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Weil Conjectures and Betti Numbers of Moduli Spaces

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Weil Conjectures and Betti Numbers of Moduli Spaces Weil conjectures and Betti numbers of moduli spaces Christopher Lazda 1 Lecture 1 Let’s start with an object you’ll hopefully all be reasonably familiar with. X s ³(s) n¡ , s C, Re(s) 1. Æ n 1 2 È ¸ Conjecture (RH). If ³(s) 0 and 0 Re(s) 1, then Re(s) 1 . Æ Ç Ç Æ 2 This has an Euler product expansion Y s 1 ³(s) (1 p¡ )¡ Æ p prime ¡ and we note that {prime numbers} {closed points of Spec(Z)} $ so we can write Y s 1 ³(s) (1 #k(x)¡ )¡ Æ x Spec(Z) ¡ 2j j where k(x) is the residue field at x. Definition. For any scheme X of finite type over Z, (i.e. any “arithmetic scheme") we define Y s 1 ³X (s) (1 #k(x)¡ )¡ Æ x X ¡ 2j j the product being taken over all closed points, and k(x) denotes the residue field. Exercise. Show that in this situation k(x) is always a finite field, so the definition makes sense. Example. Suppose that K/Q is finite, with ring of integers OK . Then taking X Spec(OK ) Æ we get Y s 1 ³Spec (s) (1 Np¡ )¡ (OK ) Æ ¡ p OK ½ which is the usual Dedekind zeta function ³K (s) of K. There is also a Riemann hypothesis 1 for ³K , again stating that zeroes of ³K with 0 Re(s) 1 satisfy Re(s) . Ç Ç Æ 2 1 ¡ ¢ Example. X AF Spec Fp t . Æ p Æ J K n 1 o © ª closed points of A irreducible polynomials over Fp Fp $ Y ds Nd ³ 1 (s) (1 p¡ )¡ AF ) p Æ d 1 ¡ ¸ 1 s where Nd # irreducible polynomials of degree d. Let’s try the following trick: set u p¡ Æ Q d Nd Æ and compute uZ0(u)/Z(u) where Z(u) (1 u )¡ . We get Æ d ¡ d dN ud u log Z(u) X d d du Æ d 1 1 u ¸ ¡ X md dNd u Æ d,m 1 ¸ X X n dNd u Æ n 1 d n ¸ j X pnun Æ n 1 ¸ n n P 1 n since (recall from Galois theory) d n dNd p . Note that p #AF (Fp ) and that we also j Æ Æ p have d p log Z(u) du Æ 1 pu ¡ 1 Z(u) ) Æ 1 pu ¡ 1 ³A1 (s) . ) Fp Æ 1 p1 s ¡ ¡ This trivially satisfies the ‘Riemann Hypothesis’, since it has no zeroes. Lemma. Let q be a prime power, and X a scheme of finite type over Fq (therefore X is a scheme of finite type over Z and has a zeta function). Write Bn #X(Fqn ). Then we have Æ Ã ! X Bn n ³X (s) exp u Æ n 1 n ¸ s where u q¡ Æ Proof. Let Nd be the number of closed points of X of degree d over Fq, i.e. such that the residue field k(x) is Fqd . Then we have the formula X dNd Bn d n Æ j since every closed point of degree d gives rise to exactly d points in X(Fqn ) for d n. Then j exactly as in the calculation we’ve just done we can show Y d Nd ³X (s) (1 u )¡ Æ d 1 ¡ ¸ s Q d Nd where u q¡ and that if we let Z(u) d 1(1 u )¡ then we have an identity of power Æ Æ ¸ ¡ series d X n u log Z(u) Bnu du Æ n 1 ¸ and the lemma follows. 2 nd Exercise. By showing that when X/Fq has dimension d,#X(Fqn ) O(q ) as d , show Æ ! 1 that ³X (s) converges for Re(s) dim X. È So zeta function of schemes over finite fields ‘encode’ the numbers of points over all finite extension fields, in that if you know one, you know the other. It’s in practise often easier to work with the power series à ! X #X(Fqn ) n Z(X, u): exp u Q u . Æ n 1 n 2 J K ¸ Note that while the ³-function ³X (s) doesn’t ‘see’ the base field Fq, the Z-function Z(X, u) s does, since we have set u q¡ . Æ 1.1 Abelian Varieties Now let’s take A/Fq an abelian variety of dimension g, that is a smooth, projective, (geo- metrically) connected group variety over Fq. We’ll let A A F Fq, and F : A A be the Æ £ q ! q-power linear Frobenius morphism, that is if we have some projective embedding A PN ! Fq q q then in homogeneous co-ordinates we have F(x0 : ... : xN ) (x0 : ... : xN ). Æ n Therefore A(Fqn ) is exactly the (closed) fixed points of F , or in other