# Lecons sur le Calcul des Coefficients. Deuxieme Partie by Denjoy A.

By Denjoy A.

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5✮ ✐s |X/Q |✳ ❝✮ g. 1✳ ❬❋♦r t❤❡ X = Spec OK ✱ ✇❤❡r❡ K ✐s ❛ ♥✉♠❜❡r ✜❡❧❞✳ v ♦❢ K ✇✐t❤ |v| = p✳ ❲❡ ♠❛② ❝❤♦♦s❡ ❢♦r E t❤❡ ●❛❧♦✐s ❝❧♦s✉r❡ ♦❢ K ❀ ✇❡ ❤❛✈❡ G = Gal(E/Q) ❀ ❧❡t ✉s ♣✉t H = Gal(E/K)✳ ❲❡ ♠❛② ✐❞❡♥t✐❢② X(Q) ✇✐t❤ G/H ✳ ▲❡t S ❜❡ t❤❡ s❡t ♦❢ ♣r✐♠❡ ❞✐✈✐s♦rs ♦❢ disc(K) ✭♦r ♦❢ disc(E) ✕ ✐t ❛♠♦✉♥ts t♦ t❤❡ s❛♠❡✮✳ ■❢ p ∈ / S ✱ ❧❡t σp ∈ G ❜❡ ✐ts ❋r♦❜❡♥✐✉s ❡❧❡♠❡♥t ✭✉♣ t♦ ❝♦♥❥✉❣❛✲ t✐♦♥✮✳ ❲❡ ❤❛✈❡ NX (p) = ϕ(σp )✳ ❚❤✐s s❤♦✇s t❤❛t t❤❡ ♠❛♣ p → NX (p) ✐s Pr♦♦❢✳ ▲❡t ✉s s✉♣♣♦s❡ ✜rst t❤❛t ❲❡ t❤❡♥ ❤❛✈❡ NX (p) ❂ ♥✉♠❜❡r ♦❢ ♣❧❛❝❡s ✸✳✹✳ ❊①❛♠♣❧❡s ♦❢ S ✲❢r♦❜❡♥✐❛♥✱ S ✲❢r♦❜❡♥✐❛♥ ❛♥❞ t❤❛t ϕ ❢✉♥❝t✐♦♥s ❛♥❞ S ✲❢r♦❜❡♥✐❛♥ s❡ts ✷✼ ✐s t❤❡ ❛ss♦❝✐❛t❡❞ ❢✉♥❝t✐♦♥✳ ❚❤✐s ♣r♦✈❡s ❛✮ ❛♥❞ ❜✮✳ ❆ss❡rt✐♦♥ ❝✮ ❢♦❧❧♦✇s ❢r♦♠ t❤❡ ❡❛s✐❧② ♣r♦✈❡❞ ❢♦r♠✉❧❛ NX (pe ) = ϕ(σpe )✳ ❙✐♥❝❡ ϕ(1) = |G/H| = [K/Q] = |X(C)|✱ t❤✐s ✐♠♣❧✐❡s t❤❡ ❛ss❡rt✐♦♥ ❛❜♦✉t f (1) ❀ ❛ s✐♠✐❧❛r ♠❡t❤♦❞ ✇♦r❦s ❢♦r f (−1)✳ ❆s ❢♦r t❤❡ ♠❡❛♥ ✈❛❧✉❡ ♦❢ f ✱ ✐t ✐s ❜② ❞❡✜♥✐t✐♦♥ t❤❡ ♠❡❛♥ ✈❛❧✉❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ϕ✱ t❤❛t ✐s ❡q✉❛❧ t♦ ✶ ❜② ❇✉r♥s✐✲ ❞❡✬s ❧❡♠♠❛ ✭s❡❡ ❡✳❣✳ ❬❙❡ ✵✷✱ ➓✷✳✶❪✮✳ ❚❤✐s ♣r♦✈❡s Pr♦♣♦s✐t✐♦♥ ✸✳✶✵ ✐♥ t❤❡ ❝❛s❡ X = Spec OK ✳ ❚❤❡ ❣❡♥❡r❛❧ ❝❛s❡ ❢♦❧❧♦✇s ❜② ❞♦✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ ♦♣❡r❛t✐♦♥s ♦♥ X ✿ • ♠❛❦✐♥❣ ✐t r❡❞✉❝❡❞ ❀ t❤✐s ❞♦❡s ♥♦t ❝❤❛♥❣❡ ❛♥② NX (pe ) ❀ • ♠❛❦✐♥❣ ✐t ♥♦r♠❛❧ ❀ t❤✐s ❝❤❛♥❣❡s NX (pe ) ❢♦r ♦♥❧② ✜♥✐t❡❧② ♠❛♥② p ❀ • ❞❡❝♦♠♣♦s✐♥❣ ✐t ✐♥t♦ ✐rr❡❞✉❝✐❜❧❡ ❝♦♠♣♦♥❡♥ts ❀ ❡❛❝❤ ❝♦♠♣♦♥❡♥t ✐s t❤❡♥ Σ✱ ✇❤❡r❡ K ✐s ❛ ♥✉♠❜❡r ✜❡❧❞ ❛♥❞ Σ ✐s ❛ ✐s♦♠♦r♣❤✐❝ t♦ X = Spec OK e e ❝❧♦s❡❞ ✜♥✐t❡ s✉❜s❡t ♦❢ Spec OK ❀ ♦♥❡ t❤❡♥ ❤❛s NX (p ) = NX (p ) ❢♦r ❛❧❧ ❧❛r❣❡ ❡♥♦✉❣❤ ♣r✐♠❡s p✳ ❈♦r♦❧❧❛r② ✸✳✶✶✳ ■❢ X ✐s ❛ s❝❤❡♠❡ ♦❢ ✜♥✐t❡ t②♣❡ ♦✈❡r t❤❡r❡ ❛r❡ ✐♥✜♥✐t❡❧② ♠❛♥② p ✇✐t❤ Pr♦♦❢✳ ❈❤♦♦s❡ ❛ ❝❧♦s❡❞ ♣♦✐♥t ♦❢ X x Z s✉❝❤ t❤❛t X/Q = ∅✱ NX (p) > 0✳ ♦❢ X/Q ❀ ✐ts ❝❧♦s✉r❡ Xx ✐♥ X ✐s ❛ s✉❜s❝❤❡♠❡ t♦ ✇❤✐❝❤ ♦♥❡ ❛♣♣❧✐❡s ♣❛rt ❝✮ ♦❢ Pr♦♣♦s✐t✐♦♥ ✸✳✶✵✳ ❍❡♥❝❡ t❤❡r❡ ❛r❡ ✐♥✜♥✐t❡❧② ♠❛♥② p ✇✐t❤ NXx (p) > 0✱ ❘❡♠❛r❦s✳ ✶✳ ❚❤❡ ❤②♣♦t❤❡s✐s ❛♥❞ ❛ ❢♦rt✐♦r✐ X/Q = ∅ NX (p) > 0✳ ✐s ❡q✉✐✈❛❧❡♥t t♦ X(Q) = ∅ ❛♥❞ t♦ X(C) = ∅✳ ✷✳ ❇② ❛ t❤❡♦r❡♠ ♦❢ ❆① ❛♥❞ ✈❛♥ ❞❡♥ ❉r✐❡s ✭❬❆① ✻✼❪✱ ❬❉r ✾✶❪✱ s❡❡ ❛❧s♦ p ✇✐t❤ NX (p) = 0 ✐s ❢r♦❜❡♥✐❛♥ ❀ > 0, ❛s ✇❛s ✜rst ♣r♦✈❡❞ ✐♥ ❬❆① ✻✼❪✳ ➓✼✳✷✳✹✮✱ t❤❡ s❡t ♦❢ ❞❡♥s✐t②✱ t❤❛t ✐s ❊①❛♠♣❧❡ ✿ ◆✉♠❜❡r ♦❢ r♦♦ts ♠♦❞ p ✐♥ ♣❛rt✐❝✉❧❛r✱ ✐t ❤❛s ❛ ♦❢ ❛ ♦♥❡✲✈❛r✐❛❜❧❡ ♣♦❧②♥♦♠✐❛❧✳ X = Z[t]/(H)✱ ✇❤❡r❡ H ✐s ❛ ♥♦♥✲③❡r♦ ❡❧❡♠❡♥t ♦❢ t❤❡ ♣♦❧②✲ Z[t]✳ ▲❡t a0 tn ❜❡ t❤❡ ❧❡❛❞✐♥❣ t❡r♠ ♦❢ H ❛♥❞ ❧❡t S ❜❡ t❤❡ s❡t ♦❢ t❤❡ ♣r✐♠❡ ❞✐✈✐s♦rs ♦❢ a0 disc(H)✳ ❚❤❡♥ ✿ ❚❤❡ ♠❛♣ p → NH (p) ✐s S ✲❢r♦❜❡♥✐❛♥✱ ❛♥❞ ✐ts ✈❛❧✉❡ ❛t 1 (r❡s♣✳ ❛t −1) ✐s t❤❡ ✽ ♥✉♠❜❡r ♦❢ ❝♦♠♣❧❡① (r❡s♣✳ r❡❛❧ ) r♦♦ts ♦❢ H ✳ ❋♦r ❡✈❡r② e 1✱ t❤❡ Ψe✲tr❛♥s❢♦r♠ ♦❢ p → NH (p) ✐s p → NH (pe )✳ ▲❡t ✉s t❛❦❡ ♥♦♠✐❛❧ r✐♥❣ NX (p) p → NX (p) ✐s ✸✳✹✳✷✳✷✳ ♠♦❞ m✳ ▲❡t X ❜❡ ❛ s❝❤❡♠❡ ♦❢ ✜♥✐t❡ t②♣❡ ♦✈❡r ♥♦t ❢r♦❜❡♥✐❛♥ ✭✉♥❧❡ss ❞✐♠ X(C) 0✮✱ Z✳ ❚❤❡ ♠❛♣ ✐❢ ♦♥❧② ❜❡❝❛✉s❡ ✐ts ✐♠❛❣❡ ✐s ✐♥✜♥✐t❡✱ ❜✉t ✐t ✐s r❡s✐❞✉❛❧❧② ❢r♦❜❡♥✐❛♥ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ s❡♥s❡ ✿ ✽ ◆♦t❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝✉r✐♦✉s ❝♦r♦❧❧❛r② ✿ t❤❡r❡ ❛r❡ ✐♥✜♥✐t❡❧② ♠❛♥② t❤❡ s❛♠❡ ♥✉♠❜❡r ♦❢ r♦♦ts ✐♥ Fp ❛s ✐♥ R✳ p s✉❝❤ t❤❛t H ❤❛s ✷✽ ✸✳ ❚❤❡ ❈❤❡❜♦t❛r❡✈ ❞❡♥s✐t② t❤❡♦r❡♠ ❢♦r ❛ ♥✉♠❜❡r ✜❡❧❞ m 1✱ Z/mZ; ✐ts f : p → NX (p) (mod m) ✐s ❛ ❢r♦❜❡♥✐❛♥ 1 ✭r❡s♣✳ ❛t −1) ✐s ❡q✉❛❧ t♦ t❤❡ ✐♠❛❣❡ ✐♥ ❝❤❛r❛❝t❡r✐st✐❝ χ(X(C)) ✭r❡s♣✳ χc (X(R)✮✮✳ ❋♦r ❡✈❡r② ✐♥t❡❣❡r t❤❡ ♠❛♣ ♠❛♣ ♦❢ P ✈❛❧✉❡ ❛t Z/mZ ♦❢ t❤❡ ❊✉❧❡r✲P♦✐♥❝❛ré ✐♥t♦ ❚❤❡s❡ st❛t❡♠❡♥ts ✇✐❧❧ ❜❡ ♣r♦✈❡❞ ✐♥ ➓✻✳✶✳✷ ❀ ♥♦t❡ t❤❛t t❤❡② ✐♠♣❧② ❚❤❡♦✲ r❡♠ ✶✳✹ ♦❢ ➓✶✳✹✳ ✸✳✹✳✸✳ ❚❤❡ p✲t❤ ❝♦❡✣❝✐❡♥t ♦❢ ❛ ♠♦❞✉❧❛r ❢♦r♠ N ✱ ❛ ✇❡✐❣❤t k > 0✱ ❛ ❉✐r✐❝❤❧❡t ❝❤❛r❛❝t❡r ε ♠♦❞ N ✱ ❛♥❞ ❛ ♠♦❞✉❧❛r ❢♦r♠ ϕ = an q n ♦♥ Γ0 (N ) ♦❢ ✇❡✐❣❤t k ❛♥❞ t②♣❡ ε✱ ❝❢✳ ❡✳❣✳ ❬❉❙ ✼✹✱ ➓✶❪✳ ❆ss✉♠❡ t❤❛t t❤❡ ❝♦❡✣❝✐❡♥ts an ♦❢ ϕ ❜❡❧♦♥❣ t♦ t❤❡ r✐♥❣ ♦❢ ✐♥t❡❣❡rs A ♦❢ s♦♠❡ ♥✉♠❜❡r ✜❡❧❞✳ ❚❤❡♥ t❤❡ ♠❛♣ p → ap ✐s r❡s✐❞✉❛❧❧② ❢r♦❜❡♥✐❛♥✱ ✐♥ ❛ ▲❡t ✉s ❝❤♦♦s❡ ❛ ❧❡✈❡❧ s✐♠✐❧❛r s❡♥s❡ ❛s ✐♥ ➓✸✳✹✳✷✳✷✳ ▼♦r❡ ♣r❡❝✐s❡❧② ✿ ❚❤❡♦r❡♠ ✸✳✶✷✳ ❋♦r ❡✈❡r② ✐♥t❡❣❡r m 1✱ SmN ✲❢r♦❜❡♥✐❛♥✱ ✇❤❡r❡ SmN ✐s t❤❡ s❡t ♦❢ ♣r✐♠❡s 1 (r❡s♣✳ ❛t −1) ✐s 2a1 ✭resp✳ 0) ✭♠♦❞ mA)✳ ❈♦r♦❧❧❛r② ✸✳✶✸✳ ❚❤❡ s❡t ♦❢ > 0✳ ❛♥❞ ✐ts ❞❡♥s✐t② ✐s p ✇✐t❤ p → ap ✭♠♦❞ mA) ✐s ❞✐✈✐❞✐♥❣ mN.

