Non-isogenous elliptic curves and hyperelliptic jacobians II

TL;DR

This paper investigates conditions under which hyperelliptic Jacobians are isogenous, revealing a link between their endomorphism algebras and roots of unity when one polynomial is irreducible and the other splits.

Contribution

It establishes a new criterion connecting isogenies of Jacobians to the presence of roots of unity in their endomorphism algebras, specifically when one polynomial is irreducible and the other splits.

Findings

If Jacobians are isogenous, then their endomorphism algebras contain a subfield of roots of unity.

The prime dividing the degree of the polynomial relates to the structure of the endomorphism algebra.

The result applies to hyperelliptic curves with specific polynomial factorization properties.

Abstract

Let $K$ be a field of characteristic different from $2$, $\bar{K}$ its algebraic closure. Let $n \ge 3$ be an odd integer. Let $f(x)$ and $h(x)$ be degree $n$ polynomials with coefficients in $K$ and without repeated roots. Let us consider genus $(n-1)/2$ hyperelliptic curves $C_f: y^2=f(x)$ and $C_h: y^2=h(x)$, and their jacobians $J(C_f)$ and $J(C_h)$, which are $(n-1)/2$-dimensional abelian varieties defined over $K$. Suppose that one of the polynomials is irreducible and the other splits completely over $K$. We prove that if $J(C_f)$ and $J(C_h)$ are isogenous over $\bar{K}$ then there is an (odd) prime $\ell$ dividing $n$ such that the endomorphism algebras of both $J(C_f)$ and $J(C_h)$ contain a subfield that is isomorphic to the field of $\ell$th roots of $1$.