I was delighted when Alain asked me to post on this blog and I came upon

the catchy title above. As a newcomer to noncommutative geometry, I am

impressed by the applications of concepts arising originally in physics

to number theory. An excellent instance of this is the expression

of the Riemann zeta function as a partition function in the work of

Bost and Connes.

For function fields over finite fields, the applications of ideas from

physics has long been a theme and I don't really have a good idea why

such things work so well except to steal a bit from Feynman: I remember

reading in one of Feynman's works his musings about how it is that

physics is able to handle so many different types of phenomena.

Feynman remarked, I believe, that this is due to the fact

that while the phenomena may be very different the differential equations

tend to be alike, thus cutting the work load greatly.

Well, to go a bit further, the

gods of mathematics were also quite frugal when they "created" mathematics.

Indeed, we see the same ideas occurring in many very different circumstances

and areas; this is in fact one of the real glories of mathematics.

For fields of finite characteristic, we see this phenomenon

very early on: Let k be a field of characteristic p and let

Fr be the p-th power morphism. It has long been known that Fr has many

similarities with differentiation D and this motivated early researchers

such as Ore. If we embed k into its perfect, one also has the

p-th root operator Fr* which is then analogous to integration. The field

of constants for D gets replaced by the fixed field of Fr, one has

adjoint operators etc.

If k is a function field over a finite field, we are free to pick

a fixed closed point \infty and view it as the "infinite prime". The

ring A of functions regular away from \infty is a Dedekind domain

with finite class and unit groups. The ring A is then, by fiat, the

"bottom" for the theory of Drinfeld A-modules. A Drinfeld A-module \phi is

essentially a representation of the ring A by polynomials in Fr; thus

given a \in A one obtains a polynomial \phi_a. The zeroes of \phi_a

then become a finite A-module which must be isomorphic

the d-th Cartesian product of A/(a) with itself; this number d is the

"rank" of the Drinfeld module.

The noncommutative algebra involved already with the simplest Drinfeld module

of them all, the "Carlitz module" (discovered by Carlitz in the 1930's),

already allowed A. Kochubei to define analogs of "creation" and

"annihilation" operators and the canonical commutation

relations of quantum mechanics.

In classical theory, function fields of course have no

bottom whereas the rational numbers are

obviously the bottom for number fields. Thus imposing a bottom allows us

to begin to model aspects of classical arithmetic in finite characteristic

that had been missed in earlier theories. In particular, due to L. Carlitz

and D. Hayes, one can create "cyclotomic" extensions of (k,\infty) based

on the torsion points of certain Drinfeld modules of rank 1.

Based on the connection Fr has with D, Drinfeld was able to produce

an analogy to the work of Krichever on KdV; thus to give an

interpretation of his modules in terms of special coherent sheaves

called "shtuka".

In the fall of 1987 Greg Anderson and Dinesh Thakur discovered a

fundamental relationship between the characteristic polynomial of

the Frobenius morphism on the Jacobian of a curve

and the type of products that arise in

the definition of characteristic p gamma functions. This arose

by analysis of a seminal example due to Robert Coleman. This and the

analogy with KdV led Greg to formulate "solitons" in characteristic

p. In turn this technology allowed Anderson, D. Brownawell and M.

Papanikolas to prove analogs of well-known transcendency conjectures

for the function field ("geometric") gamma function. This proof was

in the basic case of A=Fq[T] whereas the gamma functions exist for

all A. The difficulty is constructing the correct "Coleman functions"

in general. To solve this, Anderson reformulated things in an adelic

setting so as to be able to use harmonic analysis and, in particular,

Tate's thesis. The point being that from a Schwartz function on the

adeles one can go one way to get solitons or another to get L-functions.

Recent papers of Anderson have put the general theory (for all A)

within reach.

One therefore sees how intertwined arithmetic arising from Drinfeld

modules is with the classical (Artin-Weil) zeta function of the field

k. It is therefore natural to ask whether this function itself can be

brought directly into the set-up of Drinfeld modules. This takes us

back to Bost and Connes! Indeed, in a paper (soon to appear in the

Journal of Noncommutative Geometry), B. Jacob uses the general cyclotomic

theory of rank one Drinfeld modules mentioned above to describe

a Bost-Connes system for (k,\infty). In this case the partition function

is the Artin-Weil zeta function of k with the Euler factor at \infty

removed!

