Author: Kowalski

Some of the discussions during the Random Matrix, L-functions and primes conference reminded me of an old combinatorial question I had been struggling with around the time of my PhD thesis, because of some potential (but highly hypothetical) applications to automorphic forms. After moving to Bordeaux, I had the chance of having Laurent Habsieger as a colleague and I told him about the question, which he then solved within a few days, around April 2001. (He is currently Directeur de Recherche for the CNRS in Lyon).

Here’s the problem in question, which involves an arbitrary positive integer m:

Is it true, or not, that if N denotes any integer divisible by all integers up to m, then for any collection

$latex f(1),\ldots, f(N),\text{ with } 1\leq f(i)\leq m,$

of N integers up to m, there is a subset I of the integers up to N such that

$latex \sum_{i\in I}{f(i)}=N$

I see this as a kind of “intermediate value” property, since the sum over all integers of f(i) must be at least N. The condition that N be divisible by each integer up to m is necessary (because one can take each integer f(i) to be the same k in that range), and it is also clear that if N has the stated property, then any multiple of it also does (by splitting the range into subsets of size N).

I won’t explain the rather technical application I had in mind (which turned out to be so hypothetical as to vanish away anyway), but rather propose this as a challenge to combinatorialists (it might be already known, of course, though there would be a good chance Habsieger would have recognized it if it was standard). In fact, Habsieger’s proof shows that the answer is “Yes”, except possibly if m=5 or 6 (where the smallest N allowed, which is 60 here, might not work, and must be replaced, e.g., by 120 and 240, and their multiples, respectively). I believe these exceptions are just due to my ignorance in clearing up the corner cases in Habsieger’s argument, but in a sense I would be delighted if it were the case that the exceptions are genuine. (In fact, in my version, his argument works for m at least equal to 7, but the other potential exceptions can easily be treated by hand).

Since this is a challenge, I won’t reproduce the main ideas of the (quite elegant) argument, but the whole solution can be read in this write-up of his solution, which I have just put on the “Notes and unpublished results” section of my home page.

And now for the last day of the Conference… We started the morning with a talk of P.O. Dehaye, who explained his method (joint with Borodin) for computing averages of values of derivatives of characteristic polynomials of random unitary matrices. This approach generalizes to this case the method of Bump and Gamburd (as explained in Bump’s talk) and is based, besides partition and representation theory, on a theorem of Okounkov and Olshanski (involving shifted Schur functions). The talk ended with a suggestion that one should look closely at the Riemann zeta function (and other L-functions) from the point of view of symmetric functions of infinitely many variables, either from the Euler product over primes, or from the Hadamard product over zeros. This is quite a striking idea; certainly, from the analytic number theory point of view, primes are not really symmetric, because in most applications they do come up ordered with the information contained in their size. But maybe this ordering should be less important for certain problems — and the supply of problems in analytic number theory is not close to depletion…

The next talk, by H. Iwaniec, had been announced as Rational and p-adic zeros of ternary quadratic forms; however, as described in the previous instalment, the fall of a coin led him to change the topic and title to Asymptotic large sieve and moments of zeta and L-functions. It was a careful account of his recent work with B. Conrey and K. Soundararajan where, for the first time, an unconditional confirmation was obtained for one of the moment conjectures for families of L-functions beyond the fourth moment. More precisely, they have succeeded in proving the following asymptotic formula:

$latex \sum_{q\geq 1}f(q/Q)\int_{\mathbf{R}}\sum_{\chi}|L(1/2+it,\chi)|^6g(t)dt\sim 42abcQ^2\frac{(\log Q)^9}{9!}$

where χ runs over even primitive Dirichlet characters modulo q, and both f and g are bump functions, f being supported on [1,2] and the other analytic with rapid decay at infinity, and the constants which appear are

$latex a=\prod_p{(1-p^{-1})^5(1+5/p+5/p^2+14/p^3-15/p^4+5/p^5+4/p^6-4/p^7+1/p^8)},$

$latex b=\int{f(x)xdx},$

$latex c=\int_{\mathbf{R}}{g(t)|\Gamma(1/4+it/2)|^6dt}.$

In fact, quite spectacularly, this is obtained as a special case of a more general theorem involving both shifts and with complete lower-order terms (a polynomial in log q) and power-saving. Iwaniec explained very carefully the statement of the general conjecture which is confirmed by this result, and discussed briefly the main new technical innovation: an asymptotic large sieve formula for certain specific coefficients, which breaks the limit imposed by the usual large sieve inequality for Dirichlet characters.

