On the Values of Zeta and L-Functions: I

In this paper we begin a study of the relationship between values of zeta and L-functions on the one hand, and Euler characteristics of sheaves for the etale topology, on the other hand. We start with the following conjecture: Let F be a totally real number field, p a prime, and n a negative odd integer. Let A be the ring of integers of F, S the set of primes of F over p, X = SpecA, and X, = X S. Let j be the natural inclusion of SpecF in X, Let F be the algebraic closure of F, and GF the Galois group of F over F. Let W be the GF-module consisting of all p-power roots of unity in F, let T(W) be the Tate-module of W, and let W(m' be the GF-module W ? T(W)xm('. Then W(m' may be identified with a sheaf for the etale topology on SpecF, and we may take the direct image j* W(m' on X, If M is a finite group, let #(M) denote the number of elements of M.