ͨ਺ମ F ͷ arch.
ͳ v ʹ͓͍ͯ΋ྨࣅతͳཧ࿦͕ଘࡏ͢Δɻɹ

Θ-Link:਺ମ F ͷ bad nonarch. ͳ v ʹ͓͍ͯ Θ-link ͷ྆ଆʢʹఆٛҬͱ஋ҬʣͷͦΕͧΕͷ ‫ߏ؀‬଄ ͸ɺ‫؀‬४ಉ‫ͳͱܕ‬Βͳ͍ʢʂʣ‫ؔͰܗ‬࿈෇͚ΒΕΔɿ
 2 N
∼
× ∼
×
N
O → O ;
(Θ|l-tors ) = q j
→ qN
k

k

j=1,... ,l

def

def

ʢͨͩ͠ɺk ͸ k = Fv ͷ୅਺ดแɻ·ͨ Gv = Gal(k/k) ͱஔ͘ɻʣ
ϙΠϯτ: Θ-link ͸ɺ྆ଆͷ‫ॴہ‬ମͷઈରΨϩΞ‫ ܈‬Gv ͷؒͷಉ‫ܕ‬∼

Gv → Gv
ͱɺͦΕͧΕͷ O× ΁ͷ ࣗવͳ࡞༻ ʹؔͯ͠ ཱ྆త Ͱ͋Δɻ·ͨ਺ମ F ͷkgood nonarch./arch. ͳ v ʹ͓͍ͯ΋ɺੵެࣜ Λຬͨ͢Α͏ʹ Θ-link Λྨࣅతͳख๏Ͱఆٛ͢Δɻ
஫: ʢHodge theater ʹొ৔͢ΔΑ͏ͳʣʮந৅తͳϞϊΠυ ౳ʯΛѻ͏Α͏ʹ͠ͳ͍ͱɺlog-, Θ-link ͷΑ͏ͳʢ௨ৗͷ‫؀‬ɾεΩʔϜ࿦ͷ ‫ߏ؀‬଄ʹର͢ΔʣʮนʹোนʯΛఆٛ͢Δ͜ͱ͢ΒͰ͖ͳ͍ʂ

๬݄৽Ұʢ‫ژ‬େ਺ཧ‫ݚ‬ʣ12

஫: Ұํɺର਺ɾςʔλ֨ࢠͷ਺࿦త‫ج‬ຊ‫܈‬ɾΨϩΞ‫܈‬తͳ෦෼Ͱߏ੒͞ΕΔ étale-picture ʹొ৔͢Δର৅ͨͪ͸ɺ͜ΕΒͷʮนʯΛ͢Γൈ͚Δྗ
ʢʹʮΨϩΞಉมੑʯʹΑΓʣ͕͋ΔʂʢԼਤΛࢀরʂʣೋछྨͷ਺ֶతର৅
ந৅తͳϞϊΠυ ʹ Frobenius ‫ ܕ‬ͷର৅,਺࿦త‫ج‬ຊ‫܈‬ɾΨϩΞ‫ = ܈‬étale ‫ ܕ‬ͷର৅Λؔ࿈෇͚Δɺ༷ʑͳ‫ܗ‬ͷʮKummer ཧ࿦ʯ͸ɺIUTeich શମͷதͰ΋‫ۃ‬Ίͯॏཁͳ໾ׂΛՌͨ͢ʂ ·ͨʮFrobenius ‫  ܕ‬étale ‫ܕ‬ʯ΁ͱҠߦ͢Δૢ࡞͸ɺ‫ݹ‬యతͳ Ψ΢εੵ෼

 ∞
e

−x2

dx =

√


π

−∞



= ʮ΢ΣΠτ 12 ʯʂ∈ Q · π Z  ζ(n ∈ 2 · Z)ͷ‫ࢉܭ‬ʢʹʮσΧϧτ࠲ඪ  ‫࠲ۃ‬ඪʯʣͷɺ਺ମ্େҬతͳྨࣅ෺ ͱ‫ݟ‬Δ͜ͱ͕Ͱ͖Δʂ࣮ࡍɺ͜ͷ‫ʹࢉܭ‬ग़ͯ͘Δࣜม‫ܗ‬  −x2 −y2  −r22
2 · ( e−x dx)2 = 2 ·edx dy =e
· 2rdr dθ



2
=
d(e−r ) dθ =du dθ2
ͭ·Γɺe−r  u ͱ͍͏‫੻ح‬ʢʂʣͷΑ͏ͳ࠲ඪม‫׵‬͸ɺΘ-link ͷରԠͱ‫͕ܗ‬Α͘ࣅ͍ͯΔʂ

arith. hol.
str. Πv...

arith.
hol. str.
Πv
...

