Modularity of counting functions of convex planar polygons with rationality conditions

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.