Lectures on N_X(p) by Jean-Pierre Serre PDF

By Jean-Pierre Serre

ISBN-10: 1466501928

ISBN-13: 9781466501928

Lectures on NX(p) offers with the query on how NX(p), the variety of strategies of mod p congruences, varies with p whilst the relatives (X) of polynomial equations is mounted. whereas this kind of basic query can't have a whole resolution, it deals an outstanding celebration for reviewing numerous options in l-adic cohomology and team 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 really is computed, via closed formulae and/or utilizing effective desktops. the 1st 4 chapters disguise the preliminaries and comprise virtually no proofs. After an outline of the most theorems on NX(p), the booklet deals basic, 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 crew representations which are frequently tricky to discover within the literature, akin to the means of computing Haar measures in a compact ?-adic workforce via appearing an analogous computation in a true compact Lie team. those effects are then used to debate the prospective family among various households of equations X and Y. the writer additionally describes the Archimedean homes 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 elements, the e-book concludes with an account of the major quantity theorem and the Chebotarev density theorem in greater dimensions.  

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Extra resources for Lectures on N_X(p)

Example text

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 ).

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 .

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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Lectures on N_X(p) by Jean-Pierre Serre


by Kevin
4.3

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