There is a 60th birthday conference in honour of Frits Beukers in Utrech in July; I’m hoping to swing by there on the way to Oberwolfach. Thinking about matters Beukers made me reconsider an question that I’ve had for while.
There is a fairly well known explanation of why should be irrational (and linearly independent of ) in terms of Motives. There is also a fairly good proof that in terms of the non-vanishinjg of Borel’s regulator map on . (I guess there are also more elementary proofs of this fact.) A problem I would love to solve, however, is to show that, for all primes , the Kubota-Leopoldt -adic zeta function is non-zero. Indeed, this is equivalent to the injectivity of Soule’s regulator map
(Both these groups have rank one, and the cokernel is (at least for ) equal to by the main conjecture of Iwasawa theory.) It is somewhat of a scandal that we can’t prove that is zero or not; it rather makes a mockery out of the idea that the “main conjecture” allows us to “compute” eigenspaces of class groups, since one can’t even determine if there exists an unramified non-split extension
or not. Well, this post is about something related to this but a little different. Namely, it is about the vaguely formed following question:
What is the relationship between a real period and its -adic analogue?
Since one number is (presumably) in and the other in , it’s not entirely clear what is meant by this. So let me give an example of what I would like to understand. One could probably do this example with , but I would prefer to consider the “simpler” example of Catalan’s constant. Here
is the real Catalan’s constant, and
is the -adic analogue. (There actual definition of the Kubota-Leopoldt zeta function involves an unnatural twist so that one could conceivably say that and that the non-zero number is , but this is morally wrong, as the examples below will hopefully demonstrate. Morally, of course, they both relate to the motive .)
So what do I mean is the “relation” between and . Let me give two relations. The first is as follows. Consider the recurrence relation (think Apéry/Beukers):
It has two linearly independent solutions with and , and and . One fact concerning these solutions is that , and Moreover one has that:
The convergence is very fast, indeed fast enough to show that . What about convergence in , does it converge to the real Catalan constant? Well, a numerical test is not very promising; for example, when one gets , which isn’t anything like ; for contrast, for this value of one has , which is pretty small. There are, however, two linearly independent solutions over given analytically by
from which one can see that must converge very slowly, and indeed, one has (caveat: I have some idea on how to prove this but I’m not sure if it works or not):
So one has a naturally occurring sequence which converges to in and in . So that is some sort of “relationship” alluded to in the original question. Here’s another connection. Wadim Zudilin pointed out to me the following equality of Ramanujan:
This sum also converges -adically. So, one can naturally ask whether
(It seems to be so to very high precision.) These are not random sums at all. Indeed, they are equal to
at . Presumably, both of these connections between and must be the same, and must be related to the Picard-Fuchs equation/Gauss-Manin connection for . This reminds me of another result of Beukers in which one compares values of hypergeometric functions related to Gauss-Manin connections and elliptic curves, and finds that they converge in and for various to algebraic (although sometimes different!) values. Of course, things are a little different here, since the values are (presumably) both transcendental. Yet it would be nice to understand this better, and see to what extent there is a geometric interpretation of (say) the non-vanishing of for some odd quadratic character . Of course, one always has to be careful not to accidentally prove Leopoldt’s conjecture in these circumstances.