Is the moduli space of curves defined over the field with one element?

Question

There are various frameworks around which enlarge the category of rings to include more exotic objects such as the 'field with one element,' $\mathbb{F}_1$. While these frameworks differ in their details, there are certain things this should be true of any object that deserves to be called $\mathbb{F}_1$. For example, The algebraic K-theory of $\mathbb{F}_1$ should be sphere spectrum, and the theory of toric varieties should be defined over $\mathbb{F}_1$.

Question 1: Is there a moral reason why the moduli space of curves should (or should not) be defined over Spec $\mathbb{F}_1$?

EDIT: For anyone who would like to be more concrete, I'm happy to take the Toen-Vaquie definition of schemes over $\mathbb{F}_1$. (see arXiv:math/0509684). In this setup (and most of the other frameworks I know) an affine scheme over $\mathbb{F}_1$ is just a commutative monoid $M$. After base change to $\mathbb{Z}$ this becomes the monoid ring $\mathbb{Z}[M]$. So here is a more precise question:

Question 2: Does the moduli space of curves $\mathcal{M}_{g,n}$ (over $\mathbb{Z}$, say) admit a covering by affine charts of the form spec $\mathbb{Z}[M_i]$ for commutative monoids $M_i$? If so, can this covering be chosen so that (as in the case of toric varieties) the gluing is entirely determined by maps of monoids?

Answer 1

This "answer" will basically restate the comments of Marty and Jason Starr.

Any variety covered by schemes of the form $\mathrm{Spec}(\mathbf Z[M_i])$, or any torified variety, is rational. And indeed Severi conjectured at one point that $M_g$ is rational for any $g$! But we know a lot about the Kodaira dimension of $M_g$ by work of Harris--Mumford, Farkas, Eisenbud, Verra, ... in particular we know that Severi's conjecture is maximally false. We have $\kappa(M_g) = -\infty$ for $g \leq 16$, we don't know anything for $17 \leq g \leq 21$, and $\kappa(M_g) \geq 0$ for $g \geq 22$. In fact we know that $M_{g}$ is of general type for $g = 22$ and $g \geq 24$. But if one only wants an example of a $g$ for which $M_g$ is not rational, then I think one can find examples much earlier (maybe $g \approx 6,7$?).

See http://arxiv.org/abs/0810.0702 for a survey by Farkas.

Moreover, the cohomology of a torified scheme can only contain mixed Tate motives. In particular, this implies properties like: the number of $\mathbf F_q$-points is a polynomial function of $q$. As Marty says there are no problems in genus zero; these might be honest-to-God $\mathbf F_1$ schemes. The cohomology of $M_{1,n}$ contains only mixed Tate motives when $n \leq 10$, but for $n \geq 11$ one finds motives associated to cusp forms for $\mathrm{SL}(2,\mathbf Z)$. (The number 11 arises as one less than the smallest weight of a nonzero cusp form; the discriminant form $\Delta$.) This implies that the polynomiality behaviour changes drastically -- the number of $\mathbf F_q$-points is now given by Fourier coefficients of modular forms, which are far more complicated and contain lots of arithmetic information.

The connection with birational geometry is also very visible here. The cohomology classes on $M_{1,n}$ associated to the cusp forms of weight $n+1$ are of type $(n,0)$ and $(0,n)$ in the Hodge realization, since given such a cusp form one can explicitly write down a corresponding differential form in coordinates. So by the very definition of Kodaira dimension we can not have $\kappa = -\infty$ anymore.

Bergström computed the Euler characteristic of $M_{2,n}$ in the Grothendieck group of $\ell$-adic Galois representations by point counting techniques. That is, he found formulas for the number of $\mathbf F_q$-points of $M_{2,n}$ by working really explicitly with normal forms for hyperelliptic curves. These formulas turned out to be polynomial in $q$ for $n \leq 7$ (and conjecturally for $n \leq 9$), just as they would be if we knew that the cohomology of $M_{2,n}$ contained only mixed Tate motives. By thinking a bit about the stratification of topological type, one concludes that this holds also for $\overline M_{2,n}$ when $n \leq 7$, which is smooth and proper over the integers. Then a theorem of van den Bogaart and Edixhoven implies that the cohomology of $\overline M_{2,n}$ is all of Tate type and with Betti numbers given by the coefficients of the polynomials.

There are similar results by Bergström for $M_{3,n}$ when $n \leq 5$ and Bergström--Tommasi for $M_4$. But the general phenomenon is that increasing either $g$ or $n$ will rapidly take you out of the world of $\mathbf F_1$-schemes, at least if this is taken to mean "commutative monoids".

