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We study certain symmetric polynomials, which as very special cases includepolynomials related to the supersymmetric eight-vertex model, and otherelliptic lattice models with $\Delta=\pm 1/2$. In this paper, which is thefirst part of a series, we study the behaviour of the polynomials at specialparameter values, which can be identified with cusps of the modular group$\Gamma_0(12)$. In subsequent papers, we will show that the polynomials satisfya non-stationary Schr\"odinger equation related to theKnizhnik--Zamolodchikov--Bernard equation and that they give a four-dimensionallattice of tau functions of Painlev\'e VI.

Special polynomials related to the supersymmetric eight-vertex model. I. Behaviour at cusps