...

|
–

monoanalytic
core Gv|
arith. hol.
str. Πv–

arith.
hol. str.
Πv...

Ӊ஦ࡍλΠώϛϡʔϥʔཧ࿦΁ͷ༠ʢ͍͟ͳʣ͍ʬϨΫνϟʔϊʔτ൛ʭ13

Kummer ཧ࿦Λద༻ ͢Δओͳର৅ʢʹ Θ-link ͷࠨଆ Λࢀরʂʣɿ
 × ɹʢnonarch. v ʣ Z 2ʢbad nonarch. v ʣɹ (b) ςʔλؔ਺ͷ஋ ɹ Θ|l-tors = q jɹ (a) ୯਺‫ ܈‬ɹ O×kj=1,... ,l×ɹ (c) ਺ମ ɹ F (⊇ F ) ɹͷҰछͷʮ࣮Խʯཧ࿦ͷओͳϙΠϯτ͸ ୯਺‫ ͱ ܈‬஋‫ ܈‬Λ ෼཭ ͠ɺ(b), (c) ʹؔ࿈͢ΔϞϊΠ × ʯͱ͍͏ ෆఆੑ ͔ΒकΔ͜ͱʹ
 ʣΛɺ
ʮ Z
υʹ‫·ؚ‬ΕΔ ԁ෼෺ʢ∼= Z(1)
ʂʮԁ෼߶ੑʯ
ɹ (b) ͷ৔߹ɿΤλʔϧɾςʔλؔ਺ͷཧ࿦ ɹ =⇒ ɹ෼཭ʴԁ෼߶ੑɹ (c) ͷ৔߹ɿॳ౳త୅਺త੔਺࿦ ɹɹɹɹɹ =⇒ ɹ෼཭ʴԁ෼߶ੑ

Hodge theater ͷͦΕͧΕͷ ରশੑ ͕ಇ͘ྖҬ͸ɺ(b), (c) ͷͦΕͧΕͷ
Kummer ཧ࿦ͷ࣮ߦʹద͍ͯ͠ΔʢԼදΛࢀরʣʂ͜ͷঢ়‫گ‬͸ɺ‫ݹ‬యతͳ্൒ฏ໘ H ͷ༷ʑͳ ରশੑ ʹ෇ਵ͢Δʮؔ਺ʯͷʢΑ͘஌ΒΕ͍ͯΔʣॳ౳తͳཧ࿦ʹΑ͘ࣅ͍ͯΔʢԼදΛࢀরʣʂ͜ΕΒͷରশ
ੑ͸ Bogomolov ʹΑΔ ‫ز‬Կత Szp. ༧૝ͷผূ໌ ͷʮग़ൃ఺ʯͰ΋͋ΔʂɹɹʢΧεϓ‫ܕ‬ͷʣՃ๏తରশੑ

‫ݹ‬యతͳ
্൒ฏ໘ H

IUTeich ʹ͓͚Δ
Θ±ell NF-Hodge th.

z → z + a,z → −z + a (a ∈ R)

F±
l ରশੑ

Ճ๏తରশੑʹ෇ਵdefʮ‫ॴہ‬తͳؔ਺ʯ

௒ӽతؔ਺ q = e2πizʢϊʔυɾz → z·cos(t)−sin(t)z·sin(t)+cos(t) ,τʔϥε‫ܕ‬ͷʣ৐๏తରশੑ

z·cos(t)+sin(t)z → z·sin(t)−cos(t)Θ|l-tors ɹɹɹɹɹ

ɹ = qj2

j=1,... ,l

F
l ରশੑ

(t∈R)

৐๏తରশੑʹ෇ਵ୅਺త਺ମ F ͷ‫ʹݩ‬ର͢Δ

͢ΔʮେҬతͳؔ਺ʯ

༗ཧؔ਺ w =

def

z−i
z+i

Belyi ࣸ૾ʹΑΔදࣔ

๬݄৽Ұʢ‫ژ‬େ਺ཧ‫ݚ‬ʣ14࣮͸ɺઌ΄Ͳͷ IUTeich ͷٞ࿦͸্൒ฏ໘্ͷ‫ݹ‬యతͳ ςʔλؔ਺∞2
θ(t) =
e−πn tn=−∞ʹର͢Δ ϠίϏͷม‫ࣜ׵‬

