Applications of Non-archimedean Integration to the L-series of Τ-sheaves

Let F be a τ -sheaf. Building on previous work of Drinfeld, Anderson, Taguchi, and Wan, Böckle and Pink [BP1] develop a cohomology theory for F . In [Boc1] Böckle uses this theory to establish the analytic continuation of the L-series associated to F (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 Lseries also grow logarithmically. Our tools are Böckle’s original results, non-Archimedean integration, and the very strong estimates of Y. Amice [Am1]. Along the way, we define certain natural modules associated with non-Archimedean measures (in the characteristic 0 case as well as in characteristic p).