Families of explicitly isogenous Jacobians of variable-separated curves

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--Noguès and Couveignes' explicit classification of pairs (f,g) of polynomials such that f(x_1) - g(x_2) is reducible.