The classical product formula for number fields is a fundamental tool in arithmetic. In 1993, Pierre Colmez published a truly inspired generalization of this to the case of Grothendieck's motives. In turn, this spring Urs Hartl and Rajneesh Kumar Singh put an equally inspired manuscript on the arXiv devoted to translating Colmez into the theory of Drinfeld modules and the like. Underneath the mountains of terminology there is a fantastic similarity between these two beautiful papers and I have created a blog to bring this to the attention of the community. Please see:

https://drive.google.com/open?id=0BwCbLZazAtweamZYckpaTy15cFU

# Noncommutative Geometry

## Sunday, July 24, 2016

## Tuesday, June 14, 2016

### What is a functional equation?

Like all number theorists I am fascinated (to say the least) with the functional equation of

classical L-series. Years ago, I came up with a simple characterization of functional equations basically using only complex conjugation. This point being that, via a canonical change of variables (going back to Riemann), such L-series are, up to a nonzero scalar, given by

*real*power series with the expectation that the zeroes are also real. In characteristic p the best one can hope is also that the zeroes will be as rational as the coefficients (though this statement needs to be modified to take care of standard factorizations as well as the great generality of Drinfeld's base rings A).
For those interested, a two page pdf can be found at the following link: https://drive.google.com/open?id=0BwCbLZazAtweTmNIa1ZSc0h2UEE

## Saturday, March 26, 2016

### An indirect consequence of the famous Lucas congruence...

So, in the course of function field arithmetic, one runs into the binomial coefficients (like one does most everywhere in mathematics); or rather the coefficients modulo a prime p. The primary result about binomial coefficients modulo p is of course the congruence of Lucas. In function field arithmetic

one seems to be unable to avoid the group obtained by permuting p-adic (or q-adic) coefficients of a number. I recently discovered a congruence using these permutations and the product of two binomial coefficients that I decided to blog about. The proof is an

I put all of this into a three page pdf which, if you are interested, you can find here:

https://drive.google.com/file/d/0BwCbLZazAtweN293bkxwYUZEYVk/view?usp=sharing

one seems to be unable to avoid the group obtained by permuting p-adic (or q-adic) coefficients of a number. I recently discovered a congruence using these permutations and the product of two binomial coefficients that I decided to blog about. The proof is an

*indirect*consequence of Lucas and is perhaps more interesting than the result itself. One is then led to look for something related with the Carlitz polynomials, which are the function field analog of the binomial coefficients.I put all of this into a three page pdf which, if you are interested, you can find here:

https://drive.google.com/file/d/0BwCbLZazAtweN293bkxwYUZEYVk/view?usp=sharing

## Sunday, November 22, 2015

### Review of "Arithmetic of characteristic p special L-values" by B. Anglès and L. Taelman

Bruno Anglès and Lenny Taelman have published a profound study of finite characteristic special values as listed in the title. This appeared in

(Added 12-11-2015: A revised version of the paper by Anglès and Tavares Rebeiro cited in the review (along with a third author Tuan Ngo Dac) has been put on the arXiv as arXiv:1511.06209v2. In this new version the result is established in complete generality for all

https://drive.google.com/file/d/0BwCbLZazAtweQUJrNF8zeXNPMUk/view?usp=sharing

*Proc. London Math. Soc.*(3) 110 (2015) 1000-1032. The analogy with classical theory is clear throughout the paper, as well as some potentially interesting differences; these results bode very well for the future of the subject. I have written a long review of this paper as well as pointing to a very new one on the web. This is been submitted to Math Reviews/MathSciNet and appears here with their explicit permission.(Added 12-11-2015: A revised version of the paper by Anglès and Tavares Rebeiro cited in the review (along with a third author Tuan Ngo Dac) has been put on the arXiv as arXiv:1511.06209v2. In this new version the result is established in complete generality for all

*q*.)# arXiv:1511.06209v2

# arXiv:1511.06209v

A url for a pdf of this review ishttps://drive.google.com/file/d/0BwCbLZazAtweQUJrNF8zeXNPMUk/view?usp=sharing

## Friday, October 30, 2015

## Tuesday, August 11, 2015

### Grand Unification in the Spectral Pati-Salam Model

Last week we (Chamseddine-Connes-van Suijlekom) posted a preprint on grand unification in the spectral Pati–Salam model which I summarize here (and here).