θ(t) = t−1/2 · θ(1/t)ʢʹ͋Δҙຯɺઌ΄Ͳग़͖ͯͨ Ψ΢εੵ෼ ͷ‫ࢉܭ‬ͷ ؔ਺൛ ͱ‫ݟ‬၏ͤΔʂʣͱɺ༷ʑͳଆ໘ʹ͓͍ͯඇৗʹΑ͘ࣅ͍ͯΔʢԼදΛࢀরʣ
ʂɹ

IUTeichϠίϏͷม‫ؔʹࣜ׵‬࿈ͨ͠ཧ࿦

Τλʔϧɾςʔλؔ਺ͷ߶ੑੑ࣭Ψ΢ε෼෍ͷ
ϑʔϦΤม‫ؔ͢ʹ׵‬Δෆมੑ× ɹ
O×  Zkɹɹɹɹɹɹͱ͍͏ෆఆੑ

ϑʔϦΤม‫ʹ׵‬ग़ͯ͘Δࢦඪɹɹ (−) · eit , t ∈ Rςʔλ‫܈‬ͷ [−, −] ͷೋ࣍ੑʹΑΔ߶ੑੑ࣭ͷূ໌Ψ΢ε෼෍ͷࢦ਺ͷೋ࣍ੑʹΑΔϑʔϦΤෆมੑͷূ໌



qj

2

j=1,... ,llog-link ʹΑΔ ,  ͷճసʹςʔλؔ਺ͷΨ΢ε෼෍‫ܕ‬ల։ɹճస

 0 1
−1 0⇐⇒ t → 1t

ର͢ΔઈରԕΞʔϕϧ‫ز‬Կͷద༻

ଈͪ ∞  0 ΁ͷղੳ઀ଓઈରԕΞʔϕϧ‫ز‬ԿΞϧΰϦζϜͷ
‫ॴہ‬ɾେҬؒͷؔखੑɺ
Belyi ΧεϓԽ্൒ฏ໘શମ΁ͷղੳ઀ଓӉ஦ࡍλΠώϛϡʔϥʔཧ࿦΁ͷ༠ʢ͍͟ͳʣ͍ʬϨΫνϟʔϊʔτ൛ʭ15͜ͷ ςʔλؔ਺∞θ(t) =

2e−πn tn=−∞ʹର͢Δ ϠίϏͷม‫ࣜ׵‬

θ(t) = t−1/2 · θ(1/t)͸࣍ͷΑ͏ʹղऍ͢Δ͜ͱ΋Ͱ͖Δɿɹ ∞ ͷۙ๣ʹ͓͍֤ͯඇఆ਺߲͕
ʮ‫ݮٸ‬গؔ਺ʯͱͳΔΑ͏ͳ࿨ʹΑͬͯදࣔՄೳͰ͋Δͱ͍͏ੑ࣭͸ɺ∞  0
ͱ ղੳ઀ଓ ͯ͠΋ɺෆࢥٞ ͳ͜ͱʹ 0 ͷۙ๣ʹ͓͍ͯ΋੒ཱ͢Δʂ͜Ε͸ͪΐ͏Ͳ§1 ʹग़͖ͯͨ େҬత ͳ
 2
ʮ qj
ʯ
j=1,... ,lͷ࣮‫ʹݱ‬ରԠ͍ͯ͠Δʂ

