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

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.

💡 Research Summary

The paper addresses a strikingly strong property for genus‑2 curves over the complex numbers: the existence of a morphism of every positive integer degree n > 1 from a fixed genus‑2 curve C to a fixed elliptic curve E. The author proves that such pairs (C,E) are extremely rare – up to isomorphism there are exactly twenty.

The analysis begins with the classical observation that a degree‑2 map C → E forces the Jacobian Jac(C) to be isogenous to a product of two elliptic curves E and F, with the kernel of the isogeny given by the graph of an isomorphism ψ:E