Iwaniec mentioned that the 8th-power moment analogue is barely outside of their reach, and it might also be accessible with more work. He also very briefly explained the need for the (aesthetically displeasing) smoothing in t which is involved in the statement (and should be unnecessary): it is used to obtain cancellation in expressions of the type

$latex \left(\frac{n}{m}\right)^{it},$

unless n and m are of comparable size.

After this impressive result, the main talks of the conference concluded with a lecture by H. Montgomery who reviewed very nicely various instances of subtle combinatorics involved in performing computations of moments of arithmetic quantities of interest. One of the basic examples is found in the derivation of the expected Poisson-distribution for the number of primes in intervals of the type

$latex [n,n+c\log n]$

which Gallagher derived conditionally, based on a uniform Hardy-Littlewood conjecture for prime k-tuples (for some more on this, my paper on averages of singular series may be useful). Montgomery also discussed his work with Soundararajan on understanding the deviations between the naive probabilistic model for the number of primes in larger intervals and what is expected based on various conjectures (such as the Pair Correlation Conjecture for zeros of the Riemann zeta function).

After the free but snowy afternoon of Wednesday, and a nice conference dinner to gather new energy and strength, the conference started anew on Thursday morning. And here is what transpired that day… (I realize that those readers who only really want to know about one of the –excellent — short talks of Friday afternoon are now in a state of expectation such that even tomorrow’s relatively important event is being forgotten, but maybe this is a good thing really).

The first talk was given by M. Krishnapur and I unfortunately missed it because of teaching duties. However, reliable sources told me the talk was excellent, and he gave me a summary in the afternoon after the end of the day. It was probably the most “purely probabilistic” talk, and was devoted to ideas found in this paper of his concerning the singular values of certain random matrix-valued analytic functions, i.e., zeros of det F(z) where

$latex F(z)=\sum_{k\geq 0}{G_kz^k}$

where G_k are independent random matrices where each coefficients are themselve independent identically distributed gaussian variables. The main result is that those zeros in the unit disc form a determinantal process: this feature means that the probabilitity density for having zeros “very near” points

$latex z_1,\ \ldots,\ z_N$

is of the form

$latex \det(K(z_i,z_j))$

for some kernel function K. This is extremely useful apparently for further study of the process, and I refer to the (excellently written) introduction and Appendix 9 of Krishnapur’s paper, since I am far from being an expert here.

Next came the talk of F. Mezzadri which was again particularly fascinating (at least for me). The motivating problem he discussed was to understand the distribution of the zeros of the derivative of the characteristic polynomials of random (Haar-distributed) unitary matrices. Those, because of the Gauss-Lucas theorem, all lie inside the unit disc, and the question is to understand the distribution of the number of zeros within an annulus with a well-chosen radius close to the unit circle — normalized so as to obtain a meaningful behavior as the size N of the matrices grows. Arithmetically, this can be expected to be related to the horizontal distribution of the real parts of zeros of ζ'(s) (it is known that, under the Riemann Hypothesis, all zeros of ζ'(s) are on the right of the critical line: this is the analogue then of the Gauss-Lucas theorem).

Now this might seem like just one more question among many others, but it becomes rather more like a taunt once you look at the experimental graph

(found on page 20 of the PDF file) for the relevant density. As Mezzadri mentioned, the behavior at 0 and at infinity are fairly regular and rigorously proved, but the “middle range” is not at all theoretically understood. Of course, one can’t help noticing the two bumps there (around 1 and 3 on the graph), and clearly all self-respecting mathematicians will want to know why they are there! (It is not clear in fact that the bumps will remain in the limit N goes to infinity, but even that would require an explanation; as seen on page 29, the experimental distribution of the real parts of zeros of the derivative of the Riemann zeta function seems remarkably close, though in the expected link between the two, the heights of zeros investigated in this second graph only corresponds to matrices of size 42 instead of 100 in the first graph).

