{"id":138,"date":"2008-11-02T09:45:00","date_gmt":"2008-11-02T08:45:00","guid":{"rendered":"https:\/\/wpethzprd.ethz.ch\/kowalski\/2008\/11\/02\/random-matrices-l-functions-and-primes-what-was-said-on-wednesday\/"},"modified":"2024-02-20T13:40:57","modified_gmt":"2024-02-20T11:40:57","slug":"random-matrices-l-functions-and-primes-what-was-said-on-wednesday","status":"publish","type":"post","link":"https:\/\/blogs.ethz.ch\/kowalski\/2008\/11\/02\/random-matrices-l-functions-and-primes-what-was-said-on-wednesday\/","title":{"rendered":"Random matrices, L-functions and primes: what was said on Wednesday"},"content":{"rendered":"

After Tuesday<\/a>, 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 Z\u00fcrich in October (at least since a certain time, I guess):<\/p>\n

\"Snow<\/a><\/p>\n

(I did<\/em> mention in my post-doc marketing post <\/a>that the weather here is not always that of California…)<\/p>\n

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:<\/p>\n

(1) D. Bump explained<\/a> 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)<\/em> (equipped with Haar measure) of<\/p>\n

$latex |\\det(1-A)|^{2k}$<\/p>\n

(where k<\/em> is an integer) appears as the dimension of a representation of the unitary group U(2k)<\/em> (a representation which depends on N<\/em> 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<\/em>, Bump suggested to read A. Zelevinsky’s short book “Representations of Finite Classical Groups: A Hopf Algebra Approach”<\/em> (Springer Lecture Notes 869, 1981) — which I intend to (try to) do as soon as possible.<\/p>\n

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

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

with conductor N<\/em>, 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<\/p>\n

$latex \\tilde{\\rho}_E=\\frac{\\log N}{2\\pi}\\gamma_E$<\/p>\n

where<\/p>\n

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

and \u256c\u2502E<\/sub><\/em> 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<\/em>-function at 1\/2 could be a source of this discrepancy.<\/p>\n

(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<\/em>-functions that is often taken for granted: the link between the height T<\/em> (for the Riemann zeta function on the critical line; it would be the conductor for other families) and the size N<\/em> of matrices given by<\/p>\n

$latex N=\\frac{\\log T}{2\\pi}.$<\/p>\n

This is usually justified very quickly as “equating the mean-density of zeros” (there are N<\/em> zeros of the caracteristic polynomial on the unit circle of length 2\u00a4\u00c7<\/em>, and about log T\/2\u00a4\u00c7<\/em> zeros of \u256c\u00c2(1\/2+it)<\/em> in a vertical interval of length 1 around T<\/em>), but D. Farmer showed that one can say much more than that, and that the connection is still somewhat mysterious.<\/p>\n","protected":false},"excerpt":{"rendered":"

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 Z\u00fcrich in October (at least since a certain time, I guess): (I did mention in my … Continue reading Random matrices, L-functions and primes: what was said on Wednesday<\/span><\/a><\/p>\n","protected":false},"author":625,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[2],"tags":[],"class_list":["post-138","post","type-post","status-publish","format-standard","hentry","category-blogroll"],"_links":{"self":[{"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/posts\/138","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/users\/625"}],"replies":[{"embeddable":true,"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/comments?post=138"}],"version-history":[{"count":0,"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/posts\/138\/revisions"}],"wp:attachment":[{"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/media?parent=138"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/categories?post=138"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/tags?post=138"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}