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

Everett W. Howe: Curves of genus two with maps of every degree to a fixed elliptic curve, to appear in the proceedings of ANTS XVII.

Available as a preprint.

We show that up to isomorphism there are exactly twenty pairs (C, E), where C is a genus-2 curve over the complex numbers C, where E is an elliptic curve over 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_n², where n ranges from 2 to 59, is empty. We also produce a (very large) integer N such that the intersection of the Humbert surfaces H_n², for n in the set {2, 3, 13, N}, is empty.

(The latter statements means that no genus-2 curve has minimal maps of every degree n from 2 to 59 to an elliptic curve E, even if you let the elliptic curve vary with n; and similarly, if you let n range through the values {2, 3, 13, N}. Here, minimal means that the map does not factor through an isogeny F → E.)