Mathematical Research Letters

Volume 30 (2023)

Number 1

Non-isogenous elliptic curves and hyperelliptic jacobians

Pages: 267 – 294

DOI: https://dx.doi.org/10.4310/MRL.2023.v30.n1.a11

Author

Yuri G. Zarhin (Department of Mathematics, Pennsylvania State University, University Park, Penn., U.S.A.; and Max-Planck Institut für Mathematik, Bonn, Germany)

Abstract

Let $K$ be a field of characteristic different from $2$, $\overline{K}$ its algebraic closure. Let $n \geq 3$ be an odd prime such that $2$ is a primitive root modulo $n$. 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 reducible. We prove that if $J(C_f)$ and $J(C_h)$ are isogenous over $\overline{K}$ then both jacobians are abelian varieties of CM type with multiplication by the field of $n$th roots of $1$.

We also discuss the case when both polynomials are irreducible while their splitting fields are linearly disjoint. In particular, we prove that if $\operatorname{char}(K)=0$, the Galois group of one of the polynomials is doubly transitive and the Galois group of the other is a cyclic group of order $n$, then $J(C_f)$ and $J(C_h)$) are not isogenous over $\overline{K}$.

Received 7 May 2021

Received revised 7 January 2022

Accepted 15 February 2022

Published 21 June 2023