Gauss-Manin connection in disguise: limiting periods and quasi-Jacobi forms

Abstract

We construct a six-dimensional algebraic moduli space of elliptic curves equipped with two ordered marked points, an enhanced relative de Rham frame, and a nonzero tensor of endpoint tangents. Its normalized period map is a biholomorphism. On the classical period locus, the regular coordinates pull back under its inverse to Weierstrass, Eisenstein, and theta functions. The associated Gauss-Manin vector fields realize differential operators on quasi-Jacobi forms of integer index. To relate this elliptic space to the genus-two construction, we study a specified smoothing family over a five-dimensional space of limiting enhancements. For this family, we identify the regularized period matrix and the boundary pullback of the holomorphic part of the logarithmic connection with respect to the chosen plumbing parameter. We then compute the unit relating the algebraic smoothing parameter to the Siegel period parameter and the limits of the specified genus-two generators. Separately, we express classical Fourier-Jacobi coefficients in enhanced elliptic coordinates.