Jacobians among abelian threefolds: A formula of klein and a question of serre

In this paper we give a criterion when an indecomposable principally polarized abelian threefold (A, a) defined over a field k = ℂ is a Jacobian over k. More precisely, (A, a) is a Jacobian over k if and only if the value of a certain geometric Siegel modular form χ18(A, a) is a square over k. This answers a question of J.-P. Serre.