Before we can even attempt to describe the adelic description of the Bost-Connes Hecke algebra and its symmetries, we’d probably better recall the construction and properties of adeles and ideles. Let’s start with the p-adic numbers and its field of fractions . For p a prime number we can look at the finite rings of all integer classes modulo . If two numbers define the same element in (meaning that their difference is a multiple of ), then they certainly define the same class in any when , so we have a sequence of ringmorphisms between finite rings

The ring of p-adic integers can now be defined as the collection of all (infinite) sequences of elements with such that for all natural numbers . Addition and multiplication are defined componentswise and as all the maps are ringmorphisms, this produces no compatibility problems.

One can put a topology on making it into a compact ring. Here’s the trick : all components are finite so they are compact if we equip these sets with the discrete topology (all subsets are opens). But then, Tychonov’s product theorem asserts that the product-space with the product topology is again a compact topological space. As is a closed subset, it is compact too.

By construction, the ring is a domain and hence has a field of fraction which we will denote by . These rings give the p-local information of the rational numbers . We will now ‘glue together’ these local data over all possible prime numbers into adeles. So, forget the above infinite product used to define the p-adics, below we will work with another infinite product, one factor for each prime number.

The adeles are the restricted product of the over for all prime numbers p. By ‘restricted’ we mean that elements of are exactly those infinite vectors such that all but finitely of the components . Addition and multiplication are defined component-wise and the restriction condition is compatible with both adition and multiplication. So, is the adele ring. Note that most people call this the finite Adeles as we didn’t consider infinite places, i will distinguish between the two notions by writing adeles resp. Adeles for the finite resp. the full blown version. The adele ring has as a subring the infinite product . If you think of as a version of then corresponds to (and next time we will see that there is a lot more to this analogy).

The ideles are the group of invertible elements of the ring , that is, . That s, an element is an infinite vector with all and for all but finitely many primes we have that .

As we will have to do explicit calculations with ideles and adeles we need to recall some facts about the structure of the unit groups and . If we denote , then projecting it to the unit group of each of its components we get for each natural number n an exact sequence of groups

. In particular, we have that as the group of units of the finite field is cyclic of order p-1. But then, the induced exact sequence of finite abalian groups below splits

and as the unit group we deduce that where is the specified unique subgroup of of order p-1. All that remains is to determine the structure of . If , take and let denote the image of , then one verifies that is a cyclic generator of order of .

But then, if we denote the isomorphism between the ADDITIVE group and the MULTIPLICATIVE group by the map , then we have a compatible commutative diagram

and as this gives an isomorphism between the multiplicative group and the additive group of . In case we have to start with an element and repeat the above trick. Summarizing we have the following structural information about the unit group of p-adic integers

Because every unit in can be written as with we deduce from this also the structure of the unit group of the p-adic field

Right, now let us start to make the connection with the apparently abstract ringtheoretical post from last time where we introduced semigroup crystalline graded rings without explaining why we wanted that level of generality.

Consider the semigroup , that is all ideles with all with and with for all but finitely many primes p. Then, we have an exact sequence of semigroups

where the map is defined (with above notation) and exactness follows from the above structural results when we take .

This gives a glimpse of where we are heading. Last time we identified the Bost-Connes Hecke algebra as a bi-crystalline group graded algebra determined by a -semigroup crystalline graded algebra over the group algebra . Next, we will entend this construction starting from a -semigroup crystalline graded algebra over the same group algebra. The upshot is that we will have a natural action by automorphisms of the group on the Bost-Connes algebra. And… the group is the Galois group of the cyclotomic field extension !

But, in order to begin to understand this, we will need to brush up our rusty knowledge of algebraic number theory…

Last time (possibly with help from the survival guide) we have seen that the universal map from the modular group to its profinite completion (limit over all finite index normal subgroups ) gives an embedding of the sets of (continuous) simple finite dimensional representations

and based on the example we would like the above embedding to be dense in some kind of noncommutative analogon of the Zariski topology on .

We use the Zariski topology on as in these two M-geometry posts¹. So, what’s this idea in this special case? Let be the vectorspace with basis the conjugacy classes of elements of (that is, the space of class functions). As explained here it is a consequence of the Artin-Procesi theorem that the linear functions separate finite dimensional (semi)simple representations of . That is we have an embedding

and we can define closed subsets of as subsets of simple representations on which a set of class-functions vanish. With this definition of Zariski topology it is immediately clear that the image of is dense. For, suppose it would be contained in a proper closed subset then there would be a class-function vanishing on all simples of so, in particular, there should be a bound on the number of simples of finite quotients which clearly is not the case (just look at the quotients ).