Mezzadri then discussed the current approach he is using, with Man Yue Mo, to try to understand this middle range. This involves the Riemann-Hilbert approach and Töplitz determinants, and I will not try to say more.

The last talk of the morning was by C. Delaunay. This described his recent work with F.X Roblot (Lyon) concerning the algorithmic, heuristic (and rigorous when possible) study of those quadratic twists of a given elliptic curve E/Q which have odd rank/odd sign of the functional equation, and in particular that of the twists of rank 1 (which are expected to dominate among all those odd-rank twists). The difficulties are familiar from the study of number fields, especially real quadratic fields: assuming the Birch and Swinnerton-Dyer conjecture (for which much more is known in the rank 1 case than in the general case, from the Gross-Zagier formula and the modularity of elliptic curves over Q), the resulting formula for the central derivative of the L-function

$latex L^{\prime}(E,1)$

for a rank 1 curve involves a product of the order of the Tate-Shafarevich group (a mysterious object, if ever there was one…) and of the “regulator” which measures the size of the numerator and denominator of a generator of the Mordell-Weil group. Delaunay and Roblot have managed to suggest conjectures for moments of the regulators of odd twists of the type

$latex \sum_{-d<X}{R(E_{-d})^k}\sim c X (\log X)^{b_k}$

(sum over negative discriminants such that the rank of the twist is 1) when

$latex 0<k<1$

(why this condition? because one can expect — from Cohen-Lenstra type heuristics — that the k-th moment of the order of the Tate-Shafarevich group will have a well-defined limit here). Quite significantly, they have also succeeded in accumulating a very large amount of data that gives strong hints that this is correct. For this purpose, they produce an algorithm that computes the three invariants

$latex L^{\prime}(E_{-d},1),\ R(E_{-d}),\ \text{order of the Tate-Shafarevich group}$

in less time than typical “naive” approaches would require to compute only the first one! (This is based on the Heegner point approach and modular parameterizations, with many tricks in between).

And then came a well-deserved lunch break once more (during which, to the dismay of those readers who are waiting for an account of his announced talk on rational zeros of ternary quadratic forms, H. Iwaniec was convinced by the random fall of a biased coin to switch his topic to a presentation of his recent work with Conrey and Soundararajan on the 6th-moment of special values of Dirichlet L-functions, where the mythical number 42 first appears in a rigorous manner…).

The afternoon was devoted to two related talks by G. Ricotta and E. Royer, who presented their recent works on the 1-level density of low-lying zeros of symmetric power L-functions of classical (holomorphic) modular forms. G. Ricotta first presented the background material needed, and stated the main result, where the main feature is that a lower order term is found for this 1-level density function. This behavior is expected and analogue predictions for the Riemann zeta functions were discussed by J. Keating during his talk at the conference, but not many theoretical results are known yet in this direction, so the work of Ricotta and Royer is a very useful contribution. From the number theory point of view, note that the main novelty in this type of lower-order terms is that they definitely involve number-theoretic features, related to the distribution of primes, whereas the main term is “universal” and can be interpreted purely in terms of Random Matrices.

The second part of the talk, by E. Royer, described some features of the arguments used in the proof. His emphasis was on the fact that computations which could easily look as if they were inextricable in this setting, were in fact fairly simple — and in fact elegant — when the right point of view was taken: namely, the question is to understand various polynomial averages of local Hecke-eigenvalues

$latex \sum_{k}{c_k \lambda_f(p^k)},\text{ or }\sum_{k}{d_k\lambda_f(p)^k}.$

Such expressions are, in some sense, equivalent: each can be transformed into the other, but if one is not careful, this leads to a lot of combinatorial identities which can seem pretty obscure, when in fact the crucial insight is to use systematically the basis of the space of polynomials given by the Chebychev polynomials X_n such that

$latex X_n(\lambda_f(p))=\lambda_f(p^n)$

(for unramified primes). This is often obvious and barely worth stating for algebraists, but not at all for analytic number theorists who have not learned early enough of the representation theory of SL(2). Royer mentioned a few combinatorial identities that can be quickly derived from such considerations in a much cleaner way than the direct “binomial-coefficient” approach that is also possible (see the Appendix to the paper).

