Computing $L$-functions and semistable reduction of superelliptic curves

TL;DR

This paper provides an explicit method for analyzing the stable reduction of superelliptic curves at certain primes, enabling computation of local L-factors and conductors, which are important in number theory.

Contribution

It offers a new explicit description of stable reduction for superelliptic curves and applies it to compute local L-factors and conductors at primes.

Findings

Explicit description of stable reduction at primes with residue characteristic prime to n

Method to compute local L-factors of superelliptic curves

Calculation of conductor exponents at relevant primes

Abstract

We give an explicit description of the stable reduction of superelliptic curves of the form $y^n=f(x)$ at primes $\p$ whose residue characteristic is prime to the exponent $n$. We then use this description to compute the local $L$-factor of the curve and the exponent of conductor at $\p$.