Noncommutative geometry and the Riemann zeta function
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
![\xymatrix{\mathbb{Z}/p^n \mathbb{Z} \ar[r]^{\theta_{n+1}} \ar[d] & U_1/U_{n+1} \ar[d] \\
\mathbb{Z}/p^{n-1} \mathbb{Z} \ar[r]^{\theta_n} & U_1/U_n} \xymatrix{\mathbb{Z}/p^n \mathbb{Z} \ar[r]^{\theta_{n+1}} \ar[d] & U_1/U_{n+1} \ar[d] \\
\mathbb{Z}/p^{n-1} \mathbb{Z} \ar[r]^{\theta_n} & U_1/U_n}](/latexrender/pictures/a4271abebedd692c24e98a00ec3339d9.gif)
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…
to its profinite completion
(limit over all finite index normal subgroups
) gives an embedding of the sets of (continuous) simple finite dimensional representations
we would like the above embedding to be dense in some kind of noncommutative analogon of the
.
as in
be the vectorspace with basis the conjugacy classes of elements of
(that is, the space of class functions). As explained
separate finite dimensional (semi)simple representations 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
).
.
Even if you don’t know the formal definition of a profinte group, you know at least one example which explains the concept : the
aka the absolute Galois group. By definition it is the group of all
. 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
then classical Galois theory tells us that there is a projection
by dividing out the normal subgroup of all
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![\xymatrix{Gal \ar[rr]^{\pi_L} \ar[rd]_{\pi_K} & & G_L \ar[ld]^{\pi_{LK}} \\
& G_K &} \xymatrix{Gal \ar[rr]^{\pi_L} \ar[rd]_{\pi_K} & & G_L \ar[ld]^{\pi_{LK}} \\
& G_K &}](/latexrender/pictures/a83d2bcf00cefc1eff52e2c9bbbeafb6.gif)
and hence a better and better finite approximation
of the absolute Galois group 
. 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
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
. Finite field extensions correspond in this bijection to open subgroups and the usual normal subgroup and factor group correspondences hold!
and clearly if
we have a quotient map of finite groups
compatible with the quotient maps from ![\xymatrix{\Gamma \ar[rr]^{\pi_U} \ar[rd]_{\pi_V} & & G_U \ar[ld]^{\pi_{UV}} \\
& G_V &} \xymatrix{\Gamma \ar[rr]^{\pi_U} \ar[rd]_{\pi_V} & & G_U \ar[ld]^{\pi_{UV}} \\
& G_V &}](/latexrender/pictures/ef02f6b8a47cf14eafc287ae84d5e2b8.gif)
and groupmorphisms
we can ask for the ‘best’ group mapping to each of the
. 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 therefore we call
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. 
, 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 (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 profinite compactification 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
on these curves and their associated dessins. Because every permutation representation of 
. In this way one realizes the absolute Galois group as a subgroup of the outer automorphism group of the profinite group
. So a natural question presents itself : how are these two ‘geometrical’ objects 
we see that in this case

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!
is the generator of
. 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
(for consider
as the character of
for which there is no bound on the number of irreducibles for finite quotients, then morally the continuous simple space for the profinite completion
.
, 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
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
of a finite group
whereas there are just 194 characters to consider…
and any finite dimensional simple A-representation
the character
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![\mathfrak{g}_A = \frac{A}{[A,A]_{vect}} \mathfrak{g}_A = \frac{A}{[A,A]_{vect}}](/latexrender/pictures/4eb5d8bdfc4d737e80569afc182a7e2e.gif)
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
and therefore in this case ![\mathfrak{g}_{\C[X]} = \C[X] \mathfrak{g}_{\C[X]} = \C[X]](/latexrender/pictures/c7c00b1eb37b4ba5815bca25703e8a5a.gif)
, that is elements of the dual space
) to separate the simple representations? And, why do I (ab)use Lie-algebra notation
???