After Tuesday, naturally, came Wednesday. We had planned a half-day of talks only, with a free afternoon, and at some point it seemed like a rather poor choice since Wednesday morning saw the worse snowfall ever to happen in Zürich in October (at least since a certain time, I guess):

(I did mention in my post-doc marketing post that the weather here is not always that of California…)

Although the skies cleared up a bit in early afternoon, it was again snowing quite strongly by the time of the conference dinner in the evening. But in the morning, we had three very interesting talks:

(1) D. Bump explained the method used by A. Gamburd and himself to prove the formulas for the average over unitary matrices of values (at a fixed point) of characteristic polynomials. This proof comes historically before the probabilistic arguments that C. Hughes had described on Tuesday, and contains also a lot of interesting features from the point of view of the representation theory of unitary groups and of symmetric groups, and their inter-relations. In this setting, the average over U(N) (equipped with Haar measure) of

$latex |\det(1-A)|^{2k}$

(where k is an integer) appears as the dimension of a representation of the unitary group U(2k) (a representation which depends on N of course). This dimension is then computed using the Weyl dimension formula. As background for the type of structure that emerges with the variation in N, Bump suggested to read A. Zelevinsky’s short book “Representations of Finite Classical Groups: A Hopf Algebra Approach” (Springer Lecture Notes 869, 1981) — which I intend to (try to) do as soon as possible.

(2) The next lecture was given partly by Nina Snaith and partly by her student Duc-Khiem Huynh, and described their joint work in trying to understand some surprising features of the observed location of low-lying zeros of L-functions of elliptic curves over the rationals. More precisely, for such an L-function

$latex L(E,s)=\sum_{n\geq 1}{a_E(n)n^{-s}}$

with conductor N, it is natural from the point of view of Random Matrix Theory (to distinguish the possible symmetry types) to compute the normalized ordinate of the first zero

$latex \tilde{\rho}_E=\frac{\log N}{2\pi}\gamma_E$

where

$latex L(E,1/2+i\gamma_E)=0,\ \gamma_E\geq 0$

and γ_E is the first such ordinate of a zero. (Experimentally, of course, it is found that it is on the critical line). It turns out, experimentally, that the distribution of this low-lying zero does not at all look like what the Random Matrix model suggests, at least for currently available data (this was first described by S. J. Miller; the basic problem is that the histograms show a repulsion at the origin). The lecture was then devoted to explain how more refined models could lead to possible explanations of these features; in particular, it was suggested that the discretisation feature of the values of the L-function at 1/2 could be a source of this discrepancy.

(3) To conclude the morning, D. Farmer gave a very nice description of some issues surrounding one of the features of the correspondance between Random Matrices and L-functions that is often taken for granted: the link between the height T (for the Riemann zeta function on the critical line; it would be the conductor for other families) and the size N of matrices given by

$latex N=\frac{\log T}{2\pi}.$

This is usually justified very quickly as “equating the mean-density of zeros” (there are N zeros of the caracteristic polynomial on the unit circle of length 2π, and about log T/2π zeros of ζ(1/2+it) in a vertical interval of length 1 around T), but D. Farmer showed that one can say much more than that, and that the connection is still somewhat mysterious.

I think most people who have organized a conference (even in such outstanding conditions as provided by the Forschungsinstitut für Mathematik) will not be surprised that my promise of posting daily updates was not fulfilled. However, the web page where we have collected the PDF files with the contents of the beamer talks has been updated, and not too many are missing — and those are mostly because the speakers wanted to have a look and make a few corrections before making them available. We do not have notes, however, for those talks that were given on the blackboard or with old-fashioned slides — but I will describe all of them briefly in this post and the next ones…

In keeping with a noticeable need for rest, I will only continue my survey of the conference up to Tuesday and therefore up to N. Katz’s colloquium lecture.

