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

By Serre J.P.

ISBN-10: 1466501928

ISBN-13: 9781466501928

Lectures on NX(p) offers with the query on how NX(p), the variety of suggestions of mod p congruences, varies with p whilst the kinfolk (X) of polynomial equations is mounted. whereas this sort of common query can't have an entire resolution, it bargains a great party for reviewing numerous ideas in l-adic cohomology and team representations, offered in a context that's attractive to experts in quantity thought and algebraic geometry. besides protecting open difficulties, the textual content examines the scale and congruence homes of NX(p) and describes the ways that it truly is computed, through 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 publication deals basic, illustrative examples and discusses the Chebotarev density theorem, that's crucial in learning frobenian features and frobenian units. It additionally reports ℓ-adic cohomology. the writer is going directly to current effects on staff representations which are usually tough to discover within the literature, corresponding to the means of computing Haar measures in a compact ℓ-adic crew by means of acting an identical computation in a true compact Lie staff. those effects are then used to debate the potential family members among assorted 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 ℓ-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 better dimensions.  

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T❤❡ ❣r♦✉♣s H i (X, Z ) ❛r❡ ❞❡r✐✈❡❞ ❢r♦♠ t❤❡♠ ❜② t❤❡ ❢♦r♠✉❧❛❡ H i (X, Z ) = lim H i (X, Z/ n Z) ←− ❛♥❞ H i (X, Q ) = H i (X, Z ) ⊗Z Q = H i (X, Z )[1/ ]. ❆s ❢♦r t❤❡ ❜② ♠❡❛♥s ♦❢ ❛ Hci (X, Q )✱ ❝♦♠♣❛❝t✐✜❝❛t✐♦♥ ♦❢ X ✳ Hci (X, Z ) ❛♥❞ t❤❡② ❛r❡ ❞❡✜♥❡❞ ✐♥ ❛ s✐♠✐❧❛r ✇❛②✱ ❆ ❝♦♥s❡q✉❡♥❝❡ ♦❢ t❤❡s❡ s♦♠❡✇❤❛t ✐♥❞✐r❡❝t ❞❡✜♥✐t✐♦♥s ✐s t❤❛t ♠♦st ♦❢ t❤❡ ❜❛s✐❝ r❡s✉❧ts ✭s✉❝❤ ❛s t❤♦s❡ ♦♥ ❤✐❣❤❡r ❞✐r❡❝t ✐♠❛❣❡s✱ ♦r ❜❛s❡ ❝❤❛♥❣❡✮ ❤❛✈❡ t♦ ❜❡ ♣r♦✈❡❞ ✜rst ❢♦r t❤❡ ❝♦♥st❛♥t s❤❡❛✈❡s Z/ n Z Z ❜② ⊗Z Q ✳ ✜♥✐t❡ ❝♦♥str✉❝t✐❜❧❡ s❤❡❛✈❡s✮✱ ❛♥❞ t❤❡♥ ❡①t❡♥❞❡❞ t♦ lim✱ ←− ❛♥❞ ❡①t❡♥❞❡❞ t♦ Q ❜② ✉s✐♥❣ t❤❡ ❢✉♥❝t♦r ✭♦r✱ ♠♦r❡ ❣❡♥❡r❛❧❧②✱ ❢♦r ✉s✐♥❣ t❤❡ ❢✉♥❝t♦r ✹✳✷✳ ❆rt✐♥✬s ❝♦♠♣❛r✐s♦♥ t❤❡♦r❡♠ ❙✉♣♣♦s❡ k = R ♦r C✱ s♦ t❤❛t ks = k = C✳ ■♥ t❤❛t ❝❛s❡✱ t❤❡ C✲❛♥❛❧②t✐❝ s♣❛❝❡ X(C) ✐s ❧♦❝❛❧❧② ❝♦♠♣❛❝t ❢♦r t❤❡ ✉s✉❛❧ t♦♣♦❧♦❣②✱ ❛♥❞ ✐ts ❝♦❤♦♠♦❧♦❣② ❣r♦✉♣s H i (X(C), Q) ✇✐t❤ r❛t✐♦♥❛❧ ❝♦❡✣❝✐❡♥ts ❝❛♥ ❜❡ ❞❡✜♥❡❞ ❜② s❤❡❛❢ t❤❡♦r② ❀ ♦♥❡ i ❛❧s♦ ❣❡ts ❝♦❤♦♠♦❧♦❣② ❣r♦✉♣s ✇✐t❤ ❝♦♠♣❛❝t s✉♣♣♦rt Hc (X(C), Q).

