Are there applications of noncommutative geometry to number theory?

Question

The marriage of algebraic geometry and number theory was celebrated in the twentieth century by the school of Grothendieck. As a consequence, number theory has been completely transformed.

On the other hand, the school of Alain Connes developped a theory to include the study of non-commutative rings in algebraic geometry.

As far as I understand, the tree branched off in these two separate directions, with number theory staying mostly on the commutative side of things, for the natural reason that the rings one encounters in number theory are mostly commutative.

I'd like to know if the development of noncommutative geometry has had an impact on number theory. Has it led to concrete advancements? Has it influenced the way people think about some topics? Should I, as an aspiring number theorist, care about noncommutative geometry?

Answer 1

I know nothing about noncommutative geometry, but I had wondered this exact thing a while ago and found the following answer. It is part of a report from the BIRS Workshop on Noncommutative Geometry held at the Banff International Research Station in April 2003. The full report is available at www.pims.math.ca/birs.

Current applications and connections of noncommutative geometry to number theory can be divided into four categories.

1. The work of Bost and Connes, where they construct a noncommutative dynamical system $(B,\sigma_t)$ with partition function the Riemann zeta function $\zeta(\beta)$, where $\beta$ is the inverse temperature. They show that at the pole $\beta= 1$ there is an spontaneous symmetry breaking. The symmetry group of this system is the group of idèles which is isomorphic to the Galois group $\operatorname{Gal}(\mathbf Q^{ab}/\mathbf Q)$. This gives a natural interpretation of the zeta function as the partition function of a quantum statistical mechanical system. In particular the class field theory isomorphism appears very naturally in this context. This approach has been extended to the Dedekind zeta function of an arbitrary number field by Cohen, Harari-Leichtnam, and Arledge-Raeburn-Laca. All these results concern abelian extensions of number fields and their generalization to non-abelian extensions is still lacking.
2. The work of Connes on the Riemann hypothesis. It starts by producing a conjectural trace formula which refines the Arthur-Selberg trace formula. The main result of this theory states that this trace formula is valid if and only if the Riemann hypothesis is satisfied by all $L$-functions with Grossencharakter on the given number field $k$.
3. The work of Connes and Moscovici on quantum symmetries of the modular Hecke algebras $A(\Gamma)$ where they show that this algebra admits a natural action of the transverse Hopf algebra $\mathcal H_1$. Here $\Gamma$ is a congruence subgroup of $\text{SL}(2,\mathbf Z)$ and the algebra $A(\Gamma)$ is the crossed product of the algebra of modular forms of level $\Gamma$ by the action of the Hecke operators. The action of the generators $X, Y$ and $\delta_n$ of $\mathcal H_1$ corresponds to the Ramanujan operator, to the weight or number operator, and to the action of certain group cocycles on $GL^+(2,\mathbf Q)$, respectively. What is very surprising is that the same Hopf algebra $\mathcal H_1$ also acts naturally on the (noncommutative) transverse space of codimension one foliations.
4. Relations with arithmetic algebraic geometry and Arakelov theory. This is currently being pursued by Consani, Deninger, Manin, Marcolli and others.

Answer 2

Some of the main people of non-commutative geometry seemed at one time to be working on the field with one element. I cannot be sure about the details of said story, whether there is some connection beyond this coincidence, etc.. But it is something.

Fundamentally, the application of algebraic geometry to number theory consists of solving diophantine equations. These are ``commutative'' equations, by default. Can you imagine a noncommutative diophantine equation to make sense so easily? So if noncommutative geometry is applied into number theory, this simplistic/naive way of thinking may not apply. Maybe things like field with one element are deeper stuff, with some hopes in far future to merge NT and NCG.

There seems to be some potential links in other ways too; but my understanding is a bit hazy and I am not sure enough about those stuff to put it in writing definitively here. Maybe other better experts will come forward.