Tuesday, February 27, 2007

Physics in finite characteristic

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

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.


AC said...

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!!

Anonymous said...

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!

AC said...

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.....

AC said...

ps (for "anonyme) thus the two equations that define "prime numbers" (as spec T) are
[a,a*]=1 and ST=a a*....

Anonymous said...

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

Unknown said...

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...

Anonymous said...

Dear Professor Goss,
Thanks for your kind reply. I should say I really enjoyed your post.

yagwara said...

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.