Computing L-functions of $ \lambda $-adic representations of global function fields

TL;DR

This paper develops a systematic method to compute the coefficients and functional equation sign of L-functions associated with almost everywhere unramified lambda-adic representations over global function fields, providing explicit examples.

Contribution

It introduces a new framework for calculating L-function coefficients and signs for lambda-adic representations in the context of global function fields.

Findings

Framework successfully computes L-function coefficients.

Explicit examples demonstrate the method's effectiveness.

Sign determination aligns with theoretical expectations.

Abstract

The L-function $ L(\rho_\lambda, s) $ of an almost everywhere unramified $ \lambda $-adic representation $ \rho_\lambda $ of a global function field $ \mathbb{F}_q(C) $ is known to be a rational function in $ q^{-s} $ satisfying a functional equation up to some complex sign $ \epsilon(\rho_\lambda) $. This paper presents a systematic framework to compute the coefficients of $ L(\rho_\lambda, s) $ and its sign $ \epsilon(\rho_\lambda) $ with some explicit examples.