Narrow Results

J.-P. Serre asserted a precise form of Torelli Theorem for genus 3 curves, namely, an indecomposable principally polarized abelian threefold is a Jacobian if and only if some specific invariant is a square. We study here a three dimensional family of such threefolds, introduced by Howe, Leprevost and Poonen. By a new formulation, we link their results to Serre's assersion. Then, we recover a formula of Klein related to the question for complex threefolds. In this case the invariant is a modular form of weight 18, and the result is proved using theta functions identities.

Special arithmetic series $f(x)={\displaystyle {\sum }_{n=0}^{\infty }{c}_n{x}^n}$, whose coefficients c_n are normally given as certain binomial sums, satisfy “self-replicating” functional identities. For example, the equation