Global L-functions over function fields, by Gebhard Boeckle

In this article we prove that the global L-functions for Drinfeld-A-modules or more generally t-motives are meromorphic in a suitable sense. This was conjectured by Goss who also gave the definition of such L-functions. The result generalizes previous work of Taguchi and Wan who gave a proof for A=F_q[t]. Based on some of our results there is also a proof by Goss himself for general A.

We give two proofs of this result, the first of algebraic, the second, in the framework provided by Taguchi and Wan, of analytic nature. The algebraic one is based on the theory of crystals over function fields, as developed by R. Pink and the current author. This is used to derive results on the special values of Goss' L-functions at negative integers. A simple interpolation argument proves the asserted meromorphy. To look at such special values was suggested by Goss. Our methods also yield rather general criteria for the entireness of L-functions.

The analytic proof is based on a construction of a uniformly overconvergent v-adic phi-sheaf of rank one over Z_p which interpolates the zeta-function of the ring A at the negative integers -hn where n is a fixed positive integer and n runs through all positive integers. Here v is any place of the fraction field of A. Taguchi and Wan had done this previously for tensor powers of the Carlitz module and A=F_q[t].

The article also contains some remarks on a cohomological approach to Goss' trivial zeroes, and to L-factors at places of bad reduction.

Gebhard Boeckle <>