∞Gaussianθ

Gaussian０๬݄৽Ұʢ‫ژ‬େ਺ཧ‫ݚ‬ʣ

16§4. Ӊ஦ࡍੑͱԕΞʔϕϧ‫ز‬Կlog-link ‫ ͼٴ‬Θ-link
logv : O×
k

N

→ k,



(Θ|l-tors ) = q j2Nj=1,... ,l

∼

→ qN

͸ɺఆٛҬɾ஋Ҭͷ ‫ߏ؀‬଄ ͱ ཱ྆͠ͳ͍ ͨΊɺ‫ߏ؀‬଄ ͔Βੜ͡Δ εΩʔϜ࿦తͳ ʮ‫఺ج‬ʯ΍ɺΨϩΞ‫ ܈‬ɹʢɹ ⊆ Autﬁeld (k) ʂʂɹʣͱɺຊ࣭తʹཱ྆͠ͳ͍ʂɹͭ·Γɺlog-, Θ-link ͷʮ޲͜͏ଆʯʹҠߦ͢Δͱ͖ɺ“Πv ” ΍ “Gv ”
͸ɺந৅తͳҐ૬‫͔ͯ͠͠ͱ ܈‬ɺʮ޲͜͏ଆʯͷεΩʔϜ࿦ʹ௨༻͠ͳ͍ʂ
ʢମͷࣗ‫ݾ‬ಉ‫ʹܕ‬ΑͬͯҾ͖‫͜͞ى‬ΕΔઈରΨϩΞ‫܈‬ͷ֎෦ࣗ‫ݾ‬ಉ‫ܕ‬ͷ৔߹Λࢀরɻʣ=⇒ ఆٛҬɾ஋Ҭ૒ํͷ‫ߏ؀‬଄ͷؒͷؔ܎Λ‫͢ࢉܭ‬ΔͨΊʹ͸ɺԕΞʔϕϧ‫ز‬ԿΛ‫͢༻׆‬Δ͔͠ͳ͍ʂա‫ڈ‬ͷ࿦จͷϨϕϧͰ͍͏ͱɺ
ઈରԕΞʔϕϧ‫ز‬Կ ɹ΍ɹ Τλʔϧɾςʔλؔ਺ͷ༷ʑͳ߶ੑੑ࣭
ʹؔ͢Δ
ɾSemi-graphs of Anabelioids

ɾThe Geometry of Frobenioids I, IIɾThe Étale Theta Function ...

ɾTopics in Absolute Anab. Geo. IIIͷ݁Ռ΍ཧ࿦Λద༻͢Δ͜ͱʹΑͬͯओఆཧΛ‫݁͢ؼ‬Δɿ
ओఆཧɿɹ Θ-link ͷ ࠨล ʹରͯ͠ɺܰඍͳෆఆੑΛআ͍ͯɺӈล ͷʮҟ࣭ʯͳ ‫ߏ؀‬଄ ͔͠༻͍ͳ͍‫ݴ‬༿ʹΑΓɺ໌ࣔతͳΞϧΰϦζϜ ʹΑΔ‫ه‬ड़Λ༩͑Δ͜ͱ͕Ͱ͖Δɻղऍɿʮ‫͍ڱ‬ύΠϓʯͰ͔͠‫͍ͳ͍͕ͯͬܨ‬Α͏ͳঢ়‫گ‬ʢʹྫ͑͹ɺɹӉ஦ધʹ͍ΔӉ஦ඈߦ࢜΍஍Լͷ߭ࢁͰಇ͘࡞‫ۀ‬һ౳ʣʹ͓͍ͯɺ‫ݶ‬ΒΕͨ৘ใ Λ‫͘ݡ‬ར༻͢Δ͜ͱʹΑͬͯʮ޲͜͏ଆʯͷঢ়‫گ‬Λ෮‫͠ݩ‬ɺ೺Ѳ͢Δ͜ͱ͕Ͱ͖ΔɻӉ஦ࡍλΠώϛϡʔϥʔཧ࿦΁ͷ༠ʢ͍͟ͳʣ͍ʬϨΫνϟʔϊʔτ൛ʭ17ূ໌ͷϙΠϯτ:ɾGv  O× ͷ ίΞੑʢcoricityʣʂkɾೋछྨͷ਺ֶతର৅Λؔ࿈෇͚Δɺ༷ʑͳ‫ܗ‬ͷʮKummer ཧ࿦ʯɿʢ§3 ‫ޙ‬൒ͷղઆΛࢀরʣந৅తͳϞϊΠυ ʹ Frobenius ‫ ܕ‬ͷର৅,਺࿦త‫ج‬ຊ‫܈‬ɾΨϩΞ‫ = ܈‬étale ‫ ܕ‬ͷର৅͜͜ͰɺΨ΢εੵ෼ ͷ‫ͱࢉܭ‬ͷྨࣅʹʮ୯਺‫ ͱ ܈‬஋‫ ܈‬ͷ ෼཭ʯΛࢥ͍ग़ͦ͏ɿ

log-, Θ-link ΍ର਺ɾςʔλ֨ࢠͷఆٛɹ ←→ ɹ σΧϧτ࠲ඪઈରԕΞʔϕϧ‫ز‬Կ Λ༻͍ͨΞϧΰϦζϜʹΑΔ‫ه‬ड़ɹ ←→ ɹ ‫࠲ۃ‬ඪԁ෼෺ (∼
ͷ֬อʹ߶ੑ͕؊৺ʂɹ ←→ ɹ S1  ʹΑΔ࠲ඪม‫׵‬= Z(1))
ɾʮlog-shellʯʹʮ৐๏తͳର৅ͷͨΊͷՃ๏తͳೖΕ෺ʯlog(−) ΁ͷ࡞༻ςʔλؔ਺ͷ஋qj

