Modularity of counting functions of convex planar polygons with rationality conditions

TL;DR

This paper investigates the properties of counting functions for convex planar polygons linked to elliptic curves, focusing on their convergence, meromorphicity, and mock modularity under rationality constraints.

Contribution

It introduces new results on the convergence and modular properties of polygon counting functions with specific rationality conditions.

Findings

Counting functions are meromorphic and mock modular.

Quadratic forms related to polygon areas are analyzed for signature and rationality.

Results connect polygon counting with mirror symmetry and modular forms.

Abstract

We study counting functions of planar polygons arising from homological mirror symmetry of elliptic curves. We first analyze the signature and rationality of the quadratic forms corresponding to the signed areas of planar polygons. Then we prove the convergence, meromorphicity, and mock modularity of the counting functions of convex planar polygons satisfying certain rationality conditions on the quadratic forms.