Special L-values of t-motives: a conjecture

TL;DR

This paper proposes a conjecture relating special values of Goss L-functions at zero to the uniformization of associated abelian t-modules in the context of t-motives over function fields, supported by numerical evidence.

Contribution

It introduces a new conjecture connecting L-values of t-motives to their uniformization, extending known results for Carlitz motives.

Findings

Conjecture generalizes Anderson-Thakur theorem for Carlitz zeta values.

Provides numerical evidence supporting the conjecture.

Links special L-values to the structure of t-motives and their uniformization.

Abstract

We propose a conjecture on special values of $ L $-functions in a function field context with positive characteristic coefficients. For $ M $ a uniformizable $ t $-motive with everywhere good reduction we conjecture a relation between the value of the Goss $ L $-function $ L(M^\vee, s) $ at $ s = 0 $ and the uniformization of the abelian $ t $-module associated with $ M $. When $ M $ is a power of the Carlitz $ t $-motive the conjecture specializes to a theorem of Anderson and Thakur on Carlitz zeta values. Beyond this case we present numerical evidence.