2

j=1,... ,l log(O× )  F × ⊆ F ਺ମk

ʢBogomolov ͷূ໌ΛࢀরʂʣΛ࣮‫͢ݱ‬ΔͨΊʹ͸ɺlog-link ͷ‫͕༻׆‬ඞཁෆՄܽͰ͋Δɻ
ɹɹɽɽɽҰํɺର਺ɾςʔλ֨ࢠ ͷ ඇՄ‫׵‬ੑ ʹΑ༷ͬͯʑͳࠔ೉͕ੜ͡Δɻ
ʢ౳ࣜͰ͸ͳ͘ʂʣෆ౳ࣜ ͔͠ग़ͳ͍ʂ
ɹɹɹ =⇒ ɹ‫ޙ‬ͷʮମੵ‫ࢉܭ‬ʯͰ͸ɺओఆཧͷΞϧΰϦζϜͷ ग़ྗ ʹରͯ͠ɺମੵ‫ ࢉܭ‬Λߦ͏ͱɺ§1 Ͱղઆͨ͠
Α͏ʹ࣍ͷΑ͏ͳ‫͕݁ؼ‬ಘΒΕΔʢFaltings ʹΑΔ Mordell ༧૝ ͷূ໌ʹ
ग़ͯ͘Δɺྨମ࿦ ΍ p ਐϗοδཧ࿦ɺΞʔϕϧଟ༷ମؔ࿈ͷ୅਺‫ز‬Կ ౳Λࢀরʂʣɿ‫ܥ‬:ɹʮʢ‫ܗ͍ڧ‬ͷʣSzpiro ༧૝ʯʢ⇐⇒ ʮABC ༧૝ʯʣɻ

htE ≤ (1 + )(log-diﬀF + log-condE ) + constant͜͜ͰʮN · hLHS = hRHS ʯʢʹ Θ-linkʂʣ΍ʮN · h ≤ h + C ʯʢʹ ओఆཧʴମੵ‫ࢉܭ‬ʣͷٞ࿦ʢ§1ʣΛࢥ͍ग़ͦ͏ʂ

๬݄৽Ұʢ‫ژ‬େ਺ཧ‫ݚ‬ʣ18ઌ΄Ͳͷٞ࿦͸ɺ§3 ͷ࠷‫ʹޙ‬ղઆͨ͠ ϠίϏͷม‫ͱ ࣜ׵‬ͷྨࣅͰߟ͑Δͱɺ
༷ʑͳྨࣅ఺͕ු͔ͼ্͕Δɿ

IUTeichϠίϏͷม‫ؔʹࣜ׵‬࿈ͨ͠ཧ࿦

Ӊ஦ʹू߹ʹର͢Δ
ɹɹɹɹɹɹϥϕϧ෇͚ͷ࢓૊Έͷม‫׵‬ɹɹɹɹɹɹ

࠲ඪʹۭؒͷ఺ʹର͢Δ
ɹɹɹɹϥϕϧ෇͚ͷ࢓૊Έͷม‫׵‬ɹɹɹɹɹɹɹɹ

Ψ΢εੵ෼

log-shell log(−)

