Curves of genus two with maps of every degree to a fixed elliptic curve

TL;DR

This paper classifies genus-2 curves over complex numbers that admit maps of every degree to a fixed elliptic curve, identifying exactly twenty such pairs and analyzing their intersection properties with Humbert surfaces.

Contribution

It provides a complete classification of genus-2 curves with maps of all degrees to a fixed elliptic curve and studies the intersection of specific Humbert surfaces.

Findings

Exactly twenty such pairs (C, E) exist up to isomorphism.

The intersection of Humbert surfaces H_{n^2} for n=2 to 1811 is empty.

Characterization of genus-2 curves with universal degree maps to elliptic curves.

Abstract

We show that up to isomorphism there are exactly twenty pairs $(C,E)$, where $C$ is a genus-$2$ curve over ${\mathbf C}$, where $E$ is an elliptic curve over ${\mathbf C}$, and where for every integer $n>1$ there is a map of degree $n$ from $C$ to $E$. We also show that the intersection of the Humbert surfaces $H_{n^2}$, for $n$ ranging from 2 to 1811, is empty.