Families of explicitly isogenous Jacobians of variable-separated curves

TL;DR

This paper constructs explicit infinite families of high-genus curve pairs over number fields with Jacobians connected by isogenies of small degrees, analyzing their kernels and moduli space dimensions.

Contribution

It introduces new explicit families of isogenous Jacobians of variable-separated curves, extending previous classifications and providing detailed kernel and moduli space analyses.

Findings

Six infinite series of curve pairs constructed

Explicit isogenies with kernels and dimensions computed

Families derived from polynomial reducibility classification

Abstract

We construct six infinite series of families of pairs of curves (X,Y) of arbitrarily high genus, defined over number fields, together with an explicit isogeny from the Jacobian of X to the Jacobian of Y splitting multiplication by 2, 3, or 4. For each family, we compute the isomorphism type of the isogeny kernel and the dimension of the image of the family in the appropriate moduli space. The families are derived from Cassou--Nogu\`es and Couveignes' explicit classification of pairs (f,g) of polynomials such that f(x_1) - g(x_2) is reducible.