But then, the same holds if we replace ’simples of ‘ by ’simple components of permutation representations of ‘. This is the importance of Farey symbols to the representation problem of the modular group. They give us a manageable subset of simples which is nevertheless dense in the whole space. To utilize this a natural idea might be to ask what such a permutation representation can see of the modular group, or in geometric terms, what the tangent space is to in a permutation representation². We will call this the modular content of the permutation representation and to understand it we will have to compute the tangent quiver .

1. already, I regret terminology, I should have just called it noncommutative geometry [↩]
2. more precisely, in the ‘cluster’ of points making up the simple components of the representation representation [↩]

Even if you don’t know the formal definition of a profinte group, you know at least one example which explains the concept : the Galois group of the algebraic numbers aka the absolute Galois group. By definition it is the group of all -isomorphisms of the algebraic closure . Clearly, it is an object of fundamental importance for mathematics but in spite of this very little is known about it. For example, it obviously is an infinite group but, apart from the complex conjugation, try to give one (1!) other nontrivial element… On the other hand we know lots of finite quotients of . For, take any finite Galois extension , then its Galois group is a finite group and there is a natural onto morphism obtained by dividing out all -automorphisms of . Moreover, all these projections fit together nicely. If we take a larger Galois extension then classical Galois theory tells us that there is a projection by dividing out the normal subgroup of all -automorphisms of and these finite maps are compatible with those from the absolute Galois group, that is, for all such finite Galois extensions, the diagram below is commutative

By going to larger and larger finite Galois extensions we get closer and closer to the algebraic closure and hence a better and better finite approximation of the absolute Galois group . Still with me? Congratulations, you just rediscovered the notion of a profinite group! Indeed, the Galois group is the projective limit

over all finite Galois extensions . If the term ‘projective limit’ scares you off, it just means that all the projections coming from finite Galois theory are compatible with those coming from the big Galois group as before. That’s it : profinite groups are just projective limits of finite groups.

These groups come equipped with a natural topology : the Krull topology. Again, this notion is best clarified by considering the absolute Galois group. Now that we have we would like to extend the classical Galois correspondence between subgroups and subfields and between normal subgroups and Galois subfields. For each finite Galois extension we have a normal subgroup of finite index, the kernel of the projection map above. Let us take the set of all as a fundamental system of neighborhoods of the identity element in . This defines a topology on and this is the Krull topology. As every open subgroup has finite index it is clear that this turns into a compact topological group. Its purpose is that we can now extend the finite Galois correspondence to Krull’s Galois theorem :

There is a bijective lattice inverting Galois correspondence between the set of all closed subgroups of and the set of all subfields . Finite field extensions correspond in this bijection to open subgroups and the usual normal subgroup and factor group correspondences hold!

So far we had a mysterious group such as and reconstructed it from all its finite quotients as a projective limit. Now we can reverse the situation : suppose we have a wellknown group such as the modular group , then we can look at the set of all its normal subgroups of finite index. For each of those we have a quotient map to a finite group and clearly if we have a quotient map of finite groups compatible with the quotient maps from

For the family of finite groups and groupmorphisms we can ask for the ‘best’ group mapping to each of the compatible with the groupmaps . By ‘best’ we mean that any other group with this property will have a morphism to the best-one such that all quotient maps are compatible. This ‘best-one’ is called the projective limit

and as a profinite group it has again a Krull topology making it into a compact group. Because the modular group had quotient maps to all the we know that there must be a groupmorphism to the best-one and therefore we call the profinite compactification (or profinite completion) of the modular group.

A final remark about finite dimensional representations. Every continuous complex representation of a profinite group like the absolute Galois group has finite image and this is why they are of little use for people studying the Galois group as it conjecturally reduces the study of these representations to ‘just’ all representations of all finite groups. Instead they consider representations to other topological fields such as p-adic numbers and call these Galois representations.

For people interested in Grothendieck’s dessins d’enfants, however, continuous complex representations of the profinite compactification is exactly their object of study and via the universal map above we have an embedding

of them in all finite dimensional representations of the modular group ( and we have a similar map restricted to simple representations). I hope this clarifies a bit obscure terms in the previous post. If not, drop a comment.

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 , where K is an extension of finite type of the prime field) on (profinite) geometric fundamental groups of algebraic varieties (defined over K), and more particularly (breaking 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 profinite compactification of the modular group , whose quotient by its centre 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).

