Lectures on N_X by Jean-Pierre Serre

By Jean-Pierre Serre

Lectures on NX(p) bargains with the query on how NX(p), the variety of strategies of mod p congruences, varies with p whilst the kinfolk (X) of polynomial equations is mounted. whereas any such normal query can't have a whole resolution, it bargains a superb get together for reviewing quite a few ideas in l-adic cohomology and team representations, awarded in a context that's beautiful to experts in quantity thought and algebraic geometry.

Along with masking open difficulties, the textual content examines the scale and congruence homes of NX(p) and describes the ways that it's computed, through closed formulae and/or utilizing effective computers.

The first 4 chapters conceal the preliminaries and include virtually no proofs. After an summary of the most theorems on NX(p), the e-book deals easy, illustrative examples and discusses the Chebotarev density theorem, that is crucial in learning frobenian capabilities and frobenian units. It additionally studies ℓ-adic cohomology.

The writer is going directly to current effects on crew representations which are usually tough to discover within the literature, resembling the means of computing Haar measures in a compact ℓ-adic staff through appearing an analogous computation in a true compact Lie workforce. those effects are then used to debate the potential relatives among varied households of equations X and Y. the writer additionally describes the Archimedean homes of NX(p), an issue on which less is understood than within the ℓ-adic case. Following a bankruptcy at the Sato-Tate conjecture and its concrete elements, the e-book concludes with an account of the major quantity theorem and the Chebotarev density theorem in larger dimensions.

 

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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 ).

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).

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