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

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

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

Well, just as in the Tate module story above, we can ask about the image of rho when K is the function field of M_g and X is the universal curve. Denote the image of this Galois representation in Out(pi) by G.

Then you might ask: for a curve X over a number field K, can we give a condition guaranteeing that the image of rho_{X/K} is open in G?

The theorem of Hain and Matsumoto doesn’t quite answer that question, but it makes an impressive stride in its direction. For each positive integer m, let L_m be the mth term of the lower central series of pi (so that L^2 = [pi,pi], L^3 =[pi,[pi,pi]] and so on.) Let G_m be the image of G in Out(pi/L^{m+1}). So G_1, for instance, is just GSp_{2g}(Z_l). Now you can ask the following “big image” question for each m:

(Q): Is the image of rho_{X/K} in Out(pi/L^{m+1}) open in G_m?

Now we’ve already remarked that the question is subtle for G_1. So let’s assume that (Q) has an affirmative answer for m=1. We also take g(X) >= 3. Then Hain and Matsumoto prove:

[HM, Th 9.1 a-b]: (Q) has a positive answer for m=2 if and only if it has a positive answer for all m >=2.

You might think of this as a non-abelian analogue of the theorem (for which see [IV, 3.4, Lemma 3] of Serre’s book “Abelian l-adic Representations and Elliptic Curves”) that an l-adic Galois representation is surjective if and only if it is surjective mod l^k, for some k depending only on the target group.

But wait, there’s more! One might ask whether, as in the case of elliptic curves, there’s a geometric condition distinguishing those X where (Q) holds from those where it doesn’t. It’s not hard to check, by the way, that (Q) isn’t always the case! Once again, extra endomorphisms can force the image of Galois can be smaller; for instance, when X is a hyperelliptic curve the image of rho_{X/K} will always be smaller than the generic image. (Did you remember I said the genus of X was at least 3…?)

Now we come to the main theme of Hain and Matsumoto’s paper. Choose a basepoint x on X. Then the Abel-Jacobi map associated to X gives you an embedding of X in the abelian g-fold Jac(X). Consider the algebraic cycle Z = [X] – [-1]^*[X], where [-1] is multiplication by -1 on Jac(X). The cohomology class of Z in H^{2g-2}(J(X),Z_l(g-1)) is trivial; thus, Z has a well-defined cycle class in H^1(G_K, H^{2g-3}(J(X),Z_l(g-1)). We move this class over to H^1(G_K, H^3(J(X),Z_l(2))) by Poincare duality. Now H^3(J(X)) is just wedge^3 H^1(X), and we can put a copy of H^1(X) inside H^3(J(X)) by wedging it with the symplectic form in wedge^2 H^1(X). The quotient of H^3(J(X)) by this copy of H^1(X) is an l-adic Galois module we call V, which is pure of weight 1.

We can project our cohomology class down to V to get a class nu in H^1(G_K, V), which turns out to be independent of the choice of basepoint. Now the theorem of Hain and Matsumoto says:

[H-M, Th 9.2 c] (Q) has a positive answer for all m >= 2 if and only if it has a positive answer for m = 1 and the class nu has infinite order.

Note that this is indeed a geometric condition — at least in the sense that if it holds after replacing K with a finite extension, it holds for K. Unlike “non-CM,” though, it’s not clear to make sense of it for a curve with transcendental moduli; just one more indication that you should really think of such an object, not as a curve, but as a family of curves over number fields, the dimension of the family being the transcendence degree of the moduli. (In this sense, maybe nu should be thought of as always infinite-order when the moduli of X are transcendental, just as an elliptic curve with transcendental j-invariant is automatically non-CM…?)

I’ll leave it as an exercise to check that nu vanishes when X is hyperelliptic.

One question remains: what made me go back to Hain and Matsumoto’s paper? It was the beautiful lecture series Shou-Wu Zhang gave here at Madison last month about his work on the Arakelov-theoretic properties of the Gross-Schoen cycle. And the Gross-Schoen cycle, as Hain and Matsumoto point out (Remark 1.1 d) is just three times nu! If time permits I’ll blog about his talks, too; but maybe there’s no need, since you can always read Terry Tao’s take on a similar lecture series at UCLA.

Update:Chris Hall reminds me that in fact Serre’s open image theorem is known only when g is odd, 2, or 6; so it’s not just the g divisible by 4 that pose a problem. Chris can show that the image of the ell-adic Galois representation is large in more general dimensions under reasonable conditions: no extra endomorphisms and at least one bad fiber with toric dimension 1. Here’s the paper.