Zeroes of $L$-series in characteristic $p$

TL;DR

This paper explores the peculiar behavior of trivial zeros of $L$-series in characteristic $p$, proposing a conjecture linking their order to the sum of $p$-adic coefficients, aiming to understand the functional equations in finite characteristic.

Contribution

It introduces a conjecture connecting non-classical trivial zeroes to $p$-adic coefficients, advancing the understanding of zeroes in characteristic $p$ $L$-series.

Findings

Non-classical trivial zeroes have higher order than classical predictions.

These zeroes are correlated with integers with bounded sum of $p$-adic coefficients.

The conjecture may lead to defining the correct functional equations in finite characteristic.

Abstract

In the classical theory of $L$-series, the exact order (of zero) at a trivial zero is easily computed via the functional equation. In the characteristic $p$ theory, it has long been known that a functional equation of classical $s\mapsto 1-s$ type could not exist. In fact, there exist trivial zeroes whose order of zero is ``too high;'' we call such trivial zeroes ``non-classical.'' This class of trivial zeroes was originally studied by Dinesh Thakur \cite{th2} and quite recently, Javier Diaz-Vargas \cite{dv2}. In the examples computed it was found that these non-classical trivial zeroes were correlated with integers having {\it bounded} sum of $p$-adic coefficients. In this paper we present a general conjecture along these lines and explain how this conjecture fits in with previous work on the zeroes of such characteristic $p$ functions. In particular, a solution to this conjecture might entail finding the ``correct'' functional equations in finite characteristic.