A Riemann hypothesis and simplicity conjecture for characteristic p L-series
by
David Goss
The Ohio State University and Journal of Number Theory

Let k be a function field in one variable over the finite field with r-elements. Let \infty be a fixed place of k and A the ring of functions regular away from \infty. There are well known analogies between A and the integers Z, k with Q, etc. Associated to A one has the theory of Drinfeld modules (and their generalizations T-modules, shtuka, \Tau-sheaves etc.) in rough analogy with the classical theory of abelian varieties (resp. general motives). Let L/k be a finite extension and let E be one of the above objects. One can then define a characteristic p L-series L(E, s) as an Euler product over the finite primes of L exactly as is done in classical theory (via Frobenius morphisms, Tate modules etc.). Recently these L-series have been analytically continued in great generality through non-Archimedean integration and a cohomology theory due to R. Pink and G. Boeckle. In this talk we will explain how the use of absolute values allows us to pose analogs of the classical Generalized Riemann Hypothesis and Simplicity Conjecture for such L-series, as well as give a reformulation of the classical conjectures themselves. We will present the evidence in favor of these conjectures in finite characteristic as well as some of their function-theoretic consequences.

Date received: March 3, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cadx-49.