Sunday, July 24, 2016

A motivic product formula

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:

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:

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 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:

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 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.)



A url for a pdf of this review is

Friday, October 30, 2015

C. Armana's Formula for the coefficients of $h$

(full pdf is at

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 geometries F 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
where 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 group SU(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.
From the spectral action principle, the dynamics and interactions are described by the spectral action,
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,
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 term F0a4 in the spectral action for the spectral Pati–Salam model contains the terms
Normalizing this to give the Yang–Mills Lagrangian demands
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
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 (10111013GeV), 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 on DF, 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:
Running of coupling constants for the spectral Pati--Salam model with composite Higgs fields
Running of coupling constants for the spectral Pati–Salam model with composite Higgs fields
Running of coupling constants for the spectral Pati–Salam model with fundamental Higgs fields
Running of  coupling constants for the left-right symmetric spectral Pati--Salam model.

Running of coupling constants for the left-right symmetric spectral Pati–Salam model

In other words, we establish grand unification for all of the scenarios with unification scale of the order of 1016 GeV, thus confirming validity of the spectral action at the corresponding scale, independent of the specific form of DF.