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

TL;DR

This paper establishes a link between exceptional zeros of certain $L$-series and Bernoulli-Carlitz numbers, proving a conjecture about their non-vanishing modulo primes in the context of positive characteristic number theory.

Contribution

It proves a conjecture on the non-vanishing modulo primes of Bernoulli-Carlitz numbers and connects these to exceptional zeros of $L$-series in positive characteristic.

Findings

Proved non-vanishing modulo primes of Bernoulli-Carlitz numbers.

Established a relationship between exceptional zeros of $L$-series and Bernoulli-Carlitz numbers.

Confirmed a conjecture by Pellarin and Carlitz.

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.