Are there applications of noncommutative geometry to number theory? 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?
 A: 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. 
  
  
*
  
*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.
  
*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 $k$. 
  
*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.
  
*Relations with arithmetic algebraic geometry and Arakelov theory. This
  is currently being pursued by Consani, Deninger, Manin, Marcolli and
  others.
  

A: 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.
