Lectures on N_X(p) by Serre J.P.

By Serre J.P.

Lectures on NX(p) bargains with the query on how NX(p), the variety of suggestions of mod p congruences, varies with p whilst the relations (X) of polynomial equations is mounted. whereas one of these normal query can't have an entire solution, it deals an exceptional celebration for reviewing quite a few concepts in l-adic cohomology and staff representations, offered in a context that's attractive to experts in quantity concept 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 truly is computed, by way of closed formulae and/or utilizing effective desktops. the 1st 4 chapters disguise the preliminaries and comprise nearly no proofs. After an outline of the most theorems on NX(p), the publication deals easy, illustrative examples and discusses the Chebotarev density theorem, that's crucial in learning frobenian features and frobenian units. It additionally studies ℓ-adic cohomology. the writer is going directly to current effects on staff representations which are frequently tricky to discover within the literature, comparable to the means of computing Haar measures in a compact ℓ-adic workforce via acting the same computation in a true compact Lie crew. those effects are then used to debate the prospective kinfolk among diverse households of equations X and Y. the writer additionally describes the Archimedean homes of NX(p), a subject matter on which less is understood than within the ℓ-adic case. Following a bankruptcy at the Sato-Tate conjecture and its concrete elements, the booklet concludes with an account of the top quantity theorem and the Chebotarev density theorem in better dimensions.  