V →v ❝✮ ▲❡t B ❜❡ ❛♥ S ✲❢r♦❜❡♥✐❛♥ s✉❜s❡t ♦❢ VK S ✲❢r♦❜❡♥✐❛♥ s✉❜s❡t ♦❢ VK S ✳ S ✳ ❙❤♦✇ t❤❛t A = π(B) ✐s ❛♥ ❬❍✐♥t✳ ❆♣♣❧② ❜✮ t♦ ❛ s✉✣❝✐❡♥t ❧❛r❣❡ ❡①t❡♥s✐♦♥ ♦❢ K ❝♦♥t❛✐♥❡❞ ✐♥ KS ✱ ❛♥❞ ❝❤♦♦s❡ ψ s✉❝❤ t❤❛t fψ ✐s t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ❢✉♥❝t✐♦♥ ♦❢ B ✳ ❚❤❡♥ fϕ ✐s t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ❢✉♥❝t✐♦♥ ♦❢ A✳❪ ✸✳✹✳ ❊①❛♠♣❧❡s ♦❢ S ✲❢r♦❜❡♥✐❛♥ ❢✉♥❝t✐♦♥s ❛♥❞ S ✲❢r♦❜❡♥✐❛♥ s❡ts ■♥ t❤✐s s❡❝t✐♦♥✱ t❤❡ ❣r♦✉♥❞ ✜❡❧❞ K ✐s Q✱ s♦ t❤❛t VK ✐s t❤❡ s❡t P ♦❢ ❛❧❧ ♣r✐♠❡ ♥✉♠❜❡rs✳ ✸✳✹✳✶✳ ❉✐r✐❝❤❧❡t ❡①❛♠♣❧❡s ❚❤❡ ❢♦❧❧♦✇✐♥❣ t✇♦ ❡①❛♠♣❧❡s ❛r❡ ❡ss❡♥t✐❛❧❧② ❞✉❡ t♦ ❉✐r✐❝❤❧❡t ✭❬❉✐ ✸✾❪✮ ✿ m ❜❡ ❛♥ ✐♥t❡❣❡r > 0 ❛♥❞ ❧❡t S ❜❡ t❤❡ f : P S → (Z/mZ)× ❜❡ t❤❡ ❢✉♥❝t✐♦♥ ✸✳✹✳✶✳✶✳ ❆r✐t❤♠❡t✐❝ ♣r♦❣r❡ss✐♦♥s✳ ▲❡t s❡t ♦❢ t❤❡ ♣r✐♠❡ ❞✐✈✐s♦rs ♦❢ p → p mod m✳ ❚❤❡♥ f m✲t❤ ❝②❝❧♦t♦♠✐❝ ✜❡❧❞✳ ✐s m✳ S ✲❢r♦❜❡♥✐❛♥✱ t❤❡ r❡❧❡✈❛♥t ✜❡❧❞ ❡①t❡♥s✐♦♥ ❜❡✐♥❣ t❤❡ ❖♥❡ ❤❛s f (1) = 1 ✇❤❡r❡ f (1) ❛♥❞ f (−1) ❡①♣❧❛✐♥❡❞ ✐♥ ➓✸✳✸✳✷✳✷✳ ▲❡t ❛♥❞ f (−1) = −1, ❛r❡ t❤❡ ❡❧❡♠❡♥ts ♦❢ (Z/mZ)× ❛ss♦❝✐❛t❡❞ ✇✐t❤ f ❛s ✷✻ ✸✳ ❚❤❡ ❈❤❡❜♦t❛r❡✈ ❞❡♥s✐t② t❤❡♦r❡♠ ❢♦r ❛ ♥✉♠❜❡r ✜❡❧❞ ✸✳✹✳✶✳✷✳ ❇✐♥❛r② q✉❛❞r❛t✐❝ ❢♦r♠s✳ ▲❡t ❜✐♥❛r② q✉❛❞r❛t✐❝ ❢♦r♠✱ ✇✐t❤ B(x, y) = ax2 + bxy + cy 2 ✐♥t❡❣r❛❧ ❝♦❡✣❝✐❡♥ts✱ ✇❤♦s❡ ❜❡ ❛ ❞✐s❝r✐♠✐♥❛♥t d = b2 − 4ac ✐s ♥♦t ❛ sq✉❛r❡✳ ▲❡t S ❜❡ t❤❡ s❡t ♦❢ t❤❡ ♣r✐♠❡ ❞✐✈✐s♦rs ♦❢ d✳ ▲❡t PB ❜❡ t❤❡ s❡t ♦❢ ♣r✐♠❡s p ∈ / S t❤❛t ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❜② B ✱ ✐✳❡✳ ❛r❡ ♦❢ t❤❡ ❢♦r♠ p = B(x, y) ❢♦r s♦♠❡ x, y ∈ Z✳ ❚❤❡♥ PB ✐s S ✲❢r♦❜❡♥✐❛♥✳ ■t ✐s ♥♦t ❡♠♣t② ✐❢ t❤❡ ♦❜✈✐♦✉s ♥❡❝❡ss❛r② ❝♦♥❞✐t✐♦♥s ❛r❡ ♠❡t ✿ (a, b, c) = 1 ❛♥❞ a > 0 + ✐❢ d < 0 ❀ t❤❡ ❞❡♥s✐t② ♦❢ PB ✐s t❤❡♥ 1/h (d)✱ ❡①❝❡♣t ✐❢ B ✐s ❛♠❜✐❣✉♦✉s ✱ ✐✳❡✳ ✐♥✈❛r✐❛♥t ❜② ❛♥ ❡❧❡♠❡♥t ♦❢ GL2 (Z) ♦❢ ❞❡t❡r♠✐♥❛♥t −1✱ ✐♥ ✇❤✐❝❤ ❝❛s❡ t❤❡ + + ❞❡♥s✐t② ✐s 1/2h (d)✳ ❬❍❡r❡ h (d) ✐s t❤❡ ♥❛rr♦✇ ❝❧❛ss ♥✉♠❜❡r✱ ❝❢✳ ❬❈♦ ✾✸✱ ✺✳✷✳✼❪ ❀ ✇❤❡♥ d ✐s > 0 ✐t ♠❛② ❞✐✛❡r ❜② ❛ ❢❛❝t♦r ✷ ❢r♦♠ t❤❡ ✉s✉❛❧ ❝❧❛ss ♥✉♠❜❡r h(d)✳❪ ❚❤❡ ♣r♦♦❢s ♦❢ t❤❡s❡ st❛t❡♠❡♥ts r❡❧② ♦♥ t❤❡ st❛♥❞❛r❞ ❞✐❝t✐♦♥❛r② ❜❡t✇❡❡♥ ❜✐♥❛r② q✉❛❞r❛t✐❝ ❢♦r♠s ❛♥❞ ✐♥✈❡rt✐❜❧❡ ✐❞❡❛❧s ✐♥ q✉❛❞r❛t✐❝ r✐♥❣s ✭s❡❡ ❬❈♦ ✾✸✱ ➓✺✳✷❪✮✳ ❚❤❡ r❡❧❡✈❛♥t ●❛❧♦✐s ❡①t❡♥s✐♦♥s ❛r❡ t❤❡ ❛❜❡❧✐❛♥ ❡①t❡♥s✐♦♥s ♦❢ √ Q( d) ❦♥♦✇♥ ❛s r✐♥❣ ❝❧❛ss ✜❡❧❞s✱ ❝❢✳ ❬❈♦① ✽✾✱ ➓✾❪✳ ❊①❡r❝✐s❡✳ ❙❤♦✇ t❤❛t t❤❡ ♣r✐♠❡s r❡♣r❡s❡♥t❡❞ ❜② 2x2 + xy + 9y 2 ❤❛✈❡ ❞❡♥s✐t② ✶✴✼✳ ✸✳✹✳✷✳ ❚❤❡ ♠❛♣ p → NX (p) dim X/Q 0.

