Families of Explicit Isogenies of Hyperelliptic Jacobians

Abstract

We construct three-dimensional families of hyperelliptic curves of genus 6, 12, and 14, two-dimensional families of hyperelliptic curves of genus 3, 6, 7, 10, 20, and 30, and one-dimensional families of hyperelliptic curves of genus 5, 10 and 15, all of which are equipped with an an explicit isogeny from their Jacobian to another hyperelliptic Jacobian. We show that the Jacobians are generically absolutely simple, and describe the kernels of the isogenies. 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.