Category: Mathematics

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

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

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

and the bound

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…)

I was in Paris part of this week-end, where I had planned to attend the Bourbaki Seminar lecture of E. Breuillard on the recent works of Einsiedler, Lindenstraus, Michel and Venkatesh. Embarrassing scheduling mistakes on my part forced me to miss it, however, and to take an earlier train to Holland (where I am now, attending the conference on the analytic theory of automorphic forms coinciding with the 65th birthday of R. Bruggeman). Fortunately, I did manage on Saturday to pick up a copy of the physical Bourbaki report, so this was not entirely a disaster.

From Paris to Holland, I took the Thalys train, which is distinguished by having wireless on board and by quadrilingual announcements: French, because it starts from Paris and goes through parts of French-speaking Belgium, Dutch, because it crosses also Dutch-speaking Belgium and one branch reaches Holland, German, because of the other branch going to Köln, and finally English, for good measure.

In this multilingual environment, one notices that the names of a number of towns changes with the language; for instance, Köln is Cologne (famous for its eau) in French and English. The prize of variation on this trip was Liège, alias Luijk, alias Lüttich (alias Liege, if one wants to be picky). Hence the question of the day: which town has the most different names? (Say within same-alphabet countries, to avoid issues of transliteration).

This week, the Forschungsinstitut für Mathematik is host to a conference to honor the 61st birthday of G. Wüstholz — of course, diophantine approximation, arithmetic geometry, and related areas were very much in focus. (I write “is” because, although this is most definitely Friday evening, there are still two talks scheduled tomorrow morning).

The programme was very enticing, so I attended most of the talks, despite not knowing much about some of the topics (e.g., Arakelov geometry). Among those I found especially interesting (partly because I was at least a bit more au courant) were the following:

(1) U. Zannier explained some recent work with D. Masser which can be described as trying to understand (and devise methods to study) intersections of “sparse” sets of arithmetic interest. The concrete example he described, which had been the original question of Masser, was the following: consider the Legendre family of elliptic curves

and the points

on E_t (so there are two choices of the y-coordinates, which will not affect the question). What can one say about the set of parameters t for which both

are torsion points on the curve E_t? It is not difficult to check that for either of the two points, there are infinitely many such parameters, forming “sparse” sets, and the results (or rather, the methods) of Masser and Zannier imply, in particular, that these two have at most a finite number of intersection points, i.e., that there are only finitely many t for which both are torsion points.

One may wonder why the question should be of any interest (I personally find it very nice), but Zannier emphasized that the new techniques they had to devise were quite significant and very likely to be useful in many contexts. These techniques are quite novel in this area, and rely ultimately (and quite strikingly) on the circle of ideas that started with the 1989 work of Bombieri and Pila on the number of rational (or integral) points on transcendental curves (in the plane, say). Zannier illustrated the link with a sketch of a new proof of the original Manin-Mumford conjecture (first proved by Raynaud; it states that if an algebraic curve defined over a number field is embedded in its Jacobian variety, then the curve only contains finitely many torsion points): using a transcendental parametrization, we can see the curve as a transcendental curve

for some lattice

whereas the torsion points are the elements of

and thus have rational coordinates. The intersection is thus, in principle, similar to the situation of Bombieri-Pila. Of course, much more work is required, and Zannier said that the extension to deal with Masser’s question require rather more subtle versions of the Bombieri-PIla ideas, including the very recent ones of Pila-Wilkie where — to add more fun to the mix — logic enters the game through the consideration of transcendental varieties with graphs definable in an o-minimal structure. (About which, despite looking at the book of van den Dries, I am still terribly ignorant).

See here for the Compte Rendus note announcing this result; Zannier said the full paper is almost ready, and one can see more applications of this type of methods in this recent paper of Pila.

(2) T. Shioda gave a very nice lecture — full of beautiful examples — on recent work of his concerning the determination and structure of the (finite) set of integral sections of an elliptic surface over the projective line (see his preprint); for rational elliptic surfaces, he explained a very beautiful general description involving commutative algebra. In particular, there are then at most 240 integral sections. For instance, there are exactly 240 polynomials

such that they are points on the rational elliptic surface with equation

indeed, those points, for the height pairing, can be identified with the 240 vectors of minimal length (squared) 2 on the famous E₈ lattice.

(3) Y. Bilu explained his recently recovered-from-the-edge proof, with P. Parent, of the “split Cartan” case of the Serre uniformity question concerning the maximality of the Galois action on torsion points of prime order on elliptic curves over the rationals (see also this post for more background information– though as indicated, the first proof had a mistake, which was corrected in February–March this year, the overall strategy has remained the same). The preprint of Bilu and Parent is on arXiv.

Partly motivated as a follow-up to the Oppenheimer biography I recently read, and partly to get a clearer idea of Quantum Mechanics than I had (for my course on Spectral Theory in Hilbert Spaces, which just ended; though the posted lecture notes are a bit lagging, I will post a complete version soon…), I have just finished reading “The conceptual development of quantum mechanics“, by M. Jammer (a book which seems unfortunately to have almost no online presence; I got the reference from another book and borrowed it from the ETH library).

This was extremely interesting; the author assumed rather more knowledge of classical physics than I could claim to remember from my studies, but since I had been reading other accounts of (modern) Quantum Mechanics for mathematicians, I was not entirely lost, and I was quite fascinated. I won’t say anything about the physical aspects, but one quite amazing thing that emerges is how quickly the formalism emerged from 1925 to 1929. Not only did matrix mechanics (Heisenberg, 1925), Dirac’s formalism 1925–26), Schrödinger’s wave mechanics (1926) all come out in barely more than a year, but also Nordheim, Hilbert and von Neumann had the time to do a first mathematical reformulation in 1927 before von Neumann gave the formalization in terms of the spectral theorem for unbounded self-adjoint operators in 1929. (I note in passing that von Neumann was quite footnote-happy, at least in that paper).

Another thing I didn’t know is that the Diract δ function was first discussed by Kirchhoff in 1882 (in a paper on optics), who already explained its origin as a limit of Gaussians with variance going to 0.

During the discussion about the correct way to pronounce friable (in English) to avoid creating the impression that integers without large prime factors are to be considered as (presumably unealthy) food, it seems of some interest to indicate what type of outrageous French accent would be suitable. For this purpose, I suggest watching the very funny Bugs Bunny cartoon entitled French Rarebit (especially the part with Monsieur François and Monsieur Louie fighting over the rabbit). I can’t help picturing two irascible French mathematicians fighting over priority: “Non, non, non, Monsieur François, ze Theorem, it is mine!”, “Au contraire, Monsieur Louie, le Théorème, it is mine, not yours!”

(When I was growing up, Bugs Bunny cartoons appeared only sparingly on French television, and were a particular treat, as were the ones of Tex Avery; nowadays, I guess all of them can be found just as simply on the internet…)