T❤❡ ❣r♦✉♣s H i (X, Z ) ❛r❡ ❞❡r✐✈❡❞ ❢r♦♠ t❤❡♠ ❜② t❤❡ ❢♦r♠✉❧❛❡ H i (X, Z ) = lim H i (X, Z/ n Z) ←− ❛♥❞ H i (X, Q ) = H i (X, Z ) ⊗Z Q = H i (X, Z )[1/ ]. ❆s ❢♦r t❤❡ ❜② ♠❡❛♥s ♦❢ ❛ Hci (X, Q )✱ ❝♦♠♣❛❝t✐✜❝❛t✐♦♥ ♦❢ X ✳ Hci (X, Z ) ❛♥❞ t❤❡② ❛r❡ ❞❡✜♥❡❞ ✐♥ ❛ s✐♠✐❧❛r ✇❛②✱ ❆ ❝♦♥s❡q✉❡♥❝❡ ♦❢ t❤❡s❡ s♦♠❡✇❤❛t ✐♥❞✐r❡❝t ❞❡✜♥✐t✐♦♥s ✐s t❤❛t ♠♦st ♦❢ t❤❡ ❜❛s✐❝ r❡s✉❧ts ✭s✉❝❤ ❛s t❤♦s❡ ♦♥ ❤✐❣❤❡r ❞✐r❡❝t ✐♠❛❣❡s✱ ♦r ❜❛s❡ ❝❤❛♥❣❡✮ ❤❛✈❡ t♦ ❜❡ ♣r♦✈❡❞ ✜rst ❢♦r t❤❡ ❝♦♥st❛♥t s❤❡❛✈❡s Z/ n Z Z ❜② ⊗Z Q ✳ ✜♥✐t❡ ❝♦♥str✉❝t✐❜❧❡ s❤❡❛✈❡s✮✱ ❛♥❞ t❤❡♥ ❡①t❡♥❞❡❞ t♦ lim✱ ←− ❛♥❞ ❡①t❡♥❞❡❞ t♦ Q ❜② ✉s✐♥❣ t❤❡ ❢✉♥❝t♦r ✭♦r✱ ♠♦r❡ ❣❡♥❡r❛❧❧②✱ ❢♦r ✉s✐♥❣ t❤❡ ❢✉♥❝t♦r ✹✳✷✳ ❆rt✐♥✬s ❝♦♠♣❛r✐s♦♥ t❤❡♦r❡♠ ❙✉♣♣♦s❡ k = R ♦r C✱ s♦ t❤❛t ks = k = C✳ ■♥ t❤❛t ❝❛s❡✱ t❤❡ C✲❛♥❛❧②t✐❝ s♣❛❝❡ X(C) ✐s ❧♦❝❛❧❧② ❝♦♠♣❛❝t ❢♦r t❤❡ ✉s✉❛❧ t♦♣♦❧♦❣②✱ ❛♥❞ ✐ts ❝♦❤♦♠♦❧♦❣② ❣r♦✉♣s H i (X(C), Q) ✇✐t❤ r❛t✐♦♥❛❧ ❝♦❡✣❝✐❡♥ts ❝❛♥ ❜❡ ❞❡✜♥❡❞ ❜② s❤❡❛❢ t❤❡♦r② ❀ ♦♥❡ i ❛❧s♦ ❣❡ts ❝♦❤♦♠♦❧♦❣② ❣r♦✉♣s ✇✐t❤ ❝♦♠♣❛❝t s✉♣♣♦rt Hc (X(C), Q).

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


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