Page 33 – E. Kowalski's blog

Families of cusp forms

I do not know who first formulated consciously the idea that there should exist a good formal definition of what is a “family of $L$ -functions”, or “family of cusp forms”. Such ideas were obviously present in the late 1990’s when it was discovered that there were links between special values of $L$ -functions and statistics of random matrices in families of compact classical groups, and when this was given a rigorous expression in the case of $L$ -functions of algebraic origin over finite fields by Katz and Sarnak. Specific families of various kinds and their fundamental properties were also very clearly present in the work of Iwaniec and Sarnak on non-vanishing of central values of Hecke $L$ -functions of modular forms around the same time. However, these families were one of these mathematical objects which people are happy to recognize when they see them, but which seemed difficult to define beforehand in the right generality.

For some reason, this is a question I’ve been thinking about every once in a while since that “old” time. A paper of Conrey-Farmer-Keating-Rubinstein-Snaith proposed a good heuristic proposal of what families of $L$ -functions should be, in order to be able to make robust conjectures concerning averages of products/ratios of shifted values of these $L$ -functions. However (to my mind) this proposal did not really give any tool to attempt a general theory, and it certainly was not a Bourbakist-style definition. Around 2006, when I was finishing my (almost award-winning) book on the large sieve, I noticed a clear analogy between certain problems about families of cusp forms and the formalism I was using for sieve. This convinced me that the starting point for any good notion of “family of $L$ -functions” should be the underlying cusp forms, or rather automorphic representations when dealing with the general case, and that the first principle must be: if some collection of automorphic representations (say over the rationals) deserves the name of “family”, it must be the case that, for any prime number $p$ , the family of $p$ -components must also be well-behaved. As it turns out, this local behavior can be given a clear meaning, and this provides a “test” for any attempted definition. In fact, for classical modular forms, this good local behavior amounts to the question of equidistribution of eigenvalues of Hecke operators, which had been treated (in various generality) by Bruggeman (a bit implicitly), Sarnak, Serre, Conrey-Duke-Farmer and Royer (and a few others). This seemed a good indication that this was the right track. (To be precise, these results only handle unramified primes; especially when the local components are or can be supercuspidals, it is not so easy to get a handle on the underlying asymptotic distribution properties…)

I happened to talk about this in 2009 with P. Sarnak (during the Verbania conference) and he told me that he had also formulated a definition of families, which was sufficient to predict a “symmetry type” for any associated family of $L$ -functions (built in the Langlands style using arbitrary representations of the $L$ -group). This was explained in an unpublished letter he had written a few months ago, which is now available on his new “blog”.

We discussed the issue further in late 2009 when I was in Princeton. Later, the philosophy of “local spectral equidistribution” (as I had decided to call the idea behind the analogy with sieve) was the background of the papers I wrote with A. Saha and J. Tsimerman last year, where some special families of Siegel modular forms were considered. But it is only recently (i.e., during the blessed summer time of mellow fruitful-writingness) that I found time to attempt to write a consistent and somewhat detailed account of this point of view on families of cusp forms. This account can be found here. It is very informal, but hopefully readable. The summary of those papers with Saha and Tsimerman (in Section 13) might interest those readers who know — and like! — Böcherer’s fascinating conjecture concerning the arithmetic nature of Fourier coefficients of Siegel cuspidal eigenforms of genus 2, and their expected relation with special $L$ -values of spinor $L$ -functions.

Expanders course

I am teaching a class on expander graphs this semester, and as usual there is a web page about it. For the moment, do not be fooled by the link to “lecture notes”, since I haven’t put up anything yet (I am trying to learn the easiest way to incorporate nice pictures of graphs in LaTeX; tkz-graph and tkz-berge seem to be the best way to do that…)

The course meets only 2 hours per week (and those are ETH hours, which means 1h30 really), so it will necessarily be very restricted in what I can cover. I have decided on the content of the first two parts: a general discussion of elementary material (of course), and then a full account of the combination of the results of Helfgott and Bourgain–Gamburd, which together imply that the family of Cayley graphs of $\mathrm{SL}_2(\mathbf{F}_{p})$ , with respect to the projections modulo primes $p$ of a subset $S\subset \mathrm{SL}_2(\mathbf{Z})$ which generates a Zariski-dense subgroup of $\mathrm{SL}_2$ , is an expander family.

If there is time after this, which I hope, I haven’t quite decided what to do. One possibility would be an account of the “expander philosophy” discussed at the end of this post, another would be to discuss some applications to theoretical computer science (admittedly, this would be because I would learn about them much more than I know at the current time…)

Another exercise on compact groups

While writing the chapters about compact groups in my notes, I had a few times the impression that it would be useful to use the fact that there is a basis of conjugacy-invariant neighborhoods of 1 in such a group. This thought would then fork in two subthreads, one in which I noticed that I didn’t really see how to prove this, and the second in which I realized I didn’t need it anyway.
This happened again during the last week-end, with the difference that I thought of trying to prove this property using representation theory. I failed at first, and finally tried to look it up online to make sure the property was actually true. Indeed, it is, and I found this book, which has three proofs (of a slightly more general statement). But I have to admit to having an inferiority complex with respect to general topology: any argument that goes on for more than a few lines without being transparent tends to make me uneasy and confuse me, and this is what happened with these arguments.

