anabelian geometry

Last time we saw that a curve defined over gives rise to a permutation representation of or one of its subgroups (of index 2) or (of index 6). As the corresponding monodromy group is finite, this representation factors through a normal subgroup of finite index, so it makes sense to look at the profinite completion of , which is the inverse limit of finite groups
where N ranges over all normalsubgroups of finite index. These profinte completions are horrible beasts even for easy groups such as . Its profinite completion is

where the right hand side product of p-adic integers ranges over all prime numbers! The absolute Galois group acts on all curves defined over and hence (via the Belyi maps ans the corresponding monodromy permutation representation) there is an action of on the profinite completions of the carthographic groups.

This is what Grothendieck calls anabelian algebraic geometry

Returning to the general case, since finite maps can be interpreted as coverings over of an algebraic curve defined over the prime field itself, it follows that the Galois group of over acts on the category of these maps in a natural way.
For instance, the operation of an automorphism on a spherical map given by the rational function above is obtained by applying to the coefficients of the polynomials P , Q. Here, then, is that mysterious group intervening as a transforming agent on topologico- combinatorial forms of the most elementary possible nature, leading us to ask questions like: are such and such oriented maps ‚conjugate or: exactly which are the conjugates of a given oriented map? (Visibly, there is only a finite number of these).
I considered some concrete cases (for coverings of low degree) by various methods, J. Malgoire considered some others ‚ I doubt that there is a uniform method for solving the problem by computer. My reflection quickly took a more conceptual path, attempting to apprehend the nature of this action of G.
One sees immediately that roughly speaking, this action is expressed by a certain outer action of G on the profinite com- pactification of the oriented cartographic group , and this action in its turn is deduced by passage to the quotient of the canonical outer action of G on the profinite fundamental group of
where denotes the typical curve of genus 0 over the prime field Q, with three points re- moved.
This is how my attention was drawn to what I have since termed anabelian algebraic geometry, whose starting point was exactly a study (limited for the moment to characteristic zero) of the action of absolute Galois groups (particularly the groups Gal(K/K), where K is an extension of finite type of the prime field) on (profinite) geometric fundamental groups of algebraic varieties (defined over K), and more particularly (break- ing with a well-established tradition) fundamental groups which are very far from abelian groups (and which for this reason I call anabelian).
Among these groups, and very close to the group , there is the profinite compactification of the modular group , whose quotient by its centre ±1 contains the former as congruence subgroup mod 2, and can also be interpreted as an oriented cartographic group, namely the one classifying triangulated oriented maps (i.e. those whose faces are all triangles or monogons).

and a bit further, on page 250

I would like to conclude this rapid outline with a few words of commentary on the truly unimaginable richness of a typical anabelian group such as doubtless the most remarkable discrete infinite group ever encountered, which appears in a multiplicity of avatars (of which certain have been briefly touched on in the present report), and which from the point of view of Galois-Teichmuller theory can be considered as the fundamental ‚building block‚ of the Teichmuller tower
The element of the structure of which fascinates me above all is of course the outer action of G on its profinite compactification. By Bielyi’s theorem, taking the profinite compactifications of subgroups of finite index of , and the induced outer action (up to also passing to an open subgroup of G), we essentially find the fundamental groups of all algebraic curves (not necessarily compact) defined over number fields K, and the outer action of 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 Q) is known up to isomorphism when we know its mixed fundamental group (or what comes to the same thing, the outer action of on its profinite geometric fundamental group), we can thus say that
all algebraic curves defined over number fields are contained in the profinite compactification and in the knowledge of a certain subgroup G of its group of outer automorphisms!

To study the absolute Galois group one investigates its action on dessins denfants. Each dessin will be part of a finite family of dessins which form one orbit under the Galois action and one needs to find invarians to see whether two dessins might belong to the same orbit. Such invariants are called Galois invariants and quite a few of them are known.

Among these the easiest to compute are

• the valency list of a dessin : that is the valencies of all vertices of the same type in a dessin
• the monodromy group of a dessin : the subgroup of the symmetric group where d is the number of edges in the dessin generated by the partitions and For example, we have seen before that the two Mathieu-dessins

form a Galois orbit. As graphs (remeber we have to devide each of the edges into two and the midpoints of these halfedges form one type of vertex, the other type are the black vertices in the graphs) these are isomorphic, but NOT as dessins as we have to take the embedding of them on the curve into account. However, for both dessins the valency lists are (white) : (2,2,2,2,2,2) and (black) : (3,3,3,1,1,1) and one verifies that both monodromy groups are isomorphic to the Mathieu simple group though they are not conjugated as subgroups of .

Recently, new Galois invariants were obtained from physics. In Children’s drawings from Seiberg-Witten curves the authors argue that there is a close connection between Grothendiecks programme of classifying dessins into Galois orbits and the physics problem of classifying phases of N=1 gauge theories…

Apart from curves defined over there are other sources of semi-simple representations. We will just mention two of them and may return to them in more detail later in the course.

Sporadic simple groups and their representations There are 26 exceptional finite simple groups and as all of them are generated by two elements, there are epimorphisms and hence all their representations are also semi-simple -representations. In fact, looking at the list of ’standard generators’ of the sporadic simples

(here the conjugacy classes of the generators follow the notation of the Atlas project) we see that all but possibly one are epimorphic images of and that at least 12 of then are epimorphic images of .

Rational conformal field theories Another source of representations is given by the modular data associated to rational conformal field theories.

These representations also factor through a quotient by a finite index normal subgroup and are therefore again semi-simple -representations. For a readable introduction to all of this see chapter 6 \”Modular group representations throughout the realm\” of the book Moonshine beyond the monster the bridge connecting algebra, modular forms and physics by Terry Gannon. In fact, the whole book is a good read. It introduces a completely new type of scientific text, that of a neverending survey paper…

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