Category Archives: number theory

Group theoretic underpinnings of zeta-phenomenology

This is a follow up to the posts of Alain and Katia on the Vanderbilt
workshop this past May. It was a really great conference and I would again like
to thank the organizers for including me.

After I returned from the conference, I decided to try to write down
what I had talked about. In doing that, I finally was able to glimpse
certain underlying symmetries that I had long been looking for. I
wrote this up in a preprint that can be found here
(a slightly less clean version is in the arXiv…). I would like to explain this
preprint here; I apologize if this post runs a bit long.
Anyway, the upshot is that while classically the functional
equation can be thought of as a Z/(2) action (or a group of order 4 if
you throw in complex conjugation) in characteristic p there is
rather compelling evidence for an associated group which has the
cardinality of the continuum.

Most references not given here can be found in my preprint…

After Drinfeld’s great work introducing Drinfeld modules (called
by him “elliptic modules”) I began to try to develop the related
arithmetic. I soon learned that L. Carlitz had begun this study four
decades before! What one does is to take a complete, smooth, geometrically
curve X over the finite field Fq and then fix a place “infty”. The global
functions on the affine curve X-infty is called “A” and it plays the
role of the integers Z in the theory. The domain A is of course a Dedekind
domain and will in general have nontrivial class group.
In particular, Drinfeld (and earlier Carlitz) develops a theory of lattices associated to A and finds that one can obtain Drinfeld modules much like one obtains elliptic curves classically. The Drinfeld modules are algebraic objects and so one can discuss them over finite fields etc. Like elliptic curves, there is also a Frobenius endomorphism with acts on Tate modules (defined in a very natural way). The resulting characteristic polynomial has coefficients in A and one has the local Riemann hypothesis bounds on the absolute values (at infty) of its roots. So it really makes sense to try to create an associated theory of L-series for Drinfeld modules (and the many generalizations since devised by Drinfeld, G.Anderson, Y. Taguchi, D. Wan, G. Boeckle, R. Pink, M. Papanikolas, etc.)

Now in the 1930’s Carlitz developed a very important special
case of Drinfeld modules (called the “Carlitz module”) for A=Fq[T]. This is
a rank one object which means that the associated lattice can be
written in the form Axi where xi is a certain transcendental element
that looks suspiciously like 2pi i. Using this xi Carlitz established
a very beautiful analog of the famous formula of Euler on the values
of the Riemann zeta function at positive even integers. Indeed, he also
developed an excellent (and still quite mysterious) theory of “factorials”
for Fq[T] as well as analogs of Bernoulli numbers which are called
“Bernoulli-Carlitz elements;” they lie in Fq(T). With his incredible
combinatorial power, Carlitz then proceeded to compute the denominator
of these BC elements (all of this is in my paper); this is a “von Staudt” type
result. In particular, he presents TWO conditions for a prime to divide the
denominator. The first condition is very much like the one for classical
Bernoulli numbers. However, the second one involves the sum of the
p-adic digits of the number and seemed extremely strange into just recently.

Let k be the quotient field of A and let k_infty be the completion
at infty. So A lies discretely in k_infty with compact quotient just
as the integers Z lie in the real numbers R. Let C_infty be the
completion of the algebraic closure of k_infty equipped with its
canonical topology. So one always views C_infty as the analog of
the complex numbers except it is NOT locally compact; this is not a great
handicap and one just forges ahead.

In 1977 and 1978 I was at Princeton University (where N. Katz turned
me on to Carlitz’s series of papers) and J.-P. Serre was at the Institute.
One knows that having a polynomial be monic is a very good (but
not perfect) substitute
for having an integer be positive (so the product of two monics
is obviously monic but the sum of two monics need not be monic). If
f is a monic polynomial one can clearly raise f to the i-th power where i is any
integer. So in keeping with the spirit of Carlitz and Drinfeld, it made
sense to ask if there were any other elements s so that f^s made sense.
After discussing things with Serre, I came up with the space
S_infty defined by

S_infty:=C_infty^* times Z_p;

i.e., S_infty is the product of the nonzero elements in C_infty
with the p-adic integers.

Let me briefly explain how you can express the operation

f |—-> f^i

for a monic polynomial f of degree d and integer i in terms of S_infty. So we
pick a uniformizer pi at infty; for simplicity, let’s set
pi=1/T. Then we have obviously

f^i=(pi^{-i})^d (pi^d f)^i

= (T^i)^d (f/T^d)^i

and this corresponds to the point (pi^{-i}, i) in S_infty. So in general
for s=(x,y) in S_infty you define

f^s:=x^d (pi^d f)^y ;

the point being that (pi^d f) IS a 1-unit and so can be raised to
a p-adic power by simply using the binomial theorem.

For general A one has the nonclassical problem of having to exponentiate
nonprincipal ideals (as if the integers Z had nontrivial class group!).
It took a while but then (through discussions with Dinesh Thakur) we realized
that the above definitions naturally and easily extended to all fractional
ideals simply because the class group is finite AND the values lie
in C_infty (as opposed to the complex numbers..).

