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
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.