The paper builds on two recent discoveries in the noncommutative geometry approach to particle physics: we showed how to obtain inner fluctuations of the metric without having to assume the order one condition on the Dirac operator. Moreover the original argument by classification of finite geometriesF that can provide the fine structure of Euclidean space-time as a product M×F (where M is a usual 4-dimensional Riemannian space) has now been replaced by a much stronger uniqueness statement. This new result shows that the algebra

M2(ℍ)⊕M4(ℂ)
where ℍ
are the quaternions, appears uniquely when writing the higher analogue
of the Heisenberg commutation relations. This analogue is written in
terms of the basic ingredients of noncommutative geometry where one
takes a spectral point of view, encoding geometry in terms of operators
on a Hilbert space . In this way, the inverse line element is an unbounded self-adjoint operator D . The operator D is the product of the usual Dirac operator on M and a `finite Dirac operator’ on F , which is simply a hermitian matrix DF .
The usual Dirac operator involves gamma matrices which allow one to
combine the momenta into a single operator. The higher analogue of the
Heisenberg relations puts the spatial variables on similar footing by
combining them into a single operator Y
using another set of gamma matrices and it is in this process that the
above algebra appears canonically and uniquely in dimension 4.

This leads without arbitrariness to the Pati–Salam gauge groupSU(2)R×SU(2)L×SU(4) , together with the corresponding gauge fields and a scalar sector, all derived as inner perturbations of D .
Note that the scalar sector can not be chosen freely, in contrast to
early work on Pati–Salam unification. In fact, there are only a few
possibilities for the precise scalar content, depending on the
assumptions made on the finite Dirac operator.

tr(f(D/Λ))
where Λ is a cutoff scale and f an even and positive function. In the present case, it can be expanded using heat kernel methods,

tr(f(D/Λ))∼F4Λ4a0+F2Λ2a2+F0a4+⋯
where F4,F2,F0 are coefficients related to the function f and ak are Seeley deWitt coefficients, expressed in terms of the curvature of M
and (derivatives of) the gauge and scalar fields. This action is
interpreted as an effective field theory for energies lower than Λ .

One important feature of the spectral action is that it gives the usual Pati–Salam action with unification of the gauge couplings. Indeed, the scale-invariant termF0a4 in the spectral action for the spectral Pati–Salam model contains the terms

F02π2∫(g2L(WαμνL)2+g2R(WαμνR)2+g2(Vmμν)2).
Normalizing this to give the Yang–Mills Lagrangian demands

F02π2g2L=F02π2g2R=F02π2g2=14,
which requires gauge coupling unification. This is very similar to the case of the spectral Standard Model
where there is unification of gauge couplings. Since it is well known
that the SM gauge couplings do not meet exactly, it is crucial to
investigate the running of the Pati–Salam gauge couplings beyond the
Standard Model and to find a scale Λ where there is grand

unification:

gR(Λ)=gL(Λ)=g(Λ).
This would then be the scale at which the spectral action is valid as
an effective theory. There is a hierarchy of three energy scales: SM,
an intermediate mass scale mR where symmetry breaking occurs and which is related to the neutrino Majorana masses (1011−1013 GeV), and the GUT scale Λ .

In the paper, we analyze the running of the gauge couplings according to the usual (one-loop) RG equation. As mentioned before, depending on the assumptions onDF ,
one may vary to a limited extent the scalar particle content,
consisting of either composite or fundamental scalar fields. We will not
limit ourselves to a specific model but consider all cases separately.
This leads to the following three figures:

In other words, we establish grand unification for all of the scenarios with unification scale of the order of1016 GeV, thus confirming validity of the spectral action at the corresponding scale, independent of the specific form of DF .

The paper builds on two recent discoveries in the noncommutative geometry approach to particle physics: we showed how to obtain inner fluctuations of the metric without having to assume the order one condition on the Dirac operator. Moreover the original argument by classification of finite geometries

This leads without arbitrariness to the Pati–Salam gauge group

From the spectral action principle, the dynamics and interactions are described by the

*spectral action,*One important feature of the spectral action is that it gives the usual Pati–Salam action with unification of the gauge couplings. Indeed, the scale-invariant term

unification:

In the paper, we analyze the running of the gauge couplings according to the usual (one-loop) RG equation. As mentioned before, depending on the assumptions on

In other words, we establish grand unification for all of the scenarios with unification scale of the order of

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