The functional equation for L-functions of hyperelliptic curves

TL;DR

This paper computes and verifies the functional equations of L-functions for a broad class of hyperelliptic and superelliptic curves over , using stable reduction techniques to determine local factors and conductors.

Contribution

It provides a numerical verification of the functional equation for L-functions of hyperelliptic curves, extending previous theoretical results with extensive computations.

Findings

Numerical confirmation of the functional equation for hyperelliptic curves.

Effective computation of local L-factors and conductors at bad primes.

Application of stable reduction methods to a large class of algebraic curves.

Abstract

We compute the $L$-functions of a large class of algebraic curves, and verify the expected functional equation numerically. Our computations are based on our previous results on stable reduction to calculate the local $L$-factor and the conductor exponent at the primes of bad reduction. Most of our examples are hyperelliptic curves of genus $g\geq 2$ defined over $\mathbb{Q}$ which have semistable reduction at every prime $p$. We also treat a few more general examples of superelliptic curves.