∞2e−x dx =√π−∞ͷମੵ‫ࢉܭ‬ͷ‫࠲ۃ‬ඪʹΑΔ‫ࢉܭ‬σΟΦϑΝϯτε‫ز‬Կ΁ͷ‫ڻ‬ҟతͳԠ༻

‫ݹ‬యతςʔλؔ਺ͷ஋ͷ‫ࢉܭ‬ʹ͓͚Δ‫ڻ‬ҟతͳਫ਼౓ͷ޲্࣮͸ɺઌ΄Ͳͷෆ౳ࣜʹొ৔ͨ͠ʮʯ͸ɺ1(htE )− 2 · log(htE )ҐͷΦʔμʔʹ ཈͑Δ ͜ͱ͕Ͱ͖Δɻ͜ͷʮ 12 ʯ͸ϦʔϚϯ༧૝Λ࿈૝ͤ͞ΒΕΔ஋Ͱ͋Δ͕ɺ·͘͞͠ ϦʔϚϯ༧૝ ͱಉ͘͡ɺ
ʮ΢ΤΠτ 1/2ʯʢ஫ɿʮ΢ΣΠτʯ͸ϦʔϚϯɾθʔλ ζ(s) ͷʮsʯʣɺͭ·ΓʢTate ೧Γʹର
Ԡ͢Δʣπ ͷ੔਺ႈͰ͸ͳ͘ɺπ ͷฏํࠜ ∞2e−x dx =√π−∞ʹؔ܎͢Δ‫ݱ‬৅Ͱ͋Δɻ࣮ࡍɺઌ΄Ͳͷʮʯͷ‫Ͱࢉܭ‬͸ɺΨ΢εੵ෼΍ςʔλؔ਺ʹ‫ݱ‬ΕΔΑ͏ͳ ೋ࣍‫͕ ࣜܗ‬ग़͖ͯͯɺͦͷྔͷ࠷গ஋Λ‫ٻ‬ΊΔͱɺೋ1࣍‫ࣜܗ‬ͷࠜʹ ฏํࠜ ʹʮ(htE )− 2 ʯͱ͍͏͕ࣜൃੜ͢ΔͷͰ͋ΔɻӉ஦ࡍλΠώϛϡʔϥʔཧ࿦΁ͷ༠ʢ͍͟ͳʣ͍ʬϨΫνϟʔϊʔτ൛ʭ19p ਐ‫ॴہ‬ମ্ͷ૒‫ۂ‬త‫ۂ‬ઢ΍ C ্ͷϦʔϚϯ໘ͷ‫ز‬Կʹؔ͢Δ‫ݹ‬యతͳཧ࿦
ͱͷྨࣅͰ͍͏ͱɺʮਖ਼ଇߏ଄ͷମੵʯΛ ਖ਼ଇߏ଄ͷ֎ ʹ͋Δ࿮૊Ͱ‫͢ࢉܭ‬Δ͜ͱʹΑͬͯಘΒΕΔɺ࣍ͷΑ͏ͳʢʮ૒‫ۂ‬ੑʯΛද‫͍ͯ͠ݱ‬Δʂʣෆ౳ࣜ
͕ରԠ͍ͯ͠Δͱ‫ݟ‬၏ͤΔɿ
ɹɾɹ pTeich ʹ͓͚Δ

“Hasse invariant =1· d(Frob. lift.)”pɹɹɹͷ࣍਺ = (2g − 2)(1 − p) ≤ 0ɹɾɹ૒‫ۂ‬తͳϦʔϚϯ໘ S ্ͷ Gauss-Bonnet ͷఆཧ


0 > −

(Poincaré metric) = 4π(1 − g)S————————————————————————————————
࠷‫ʹޙ‬ɺʮIU ‫ز‬Կͷ৺ʯʹɹʮ௨ৗͷεΩʔϜ࿦͕༗ޮͰ͸ͳ͍Α͏ͳ ‘૊߹ͤ࿦త’ ͳઃఆʹɹ͓͍ͯɺ௨ৗͷ εΩʔϜ࿦ ʹ ώϯτ Λಘͨߏ੒Λߦͳ͍ɺ௨ৗ
ɹͷεΩʔϜ࿦Λ͋Δఔ౓ ۙࣅ ͢Δ͜ͱʹΑͬͯ ඇࣗ໌ ͳ݁ՌΛɹग़͢ʯͱ͍͏ߟ͑ํͷ΋͏ҰͭͷʢΑΓ ॳ౳త ͳʣྫͱͯ͠
૊߹ͤ࿦తԕΞʔϕϧ‫ز‬Կ  GT ‫ؔ͢ʹ ܈‬Δ༷ʑͳ݁Ռ
ͱ͍͏ྫ͕ଘࡏ͢Δ͜ͱΛࢦఠ͍ͨ͠ɻ͜ΕΒͷ݁Ռͷझࢫ͸ɺGT ‫͕܈‬ʮGQͱಉ‫͋Ͱܕ‬Δʯ͜ͱΛࣔ͢ ୅ΘΓ ʹɺGT ‫ ͕܈‬GQ ͱʮྨࣅతͳੑ࣭ʯΛຬͨ͢͜ͱΛࣔ͢͜ͱʹ͋Δɻ