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

TL;DR

The paper develops invariants from Siegel modular forms to identify Jacobians among abelian varieties, providing new proofs and answering Serre's question for genus 3, with potential extensions beyond that.

Contribution

It introduces a method to associate classical invariants to abelian varieties using modular forms, enabling Jacobian detection and linking to classical invariants, and addresses Serre's question for genus 3.

Findings

A new proof of Klein's formula relating 8 to plane quartic discriminant.

A criterion to determine Jacobians over a field by checking if 8 value is a square.

Insights into extending the approach for higher genus g>3.

Abstract

Let k be a field and f be a Siegel modular form of weight h \geq 0 and genus g>1 over k. Using f, we define an invariant of the k-isomorphism class of a principally polarized abelian variety (A,a)/k of dimension g. Moreover when (A,a) is the Jacobian of a smooth plane curve, we show how to associate to f a classical plane invariant. As straightforward consequences of these constructions, when g=3 and k is a subfield of the complex field, we obtain (i) a new proof of a formula of Klein linking the modular form \chi_{18} to the square of the discriminant of plane quartics ; (ii) a proof that one can decide when (A,a) is a Jacobian over k by looking whether the value of \chi_{18} at (A,a) is a square in k. This answers a question of J.-P. Serre. Finally, we study the possible generalizations of this approach for g>3.