Curves of genus $2$ with split Jacobian

Abstract:

We say that an algebraic curve has split jacobian if its jacobian is isogenous to a product of elliptic curves. If $X$ is a curve of genus $2$, and $f:X \to E$ a map from $X$ to an elliptic curve, then $X$ has split jacobian. It is not true that a complement to $E$ in the jacobian of $X$ is uniquely determined, but, under certain conditions, there is a canonical choice of elliptic curve $E’$ and algebraic $f:X \to E’$, and we give an algorithm for finding that curve. The construction works in any characteristic other than two. Applications of the algorithm are given to give explicit examples in characteristics $0$ and $3$.