Lectures on N_X(p) by Jean-Pierre Serre

By Jean-Pierre Serre

Lectures on NX(p) bargains with the query on how NX(p), the variety of options of mod p congruences, varies with p while the relatives (X) of polynomial equations is mounted. whereas one of these basic query can't have an entire solution, it deals a great get together for reviewing quite a few recommendations in l-adic cohomology and staff representations, provided in a context that's beautiful to experts in quantity conception and algebraic geometry. besides protecting open difficulties, the textual content examines the dimensions and congruence homes of NX(p) and describes the ways that it really is computed, via closed formulae and/or utilizing effective pcs. the 1st 4 chapters disguise the preliminaries and include virtually no proofs. After an summary of the most theorems on NX(p), the ebook deals easy, illustrative examples and discusses the Chebotarev density theorem, that's crucial in learning frobenian services and frobenian units. It additionally experiences ?-adic cohomology. the writer is going directly to current effects on workforce representations which are usually tricky to discover within the literature, resembling the means of computing Haar measures in a compact ?-adic team by way of acting the same computation in a true compact Lie crew. those effects are then used to debate the prospective family members among varied households of equations X and Y. the writer additionally describes the Archimedean houses of NX(p), a subject on which less is understood than within the ?-adic case. Following a bankruptcy at the Sato-Tate conjecture and its concrete features, the ebook concludes with an account of the top quantity theorem and the Chebotarev density theorem in greater dimensions.  

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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.

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