Gromov-Witten theory of abelian varieties in families and modular forms

Abstract

This is the first paper in a series on the Gromov-Witten theory of the universal abelian variety over the moduli space of principally polarized abelian varieties of dimension $h$. We conjecture that the generating series of Gromov-Witten classes, when summed over the degree against the principal polarization, is a cycle-valued quasimodular form for $\mathrm{SL}_2(\mathbb{Z})$ and satisfy a holomorphic anomaly equation. These conjectures generalize the quasimodularity of the Gromov-Witten theory of elliptic curves to higher dimension and raise interesting questions regarding enumerative mirror symmetry for abelian varieties. In genus $1$ it specializes to a conjecture of Greer and Lian which was proven by Iribar Lopez after tautological projection. We also discuss a special family of abelian varieties with a conjectural relation to Siegel quasimodular forms of higher genus. The main result of the paper is a proof of the conjectures in genus $2$ after tautological projection. For that we introduce quotient Gromov-Witten invariants which are indexed by the characteristic polynomial of the curve class and are shown to determine all descendent Gromov-Witten invariants satisfying a degree conditions. We then give an explicit formula for all genus $2$ quotient invariants after tautological projection as the Shimura lift of the product of two Eisenstein series. The formula is based on a curious modular identity derived in a joint appendix with Brandon Williams.

Disclosure

“ng the SwissMAP workshop Moduli of curves and abelian varieties in Les Diablerets, May 2026, the task to find a proof was submitted as a problem to the AI-benchmark site IMProofBench [44]. With further support of Johannes Schmitt the model ChatGPT 5.5 (using an early version of the harness developed in [45]) provided then a 14-page document claiming a proof. We did not fact check the AI document, but after understanding the proof strategy, we then wrote our own cleaner, but essentia”

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Classification

Drafting a complete proof for author revision

Multiplier

9

Verified

2026-08-23T05:20:29.703933Z

Structural counts