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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”?

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