words, if we con- sider the isogeny n ©n id F : A A Æ ¡ ! n then A(Fqn ) ker(©n)(Fq). Since id F separable, we therefore have Æ ¡ n #A(Fqn ) deg(id F ). Æ ¡ n To calculate this, we’ll introduce the Tate module of A. Let ` - q be a prime, so that A[` ] is n2g an étale group scheme over Fq of order ` , and n n 2g A[` ](Fq) (Z/` Z) Æ» (non-canonically). The multiplication by ` induces transition maps n n 1 [`]: A[` ](Fq) A[` ¡ ](Fq) ! so we can form the inverse limit n T (A): lim A[` ](Fq), ` Æ Ã¡¡n this is a free Z -module of rank 2g, and every endomorphism à End(A) of A induces an ` 2 endomorphism à of T (A), and hence of V (A): T (A) Z Q . ` ` ` Æ ` ­ ` ` Theorem (See Mumford’s “Abelian Varieties”). For every endomorphism à End(A) there 2 exists a unique polynomial P (t) Z[t] (the characteristic polynomial of Ã) such that for all à 2 integers n we have P (n) deg([n] Ã). Ã Æ ¡ 3 Moreover for any ` - q we have that P (t) det(t à ) Ã Æ ¡ ` is the characteristic polynomial of Ã` acting on V`(A). We’re now going to take à F, the Frobenius endomorphism, this tells us that if we let Æ 2g Y PF (t) (t ! j) Æ i 1 ¡ Æ be the characteristic polynomial of F acting on V (A), and n 1, then ` ¸ 2g Y n PF n (t) (t ! j ) Æ i 1 ¡ Æ n is the characteristic polynomial of F acting on V`(A) and hence we can count the number of Fqn -rational points on A as 2g n Y n #A(Fqn ) deg(id F ) PF n (1) (1 ! j ) Æ ¡ Æ Æ i 1 ¡ Æ X n X n n n n 1 ! ! ! ... ! ...! . Æ ¡ j1 Å j1 j2 ¡ Å j1 j2g j1 j1, j2 Actually, some simple linear algebra tells us that X n n ! ...! j1 jk j1,..., jk n k is exactly the trace of F acting on ¤ V`(A), so we can write 2g X k n k #A(Fqn ) ( 1) Tr(F ¤ V`(A)). Æ k 0 ¡ j Æ Substituting this in to the expression for Z(A, u) gives us k 1 à !( 1) Å 2g n ¡ Y X n k u Z(A, u) exp Tr(F ¤ V`(A) Æ k 0 ¡ n 1 j n Æ ¸ Lemma. Let V be a finite dimensional vector space over a field K, and ' : V V and ! endomorphism. Then à n ! X n u det¡id 'u¢ exp Tr(' ) . ¡ Æ ¡ n 1 n ¸ Proof. Exercise (hint: express both sides in terms of the eigenvalues of '). So we can therefore write the Z-function of A as 2g k 1 Y k ( 1) Å Z(A, u) det(id Fu ¤ V`(A)) ¡ Æ k 0 ¡ j Æ P1(t)...P2g 1(u) ¡ Æ P0(u)...P2g(u) 4 where k Pk(t) det(id Fu ¤ V (A)) Z[u] Æ ¡ j ` 2 Q can be written as (1 ®iku) with ®ik running over things of the form ! j ...! j . i ¡ 1 k 1/2 Theorem (Riemann Hypothesis). Each ! j has norm q . s Exercise. What does this tell us about the zeroes of ³A(s) Z(A, q¡ )? Æ Exercise. We expect a functional equation g 1 Z(A,(q u)¡ ) Z(A, u) Æ can you interpret this in terms of a statement about the ! j? 2 Lectures 2 & 3 James told you about the Weil conjectures, and said that a ‘good’ cohomology theory for varieties over Fq would imply them more or less formally. Let’s look at that in a bit more detail. Weil cohomology theories Definition. Fix an algebraically closed field k and let Vk be the category of smooth, pro- jective, connected varieties over k, that is non-singular, connected varieties cut out by ho- mogenous equations in some PN (k). Let K be a field of characteristic 0. A K-valued Weil cohomology theory is a contravariant functor H¤( ): Vk {graded vector spaces over K} ¡ ! X H¤(X) 7! f : X Y f ¤ : H¤(Y ) H¤(X) ! 7! ! together with a graded commutative product i j i j : H (X) K H (X) H Å (X) [ ­ ! a trace morphism Tr : H2dim X (X) K ! and a cycle class map cl : Cr(X) H2r(X) ! such that: i) Hi(X) is finite dimensional, and zero outside the range [0,2dim X]. ii) (Künneth formula) if we let p1, p2 : X Y X,Y denote the two projections, then the £ ! natural map H¤(X) K H¤(Y ) H¤(X Y ) ­ ! £ ® ¯ ® ¯ : p¤(®) p¤(¯): ­ 7! Æ 1 [ 2 is an isomorphism.
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