(1) We started with Z. Rudnick’s discussion of his recent works with Kurlberg and with Faifman concerning certain cases where it is possible to consider a limit of curves over finite fields with increasing genus without needing to first have the finite base field become larger and larger (the “honest” limit if one is most concerned with analogies with the number field case, and in particular with the Riemann zeta function). This concerns the family of hyperelliptic curves, and the reason it is easier to control is related to their interpretation as families of quadratic L-functions with increasing conductor — so the analogy pays off, in fact, partly because such families have already been extensively studied over number fields!

(2) The next talk was given by D. Ulmer, who explained how to produce many examples (even parameterized families) of elliptic curves over function fields over finite fields with explicit points of infinite order generating a finite index subgroup of the Mordell-Weil group, such examples having unbounded rank (a feat which no one knows how to do over number fields). This involved some discussion of rather lovely algebraic geometry, and leads to the hope that such examples will be useful for numerical experiments in testing conjectures of various types.

(3) Following this was a talk by A. Gamburd on his recent work with Bourgain and Sarnak concerning the use of expanding Cayley graphs in attacking sieve problems of “hyperbolic” nature, and also how they managed to obtain the necessary tremendous expansion (pun intended) of the number of situations where expanders arise from reductions modulo primes of a subgroup of SL(2,Z) satisfying the minimal assumption that it is Zariski dense in SL(2). This involves a lot of insights from additive combinatorics (sum-product estimates), and the breakthrough of Helfgott concerning growth of subsets of SL(2,F_p) under products.

(4) After a well-deserved lunch break, the lecture of P. Biane (“Brownian motion on matrix spaces and symmetric spaces, and some connections with Riemann zeta function”) gave a very nice introduction to an interpretation of the statistics of eigenvalues of random unitary matrices based on Brownian motion conditioned (in a suitable way) to remain within certain domains (Weyl chambers or alcoves). This was something I was not at all aware of (and I think the same was true for many of the number-theorists in the audience), and I think this was an excellent example of the type of “food for thought” we hoped that this conference could provide. We suspect that those eigenvalues are highly relevant in analytic number theory, but we do not know at the moment how this comes about, and any insight that may lead to a different way of thinking of the problem seems beneficial. [Update (11/11/08): P. Biane has just posted on arXiv a paper about the topic of his talk].

(5) There followed an equally insightful and enthusiastic lecture of C. Hughes, presenting the recent probabilistic interpretation of Haar measure on unitary matrices due to P. Bourgade, A. Nikeghbali, M. Yor and himself. This interpretation leads to a decomposition of the value of the characteristic polynomial (at a given point of the unit circle) as a product of fairly simple independent random variables, and from there it is easy to get either new proofs of known results concerning the distribution of those values (e.g., the moments of the values, first computed by Keating and Snaith), or new results which seem hard to derive from the other approaches.

(6) And the final lecture was N. Katz’s colloquium, “Some simple things we don’t known”. The type of question considered was the following: suppose you play a two person game where one person selects an integral polynomial f (of some odd degree 2g+1), without saying which polynomial it is, nor even what g is. Suppose further that this first player then gives to the second player two pieces of data: (1) an integer N such that all prime divisors of the discriminant of f divide N; (2) for every prime not dividing N, the number n_p of solutions (x,y) of

$latex y^2=f(x)\text{ modulo } p,\text{ where } (x,y)\in \mathbf{F}_p^2.$

Then the second player’s goal is to compute the integer g (which is the genus of the underlying hyperelliptic curve). As Katz explained, this should be possible, conjecturally, by computing

$latex \limsup_{p\rightarrow+\infty}{\left|\frac{n_p-p}{\sqrt{p}}\right|},$

because this quantity should be exactly 2g. However, although we know since A. Weil’s proof of the Riemann Hypothesis for curves that

$latex \limsup_{p\rightarrow+\infty}{\left|\frac{n_p-p}{\sqrt{p}}\right|}\leq 2g,$

obtaining equality is intimately related with particular cases of the generalized Sato-Tate conjecture, and this is unknown for any polynomial f with g>1 (except for very special exceptions, having to do with curves whose jacobians are isogenous to a product of the same elliptic curve)!