Applications of non-Archimedean integration to the L-series of τ-sheaves

Let F̲ be a τ-sheaf. Building on previous work of Drinfeld, Anderson, Taguchi, and Wan, Böckle and Pink (A cohomological theory of crystals over function fields, in preparation), develop a cohomology theory for F̲. Böckle (Math. Ann. 323 (2002) 737) uses this theory to establish the analytic continuation of the L-series associated to F̲ (which is a characteristic p-valued “Dirichlet series”) and the logarithmic growth of the degrees of its special polynomials. In this paper, we shall show that this logarithmic growth is all that is needed to analytically continue the original L-series as well as all associated partial L-series. Moreover, we show that the degrees of the special polynomials attached to the partial L-series also grow logarithmically. Our tools are Böckle's original results, non-Archimedean integration, and the very strong estimates of Amice (Bull. Soc. Math. France 92 (1964) 117). Along the way, we define certain natural modules associated with non-Archimedean measures (in the characteristic 0 case as well as in characteristic p).