## Hain-Matsumoto, “Galois actions on fundamental groups of curves…”

I recently had occasion to spend some time with Richard Hain and Makoto Matsumoto’s 2005 paper “Galois actions on fundamental groups and the cycle C – C^-,” which I’d always meant to delve into.  It’s really beautiful!  I cannot say I’ve really delved — maybe something more like scratched — but I wanted to share some very interesting things I learned.

Serre proved long ago that the image of the l-adic Galois representation on an elliptic curve E/Q is open in GL_2(Z_l), so long as E doesn’t have CM.  This is a geometric condition on E, which is to say it only depends on the basechange of E to an algebraic closure of Q, or even to C.

What’s the analogue for higher genus curves X?  You might start by asking about the image of the Galois representation G_Q -> GSp_2g(Z_l) attached to the Tate module of the Jacobian of X.  This image lands in GSp_{2g}(Z_l).  Just as with elliptic curves, any extra endomorphisms of Jac(X) may force the image to be much smaller than GSp_{2g}(Z_l).  But the question of whether the image of rho must be open in GSp_2g(Z_l) whenever no “obvious” geometric obstruction forbids it is difficult, and still not completely understood.  (I believe it’s still unknown when g is a multiple of 4…?)  One thing we do know in general, though, is that when X is the generic curve of genus g (that is, the universal curve over the function field Q(M_g) of M_g) the resulting representation

$\rho^{univ}: G_{Q(M_g)} \rightarrow GSp_{2g}(\mathbf{Z}_\ell)$

is surjective.

Hain and Matsumoto generalize in a different direction.  When X is a curve of genus greater than 1 over a field K, the Galois group of K acts on more than just the Tate modules (or l-adic H_1) of X; it acts on the whole pro-l geometric fundamental group of X, which we denote pi.  So we get a morphism

$\rho_{X/K}: G_K \rightarrow Aut(\pi)$

What does it mean to ask this representation to have “big image”?

## “Every curve is a Teichmuller curve,” or “Why SL_2(Z) has the congruence subgroup property.”

Teichmüller curve in M_g, the moduli space of genus-g curves, is an algebraic curve V in M_g such that the inclusion V -> M_g induces an isometry between the constant-curvature metric on V and the restriction of the Teichmüller metric on M_g.

Alternatively:  the cotangent bundle of M_g, considered as a real manifold, admits a natural action of SL_2(R); the orbits are all copies of SL_2(R) / SO(2), or the upper half-plane.  Most of the time, when you project that hyperbolic plane H down to M_g, you get a dense orbit that wanders all over M_g.  But every once in a while, the fibers of the map H -> M_g are a lattice in H, and the image is actually an algebraic curve; that, again, is a Teichmüller curve.

Teichmüller curves are the subject of lots of recent research; for now, let me just say that they are interestingly canonical curves inside M_g.  Matt Bainbridge proved strong results about their intersection numbers in Hilbert modular surfaces.  McMullen classified Teichmuller curves in M_2, giving a very nice algebraic description of the 1-parameter families of genus-2 curves parametrized by Teichmüller curves.  (As far as I know, there’s no such description in higher genus.)  In a recent note, McMullen proved that they are all defined over number fields.

This leads one to ask:  which curves defined over algebraic number fields are Teichmüller curves?  This is the subject of a paper Ben McReynolds and I just posted to the arXiv, “Every curve is a Teichmüller curve.”  The title should be read birationally; what we prove is that every curve X over an algebraic number field is birational (over C) to a Teichmüller curve in some M_g.  (In the posted version, we prove the slightly weaker statement that X is birational to a Teichmüller curve in M_{g,n}), but we’ve since tweaked the argument to get the closed-surface version.)

So why does SL_2(Z) have the congruence subgroup property?  Especially given that it, y’know, doesn’t?

Here’s what I mean.  Let Gamma_{g,n} be the mapping class group of a genus-g surface with n punctures.  Then Gamma_{g,n} acts as a group of outer automorphisms of the fundamental group pi_{g,n} of the surface; and from this, you get an action of Gamma_{g,n} on the finite set

Hom(pi_{g,n},G)/~

where G is a finite group and ~ is conjugacy.