So one can now proceed easily to define L-series in great generality
by using Euler-products over the primes of A. One always obtains *families* of entire power series in 1/x, where y is the parameter; thus one can certainly talk about the order of zero at a point s in S_infty, etc. The proof that we obtain such families uses the cohomology of certain “crystals” associated to Drinfeld modules etc., by
G. Boeckle and R. Pink (see e.g., Math. Ann. 323, (2002) 737-795). The idea is thatwhen
y is a negative integer the resulting function in 1/x is a *polynomial* that can be computed cohomologically. Boeckle then shows that the *degree in 1/x* of these polynomials grows *logarithmically* with y (of course logarithmic growth is a standard theme of classical L-series). This, combined with standard and powerful results in nonArchimedean analysis, due to Amice, gives the analytic continuation.

Of course then a reasonable question arises: where is the functional equation?
It turns out that the evidence for *many* functional equations was
there all the time. However, the case A=Fq[T], which is the easiest
to compute with, is misleading (just as the classical zeta function of
the projective line over Fq is misleading; looking only at this function
one might suppose that ALL classical zeta functions of curves/Fq
have no zeroes….). It is only recently that calculations due to Dinesh Thakur and Javier Diaz-Vargas with more general A have given us the correct hints.

Indeed, for general A one writes down the analog of the Riemann zeta
function as

zeta(s):=sum_I I^{-s}

for s in S_infty. When A=Fq[T] one has the results of Carlitz mentioned
above at the positive integers i where i is divisible by (q-1). At the negative integers divisible by (q-1) one has “trivial zeroes” which in this case are simple.

So the obvious thing to do is to try to emulate Euler’s fabulous discovery of the functional equation of the Riemann zeta function from knowledge only of special values (as in my preprint or, better, the wonderful paper of Ayoub referenced there!). However, this never worked (and one can immediately see problems when q is not 3) and so we were left looking for other ideas.

In retrospect, one reason that a direct translation of Euler’s ideas did not work was that at the positive integers, one obtains Bernoulli-Carlitz *ideals* not values. Indeed, as in my paper, Carlitz’s notion of factorial makes sense for all A *but* only as an ideal of A, not a value; so when one multiplies by this factorial, one must do it in the group of ideals and we are out of the realm of values alone.

In the mid 1990’s, there was some essential progress made by Dinesh Thakur.
Dinesh decided to look at trivial zeroes for more general A than just
Fq[T]. He was able to do some calculations in a few cases; these calculations
were then much more recently extended by Javier Diaz-Vargas. What these
two found intrigued me greatly: If one looks at the values i where the
trivial zero at -i has order strictly greater than the obvious classical looking
lower bound (this is the “non-classical set”) one finds that this set appears
to consist of integers with *bounded* sum of q-adic digits!

These inspired calculations of Thakur and Diaz-Vargas thrilled me and
vexed me at the same time! On the one hand, they are so obviously
p-adic that they guarantee we are looking at very new ideas, but on
the other hand I wanted to know just what these ideas might be!

Now first of all, these calculations really do tell us that some sort
of functional equation should be lurking about. Indeed, classically
the order of special values falls out of the functional equation. As
the calculations of Thakur and Diaz-Vargas are only hints; one will need
other techniques to make them truly theorems.

Still I wanted to do better. The set of integers i with bounded sum of
q-adic digits is remarkable. One can take one such i and torture its q-adic
digits in many ways and still stay in the set! It finally dawned on me
that all of these “tortures” really form a group and that this group
replaces the Z/(2) group of classical arithmetic.

So here is the definition of the group S_{(q)}. It consists of
all permutations of the q-adic digits of a p-adic number; you just
reshuffle them in any way you would like! Surprisingly, this shuffling
is continuous p-adically and so we obtain a group of homeomorphisms of
Z_p. This group is obviously huge and indeed its cardinality is that of the
continuum. And, clearly, this group permutes the set of i with bounded sum of q-adic digits etc.

One also sees that these permutations stabilize both the positive
and negative integers and also stabilizes the classes modulo (q-1).

The key point then is a refinement of the observations of
Thakur and Diaz-Vargas:

The order of the trivial zero at -i is an invariant of the action of S_{(q)}.

Again, this is just an observation (which is easily seen to be a theorem
in the A=Fq[T] case as there one only worries about whether i is divisible
by q-1 or not!). But it seems to point the way to deeper structure.
In fact, the special values appear to “know” that they lie on a family of
functions and this large automorphism group may help us control the family…

Finally, this all relates back to Carlitz’s von Staudt result: It turns out that
the divisibility of the denominator of Carlitz’s von Staudt result is also
an invariant of subgroups of S_{(q)}. This is really mysterious: On the
one had, we have invariants related to zeroes of function and yet on the
other hand we have invariants related to objects made up from special
values. I don’t have any good explanation for this at this point. Nor
can I guess, like Euler did, as to the exact form a global statement should take….

There is an associated theory of modular forms on Drinfeld’s upper half space. In the past few years, great progress has been made on these forms by Gebhard Boeckle using the techniques mentioned above. There is a great deal of mystery in his results and perhaps these mysteries are related to the huge group of symmetries that now seems to underlie the theory.

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.