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Deligne's Proof of the Weil-Conjecture

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Deligne's Proof of the Weil-Conjecture Deligne's Proof of the Weil-conjecture Prof. Dr. Uwe Jannsen Winter Term 2015/16 Inhaltsverzeichnis 0 Introduction 3 1 Rationality of the zeta function 9 2 Constructible sheaves 14 3 Constructible Z`-sheaves 21 4 Cohomology with compact support 24 5 The Frobenius-endomorphism 27 6 Deligne's theorem: Formulation and first reductions 32 7 Weights and determinant weights 36 8 Cohomology of curves and L-series 42 9 Purity of real Q`-sheaves 44 10 The formalism of nearby cycles and vanishing cycles 46 11 Cohomology of affine and projective spaces, and the purity theorem 55 12 Local Lefschetz theory 59 13 Proof of Deligne's theorem 74 14 Existence and global properties of Lefschetz pencils 86 0 Introduction The Riemann zeta-function is defined by the sum and product X 1 Y 1 ζ(s) = = (s 2 C) ns 1 − p−s n≥1 p which converge for Re(s) > 1. The expression as a product, where p runs over the rational prime numbers, is generally attributed to Euler, and is therefore known as Euler product formula with terms the Euler factors. Formally the last equation is easily achieved by the unique decomposition of natural numbers as a product of prime numbers and by the geometric series expansion 1 1 X = p−ms : 1 − p−s m=0 The { to this day unproved { Riemann hypothesis states that all non-trivial zeros of ζ(s) 1 should lie on the line Re(s) = 2 . This is more generally conjectured for the Dedekind zeta functions X 1 Y 1 ζ (s) = = : K Nas 1 − Np−s a⊂OK p Here K is a number field, i.e., a finite extension of Q, a runs through the ideals =6 0 of the ring OK of the integers of K, p runs through the prime ideals =6 0, and Na = jOK =aj, where jMj denotes the cardinality numbers of a finite set M. Artin examined the analogue for global function fields. Let Fq be a finite field with q elements, q a power of a prime p Q corresponds to Fq(t) (the rational function field), Z corresponds to Fq[t] (the polynomial ring in a variable), and on considers the analogous functions: X 1 Y 1 = ; Nas 1 − Np−s a⊂Fq[t] p where again a and p runs through the non-trivial ideals and the prime ideals of Fq[t] re- spectively, and where Na = jFq[t]=aj. Similarly one can examine global function fields K, i.e., finite extensions to Fq(t). However the ring Fq[t] is no longer defined by the field Fq(t), Z Q F 1 ⊆ F as it was for in ; one could also consider q[ t ] q(t). This is even more the case for the general fields K, because these no longer contain Fq(t) canonically. It is better and more canonical to consider the uniquely determined smooth projective curve X over Fq with function field K and to define Y 1 Y 1 ζ (s) = ζ(X; s) = = : K 1 − (Nx)−s 1 − q−deg(x)s x2X0 x2X0 Here X0 denotes the set of the closed points of X, and for x 2 X0 Nx = jk(x)j is the (finite) cardinality of the residue field k(x) of x. With deg(x) = [k(x): Fq] we apparently 3 have Nx = qdeg(x), and therefore the last equality. These points here are meant in a scheme- theoretic sense: Observe that for an affine open part U = SpecR ⊂ X the points x 2 U correspond to the prime ideals p of R, and that k(x) is the quotient field of R=p. The finite points x correspond to the maximal ideals; for these one has Nx = jR=pj, and one obtains a similar setting as above. By the last formula one has ζ(X; s) = Z(X; q−s), where Y 1 Z(X; T ) = 2 Z[[T ]] : 1 − T deg(x) x2X0 From this one obtains the equality of formal power series P P P1 − − deg(x) T deg(x)·n log Z(X; T ) = log(1 T ) = n x2X x2X n=1 P1 P P1 T m T m j F m j = ( deg(x)) m = X( q ) m ; m=1 deg(x)jm m=1 where X(Fqm ) is the set of the Fqm -rational points of X over Fq: In fact for every x 2 X0 with deg(x)jm there exist exactly as many Fqm -rational points, as there exist Fq-linear embeddings κ(x) ,! Fqm , and their quantity is deg(x). F P1 We consider an example. The smooth projective curve with function field q(t) is Fq , the