By a congruence subgroup of Gamma_{g,n} let’s mean a stabilizer in this action.  Why this definition?  Well, when g = 1, n = 0, and G = Z/NZ, the stabilizer is just the standard congruence subgroup Gamma_0(N).  And you can easily check that the class of congruence subgroups of Gamma_{1,0} is cofinal with the usual class of congruence subgroups in SL_2(Z).

Now Gamma_{1,1} is also isomorphic to SL_2(Z), but the notion of “congruence subgroup of SL_2(Z)” afforded by this isomorphism is much more general than the usual one.  So much so that one gets the following, which is really the main point of my paper with Ben:

Every finite-index subgroup of Gamma_{1,1} containing the center and contained in Gamma(2) is a congruence subgroup.

It turns out that the finite covers of the moduli space M_{1,1} corresponding to such finite-index subgroups are always Teichmüller curves; since, by Belyi’s theorem, every curve over a number field can be so expressed, we get the desired result.

The italicized assertion above can be thought of as a very strong kind of “congruence subgroup property.”  Of course, CSP usually refers to the property that every finite-index subgroup contains a principal congruence subgroup.  That finite-index subgroups Gamma_{1,1} (and even Gamma_{1,n}) always contain congruence subgroups as defined above is a theorem of Asada, and it’s conjectured to be true for all g,n.  But the statement that every finite-index subgroup of a mapping class group is a congruence subgroup on the nose is substantially stronger, and I imagine it’s true only for (1,1) and the closely related case (0,4), which was proved, in somewhat different language, in the paper “Every curve is a Hurwitz space,” by Diaz, Donagi, and Harbater.  Our argument is very much inspired by theirs — it was to emphasize this debt that we gave our paper more or less the same title.

## Positive motivic measures are counting measures

A new, very short paper with Michael Larsen, “Positive motivic measures are counting measures” is up on the arXiv today.  I thought I’d say a bit here about where the problem came from, since we don’t do so in the paper.

In the project with Akshay that I talked about at the recent Columbia conference on rational curves on varieties, one thing you do is compute estimates for |M_n(F_q)|, where M_n is the moduli space of algebraic maps of degree n from P^1 to some fixed target variety X, and F_q is a finite field.  These inequalities turn out to be very nicely uniform in q, which leads one naturally to ask; do the proofs actually give “motivic estimates” for the class [M_n] in the Grothendieck ring K_0(Var_K), for various non-finite fields K?

Well, what does it mean for one element r of a ring R to “estimate” another element s?  It might mean that r-s is rather deep in some natural filtration on R.  Those don’t seem to be the kind of estimates our methods provide; rather, they say something more like

(r-s)^2 <= B

where B is some fixed element of K_0(Var_K).  But what does “<=” mean?  Well, it means that B – (r-s)^2 is nonnegative.  And what does “nonnegative” mean?  That’s the question.  What the proof really gives is that B – (r-s)^2 lies in a certain semiring N of “nonnegative motives” in K_0(Var_K).  Let’s not be too precise about what N is; let’s just say that it includes [V] for every variety V, and it has the property that |n(F_q)| >=0 for all q, whenever n lies in N.  In particular, that means that

(|r(F_q)| – |s(F_q)|)^2 <= B(F_q)

so that, on the level of counting points, s(F_q) really is a good estimate for r(F_q).

So one might ask:  are there other interesting positive motivic measures — that is, homomorphisms

f: K_0(Var_K) -> reals

which take N to nonnegative reals?  If so, f(s) would be a good estimate for f(r).

And the point of this note with Larsen is to say, with some regret, no — any motivic measure which assigns nonnegative values to the classes of varieties is in fact just counting points over some finite field.  Which sort of kills in its crib the initial hope of some exciting world of “motivic inequalities.”

Of course, the reals are not the only ordered ring!  As Bjorn Poonen pointed out to me, for a general field K you can find an ordered ring A and a measure

g: K_0(Var_K) -> A

which is positive in the sense that g sends every variety to an element of A greater than or equal to 0; these come from big ultraproducts of counting measures of different finite fields.  Whether these measures are “interesting” I’m not sure.