Polynomials Defining Teichmüller Curves and Their Factorizations mod <i>p</i>

We investigate the arithmetic nature of Teichmüller curves in the spirit of the theory developed for classical modular curves. We focus on the Weierstrass curves for square discriminants introduced in [McMullen 05] which give arithmetic Teichmüller curves in genus two. For these curves, we introduce an analog of the classical modular polynomials. These Weierstrass polynomials have integer coefficients and give defining equations for Weierstrass curves. We formulate, and provide evidence towards, several conjectures concerning the factorizations of Weierstrass polynomials in positive characteristic similar to the factorizations for modular polynomials discovered by Kronecker and Deuring.