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Computing L-functions of $ λ$-adic representations of global function fields

David Kurniadi Angdinata

TL;DR

This work addresses the computation of L-functions for λ-adic representations over global function fields, showing that $L(\rho_\lambda,s)$ is a rational function in $q^{-s}$ satisfying a functional equation with sign $\epsilon(\rho_\lambda)$. It develops an explicit, cohomology-informed framework based on the formal L-function $\mathcal{L}(\rho_\lambda,T)$, local Euler factors, and global invariants, and provides concrete algorithms (Rationality, Functional equation, and epsilon-factor procedures) with complexity analyses. The authors implement these methods in Magma and illustrate them with detailed examples for trivial representations, elliptic curves, Dirichlet characters, and other motivic objects, validating the approach against known cases and revealing the practical feasibility of L-function computations in characteristic $p$. The results extend analogue computational capabilities to global function fields, contributing to experimental aspects of the Langlands program and offering a toolkit for exploring arithmetic of function-field L-functions. The framework also sets the stage for future work on tensor products, non-standard endomorphisms, and Drinfeld-module–driven L-functions.

Abstract

The L-function $ L(ρ_λ, s) $ of an almost everywhere unramified $ λ$-adic representation $ ρ_λ$ 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 $ ε(ρ_λ) $. This paper presents a systematic framework to compute the coefficients of $ L(ρ_λ, s) $ and its sign $ ε(ρ_λ) $ with some explicit examples.

Computing L-functions of $ λ$-adic representations of global function fields

TL;DR

This work addresses the computation of L-functions for λ-adic representations over global function fields, showing that is a rational function in satisfying a functional equation with sign . It develops an explicit, cohomology-informed framework based on the formal L-function , local Euler factors, and global invariants, and provides concrete algorithms (Rationality, Functional equation, and epsilon-factor procedures) with complexity analyses. The authors implement these methods in Magma and illustrate them with detailed examples for trivial representations, elliptic curves, Dirichlet characters, and other motivic objects, validating the approach against known cases and revealing the practical feasibility of L-function computations in characteristic . The results extend analogue computational capabilities to global function fields, contributing to experimental aspects of the Langlands program and offering a toolkit for exploring arithmetic of function-field L-functions. The framework also sets the stage for future work on tensor products, non-standard endomorphisms, and Drinfeld-module–driven L-functions.

Abstract

The L-function of an almost everywhere unramified -adic representation of a global function field is known to be a rational function in satisfying a functional equation up to some complex sign . This paper presents a systematic framework to compute the coefficients of and its sign with some explicit examples.
Paper Structure (15 sections, 11 theorems, 63 equations, 4 algorithms)

This paper contains 15 sections, 11 theorems, 63 equations, 4 algorithms.

Table of Contents

  1. Introduction
  2. l-adic representations
  3. Local invariants
  4. Global invariants
  5. L-functions
  6. Computing L-functions
  7. Rationality
  8. Functional equation
  9. epsilon factors
  10. l-adic representations
  11. Example computations
  12. Trivial representations
  13. Elliptic curves
  14. Dirichlet characters
  15. Other examples

Key Result

Theorem 1.1

Let $\rho_\lambda$ be an almost everywhere unramified $\lambda$-adic representation of a global function field $K$. Assume further that $\rho_\lambda$ is self-dual of weight $w(\rho_\lambda)$ and sign $c(\rho_\lambda) \in \{\operatorname{id}, \operatorname{cc}\}$.

Theorems & Definitions (34)

  • Theorem 1.1: Corollary \ref{['cor:rationalfunction']}, Corollary \ref{['cor:functionalequation']}, Corollary \ref{['cor:computeepsilon']}
  • Example 2.1
  • Example 2.2
  • Proposition 2.3
  • proof
  • Example 2.4
  • Example 2.5
  • Proposition 2.6
  • proof
  • Theorem 2.7
  • ...and 24 more