Explicit computations of Serre’s obstruction for genus-3 curves and application to optimal curves

Let k be a field of characteristic other than 2. There can be an obstruction to a principally polarized abelian threefold (A,a) over k, which is a Jacobian over $\bar {k}$ , being a Jacobian over k; this can be computed in terms of the rationality of the square root of the value of a certain Siegel modular form. We show how to do this explicitly for principally polarized abelian threefolds which are the third power of an elliptic curve with complex multiplication. We use our numerical results to prove or refute the existence of some optimal curves of genus 3.