Lectures on N_X(p) by Jean-Pierre Serre

By Jean-Pierre Serre

Lectures on NX(p) offers with the query on how NX(p), the variety of recommendations of mod p congruences, varies with p whilst the relatives (X) of polynomial equations is fastened. whereas this type of basic query can't have an entire solution, it bargains an excellent get together for reviewing quite a few innovations in l-adic cohomology and staff representations, provided in a context that's beautiful to experts in quantity idea and algebraic geometry. in addition to overlaying open difficulties, the textual content examines the dimensions and congruence houses 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 hide the preliminaries and comprise virtually no proofs. After an outline of the most theorems on NX(p), the booklet deals easy, illustrative examples and discusses the Chebotarev density theorem, that's crucial in learning frobenian capabilities and frobenian units. It additionally studies ?-adic cohomology. the writer is going directly to current effects on staff representations which are frequently tricky to discover within the literature, similar to the means of computing Haar measures in a compact ?-adic workforce by way of acting the same computation in a true compact Lie workforce. those effects are then used to debate the prospective family 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 e-book concludes with an account of the leading quantity theorem and the Chebotarev density theorem in greater dimensions.  

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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.

R❡ ❛r❡ ♥❛t✉r❛❧ ♠❛♣s ✭❞✉❡ t♦ t❤❡ ❢❛❝t t❤❛t t❤❡ ✉s✉❛❧ t♦♣♦❧♦❣② ✐s ✜♥❡r t❤❛♥ t❤❡ ét❛❧❡ ♦♥❡✮ ✿ H i (X, Q ) → H i (X(C), Q) ⊗ Q ❛♥❞ Hci (X, Q ) → Hci (X(C), Q) ⊗ Q . , xqn ). k ✲♣♦✐♥t x ♦❢ X ✐s k ✲r❛t✐♦♥❛❧ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐t ✐s ✜①❡❞ ✉♥❞❡r F ✳ ❙✐♠✐❧❛r❧②✱ ✐❢ m ✐s ❛♥ ✐♥t❡❣❡r > 0✱ ❛♥❞ ✐❢ km ❞❡♥♦t❡s t❤❡ s✉❜❡①t❡♥s✐♦♥ ♦❢ k ♦❢ ❞❡❣r❡❡ m ♦✈❡r k ✱ t❤❡♥ X(km ) ✐s t❤❡ s✉❜s❡t m ♦❢ X(k) ♠❛❞❡ ✉♣ ♦❢ t❤❡ ♣♦✐♥ts ✜①❡❞ ✉♥❞❡r t❤❡ m✲t❤ ✐t❡r❛t❡ F ♦❢ F ✳ ❚❤❡ ♠♦r♣❤✐s♠ F : X → X ✐s ♣r♦♣❡r ❀ ❤❡♥❝❡ ✐t ❛❝ts ❜② ❢✉♥❝t♦r✐❛❧✐t② ♦♥ i t❤❡ ❝♦❤♦♠♦❧♦❣② s♣❛❝❡s Hc (X, Q )✱ ✇❤❡r❡ ✐s ❛♥② ♣r✐♠❡ ♥✉♠❜❡r = p✳ ▲❡t ✉s ❞❡♥♦t❡ ❜② Tri (F ) t❤❡ tr❛❝❡ ♦❢ t❤✐s ❡♥❞♦♠♦r♣❤✐s♠✱ ❛♥❞ ❞❡✜♥❡ ✿ ❖♥❡ ♦❢ ✐ts ♠❛✐♥ ♣r♦♣❡rt✐❡s ✐s t❤❛t ❛ (−1)i Tri (F ).

R❡ ❛r❡ ♥❛t✉r❛❧ ♠❛♣s ✭❞✉❡ t♦ t❤❡ ❢❛❝t t❤❛t t❤❡ ✉s✉❛❧ t♦♣♦❧♦❣② ✐s ✜♥❡r t❤❛♥ t❤❡ ét❛❧❡ ♦♥❡✮ ✿ H i (X, Q ) → H i (X(C), Q) ⊗ Q ❛♥❞ Hci (X, Q ) → Hci (X(C), Q) ⊗ Q . , xqn ). k ✲♣♦✐♥t x ♦❢ X ✐s k ✲r❛t✐♦♥❛❧ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐t ✐s ✜①❡❞ ✉♥❞❡r F ✳ ❙✐♠✐❧❛r❧②✱ ✐❢ m ✐s ❛♥ ✐♥t❡❣❡r > 0✱ ❛♥❞ ✐❢ km ❞❡♥♦t❡s t❤❡ s✉❜❡①t❡♥s✐♦♥ ♦❢ k ♦❢ ❞❡❣r❡❡ m ♦✈❡r k ✱ t❤❡♥ X(km ) ✐s t❤❡ s✉❜s❡t m ♦❢ X(k) ♠❛❞❡ ✉♣ ♦❢ t❤❡ ♣♦✐♥ts ✜①❡❞ ✉♥❞❡r t❤❡ m✲t❤ ✐t❡r❛t❡ F ♦❢ F ✳ ❚❤❡ ♠♦r♣❤✐s♠ F : X → X ✐s ♣r♦♣❡r ❀ ❤❡♥❝❡ ✐t ❛❝ts ❜② ❢✉♥❝t♦r✐❛❧✐t② ♦♥ i t❤❡ ❝♦❤♦♠♦❧♦❣② s♣❛❝❡s Hc (X, Q )✱ ✇❤❡r❡ ✐s ❛♥② ♣r✐♠❡ ♥✉♠❜❡r = p✳ ▲❡t ✉s ❞❡♥♦t❡ ❜② Tri (F ) t❤❡ tr❛❝❡ ♦❢ t❤✐s ❡♥❞♦♠♦r♣❤✐s♠✱ ❛♥❞ ❞❡✜♥❡ ✿ ❖♥❡ ♦❢ ✐ts ♠❛✐♥ ♣r♦♣❡rt✐❡s ✐s t❤❛t ❛ (−1)i Tri (F ).

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