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 A\xi where \xi is a certain transcendental element

that looks suspiciously like 2\pi 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.