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

By Denjoy A.

Show description

Read or Download Lecons sur le Calcul des Coefficients. Deuxieme Partie PDF

Similar elementary books

Notes on Rubik's Magic Cube

Notes on Rubik's 'Magic dice'

Beginner's Basque

This name features a e-book and a pair of audio CDs. Basque is the language spoken via the Basque those who stay within the Pyrenees in North principal Spain and the adjacent zone of south west France. it's also spoken by way of many immigrant groups world wide together with the USA, Venezuela, Argentina, Mexico and Colombia.

Elementary Algebra

User-friendly Algebra is a piece textual content that covers the conventional subject matters studied in a contemporary easy algebra direction. it truly is meant for college students who (1) haven't any publicity to ordinary algebra, (2) have formerly had an uncongenial event with common algebra, or (3) have to overview algebraic thoughts and methods.

Additional resources for Lecons sur le Calcul des Coefficients. Deuxieme Partie

Sample text

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.

Download PDF sample

Rated 4.63 of 5 – based on 38 votes