Dens(F ) = 21 dens(F ) ❀ ❧♦♦s❡❧② s♣❡❛❦✐♥❣✱ t❤❡ ❝♦♥❞✐t✐♦♥ p ∈ PX ✐s ✐♥❞❡♣❡♥❞❡♥t ♦❢ ❛♥② ❢r♦❜❡♥✐❛♥ ❝♦♥❞✐t✐♦♥✳ ❈❤❛♣t❡r ✹ ❘❡✈✐❡✇ ♦❢ ✲❛❞✐❝ ❝♦❤♦♠♦❧♦❣② ❚❤❡ r❡s✉❧ts s✉♠♠❛r✐③❡❞ ✐♥ t❤✐s ❝❤❛♣t❡r ✭❡①❝❡♣t t❤♦s❡ ♦❢ ➓✹✳✻✮ ❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥ t❤❡ t❤r❡❡ ✈♦❧✉♠❡s ♦❢ ❙●❆ r❡❧❛t✐✈❡ t♦ ét❛❧❡ ❝♦❤♦♠♦❧♦❣② ✿ ❬❙●❆ ✹❪✱ ❬❙●❆ 4 12 ❪ ❛♥❞ ❬❙●❆ ✺❪✱ t♦❣❡t❤❡r ✇✐t❤ ❉❡❧✐❣♥❡✬s ♣❛♣❡rs ❬❉❡ ✼✹❪ ❛♥❞ ❬❉❡ ✽✵❪ ♦♥ ❲❡✐❧✬s ❝♦♥❥❡❝t✉r❡s✳ ❋♦r ❛ s❤♦rt❡r ❛❝❝♦✉♥t ✭✇✐t❤ ♦r ✇✐t❤♦✉t ♣r♦♦❢s✮✱ s❡❡ ❡✳❣✳ ❬❋❑ ✽✽❪✱ ❬❑❛ ✾✹❪✱ ❬❑❛ ✵✶❜❪ ♦r ❬▼✐ ✽✵❪✳ ✹✳✶✳ ❚❤❡ ✲❛❞✐❝ ❝♦❤♦♠♦❧♦❣② ❣r♦✉♣s ks ❛♥❞ ❧❡t Γk = Gal(ks /k).

5✮ ✐s |X/Q |✳ ❝✮ g. 1✳ ❬❋♦r t❤❡ X = Spec OK ✱ ✇❤❡r❡ K ✐s ❛ ♥✉♠❜❡r ✜❡❧❞✳ v ♦❢ K ✇✐t❤ |v| = p✳ ❲❡ ♠❛② ❝❤♦♦s❡ ❢♦r E t❤❡ ●❛❧♦✐s ❝❧♦s✉r❡ ♦❢ K ❀ ✇❡ ❤❛✈❡ G = Gal(E/Q) ❀ ❧❡t ✉s ♣✉t H = Gal(E/K)✳ ❲❡ ♠❛② ✐❞❡♥t✐❢② X(Q) ✇✐t❤ G/H ✳ ▲❡t S ❜❡ t❤❡ s❡t ♦❢ ♣r✐♠❡ ❞✐✈✐s♦rs ♦❢ disc(K) ✭♦r ♦❢ disc(E) ✕ ✐t ❛♠♦✉♥ts t♦ t❤❡ s❛♠❡✮✳ ■❢ p ∈ / S ✱ ❧❡t σp ∈ G ❜❡ ✐ts ❋r♦❜❡♥✐✉s ❡❧❡♠❡♥t ✭✉♣ t♦ ❝♦♥❥✉❣❛✲ t✐♦♥✮✳ ❲❡ ❤❛✈❡ NX (p) = ϕ(σp )✳ ❚❤✐s s❤♦✇s t❤❛t t❤❡ ♠❛♣ p → NX (p) ✐s Pr♦♦❢✳ ▲❡t ✉s s✉♣♣♦s❡ ✜rst t❤❛t ❲❡ t❤❡♥ ❤❛✈❡ NX (p) ❂ ♥✉♠❜❡r ♦❢ ♣❧❛❝❡s ✸✳✹✳ ❊①❛♠♣❧❡s ♦❢ S ✲❢r♦❜❡♥✐❛♥✱ S ✲❢r♦❜❡♥✐❛♥ ❛♥❞ t❤❛t ϕ ❢✉♥❝t✐♦♥s ❛♥❞ S ✲❢r♦❜❡♥✐❛♥ s❡ts ✷✼ ✐s t❤❡ ❛ss♦❝✐❛t❡❞ ❢✉♥❝t✐♦♥✳ ❚❤✐s ♣r♦✈❡s ❛✮ ❛♥❞ ❜✮✳ ❆ss❡rt✐♦♥ ❝✮ ❢♦❧❧♦✇s ❢r♦♠ t❤❡ ❡❛s✐❧② ♣r♦✈❡❞ ❢♦r♠✉❧❛ NX (pe ) = ϕ(σpe )✳ ❙✐♥❝❡ ϕ(1) = |G/H| = [K/Q] = |X(C)|✱ t❤✐s ✐♠♣❧✐❡s t❤❡ ❛ss❡rt✐♦♥ ❛❜♦✉t f (1) ❀ ❛ s✐♠✐❧❛r ♠❡t❤♦❞ ✇♦r❦s ❢♦r f (−1)✳ ❆s ❢♦r t❤❡ ♠❡❛♥ ✈❛❧✉❡ ♦❢ f ✱ ✐t ✐s ❜② ❞❡✜♥✐t✐♦♥ t❤❡ ♠❡❛♥ ✈❛❧✉❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ϕ✱ t❤❛t ✐s ❡q✉❛❧ t♦ ✶ ❜② ❇✉r♥s✐✲ ❞❡✬s ❧❡♠♠❛ ✭s❡❡ ❡✳❣✳ ❬❙❡ ✵✷✱ ➓✷✳✶❪✮✳ ❚❤✐s ♣r♦✈❡s Pr♦♣♦s✐t✐♦♥ ✸✳✶✵ ✐♥ t❤❡ ❝❛s❡ X = Spec OK ✳ ❚❤❡ ❣❡♥❡r❛❧ ❝❛s❡ ❢♦❧❧♦✇s ❜② ❞♦✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ ♦♣❡r❛t✐♦♥s ♦♥ X ✿ • ♠❛❦✐♥❣ ✐t r❡❞✉❝❡❞ ❀ t❤✐s ❞♦❡s ♥♦t ❝❤❛♥❣❡ ❛♥② NX (pe ) ❀ • ♠❛❦✐♥❣ ✐t ♥♦r♠❛❧ ❀ t❤✐s ❝❤❛♥❣❡s NX (pe ) ❢♦r ♦♥❧② ✜♥✐t❡❧② ♠❛♥② p ❀ • ❞❡❝♦♠♣♦s✐♥❣ ✐t ✐♥t♦ ✐rr❡❞✉❝✐❜❧❡ ❝♦♠♣♦♥❡♥ts ❀ ❡❛❝❤ ❝♦♠♣♦♥❡♥t ✐s t❤❡♥ Σ✱ ✇❤❡r❡ K ✐s ❛ ♥✉♠❜❡r ✜❡❧❞ ❛♥❞ Σ ✐s ❛ ✐s♦♠♦r♣❤✐❝ t♦ X = Spec OK e e ❝❧♦s❡❞ ✜♥✐t❡ s✉❜s❡t ♦❢ Spec OK ❀ ♦♥❡ t❤❡♥ ❤❛s NX (p ) = NX (p ) ❢♦r ❛❧❧ ❧❛r❣❡ ❡♥♦✉❣❤ ♣r✐♠❡s p✳ ❈♦r♦❧❧❛r② ✸✳✶✶✳ ■❢ X ✐s ❛ s❝❤❡♠❡ ♦❢ ✜♥✐t❡ t②♣❡ ♦✈❡r t❤❡r❡ ❛r❡ ✐♥✜♥✐t❡❧② ♠❛♥② p ✇✐t❤ Pr♦♦❢✳ ❈❤♦♦s❡ ❛ ❝❧♦s❡❞ ♣♦✐♥t ♦❢ X x Z s✉❝❤ t❤❛t X/Q = ∅✱ NX (p) > 0✳ ♦❢ X/Q ❀ ✐ts ❝❧♦s✉r❡ Xx ✐♥ X ✐s ❛ s✉❜s❝❤❡♠❡ t♦ ✇❤✐❝❤ ♦♥❡ ❛♣♣❧✐❡s ♣❛rt ❝✮ ♦❢ Pr♦♣♦s✐t✐♦♥ ✸✳✶✵✳ ❍❡♥❝❡ t❤❡r❡ ❛r❡ ✐♥✜♥✐t❡❧② ♠❛♥② p ✇✐t❤ NXx (p) > 0✱ ❘❡♠❛r❦s✳ ✶✳ ❚❤❡ ❤②♣♦t❤❡s✐s ❛♥❞ ❛ ❢♦rt✐♦r✐ X/Q = ∅ NX (p) > 0✳ ✐s ❡q✉✐✈❛❧❡♥t t♦ X(Q) = ∅ ❛♥❞ t♦ X(C) = ∅✳ ✷✳ ❇② ❛ t❤❡♦r❡♠ ♦❢ ❆① ❛♥❞ ✈❛♥ ❞❡♥ ❉r✐❡s ✭❬❆① ✻✼❪✱ ❬❉r ✾✶❪✱ s❡❡ ❛❧s♦ p ✇✐t❤ NX (p) = 0 ✐s ❢r♦❜❡♥✐❛♥ ❀ > 0, ❛s ✇❛s ✜rst ♣r♦✈❡❞ ✐♥ ❬❆① ✻✼❪✳ ➓✼✳✷✳✹✮✱ t❤❡ s❡t ♦❢ ❞❡♥s✐t②✱ t❤❛t ✐s ❊①❛♠♣❧❡ ✿ ◆✉♠❜❡r ♦❢ r♦♦ts ♠♦❞ p ✐♥ ♣❛rt✐❝✉❧❛r✱ ✐t ❤❛s ❛ ♦❢ ❛ ♦♥❡✲✈❛r✐❛❜❧❡ ♣♦❧②♥♦♠✐❛❧✳ X = Z[t]/(H)✱ ✇❤❡r❡ H ✐s ❛ ♥♦♥✲③❡r♦ ❡❧❡♠❡♥t ♦❢ t❤❡ ♣♦❧②✲ Z[t]✳ ▲❡t a0 tn ❜❡ t❤❡ ❧❡❛❞✐♥❣ t❡r♠ ♦❢ H ❛♥❞ ❧❡t S ❜❡ t❤❡ s❡t ♦❢ t❤❡ ♣r✐♠❡ ❞✐✈✐s♦rs ♦❢ a0 disc(H)✳ ❚❤❡♥ ✿ ❚❤❡ ♠❛♣ p → NH (p) ✐s S ✲❢r♦❜❡♥✐❛♥✱ ❛♥❞ ✐ts ✈❛❧✉❡ ❛t 1 (r❡s♣✳ ❛t −1) ✐s t❤❡ ✽ ♥✉♠❜❡r ♦❢ ❝♦♠♣❧❡① (r❡s♣✳ r❡❛❧ ) r♦♦ts ♦❢ H ✳ ❋♦r ❡✈❡r② e 1✱ t❤❡ Ψe✲tr❛♥s❢♦r♠ ♦❢ p → NH (p) ✐s p → NH (pe )✳ ▲❡t ✉s t❛❦❡ ♥♦♠✐❛❧ r✐♥❣ NX (p) p → NX (p) ✐s ✸✳✹✳✷✳✷✳ ♠♦❞ m✳ ▲❡t X ❜❡ ❛ s❝❤❡♠❡ ♦❢ ✜♥✐t❡ t②♣❡ ♦✈❡r ♥♦t ❢r♦❜❡♥✐❛♥ ✭✉♥❧❡ss ❞✐♠ X(C) 0✮✱ Z✳ ❚❤❡ ♠❛♣ ✐❢ ♦♥❧② ❜❡❝❛✉s❡ ✐ts ✐♠❛❣❡ ✐s ✐♥✜♥✐t❡✱ ❜✉t ✐t ✐s r❡s✐❞✉❛❧❧② ❢r♦❜❡♥✐❛♥ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ s❡♥s❡ ✿ ✽ ◆♦t❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝✉r✐♦✉s ❝♦r♦❧❧❛r② ✿ t❤❡r❡ ❛r❡ ✐♥✜♥✐t❡❧② ♠❛♥② t❤❡ s❛♠❡ ♥✉♠❜❡r ♦❢ r♦♦ts ✐♥ Fp ❛s ✐♥ R✳ p s✉❝❤ t❤❛t H ❤❛s ✷✽ ✸✳ ❚❤❡ ❈❤❡❜♦t❛r❡✈ ❞❡♥s✐t② t❤❡♦r❡♠ ❢♦r ❛ ♥✉♠❜❡r ✜❡❧❞ m 1✱ Z/mZ; ✐ts f : p → NX (p) (mod m) ✐s ❛ ❢r♦❜❡♥✐❛♥ 1 ✭r❡s♣✳ ❛t −1) ✐s ❡q✉❛❧ t♦ t❤❡ ✐♠❛❣❡ ✐♥ ❝❤❛r❛❝t❡r✐st✐❝ χ(X(C)) ✭r❡s♣✳ χc (X(R)✮✮✳ ❋♦r ❡✈❡r② ✐♥t❡❣❡r t❤❡ ♠❛♣ ♠❛♣ ♦❢ P ✈❛❧✉❡ ❛t Z/mZ ♦❢ t❤❡ ❊✉❧❡r✲P♦✐♥❝❛ré ✐♥t♦ ❚❤❡s❡ st❛t❡♠❡♥ts ✇✐❧❧ ❜❡ ♣r♦✈❡❞ ✐♥ ➓✻✳✶✳✷ ❀ ♥♦t❡ t❤❛t t❤❡② ✐♠♣❧② ❚❤❡♦✲ r❡♠ ✶✳✹ ♦❢ ➓✶✳✹✳ ✸✳✹✳✸✳ ❚❤❡ p✲t❤ ❝♦❡✣❝✐❡♥t ♦❢ ❛ ♠♦❞✉❧❛r ❢♦r♠ N ✱ ❛ ✇❡✐❣❤t k > 0✱ ❛ ❉✐r✐❝❤❧❡t ❝❤❛r❛❝t❡r ε ♠♦❞ N ✱ ❛♥❞ ❛ ♠♦❞✉❧❛r ❢♦r♠ ϕ = an q n ♦♥ Γ0 (N ) ♦❢ ✇❡✐❣❤t k ❛♥❞ t②♣❡ ε✱ ❝❢✳ ❡✳❣✳ ❬❉❙ ✼✹✱ ➓✶❪✳ ❆ss✉♠❡ t❤❛t t❤❡ ❝♦❡✣❝✐❡♥ts an ♦❢ ϕ ❜❡❧♦♥❣ t♦ t❤❡ r✐♥❣ ♦❢ ✐♥t❡❣❡rs A ♦❢ s♦♠❡ ♥✉♠❜❡r ✜❡❧❞✳ ❚❤❡♥ t❤❡ ♠❛♣ p → ap ✐s r❡s✐❞✉❛❧❧② ❢r♦❜❡♥✐❛♥✱ ✐♥ ❛ ▲❡t ✉s ❝❤♦♦s❡ ❛ ❧❡✈❡❧ s✐♠✐❧❛r s❡♥s❡ ❛s ✐♥ ➓✸✳✹✳✷✳✷✳ ▼♦r❡ ♣r❡❝✐s❡❧② ✿ ❚❤❡♦r❡♠ ✸✳✶✷✳ ❋♦r ❡✈❡r② ✐♥t❡❣❡r m 1✱ SmN ✲❢r♦❜❡♥✐❛♥✱ ✇❤❡r❡ SmN ✐s t❤❡ s❡t ♦❢ ♣r✐♠❡s 1 (r❡s♣✳ ❛t −1) ✐s 2a1 ✭resp✳ 0) ✭♠♦❞ mA)✳ ❈♦r♦❧❧❛r② ✸✳✶✸✳ ❚❤❡ s❡t ♦❢ > 0✳ ❛♥❞ ✐ts ❞❡♥s✐t② ✐s p ✇✐t❤ p → ap ✭♠♦❞ mA) ✐s ❞✐✈✐❞✐♥❣ mN.