However, the noncommutative geometry does not stop with recapturing

the classical zeta-function of k. Indeed, encoding the characteristic

polynomials of the Frobenius morphism leads naturally into characteristic

p valued L-series; for instance one can (beginning with Carlitz) prove

analogs here of Euler's results on the values at positive even

integers of Riemann's zeta function. In the Journal of Number Theory 123

(2007), C. Consani and M. Marcolli translate the machinery of Bost-Connes

into characteristic p analysis and thereby express the characteristic p

zeta function as a partition function!

Finally, Papanikolas reappears at this stage. Indeed, Matt has developed

the correct Tannakian theory in this situation and thus also the

appropriate geometric Galois groups. Using Matt's technology,

Chieh-Yu Chang and Jing Yu have recently established that the

above mentioned zeta values ONLY satisfy the algebraic relations

given by the analog of Euler's result AND the obvious one coming

from the p-th power mapping!

Well that finishes my necessarily very incomplete first post. This clearly

represents my take on things. It would be fabulous to hear, in their

own voices, from the other people involved with these results. No one

viewpoint ever describes everything when it comes to number theory

(and physics too?)!

ps: In a previous post, Alain gave the url for Serre's talk at Harvard

on "how to write mathematics badly." Fortunately about 30 years ago Serre

took me aside and gave me a talk on how to write math well! In the

early 90's I wrote these hints down and got Serre's opinion on them.

Then in 1998 I incorporated some input from E.G. Dunne and P.Vojta.

Finally, when I told Serre that I was going to mention these hints

on this blog, he sent me another minor change... Anyway, those

interested may find these hints here.

## 8 comments:

First, thanks to David for this great post. That's really what we

are aiming at in this blog and we also hope for some feedback since

the main idea is to provide a space of "freedom" for interactions. I

just want to add to David's post a simple fact that explains how the

prime numbers appear naturally in the right column (the q-number

column) of the parallel texts of Masoud's post. We already saw that

Dirac, in his first paper on second quantization, came up with a

very simple equation [a,a*]=1, that gives an operator : a a* with

spectrum the set of positive integers. Also second quantization is

an "operation" on q-numbers and simply consists in replacing the

Hilbert space H by the symmetric power SH which is the direct sum of

all S^n H. This is a functorial operation which means that to an

operator T in H corresponds naturally an operator ST in SH. We now take the "a" of Dirac and

look at the following equation:

ST = a a*

which is an equation in T.

The main point then is that 1) This

equation has a solution 2) The spectrum of the solution T is ........... the set of prime numbers!!

This is a nice blog. A lot of things is dicussed but we need more details and a bit slower pace please! Very nice post by David Goss, but I hope in his later posts professor Goss will explain the terms and ideas in this post in greater details. I have one question regarding professor Connes' comment. From what I know given any set of numbers it is easy to construct a diagonal operator whose spectrum is the given set(multiplication operator). So what is so special

about this operator whose spectrum is the set of prime numbers? Thanks again!

Anonyme: what is so special is that we did not have to define prime numbers: they come out naturally from the solution T of the equation ST= a a*. You would have a hard time finding such a simple equation for "any" subset of R.....

ps (for "anonyme) thus the two equations that define "prime numbers" (as spec T) are

[a,a*]=1 and ST=a a*....

thanks a lot; it makes more sense now. Would be nice to see this idea applied to number theory somehow.

Dear Anon: I will be happy to expand

on anything I wrote as best I can.

Please feel free to contact me

directly if you would like...

David

Dear Professor Goss,

Thanks for your kind reply. I should say I really enjoyed your post.

I have a few suggested additions to the "how to write mathematics well":

It is rarely necessary to use a bullet list with more than one level, and never necessary to use one with more than two levels (see Edward Tufte,

The Cognitive Style of Power Point).Each item of the same depth in the list should be semantically equivalent, e.g. a statement should not be followed by a topic heading; a verb should not be followed by a noun.

Conversely, semantically linked items should not be split across depths, e.g. an "if..." sentence in a sublist should not be followed by a corresponding "conversely..." sentence in a sub-sublist.

Sorry, there was a pedantic monkey on my back. As to the main post, thanks, this was really very helpful! I was vaguely aware of these ideas but had no idea how they all fit together. It will be much easier to do some reading now that I have a sense of where it's all been going. This sort of informal information is exactly what a blog is good for.

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