Counting Hyperelliptic curves on Abelian surfaces with Quasi-modular forms

TL;DR

This paper derives a generating function for counting hyperelliptic curves on polarized Abelian surfaces, demonstrating that these counts are quasi-modular forms and connecting them to MacMahon's divisor functions.

Contribution

It introduces a novel generating function for hyperelliptic curves on Abelian surfaces and proves their quasi-modularity, linking algebraic geometry with modular form theory.

Findings

Generated explicit formulas using crepant resolution and Yau-Zaslow

Established that the counting functions are quasi-modular forms

Connected curve counting to MacMahon's sum-of-divisors functions

Abstract

In this paper we produce a generating function for the number of hyperelliptic curves (up to translation) on a polarized Abelian surfaces using the crepant resolution conjecture and the Yau-Zaslow formula. We present a formula to compute these in terms of MacMahon's generalized sum-of-divisors functions, and prove that they are quasi-modular forms.