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

摘要

Let (A, α) be an indecomposable principally polarized abelian threefold defined over a field k ? C. Using a certain geometric Siegel modular form X18 on the corresponding moduli space, we prove that (A, a) is a Jacobian over k if and only if X18(A, α) 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 X18(Jac C, j) is equal to the square of the discriminant of C.