Root number of the Jacobian of $y^2=x^p+a$

TL;DR

This paper explicitly calculates the root number of the Jacobian of a specific hyperelliptic curve over rationals, focusing on the local root number at the prime of wild ramification, advancing understanding of its arithmetic properties.

Contribution

It provides an explicit determination of the root number for Jacobians of hyperelliptic curves of the form y^2=x^p+a, especially at the prime p with wild ramification.

Findings

Explicit formulas for the root number at p

Analysis of wild ramification effects

Enhanced understanding of Jacobian arithmetic

Abstract

Let $C/\mathbb{Q}$ be a hyperelliptic curve with an affine model of the form $y^2=x^p+a$. We explicitly determine the root number of the Jacobian of $C$, with particular focus on the local root number at $p$ where $C$ has wild ramification.