# Abelian $\ell$-adic representations in $\widehat{\mathrm{SL}(2,\mathbb{Z})}$

$$\DeclareMathOperator\SL{SL}\DeclareMathOperator\Gal{Gal}\newcommand{\Z}{\mathbb{Z}}$$In Grothendieck's Esquisse he claims that the action of $$\Gal(\mathbb Q)\to\text{Out}(\pi_1(M_{1,1})=\text{Out}(\widehat{\SL(2,\Z)})$$ obtained from the homotopy exact sequence of the étale fundamental group contains, after abelianization, all abelian $$\ell$$-adic representations defined by Jacobians and generalised Jacobians of algebraic curves defined over number fields. My questions (which might be hopelessly broad and misguided) are the following:

1. How can one show this?
2. Has any work been done extending this idea? A list of references is most welcome.

The original text is the following paragraph:

...from the point of view of Galois-Teichmüller theory, $$\SL(2,\Z)$$ can be considered as the fundamental “building block” of the "Teichmüller tower". The element of the structure of $$\SL(2,\Z)$$ which fascinates me above all is of course the outer action of $$\Gal(\mathbb Q)$$ on its profinite compactification. By Bielyi’s theorem, taking the profinite compactifications of subgroups of finite index of $$\SL(2,\Z)$$, and the induced outer action (up to also passing to an open subgroup of $$\Gal(\mathbb Q)$$), we essentially find the fundamental groups of all algebraic curves (not necessarily compact) defined over number fields $$K$$, and the outer action of $$\Gal(K)$$ on them – at least it is true that every such fundamental group appears as a quotient of one of the first groups (*). Taking the “anabelian yoga” (which remains conjectural) into account, which says that an anabelian algebraic curve over a number field $$K$$ (finite extension of $$\mathbb Q$$) is known up to isomorphism when we know its mixed fundamental group (or what comes to the same thing, the outer action of $$\Gal(K)$$ on its profinite geometric fundamental group), we can thus say that all algebraic curves defined over number fields are “contained” in the profinite compactification $$\widehat{\SL(2, \Z)}$$, and in the knowledge of a certain subgroup $$\Gal(\mathbb Q)$$ of its group of outer automorphisms! Passing to the abelianisations of the preceding fundamental groups, we see in particular that all the abelian $$\ell$$-adic representations dear to Tate and his circle, defined by Jacobians and generalised Jacobians of algebraic curves defined over number fields, are contained in this single action of $$\Gal(\mathbb Q)$$ on the anabelian profinite group $$\widehat{\SL(2, \Z)}$$!

What Grothendieck is doing to get all the Galois representations is:

1) Take a finite index subgroup of $\pi_1(\mathcal M_{1,1})$ that is stable by the $\operatorname{Gal}(\overline{\mathbb Q}|\mathbb Q)$ action.

2) Take the maximal pro-$\ell$ quotient of its abelianization.

3) Look at the action of $\operatorname{Gal}(\overline{\mathbb Q}|\mathbb Q)$ on this $\mathbb Z_\ell$ module.

The key points are, in step 1, that every such subgroup defines an algebraic curve over $\mathbb Q$, in step 2, that the maximal pro-$\ell$ quotient of the subgroup is the Tate module of its Jacobian, and in step 3, that the action of $\operatorname{Gal}(\overline{\mathbb Q}|\mathbb Q)$ by conjugation is the same action defined by Tate.

This all follows from the basic categorical theory of the \'{e}tale fundamental group as worked out by Grothendieck.

The only missing step is that every algebraic curve defined over $\mathbb Q$ arises this way. This is Belyi's theorem.

In terms of your second question, here are some things that pop into my head. First, the fact that abelianizations of finite index subgroups of $\pi_1$ are homology of covers has been crucial in all of the progress on Grothendieck's anabelian conjecture for curves (that a hyperbolic curve is determined by its fundamental group) http://www.math.sci.osaka-u.ac.jp/~nakamura/zoo/rhino/NTM300.pdf. In addition, nilpotent quotients of fundamental groups have been connected to homology in works like Deligne's three points on $\mathbb P^1$ paper and the work coming out of that.

However, I don't know any work that has been successful in using Grothendieck's observation to prove something deep about the Tate modules of Jacobians of curves. The information we get about them from other sources seems much easier to use.