By Jean-Pierre Serre

**Lectures on N _{X}**

**(p)**offers with the query on how N

_{X}(p), the variety of recommendations of mod p congruences, varies with p whilst the relations (X) of polynomial equations is mounted. whereas the sort of common query can't have an entire solution, it bargains a great get together for reviewing quite a few ideas in l-adic cohomology and staff representations, offered in a context that's attractive to experts in quantity concept and algebraic geometry.

Along with overlaying open difficulties, the textual content examines the scale and congruence homes of N_{X}(p) and describes the ways that it truly is computed, through closed formulae and/or utilizing effective computers.

The first 4 chapters conceal the preliminaries and comprise virtually no proofs. After an outline of the most theorems on N_{X}(p), the booklet deals uncomplicated, illustrative examples and discusses the Chebotarev density theorem, that's crucial in learning frobenian capabilities and frobenian units. It additionally stories ℓ-adic cohomology.

The writer is going directly to current effects on staff representations which are usually tough to discover within the literature, equivalent 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 relatives among diversified households of equations X and Y. the writer additionally describes the Archimedean houses of N_{X}(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 publication concludes with an account of the top quantity theorem and the Chebotarev density theorem in larger dimensions.

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**Sample text**

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).

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 ).

X ✐s ❣❡♦♠❡tr✐❝❛❧❧② ✐rr❡❞✉❝✐❜❧❡✱ e = 1✱ ❛♥❞ ✇❡ ❣❡t t❤❡ ❜♦✉♥❞ ❚❤✐s ✐s ❡s♣❡❝✐❛❧❧② ✉s❡❢✉❧ ✇❤❡♥ IX i ❜❡❝❛✉s❡ 1 |NX (q) − q d | (B − 1)q d− 2 .