Communications in Number Theory and Physics

Volume 8 (2014)

Number 2

Counting hyperelliptic curves on an Abelian surface with quasi-modular forms

Pages: 243 – 293



Simon C. F. Rose (Department of Math and Statistics, Queen’s University. Kingston, Ontario, Canada; and Fields Institute, University of Toronto, Ontario, Canada)


In this paper, we produce a generating function for the number of hyperelliptic curves (up to translation) on a polarized Abelian surface 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.

Full Text (PDF format)