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

Modularity of counting functions of convex planar polygons with rationality conditions