Random matrices, L-functions and primes, III

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,Fp) 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 np of solutions (x,y) of

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

\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

\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)!

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Kowalski

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

One thought on “Random matrices, L-functions and primes, III”

  1. Dear Sir,
    I wonder if you would be kind enough to help me with this problem: it’s certainly an old one. Take m to be some modulus and say 0<= r 1 – as we have for d(n) and the like.

    I can’t for the life of me make any progress here. Oddly enough this function does not appear in Apostol’s Vol I. Perhaps it is somewhat trivial.

    Many thanks for your time,

    Sincerely,
    Arun

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