Generalized Tate curve and integral Teichmüller modular forms

In this paper, we establish a higher genus version of the Tate (elliptic) curve, i.e., for a given degenerate curve, we construct its universal deformation defined over the ring of formal power series of certain deformation parameters whose coefficients are in a finitely generated z-algebra described by the dual graph of the degenerate curve. Further, we study automorphic forms on the moduli space of algebraic curves (which we call Teichmüller modular forms) by evaluating these forms on the generalized Tate curve. In particular, we show that integral Teichmüller modular forms of given degree make a finitely generated ring, and we describe the structure of this ring in degree 2 and 3 cases.