Teichmuller modular forms of degree 3.

Introduction. Teichmuller modular forms (of degree g) are introduced in [Ic2] as global sections of automorphic line bundles on the moduli space of algebraic curves (of genus g). A fundamental problem is to study the structure of the rings of Teichmuller modular forms. By restricting to the jacobian locus, we have the canonical homomorphism from the space of Siegel modular forms of degree g to that of Teichmuller modular forms of degree g. This homomorphism is not injective if g [greater than or equal to] 4, and in [Ic1], the author characterizes Siegel modular forms belonging to this kernel by their Fourier coefficients. Moreover, the above homomorphism is, in general, not surjective. For example, as seen in [Ic2] and as we will see below, there exists a Teichmuller modular form [[micro].sub.3,9] of degree 3 and weight 9 which is not induced from Siegel modular forms. The aim of this paper is to show that all Teichmuller modular forms of degree 3 over fields of characteristic 0 are generated by Siegel modular forms of degree 3 and the above [[micro].sub.3,9]. This fact is mainly due to that the Torelli map in the genus 3 case is the double covering over fields of characteristic [not equal to] 2, and to complete the proof, we need Igusa's result [Ig] on the ring of Siegel modular forms of degree 3 over C.

1. Review of moduli theory. Here we recall a summary of moduli theory on algebraic curves and abelian schemes. Let g and n be positive integers, and let [M.sub.g,n] denote the moduli stack of smooth and proper curves of genus g with symplectic level n structures ([D-M]). Let [pi] : C [right arrow] [M.sub.g,n] be the universal curve, and let [lambda] be the invertible sheaf [[lambda].sup.g][pi]*([[omega].sub.C/Mg,n]) on [M.sub.g,n]. Let [X.sub.g,n] denote the moduli stack of …