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Example text

Tr(F ) = i ❚❤✐s ✐s t❤❡ ▲❡❢s❝❤❡t③ ♥✉♠❜❡r ♦❢ F✱ r❡❧❛t✐✈❡ t♦ t❤❡ ✲❛❞✐❝ ❝♦❤♦♠♦❧♦❣② ✇✐t❤ ♣r♦♣❡r s✉♣♣♦rt✳ ❆ ♣r✐♦r✐✱ ✐t ❞❡♣❡♥❞s ♦♥ t❤❡ ❝❤♦✐❝❡ ♦❢ ✳ ■♥ ❢❛❝t✱ ✐t ❞♦❡s ♥♦t✱ ❜❡❝❛✉s❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡s✉❧t ♦❢ ●r♦t❤❡♥❞✐❡❝❦ ✭❬●r ✻✹❪✱ s❡❡ ❛❧s♦ ❬❙●❆ 4 21 ✱ ♣✳✽✻✱ t❤✳✸✳✷❪✮ ✿ ❚❤❡♦r❡♠ ✹✳✷✳ Tr(F ) = |X(k)|. ❚❤✐s ❛❧s♦ ❛♣♣❧✐❡s t♦ t❤❡ ✜♥✐t❡ ❡①t❡♥s✐♦♥s ♦❢ ❈♦r♦❧❧❛r② ✹✳✸✳ Tr(F m ) = |X(km )| ❢♦r ❡✈❡r② k✳ ❍❡♥❝❡ ✿ m 1✳ ❘❡♠❛r❦s✳ ✶✮ ❙✐♥❝❡ F :X→X ✐s ❛ r❛❞✐❝✐❛❧ ♠♦r♣❤✐s♠✱ ✐t ✐s ❛♥ ❤♦♠❡♦♠♦r♣❤✐s♠ ❢♦r t❤❡ ét❛❧❡ t♦♣♦❧♦❣②✳ ❍❡♥❝❡ ❡✈❡r② ❡✐❣❡♥✈❛❧✉❡ ♦❢ F ♦♥ Hci (X, Q ) ✐s ♥♦♥✲③❡r♦ ❀ ❢♦r ❛ ♠♦r❡ ♣r❡❝✐s❡ st❛t❡♠❡♥t✱ s❡❡ ❚❤❡♦r❡♠ ✹✳✺ ❜❡❧♦✇✳ ✷✮ ❚❤❡ t❤❡♦r❡♠ ♣r♦✈❡❞ ❜② ●r♦t❤❡♥❞✐❡❝❦✱ ❧♦❝✳❝✐t✳✱ ✐s ♠♦r❡ ❣❡♥❡r❛❧ t❤❛♥ ❚❤❡♦r❡♠ ✹✳✷ ✿ ✐t ❛♣♣❧✐❡s t♦ ❡✈❡r② ❝♦♥str✉❝t✐❜❧❡ ❛s ❛ s✉♠ ♦❢ ❧♦❝❛❧ tr❛❝❡s ❛t t❤❡ ♣♦✐♥ts ♦❢ ✸✮ ❆ss✉♠❡ k = Fp ✱ Q ✲s❤❡❛❢✱ ❛♥❞ ❣✐✈❡s Tr(F ) X(k)✳ t♦ s✐♠♣❧✐❢② ♥♦t❛t✐♦♥s✳ ❚❤❡♥ ❈♦r♦❧❧❛r② ✹✳✸ ✐s ❡q✉✐✈❛✲ ❧❡♥t t♦ s❛②✐♥❣ t❤❛t t❤❡ ❉✐r✐❝❤❧❡t s❡r✐❡s ❞❡♥♦t❡❞ ❜② ζX,p (s) ✐♥ ➓✶✳✺ ✐s ❡q✉❛❧ ✸✹ ✹✳ ❘❡✈✐❡✇ ♦❢ −s F |Hci (X, Q i det(1 − p ▼♦r❡♦✈❡r✱ ♦♥❡ ❤❛s t♦ i+1 ))(−1) NX (pe ) = ✲❛❞✐❝ ❝♦❤♦♠♦❧♦❣② ✱ ✇❤✐❝❤ ✐s ❛ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥ ♦❢ p−s ✳ (−1)i Tri (F e ) i e ∈ Z ✭❛♥❞ ♥♦t ♠❡r❡❧② ❢♦r e 1✮✳ ■♥ ♣❛rt✐❝✉❧❛r✱ NX (p0 ) ✐s ❡q✉❛❧ t♦ i i (−1) dim Hc (X, Q )✱ ✇❤✐❝❤ ✐s t❤❡ ❊✉❧❡r✲P♦✐♥❝❛ré ❝❤❛r❛❝t❡r✐st✐❝ ♦❢ X ✳ ❢♦r ❡✈❡r② i ✹✳✹✳ ❚❤❡ ❝❛s❡ ♦❢ ❛ ✜♥✐t❡ ✜❡❧❞ ✿ t❤❡ ❣❡♦♠❡tr✐❝ ❛♥❞ t❤❡ ❛r✐t❤♠❡t✐❝ ❋r♦❜❡♥✐✉s ❑❡❡♣ t❤❡ ♥♦t❛t✐♦♥ ♦❢ ➓✹✳✸✳ ❚❤❡ ●❛❧♦✐s ❣r♦✉♣ i ❝♦❤♦♠♦❧♦❣② ❣r♦✉♣ Hc (X, Q ♦❢ Γk )✳ Γk = Gal(k/k) ❛❝ts ♦♥ ❡❛❝❤ σ = σq σ ✮✱ t❤❛t ✐s ❝❛❧❧❡❞ t❤❡ ❛r✐t❤✲ ■♥ ♣❛rt✐❝✉❧❛r✱ t❤❡ ♥❛t✉r❛❧ ❣❡♥❡r❛t♦r ❛❝ts ❜② ❛♥ ❛✉t♦♠♦r♣❤✐s♠ ✭st✐❧❧ ❞❡♥♦t❡❞ ❜② ♠❡t✐❝ ❋r♦❜❡♥✐✉s ❛✉t♦♠♦r♣❤✐s♠ ✐♥ ♦r❞❡r t♦ ❞✐st✐♥❣✉✐s❤ ✐t ❢r♦♠ t❤❡ ❣❡♦♠❡tr✐❝ ❋r♦❜❡♥✐✉s F ❞❡✜♥❡❞ ❛❜♦✈❡✳ ❚❤❡s❡ t✇♦ ❦✐♥❞ ♦❢ ✏❋r♦❜❡♥✐✉s ❛✉t♦♠♦r♣❤✐s♠s✑ ❛r❡ r❡❧❛t❡❞ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ s✐♠♣❧❡ r❡s✉❧t ✭s❡❡ ❬❙●❆ ✺✱ ♣✳✹✺✼❪✱ ♦r ❬❑❛ ✾✹✱ ✷✹✲✷✺❪✮ ✿ ❚❤❡♦r❡♠ ✹✳✹✳ ❚❤❡ ❛r✐t❤♠❡t✐❝ ❋r♦❜❡♥✐✉s ❛♥❞ t❤❡ ❣❡♦♠❡tr✐❝ ❋r♦❜❡♥✐✉s ❛r❡ ✐♥✈❡rs❡s ♦❢ ❡❛❝❤ ♦t❤❡r✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡ σ(F ξ) = F (σξ) = ξ ❢♦r ❡✈❡r② ❆ s✐♠✐❧❛r r❡s✉❧t ❤♦❧❞s ❢♦r t❤❡ ❝♦❤♦♠♦❧♦❣② ❣r♦✉♣s ξ ∈ Hci (X, Q )✳ H i (X, Q )✱ ❜✐tr❛r② s✉♣♣♦rt ✭❛♥❞ ❛❧s♦ ❢♦r t❤❡ ❝♦❤♦♠♦❧♦❣② ✇✐t❤ ❝♦❡✣❝✐❡♥ts ✐♥ ❊①❛♠♣❧❡✳ ❙✉♣♣♦s❡ t❤❛t ❚❛t❡ Q X ✐s ❛♥ ❛❜❡❧✐❛♥ ✈❛r✐❡t② ♦✈❡r ✇✐t❤ ❛r✲ Z/ n Z✮✳ k ✱ ❛♥❞ ❧❡t V (X) ❜❡ ✐ts ✲♠♦❞✉❧❡✳ ❬❘❡❝❛❧❧ t❤❛t V (X) = Q ⊗ lim X[ n ]✱ ✇❤❡r❡ X[ n ] ✐s t❤❡ ❣r♦✉♣ ♦❢ t❤❡ n ✲❞✐✈✐s✐♦♥ ←− ♣♦✐♥ts ♦❢ X(k)✱ ✐✳❡✳ t❤❡ ❦❡r♥❡❧ ♦❢ n : X(k) → X(k✮ ❀ ✐t ✐s ❛ Q ✲✈❡❝t♦r s♣❛❝❡ ♦❢ ❞✐♠❡♥s✐♦♥ 2dim X ✳❪ ❚❤❡ ❋r♦❜❡♥✐✉s ❡♥❞♦♠♦r♣❤✐s♠ ♠❡t✐❝ ❋r♦❜❡♥✐✉s F s F : X → X ❛❝ts ♦♥ V (X) ❀ t❤❡ ❛r✐t❤✲ ❛❧s♦ ❛❝ts✱ ❛♥❞ ✐ts ❛❝t✐♦♥ ✐s t❤❡ s❛♠❡ ❛s t❤❡ ❛❝t✐♦♥ ♦❢ F ❛♥❞ s ❛❝t ✐♥ t❤❡ s❛♠❡ ✇❛② ♦♥ X(k)✮✳ ❚❤❡ ✜rst ❝♦❤♦♠♦❧♦❣② H 1 (X, Q ) ✐s t❤❡ ❞✉❛❧ ♦❢ V (X) ❀ t❤❡ ❛❝t✐♦♥ ♦❢ F ♦♥ ✐t ✐s ❞❡✜♥❡❞ ❜② ❢✉♥❝t♦r✐❛❧✐t②✱ ✐✳❡✳ ❜② tr❛♥s♣♦s✐t✐♦♥ ❀ t❤❡ ❛❝t✐♦♥ ♦❢ s ✐s ❞❡✜♥❡❞ ❜② tr❛♥s♣♦rt ♦❢ ✭❜❡❝❛✉s❡ ❣r♦✉♣ str✉❝t✉r❡✱ ✐✳❡✳ ❜② ✐♥✈❡rs❡ tr❛♥s♣♦s✐t✐♦♥✳ ❚❤✐s ❡①♣❧❛✐♥s ✇❤② t❤❡ t✇♦ ❛❝t✐♦♥s ❛r❡ ✐♥✈❡rs❡ ♦❢ ❡❛❝❤ ♦t❤❡r✳ ✸ ❲❤❛t ✸ t❤✐s ❡①❛♠♣❧❡ s✉❣❣❡sts ✐s t❤❛t✱ ✐❢ ét❛❧❡ t♦♣♦❧♦❣② ✇❡r❡ ❡①♣r❡ss❡❞ ✐♥ t❡r♠s ♦❢ ❤♦♠♦❧♦❣② ✐♥st❡❛❞ ♦❢ ❝♦❤♦♠♦❧♦❣②✱ t❤❡ t✇♦ t②♣❡s ♦❢ ❋r♦❜❡♥✐✉s ✇♦✉❧❞ ❜❡ t❤❡ s❛♠❡✳ ✹✳✺✳ ✳ ❚❤❡ ❝❛s❡ ♦❢ ❛ ✜♥✐t❡ ✜❡❧❞ ✿ ❉❡❧✐❣♥❡✬s t❤❡♦r❡♠s ✸✺ ✹✳✺✳ ❚❤❡ ❝❛s❡ ♦❢ ❛ ✜♥✐t❡ ✜❡❧❞ ✿ ❉❡❧✐❣♥❡✬s t❤❡♦r❡♠s ❲❡ ❦❡❡♣ t❤❡ ♥♦t❛t✐♦♥ ❛♥❞ ❤②♣♦t❤❡s❡s ♦❢ ➓✹✳✹ ❛❜♦✈❡✳ q ✲❲❡✐❧ ✐♥t❡❣❡r ♦❢ ✇❡✐❣❤t w ∈ N ✐s ❛♥ ❛❧❣❡❜r❛✐❝ ✐♥t❡❣❡r α |ι(α)| = q w/2 ❢♦r ❡✈❡r② ❡♠❜❡❞❞✐♥❣ ι : Q(α) → C✳ ❋♦r ✐♥st❛♥❝❡ ❛ ❘❡❝❛❧❧ t❤❛t ❛ s✉❝❤ t❤❛t q ✲❲❡✐❧ ✐♥t❡❣❡r ♦❢ ✇❡✐❣❤t ✵ ✐s ❛ r♦♦t ♦❢ ✉♥✐t② ✭❑r♦♥❡❝❦❡r✮✳ q ✲❲❡✐❧ ✐♥t❡❣❡r ♦❢ ✇❡✐❣❤t ✇❡✐❣❤t w r❡❧❛t✐✈❡❧② t♦ q ✑✳ ❘❡♠❛r❦✳ ■♥ ❉❡❧✐❣♥❡ ❬❉❡ ✽✵✱ ➓✶✳✷✳✶❪✱ ✇❤❛t ✇❡ ❝❛❧❧ ❛ w ✐s ❝❛❧❧❡❞ ✏ ❛♥ ❛❧❣❡❜r❛✐❝ ✐♥t❡❣❡r t❤❛t ✐s ♣✉r❡ ♦❢ ❚❤❡♦r❡♠ ✹✳✺✳ ✭❉❡❧✐❣♥❡✮ ▲❡t d = dim X ✳ α ♦❢ t❤❡ ❣❡♦♠❡tr✐❝ ❋r♦❜❡♥✐✉s F ❛❝t✐♥❣ ♦♥ Hci (X, Q ) ✐s ❛ q ✲❲❡✐❧ ✐♥t❡❣❡r ♦❢ ✇❡✐❣❤t i ; ✐❢ i d✱ t❤❡♥ α ✐s ❞✐✈✐s✐❜❧❡ ❜② q i−d .

1✱ ❡①❡r❝✳❪✳ ✷✹ ✸✳ ❚❤❡ ❈❤❡❜♦t❛r❡✈ ❞❡♥s✐t② t❤❡♦r❡♠ ❢♦r ❛ ♥✉♠❜❡r ✜❡❧❞ Ψef ✳ ❇② ❞❡✜♥✐t✐♦♥✱ ✇❡ ❤❛✈❡ Ψef (v) = f (σve ) S ✱ ✇❤❡r❡ σve ❞❡♥♦t❡s t❤❡ e✲t❤ ♣♦✇❡r ♦❢ t❤❡ ❋r♦❜❡♥✐✉s ❢♦r ❡✈❡r② v ∈ VK e e ❡❧❡♠❡♥t σv ✳ ❲❡ ❤❛✈❡ Ψ f (1) = f (1), Ψ f (−1ι ) = f (1) ✐❢ e ✐s ❡✈❡♥✱ ❛♥❞ e Ψ f (−1ι ) = f (−1ι ) ✐❢ e ✐s ♦❞❞✳ ✇❤✐❝❤ ✇❡ s❤❛❧❧ ❞❡♥♦t❡ ❜② ✸✳✸✳✸✳✹✳ ❇❛s❡ ❝❤❛♥❣❡✳ ▲❡t ❤❡♥❝❡ ✉♥r❛♠✐✜❡❞ ♦✉ts✐❞❡ K S✱ K ❝♦♥t❛✐♥❡❞ ✐♥ KS ✱ GS = Gal(KS /K ) ❜❡ t❤❡ ❝♦rr❡s♣♦♥✲ ❜❡ ❛ ✜♥✐t❡ ❡①t❡♥s✐♦♥ ♦❢ ❛♥❞ ❧❡t ΓS ✳ S→ Ω ❞✐♥❣ s✉❜❣r♦✉♣ ♦❢ f : VK ❜❡ ❛♥ S ✲❢r♦❜❡♥✐❛♥ ♠❛♣✱ ❛♥❞ ❧❡t S ❜❡ t❤❡ ✐♥✈❡rs❡ ✐♠❛❣❡ ♦❢ S ✐♥ VK ✳ ❚❤❡r❡ ✐s ❛ ✉♥✐q✉❡ S ✲❢r♦❜❡♥✐❛♥ ♠❛♣ f : VK S →Ω S ❤❛s s✉❝❤ t❤❛t ϕf : GS → Ω ✐s t❤❡ r❡str✐❝t✐♦♥ ♦❢ ϕf t♦ GS ✳ ■❢ v ∈ VK ✐♠❛❣❡ v ✐♥ VK S ✱ ❛♥❞ ✐❢ e ✐s t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ r❡s✐❞✉❡ ❞❡❣r❡❡✱ ✇❡ ❤❛✈❡ ✿ ▲❡t f (v ) = ϕf (σv ) = ϕf (σve ) = Ψef (v).

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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