The above text is taken from Alexander Grothendieck’s visionary text Sketch of a Programme. He was interested in the permutation representations of the modular group as they correspond via Belyi-maps and his own notion of dessins d’enfants to smooth projective curves defined over . One can now study the action of the absolute Galois group on these curves and their associated dessins. Because every permutation representation of factors over a finite quotient this gives an action of the absolute Galois group as automorphisms on the profinite compactification

where the limit is taken over all finite index normal subgroups . In this way one realizes the absolute Galois group as a subgroup of the outer automorphism group of the profinite group . As a profinite group is a compact topological group one should study its continuous finite dimensional representations which are precisely those factoring through a finite quotient. In the case of the simple continuous representations are precisely the components of the permutation representations of the modular group. So in a sense, anabelian geometry is the study of these continuous simples together wirth the action of the absolute Galois group on it.

In noncommutative geometry we are interested in a related representation theoretic problem. We would love to know the simple finite dimensional representations of the modular group as this would give us all simples of the three string braid group . So a natural question presents itself : how are these two ‘geometrical’ objects (anabelian) and (noncommutative) related and can we use one to get information about the other?

This is all rather vague so far, so let us work out a trivial case to get some intuition. Consider the profinite completion of the infinite Abelian group

As all simple representations of an Abelian group are one-dimensional and because all continuous ones factor through a finite quotient we see that in this case

is the set of all roots of unity. On the other hand, the simple representations of are also one-dimensional and are determined by the image of the generator so

Clearly we have an embedding and the roots of unity are even dense in the Zariski topology. This might look a bit strange at first because clearly all roots of unity lie on the unit circle which ’should be’ their closure in the complex plane, but that’s because we have a real-analytic intuition. Remember that the Zariski topology of is just the cofinite topology, so any closed set containing the infinitely many roots of unity should be the whole space!

Let me give a pedantic alternative proof of this (but one which makes it almost trivial that a similar result should be true for most profinite completions…). If is the generator of then the different conjugacy classes are precisely the singletons . Now suppose that there is a polynomial vanishing on all the continuous simples of then this means that the dimensions of the character-spaces of all finite quotients should be bounded by (for consider as the character of ), which is clearly absurd.

Hence, whenever we have a finitely generated group for which there is no bound on the number of irreducibles for finite quotients, then morally the continuous simple space for the profinite completion

should be dense in the Zariski topology on the noncommutative space of simple finite dimensional representations of . In particular, this should be the case for the modular group .

There is just one tiny problem : unlike the case of for which this space is an ordinary (ie. commutative) affine variety , what do we mean by the “Zariski topology” on the noncommutative space ? Next time we will clarify what this might be and show that indeed in this case the subset

will be a Zariski closed subset!

Last time we introduced the tangent quiver of an affine algebra A to be a quiver on the isoclasses of simple finite dimensional representations. When is the coordinate ring of an affine variety, these vertices are just the points of the variety and this set has the extra structure of being endowed with the Zariski topology. For a general, possibly noncommutative algebra, we would like to equip the vertices of also with a topology.

In the commutative case, the Zariski topology has as its closed sets the common zeroes of a set of polynomials on , so we need to generalize the notion of ‘functions’ the the noncommutative world. The NC-mantra states that we should view the algebra A as the ring of functions on a (usually virtual) noncommutative space. And, face it, for a commutative variety the algebra does indeed do the job. Still, this is a red herring.

Let’s consider the easiest noncommutative case, that of the group algebra of a finite group . In this case, the vertices of the tangent quiver are the irreducible representations of and no sane person would consider the full group algebra to be the algebra of functions on this set. However, we do have a good alternative in this case : characters which allow us to separate the irreducibles and are a lot more manageable than the full group algebra. For example, if is the monster group then the group algebra has dimension approx whereas there are just 194 characters to consider…

But, can we extend characters to arbitrary noncommutative algebras? and, more important, are there enough of these to separate the simple representations? The first question is easy enough to answer, after all characters are just traces so we can define for every element and any finite dimensional simple A-representation the character

where is the matrix describing the action of a on S. But, you might say, characters are then just linear functionals on the algebra A so it is natural to view A as the function algebra, right? Wrong! Traces have the nice property that and so they vanish on all commutators of A, so characters only carry information of the quotient space

where is the vectorspace spanned by all commutators (and not the ideal…). If one is too focussed on commutative geometry one misses this essential simplification as clearly for being a commutative algebra,

and therefore in this case

Ok, but are there enough characters (that is, linear functionals on , that is elements of the dual space ) to separate the simple representations? And, why do I (ab)use Lie-algebra notation to denote the vectorspace ???