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.