So I tried again to use representations, and indeed it works! Here’s the idea: the regular representation $\rho$ of $G$ on $H=L^2(G)$ (with respect to Haar measure) is faithful, and gives a continuous injection
$G\rightarrow U(H)$
where the unitary group $U(H)$ is given the strong operator topology (if $G$ is infinite, it is not continuous for the operator norm topology). Hence this map is a homeomorphism onto its image (we use here the compactness of $G$ ). What we gain from seeing $G$ in this way as a subgroup of the unitary group of a (typically) infinite-dimensional space, and with a “strange” topology to boot, is a concrete description of a basis of open neighborhoods of 1, which is open to further manipulations.

Indeed, from the definition of the strong topology on $U(H)$ , any open neighborhood $U$ of 1 contains a finite intersection of sets of the type
$U_{f,\epsilon}=\{g\in G\,\mid\, \|\rho(g)f-f\|\text{\textless} \epsilon\}$
where $f\in H$ has norm 1 and $\epsilon\text{\textgreater} 0$ . It is then easy to see, using unitarity, that the conjugacy-invariant subset
$V_{f,\epsilon}=\bigcap_{x\in G}{x^{-1}U_{f,\epsilon}x}\subset U_{f,\epsilon}$
is equal to
$V_{f,\epsilon}=\{g\in G\,\mid\, \|\rho(g)\varphi-\varphi\|\text{\textless} \epsilon\text{ for all }\varphi\in A_f\}$
where
$A_f=\{\rho(x)f\,\mid\, x\in G\}\subset H$
is the orbit of $f$ under $G$ . But the strong continuity of $\rho$ implies that the orbit map $g\mapsto \rho(g)f$ is continuous for fixed $f$ , so that $A_f$ is compact in $H$ (as image of a compact set under a continuous map; here $H$ has the usual norm topology).
It is then a quick application of a standard compactness argument and splitting of epsilons to check that $V_{f,\epsilon}$ is a neighborhood of 1 contained in $U_{f,\epsilon}$ , and intersecting finitely many of these, we see that $U$ contains indeed a conjugacy invariant neighborhood of 1…

I really can’t say that it is simpler than the purely topological argument (mostly because one needs to know about Haar measure!), but I find this rather nice as an exercise. It illustrates how representation theory can be useful to study a general compact group, at a very basic level, and also shows that it may be useful to embed (or project, with the orbit map) a compact group into complicated-looking infinite-dimensional beasts… (Of course, if $G$ has a faithful finite-dimensional representation, this can be used instead of $\rho$ , but the purely topological argument becomes also much simpler.)

With many thanks to Dickinson State College

I said in the last post that I didn’t know anything about Arthur Schuster before reading his quote on hearing the shape of a bell. Actually that was not quite true: he is mentioned three times in the book of Max Jammer on the history of Quantum Mechanics, and one reference leads to the source of the quote. This was a report for the 1882 meeting of the British Association for the Advancement of Science, held in Southampton in August 1882. The full report of the meeting can be read online, and Schuster’s paper starts on page 120, the baffled skillful mathematician appearing at the end of this first page (I’ve also prepared a PDF of these two pages).

Now I have to thank whoever decided to withdraw Max Jammer’s book from the library of Dickinson State University (née Dickinson State College), which is where the copy I recently got from BetterWorldBooks came from…

The holder for the library slip

is still in the book, and it was apparently only borrowed twice, once in 1978 and once in 1982 (or maybe 1992).

Shapes of bells and shapes of drums

Who first asked “Can one hear the shape of a drum?” When I heard this question, it was attributed to Mark Kac, who wrote (in 1966) a famous paper in the American Mathematical Monthly with this title (this must have been around the time I first heard of Property ]T[ and of expanders, and — name-dropping warning — it might have been from any of Y. Colin de Verdière, G. Besson, P. Bérard or C. Bavard). I have to admit that I only had the curiosity to look at this paper two or three years ago, and I was amused to see the following quote (on top of page 3):

I first heard the problem posed this way some ten years ago from Professor Bochner. Much more recently, when I mentioned it to Professor Bers, he said, almost at once, “You mean, if you had perfect pitch could you find the shape of a drum.”

So the mathematical question goes back further than Kac, and the elegant turn of phrase is really due to someone else!

But this post is motivated by another quote, which I read today in Appendix F of Schlomo¹ Sternberg’s book “Group theory and physics”, where he gives a quick survey of 19th Century spectroscopy. The word itself was introduced in 1882 by Arthur Schuster, who goes on to say (top of page 395 in Sternberg’s book, italics mine):

To find out the different tunes sent out by a vibrating system is a problem which may or may not be solvable in certain special cases, but it would baffle the most skillful mathematician to solve the inverse problem and to find out the shape of a bell by means of the sounds which it is capable of giving out. And this is the problem which ultimately spectroscopy hopes to solve in the case of light. In the meantime we must welcome with delight even the smallest step in the right direction.

I didn’t know Schuster before, but he seems to have been a very interesting character, who pioneered the use of harmonic analysis and probability to detect (non)-periodicity in certain data sets (e.g., dates of earthquakes, in 1897), and also speculated about “antimatter” (using that word) in 1898…

(I haven’t found yet the exact source quoted by Sternberg, who doesn’t have a bibliography for this Appendix…)

Footnote: ¹ No relation to Schlomo Cohen.