Exceptional Zeros of $L$-series and Bernoulli-Carlitz Numbers

Abstract

Bernoulli-Carlitz numbers were introduced by L. Carlitz in 1935, they are the analogues in positive characteristic of Bernoulli numbers. We prove a conjecture formulated by F. Pellarin and the first author on the non-vanishing modulo a given prime of families of Bernoulli-Carlitz numbers. We then show that the "exceptional zeros" of certain $L$-series are intimately connected to the Bernoulli-Carlitz numbers.

with an appendix by B. Anglès, D. Goss, F. Pellarin and F. Tavares Ribeiro : "The digit principle and derivatives of certain $L$-series"