Contents Online
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
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 $\overline{K}$.
To the memory of Yuri Ivanovich Manin
The author was partially supported by Simons Foundation Collaboration grant #585711.
Received 7 May 2021
Received revised 7 January 2022
Accepted 15 February 2022
Published 21 June 2023