Jacobians among Abelian threefolds: A formula of Klein and a question of Serre

Let $(A,a)$ be an indecomposable principally polarized abelian threefold defined over a field $k \subset \CC$. Using a certain geometric Siegel modular form $\chi_{18}$ on the corresponding moduli space, we prove that $(A,a)$ is a Jacobian over $k$ if and only if $\chi_{18}(A, a)$ is a square over $k$. This answers a question of J.-P. Serre. Then, via a natural isomorphism between invariants of ternary quartics and Teichmüller modular forms of genus $3$, we obtain a simple proof of Klein formula, which asserts that $\chi_{18}(\Jac C, j)$ is equal to the square of the discriminant of $C$.

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Published 1 January 2010