# Random matrices, L-functions and primes: what was said on Wednesday

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

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

(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

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

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

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

where

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

and γE 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

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

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 , 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.