Mathematical Research Letters

Volume 25 (2018)

Number 1

Artin conjecture for $p$-adic Galois representations of function fields

Pages: 143 – 157



Ruochuan Liu (Beijing International Center for Mathematical Research, Peking University, Beijing, China)

Daqing Wan (Department of Mathematics, University of California at Irvine)


For a global function field $K$ of positive characteristic $p$, we show that Artin’s entireness conjecture for L-functions of geometric padic Galois representations of $K$ is true in a non-trivial $p$-adic disk but is false in the full $p$-adic plane. In particular, we prove the non-rationality of the geometric unit root L-functions.

Received 11 January 2016

Accepted 3 December 2016

Published 4 June 2018