We show that up to isomorphism there are exactly twenty pairs $(C, E)$, where $C$ is a genus-2 curve over the complex numbers, where $E$ is an elliptic curve over the complex numbers, and where for every integer $n > 1$ there is a map of degree $n$ from $C$ to $E$. For example, if $C$ is the curve $y^2 = x^5 + 5 x^3 + 5 x$, and if $E$ is an elliptic curve with CM by the order of discriminant -20, then $(C, E)$ is such a pair.

On the other hand, we produce some finite sets $S$ of integers such that if $C$ is a genus-2 curve in characteristic 0, and if for every $n$ in $S$ there is an elliptic curve $E_n$ and a degree-$n$ map $\phi_n$ from $C$ to $E_n$, then for at least one of these $n$, the map $\phi_n$ factors through a nontrivial isogeny.

[pre-talk at 3:00PM]