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.

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In this paper, we determine a primitive Teichmu ller modular form of degree g 3 over Z obtained from dividing the product of even theta constants by a certain integer, and we study this root as a Teichmu ller modular form over Q. 1996Academic Press, Inc.

Using the moduli theory of abelian varieties and a recent result of Böcherer-Nagaoka on lifting of the generalized Hasse invariant, we show congruences between the weights of Siegel modular forms with congruent Fourier expansions. This result implies that the weights of p-adic Siegel modular forms are well defined.