Characteristic polynomials of automorphisms of hyperelliptic curves

TL;DR

This paper characterizes the relationship between automorphisms of hyperelliptic curves and their Jacobians, providing explicit formulas for the characteristic polynomial based on the automorphism orders and genus.

Contribution

It establishes a complete determination of the characteristic polynomial of automorphisms of hyperelliptic curves' Jacobians using the triple (g, n, n'), with explicit formulas for all cases.

Findings

The triple (g, n, n') determines the characteristic polynomial except in a specific even case.

Explicit formulas for the characteristic polynomial are provided for all cases.

The paper clarifies the influence of automorphism orders on Jacobian automorphisms.

Abstract

Let alpha be an automorphism of a hyperelliptic curve C of genus g, and let alpha' be the automorphism of P^1 induced by alpha. Let n be the order of alpha and let n' be the order of alpha'. We show that the triple (g,n,n') completely determines the characteristic polynomial of the automorphism alpha^* of the Jacobian of C, unless n is even, n=n', and (2g+2)/n is even, in which case there are two possibilities. We give explicit formulas for the characteristic polynomial in all cases.