Automorphic forms, R. Bruggeman’s 65th birthday, and silly conjectures

As I already mentioned, I was last week at the conference organized to honor R. Bruggeman’s 65th birthday — though both organizers and honoree claimed this was a pretext to have a conference on the analytic theory of automorphic forms in Holland.

It was a very enjoyable week, and an excellent occasion to learn more about some of the newer brands of automorphic objects which have become popular, but which I don’t really understand yet (I must admit I couldn’t tell the difference between a weak harmonic Maass form, a mock theta function, a mixed modular form, or a weak harmonic Jacobi form…). Since I used a beamer presentation for my talk on families of L-functions and cusp forms, here is a link to it. Although it is almost completely “philosophy”, and does not discuss any new result, some of the points, it may be of some interest (at some point, I will probably write a more complete post on the questions which are raised there).

The workshop was held in a conference center located about 10 minutes from Utrecht. This was a very nice location, and since everyone (except for local people) stayed in the same place, and the coffee was as plentiful as I’ve ever seen (and free), the evenings were quite social affairs. During the Thursday evening barbecue, the question arose (at least at my table) of determining which was the silliest conjecture that people had seriously spent time on. Various glorious names were suggested by participants: Goldbach, Fermat, ABC, etc (I will of course hide the identity of those who made these propositions…) A semi-popular favorite was the Lehmer conjecture which claims that

\tau(n)\not=0,\quad\text{ for } n\geq 1,

i.e., the non-vanishing of the Ramanujan tau-function. Recall that the latter is defined by the formal power-series expansion

\sum_{n\neq 1}{\tau(n)q^n}=q\prod_{n\geq 1}{(1-q^n)^{24}},

and that Ramanujan had conjectured some of its remarkable properties, including the multiplicativity

\tau(mn)=\tau(m)\tau(n)\quad\text{ if } m \text{ and } n \text{ are coprime},

and the bound

|\tau(n)|\leq n^{11/2}d(n)

where d(n) is the number of positive divisors of n. The latter (which is generalized to the “Ramanujan-Petersson conjecture”) was proved by Deligne as a very deep consequence of the Riemann Hypothesis over finite fields.

As a matter of fact, this conjecture of Lehmer had been the topic of one of the morning lectures; E. Bannai had explained his work with T. Miezaki, which gives an interpretation of the Lehmer conjecture in terms of properties of the E8 lattice and spherical design properties of its shells. This suggests that the conjecture is more than a random guess that has every chance to be true, but for no good reason.

In the end, I believe a consensus arose that, at least, the following strengthening of the Lehmer conjecture is an extremely silly question:

“Conjecture”. The tau function is injective.

(this is stronger than Lehmer’s conjecture because if some τ(n) is zero, then by multiplicativity, many others will also be).

This is a question which I had raised (to myself) after reading a paper of Garaev, Garcia and Konyagin which shows, using quite clever arguments, that the Ramanujan function takes “many” different values; at the time I checked it was valid for the largest table of tau that I could find by a quick googling. If there are bigger ones now easily available and obvious counterexamples, I will of course emphasize that this was just a random guess that had every chance to be true for no good reason.

(Note that the example of Hardy apologizing for discussing the tau-function, as seemingly part of the “backwaters of mathematics”, means one must be careful with judgments of value about mathematical problems based on one’s current understanding…)

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Kowalski

I am a professor of mathematics at ETH Zürich since 2008.

7 thoughts on “Automorphic forms, R. Bruggeman’s 65th birthday, and silly conjectures”

  1. Will Sarnak’s letter be made available at some point?

    One interesting problem, perhaps implicit in your lecture, and certainly raised at the NYU higher rank groups conference last year, is the following naive question: How many automorphic forms on GL(n)/F have analytic conductor < X? Even for F=Q and n=2, it does not seem so easy to get an asymptotic formula, though perhaps it could be derived “by brute force” from known counting results for holomorphic and Maass forms of varying level and archimedian parameter.

  2. I’m sure Sarnak will put the letter on his preprint/letters site at some point (though I don’t know when).

    The question you raise is indeed something that is clearly needed to really start up the general theory; Sarnak mentioned it in Verbania during the RH 150 conference, and said it was likely to come cleanly out of the ongoing work of Lapid and Müller on the Trace Formula. For the case of GL(2) forms over the rationals, apparently it should indeed be possible to do things as you suggest (in particular he said Brumley’s thesis contained a result close to this, though that part hasn’t appeared yet in the desired asymptotic form).

  3. Yeah, it seems highly plausible to me that one could cook up a test function in the trace formula whose trace on a given pi is precisely Conductor(pi)^{-s} ; the trace formula would then hand you a “height zeta function” for analytic conductors, and conceivably the geometric side would tell you about its poles and their residues.

  4. One has to be careful in interpreting Hardy’s comment. I don’t think he says that the tau function is part of the backwaters of mathematics (although I think that he has been mischaracterized as saying this). Rather, he says that one might think that Ramanujan had entered the backwaters of mathematics, but then goes on to explain that in fact he has not.

  5. I believe the 2008 Sarnak letter referred to is “Definition of Families of L-Functions” and can be found here

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