Prym varieties of \'etale covers of hyperelliptic curves

TL;DR

This paper investigates Prym varieties of étale covers of hyperelliptic curves, focusing on the degree of the canonical isogeny and identifying cases where it is an isomorphism, especially for degree 4 covers.

Contribution

It computes the degree of the canonical isogeny for all cases and establishes that only degree 4 covers yield an isomorphism.

Findings

The isogeny degree varies depending on the cover degree.

Only degree 4 covers produce an isomorphism.

Provides explicit calculations for the remaining cases.

Abstract

It is well known that the Prym variety of an \'etale cyclic covering of a hyperelliptic curve is isogenous to the product of two Jacobians. Moreover, if the degree of the covering is odd or congruent to 2 mod 4, then the canonical isogeny is an isomorphism. We compute the degree of this isogeny in the remaining cases and show that only in the case of coverings of degree 4 it is an isomorphism.