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AbstractWe 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.

ABSTRACTWe describe a method to compute the norm on the cotangent space to the moduli space of Riemann surfaces associated to the Finsler Teichmüller metric. Our method involves computing the periods of abelian double covers and is easy to implement for Riemann surfaces presented as algebraic curves using existing tools for approximating period matrices of plane algebraic curves. We illustrate our method by depicting the unit sphere in the cotangent space to moduli space at a particular surface of genus zero with five punctures and by corroborating the proof of a theorem of Royden’s for our example.