# What automorphic forms are expected to occur in the zeta function of moduli space of curves?

Assume $$g \geq 1$$ and $$n \geq 0$$, the moduli stack $${\mathcal {M}}_{g,n}$$ classifies families of smooth projective curves of genus $$g$$ with $$n$$ marked points , together with their isomorphisms. It has a model over $$\mathbb Z$$ determined by moduli problems, and there are compactification $$\overline {{\mathcal {M}}}_{{g,n}}$$ using stable curves, which is proper over $$\mathbb Z$$. And with enough marked points or higher genus, these models seem to be (essentially) smooth, and one can also talk about coarse moduli spaces.

It's interesting to consider the Hasse-Weil zeta function of moduli space of curves (there is a notion of field valued points on an algebraic stack, although it may not agree with the naive one), so by Langland's philosophy it seems to be an automorphic $$L$$ function. So the question is, for fixed $$g$$ and $$n$$, what kind of irreducible automorphic representations are expected to contribute to this $$L$$ function?

Motivation: the Euler characteristic of the moduli space of curves has something to do with special value of zeta functions, see "The Euler characteristic of the moduli space of curves “ by J. Harer and D. Zagier. So it's natural to expect some arithmetic properties of these moduli spaces.

I can tell you the complete answer for $$g \leq 2$$:

• When $$g=0$$ and $$n \geq 3$$ no nontrivial automorphic forms appear.
• When $$g=1$$ and $$n \geq 1$$ the class of automorphic forms which appear are exactly the cusp forms for $$\mathrm{SL}(2,\mathbf Z)$$. (Deligne, Birch)
• When $$g=2$$ and $$n \geq 0$$ the automorphic forms are precisely the cusp forms for $$\mathrm{SL}(2,\mathbf Z)$$ and the vector-valued Siegel cusp forms for $$\mathrm{Sp}(4,\mathbf Z)$$.

(In none of the cases will it make a difference for the answer above whether you work on the open moduli space or its compactification.) The $$g=2$$ story is spread across several of my papers e.g. arXiv:1310.7369 and arXiv:1310.2508.

One general statement you can make is that since $$\overline M_{g,n}$$ is a smooth proper stack over $$\mathbf Z$$, and $$M_{g,n}$$ is the complement of a normal crossing divisor over $$\mathbf Z$$, there is a strong condition on the automorphic forms involved: they must have "conductor one". Or, the $$\ell$$-adic Galois representations are unramified everywhere and crystalline at $$\ell$$. This is why in $$g=1$$ you only get cusp forms for the full modular group and never a congruence subgroup, and again for $$g=2$$ you only get the full group $$\mathrm{Sp}(4,\mathbf Z)$$.

There has been great progress in the past five years or so on the problem of enumerating automorphic representations of conductor one, by Chenevier, Renard, Taïbi, Lannes; perhaps others should be mentioned here as well. They can now tell you unconditional answers to things like "Up to some given dimension and weight, what is the complete list of automorphic representations of conductor one? What are their Hodge numbers?

There is a long-running project in experimental mathematics by Bergström, Faber and Van der Geer trying to study the automorphic forms appearing for $$g=3$$ and $$n \geq 0$$ by counting points on these moduli spaces over finite fields and interpreting the results. Unfortunately most of this work is still unpublished. They find that the class of automorphic forms they find is strictly larger than the class of Siegel cusp forms for $$\mathrm{Sp}(2g,\mathbf Z)$$ for $$g=1,2,3$$. Some of the "new" Galois representations they find should be related to Ichikawa's Teichmüller modular forms, but I'm not sure that they expect all the Galois representations to be accounted for in this way. This is based on things like them reaching $$n=17$$ (IIRC) where they expected to find for the first time a class in Betti cohomology given by a Teichmüller modular form. And then in the point counts they find some unknown six-dimensional Galois representations for exactly this $$n$$, transforming according to the right representation of the symmetric group $$S_{17}$$. So they expect this to be a six-dimensional "motive" "attached" to this Teichmüller modular form. And then they look in the tables of Chenevier et al and find that for precisely this dimension and weight they should actually find a nontrivial automorphic representation of conductor one (maybe it was a representation of $$\mathrm{SO}(7)$$ in this case?) and its Hodge numbers match up beautifully with various restrictions coming from geometry. In particular, assuming it all matches up they now have computed lots of traces of Frobenius on the Galois representation which should be attached to this automorphic representation which was just abstractly known to exist. And then they have repeated this for higher values of $$n$$. But from what they say they don't seem to find many patterns and I don't think they have any real conjecture of what the general picture should look like.

So already the $$g=3$$ case is a mystery and the higher genus situation even moreso.

• Thank you! That's a very good answer! Before asking the question I only find little related material online. I hope the problem for large $g$ (general type case) can be solved one day.. Jan 24, 2019 at 15:05
• Thank you for the great answer! Do you know if the Galois representations associated to these automorphic representations of $SO(7)$ have been constructed? These genuine Teichmuller forms should live on the quartic locus (complement of the hyperelliptic locus) on $M_3$ which admits an occult period map to a certain unitary Shimura variety (c.f. arxiv.org/abs/1203.1272), do you know if this gives an automorphic description of Teichmuller modular forms? Jan 25, 2019 at 14:36
• That's a nice comment. I don't know what an occult period map is and I haven't heard them talk about it, but I'll look at it. Jan 26, 2019 at 5:46