one-dimensional projective space over Fq. Geometrically, i.e., scheme theoretically we have P1 [ F 1 F Fq = U1 U2, with U1 = Spec q[t] = AFq (the one-dimensional affine space over q) −1 −1 and U2 = SpecFq[t ] (the affine space with the coordinate t ). Then we have U1 \ U2 = −1 −1 00 SpecFq[t; t ] and U1 n U2 = point t = 0 and U2 n U1 = point t = 0 (\t = 1 ). Since 1 1 ∼ F m F F m F F F m F m AFq ( q ) = Hom q (Spec q ; AFq ) = Hom q ( q[t]; q ) = q (the last bijection sends a ring homomorphism ' to '(t)), one obtains 1 m jP F m j Fq ( q ) = q + 1 : This also follows from the known description of points 1 2 × P F m F m n f g F m Fq ( q ) = (( q ) 0 )= q = f[a0 : a1]jai 2 Fqm ; not both zerog = f[1 : a1]ja1 2 Fqm g [ f[0 : 1]g : a1 By choosing the coordinate t = , the first set of the union is of course U1(Fqm ), and [0 : 1] a0 is the point \t = 100. With this we now calculate P1 Z(P1 ; t) = exp( (1 + qm) T m ) Fq m m=1 P1 P1 T m · (qT )m 1 = exp( m ) exp( m )) = − − : m=1 m=1 (1 T )(1 qT ) In particular, this is a rational function! More generally one can show the following result which goes back to E. Artin and F.K. Schmidt: for a smooth projective (geometrically irreducible) curve X of genus g over Fq one has: P (T ) Z(X; T ) = ; (1 − T )(1 − qT ) 4 where P (T ) is a polynomial of degree 2g in Z[T ], with constant coefficient 1. Furthermore Hasse (for g = 1, as well as for elliptic curves) and Weil (for arbitrary g) proved that the −s 1 −s zeros of P (q ) lie on the line Re(s) = 2 . Applied to ζ(X; s) = Z(X; q ) this proves the analogoue (conjectured by Artin) of the Riemann hypothesis in the case of function fields. We change the interpretation. Write Y2g P (T ) = (1 − αiT ); with αi 2 Q ⊂ C ; i=1 where Q denotes the algebraic closure of Q in C. For a complex number s we obviously have −s −s P (q ) = 0 if and only if there is an i with αi · q = 1. In this case we furthermore have 1 1 Re(s) = ()j α j= q 2 : 2 i A. Weil now observed, that the definition of the zeta-function makes sense for arbitrary varieties over Fq, and after calculation of these in several non-trivial cases ([Weil]) stated the following conjectures. Weil-conjecture (proved by Deligne in 1973): Let X be a geometric irreducible smooth projective variety Fq. Define 1 X T n Z(X; T ) = exp( jX(F n )j ) 2 Q[[T ]] : q n n=1 Then the following holds I: Z(X; T ) is rational, i.e., in Q(T ). (In particular, this implies the existence of a meromorphic continuation of the zeta-function ζ(X; s) = Z(X; q−s), for which the series initially only converges for Re(s) >> 0). II: One has a functional equation 1 dE E Z(X; ) = ±q 2 T Z(X; T ) ; qdT where d = dim X is the dimension of X and E = (∆ · ∆) is the selfintersection number of the diagonal ∆ on X × X. For the zeta-function in s this means E( d −s) ζ(X; d − s) = ±q 2 ζ(X; s) : For a curve of genus g it is easily shown that E = 2 − 2g, and one obtains the classical type of functional equation, which relates s and 1 − s). III: One has P (T )P (T ) :::P − (T ) Z(X; T ) = 1 3 2d 1 ; P0(T )P2(T ) :::P2d(T ) d where P0(T ) = 1−T;P2d(T ) = 1−q T , and generally Pi(X) 2 Z[T ] with constant coefficient 1. Moreover one has Ybi − (i) C Pi(T ) = (1 αj T ) in [T ] ; j=1 5 with (i) i j j 2 αj = q for all j This is the most difficult part - the analogue of the Riemann hypothesis for arbitrary dimen- sions. IV: If X is obtained by reduction mod p (i.e., mod p for a prime ideal pjp) of a variety over a number field K ⊆ C, then bi = deg Pi is equal to the i-th Bettinumber of X(C) (the dimension of the i-th singular homology group of X(C)). We add two remarks. Property III implies that there is no cancellation between the Pi(T ), so they are determined by Z(X; T ). In IV arithmetic properties are linked with topological invariants in an interesting way. For example, if X is a curve of genus g over Q, then X(C) is a Riemannian surface with \g handles", and therefore b0 = 1 = b2; b1 = 2g (this coincides with the results of Hasse and Weil).
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