Families of absolutely simple hyperelliptic jacobians

TL;DR

This paper proves that certain hyperelliptic Jacobians have only trivial endomorphisms over algebraic closures, under conditions on the Galois group of the defining polynomial and the degree of the polynomial.

Contribution

It establishes new conditions ensuring the absolute simplicity of hyperelliptic Jacobians based on Galois group properties and polynomial degree.

Findings

Jacobian has no nontrivial endomorphisms under specified conditions

Results apply to even degree polynomials greater than 8

Extends previous work to new cases of polynomial degrees

Abstract

We prove that the jacobian of a hyperelliptic curve $y^2=(x-t)h(x)$ has no nontrivial endomorphisms over an algebraic closure of the ground field $K$ of characteristic zero if $t \in K$ and the Galois group of the polynomial $h(x)$ over $K$ is "very big" and $deg(h)$ is an even number >8. (The case of odd $deg(h)>3$ follows easily from previous results of the author.)