Do higher genus Teichmuller modular forms have, or are they expected to have, implications for number theory that generalize the sorts of results that flow from the study of classical modular forms?



Teichmuller modular forms and number theory
@David Hansen: Teichmuller modular forms are basically the natural analogue of Siegel modular forms when one considers sections of line bundles on $M\_g$ instead of $A\_g$. Search for papers by Ichikawa.


I don't know very much about Teichmuller modular forms but my advisor knows a bit about them and from what I can tell from conversations with him, pretty much nothing at all is known about this question. I can think of some plausible reasons why this is so. (This answer is probably a bit naive, my apologies.)


One is that the difference between Siegel modular forms and Teichmuller modular forms does not become visible until genus three. For genus one and two the map from $M\_g$ to $A\_g$ is an isomorphism and an open immersion respectively, and all Teichmuller modular forms are just pullbacks of forms from $A\_1$ and $A\_2$. In genus three the Torelli map is an open immersion on coarse moduli spaces but on the level of stacks it is a ramified double cover, which makes it plausible that there should be more functions on $M\_3$ than on $A\_3$. And indeed the ring of Teichmuller modular forms in genus three is obtained from the ring of Siegel modular forms by adjoining a square root of $\chi\_{18}$, which vanishes exactly at the hyperelliptic locus i.e. the branch locus of the Torelli map. Anyway, the point is that already Siegel modular forms in higher genera are not so well understood compared to the rich theory we have in low genus. In particular we know extremely few examples of genuine Teichmuller modular forms.


Another more serious reason is that it is not clear how any of the standard tools for studying ordinary modular forms can be applied to the Teichmuller case. As a very basic example, it is for instance completely open if or how one can define a notion of Hecke operators acting on Teichmuller modular forms: double cosets in the mapping class group are a scary prospect, and there seems to be no analogue of the standard Hecke correspondences in terms of cyclic subgroups of order p. What's worse is that none of the standard automorphic/Langlands etc tools seem to be applicable to the study of Teichmuller modular forms. Unlike Siegel's upper half space, Teichmuller space is not a symmetric space. If we believe in the Langlands philosophy then there should be a reductive group somewhere that the Teichmuller modular forms "come from", but there is no natural reductive group anywhere in sight. 


Anyway, to actually answer your question: there probably is some connection between Teichmuller modular forms and number theory. In particular it seems that there are $\ell$-adic Galois representations naturally attached to Teichmuller modular forms. There are just no tools to study these representations.