However, I don't know enough $\mathbf F_1$--geometry to say what the answer is if one takes $\mathbf F_1$-schemes in the sense of Borger. The first nontrivial case to answer would be: are the motives associated to cusp forms for $\mathrm{SL}(2,\mathbf Z)$ defined over $\mathbf F_1$ in his set-up? To be clear, I don't believe that this is true, but I know almost nothing about $\lambda$-schemes.

Let me also make two small remarks: (i) Everything I have written above is necessary conditions for being defined over $\mathbf F_1$. (ii) The fact that $M_{g,n}$ is smooth over the integers says that the cohomology of $M_{g,n}$ is at least quite "special", even though it is not defined over $\mathbf F_1$. Smoothness is a strong restriction on the Galois representations that can occur in the cohomology.

Answer 2

Here a few remarks from the $\Lambda$ point of view on $\mathbf{F}_1$. (There we say a scheme is defined over $\mathbf{F}_1$ if it admits a $\Lambda$-structure. If the scheme is flat over $\mathbf{Z}$, a $\Lambda$-structure is equivalent to a commuting family of endomorphisms $\psi_p$, one for each prime number $p$, such that each $\psi_p$ agrees with the Frobenius map modulo $p$.)

I prove in http://arxiv.org/abs/0906.3146 that if a scheme of finite type over $\mathbf{Z}$ is defined over $\mathbf{F}_1$, then its motive is pure Tate (or rather becomes so after base change to some cyclotomic field - it's actually false in general without this qualification). So as in Dan Petersen's answer, $M_{g,n}$ won't admit a $\Lambda$-structure unless the pair $(g,n)$ is sufficiently small.

I haven't really thought about the small cases, except just a bit when $(g,n)=(1,1)$. Then $M_{g,n}$ is just the affine line, so it has many $\Lambda$-structures. As far as I know, you can't really say that they have any meaning, but I suspect there is still something interesting to say.

1) There is a $\Lambda$-structure on the completion of $\bar{M}_{1,1}$ at the point at infinity which does have a meaning. The completion is $\mathbf{Z}[[q-1]]$, and the $\Lambda$-structure is defined by $\psi_p(q)=q^p$, for all $p$. The interest in this is that this $\Lambda$-structure prolongs to a $\Lambda$-structure on the universal generalized elliptic curve over $\mathbf{Z}[[q-1]]$ (i.e. the Tate curve). From what I remember, the idea is that a point $z\in\mathbf{C}^*/q^{\mathbf{Z}}$ should be mapped to its image $z^p\in\mathbf{C}^*/q^{p\mathbf{Z}}$. Making this rigorous is not hard, though of course you have to have an actual definition of the Tate curve over $\mathbf{Z}[[q-1]]$, a point that is often glided over.

What I think could be interesting is to study the extent to which these $\Lambda$-structures extend to the whole moduli space. Here is one possible interpretation. A $\Lambda$-structure on a scheme $X$ is equivalent to a section of the canonical projection $W_*(X)\to X$ satisfying a certain associativity property. ($W_*$ is the right adjoint to the Witt vector functor $W^*$. It is an arithmetic analogue of a jet space.) Now consider the map $T\to X$, where $T$ is the Tate curve above and $X$ is the total space of the universal elliptic curve. (This is a stack. One would have to check that all these $\Lambda$ and $W$ concepts make sense for stacks.) Then the projection $W_*(X)\to X$ has a natural section over $T$. Then we can ask: what is the Zariski closure $Z$ of this section in $W_*(X)$? The extent to which $Z$ fails to be a section of $W_*(X)\to X$ should be some measure of the extent to which the $\Lambda$-structure on $T$ fails to extend to one on $X$.

2) One could try to do something similar at CM points, instead of at the boundary of the moduli space. CM elliptic curves admit a certain generalized version of a $\Lambda$-structure. (See the end of my paper cited above.) Do these generalized $\Lambda$-structures extend to the formal neighborhood of the elliptic curve in the universal elliptic curve? If so, one could look at closures of formal sections (as above) to see the extent to which they fail to extend to generalized $\Lambda$-structures on the entire universal elliptic curve. For all this, you'd have to fix an imaginary quadratic field, so all this is probably less fundamental than in 1).

3) Thomas Scanlon tried to convince me a few years ago that the Hecke correspondences have some kind of $\Lambda$ nature. From what I remember, there were certain $\Lambda$-schemes closely related to modular varieties which are not of finite type but which are still finite-dimensional in some model-theoretic sense. I never got around to understanding what he meant (though I always intended to). Perhaps it is related to what I wrote above.

So there are a few pieces of evidence that something is going on with $\Lambda$-structures and modular curves, though it remains to be seen if it's really an identifiable phenomenon and, if so, how interesting it is.