Unit L-functions and a conjecture of Katz

Abstract

Let f: X -> Y be a separated morphism of schemes of finite type over a finite field of characteristic p, let Lambda be an artinian local Z_p-algebra with finite residue field, let m be the maximal ideal of Lambda, and let L^\bullet be a bounded constructible complex of sheaves of finite free Lambda-modules on the \'etale site of Y. We show that the ratio of L-functions L(X,L^\bullet)/L(Y,f_! L^\bullet), which is a priori an element of 1+T Lambda[[T]], in fact lies in 1+ m T Lambda [T]. This implies a conjecture of Katz predicting the location of the zeroes and poles of the L-function of a p-adic \'etale lisse sheaf on the closed unit disk in terms of \'etale cohomology with compact support.