Factorization of Coefficients for Exponential and Logarithm in Function Fields

TL;DR

This paper investigates the factorization of coefficients in exponential and logarithm series of Hayes modules over elliptic and hyperelliptic curves, enabling new $v$-adic convergence results and demonstrating the log-algebraicity of certain Goss $L$-values.

Contribution

It introduces a method to factorize coefficients of exponential and logarithm series for Hayes modules over specific curves, leading to new convergence and algebraicity results.

Findings

Established $v$-adic convergence of exponential and logarithm series.

Proved $v$-adic Goss $L$-values are log-algebraic for certain characters.

Provided a factorization technique for coefficients in function field settings.

Abstract

Let $X$ be an elliptic curve or a ramifying hyperelliptic curve over $\mathbb F_q$. We will discuss how to factorize the coefficients of the exponential and logarithm series for a Hayes module over such a curve. This allows us to obtain $v$-adic convergence results for such exponential and logarithm series, for $v$ a 'finite' prime. As an application, we can show that the $v$-adic Goss $L$-value $L_v(1, \Psi)$ is log-algebraic for suitable characters $\Psi$.