Peter Sarnak

Extremals of determinants of Laplacians

On etudie le determinant associe au laplacien en fonction de la metrique sur une surface donnee et en particulier ses valeurs extremes quand la metrique est bien restreinte

The behaviour of eigenstates of arithmetic hyperbolic manifolds

In this paper we study some problems arising from the theory of Quantum Chaos, in the context of arithmetic hyperbolic manifolds. We show that there is no strong localization (“scarring”) onto totally geodesic submanifolds. Arithmetic examples are given, which show that the random wave model for eigenstates does not apply universally in 3 degrees of freedom.

Low lying zeros of families of L-functions

In Iwaniec-Sarnak [IS] the percentages of nonvanishing of central values of families of GL_2 automorphic L-functions was investigated. In this paper we examine the distribution of zeros which are at or neat s=1/2 (that is the central point) for such families of L-functions. Unlike [IS], most of the results in this paper are conditional, depending on the Generalized Riemann Hypothesis (GRH). It is by no means obvious, but on the other hand not surprising, that this allows us to obtain sharper results on nonvanishing.

Zeroes of zeta functions and symmetry

Hilbert and Polya suggested that there might be a natural spectral interpretation of the zeroes of the Riemann Zeta function. While at the time there was little evidence for this, today the evidence is quite convincing. Firstly, there are the “function field” analogues, that is zeta functions of curves over finite fields and their generalizations. For these a spectral interpretation for their zeroes exists in terms of eigenvalues of Frobenius on cohomology. Secondly, the developments, both theoretical and numerical, on the local spacing distributions between the high zeroes of the zeta function and its generalizations give striking evidence for such a spectral connection. Moreover, the low-lying zeroes of various families of zeta functions follow laws for the eigenvalue distributions of members of the classical groups. In this paper we review these developments. In order to present the material fluently, we do not proceed in chronological order of discovery. Also, in concentrating entirely on the subject matter of the title, we are ignoring the standard body of important work that has been done on the zeta function and L-functions. 1. The Montgomery-Odlyzko Law We begin with the Riemann Zeta function and some phenomenology associated with it.

Explicit expanders and the Ramanujan conjectures

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Determinants of Laplacians

The determinant of the Laplacian on spinor fields on a Riemann surface is evaluated in terms of the value of the Selberg zeta function at the middle of the critical strip. A key role in deriving this relation is played by the Barnes double gamma function.

ON SELBERG'S EIGENVALUE CONJECTURE

Let Γ ⊂ SL 2(Z) be a congruence subgroup, and λ0 = 0 3/16. Iwaniec ([I]) showed that for almost all Hecke congruence groups Γ0(p) with a certain multiplier χ p , one has λ1(Γ0(p), χ p ) ≥ 44/225 = 0.19555…. In [I], he also established a density theorem for possible exceptional eigenvalues as above, which while not giving any improvement on 3/16 for an individual Γ is sufficiently strong to substitute for Selberg’s conjecture in many applications to number theory. Selberg’s conjecture is the archimedean analogue of the “Ramanujan Conjectures” on the Fourier coefficients of Maass forms. For these, much progress has been made in improving the relevant estimates, beginning with Serre ([Ser]) and later on Shahidi ([Sh2]) and Bump-Duke-Hoffstein-Iwaniec ([BDHI]). In this paper we restore the balance and establish in part for the archimedean place what is known at the finite places. The method on the face of it is quite different, but the quality of the results coincide (the reason will be made clear later).

Chebyshev's bias

The title refers to the fact, noted by Chebyshev in 1853, that primes congruent to 3 modulo 4 seem to predominate over those congruent to 1. We study this phenomenon and its generalizations. Assuming the Generalized Riemann Hypothesis and the Grand Simplicity Hypothesis (about the zeros of the Dirichlet L-function), we can characterize exactly those moduli and residue classes for which the bias is present. We also give results of numerical investigations on the prevalence of the bias for several moduli. Finally, we briefly discuss generalizations of the bias to the distribution to primes in ideal classes in number fields, and to prime geodesics in homology classes on hyperbolic surfaces.

Perspectives on the Analytic Theory of L-Functions

To the general mathematician L-functions might appear to be an esoteric and special topic in number theory. We hope that the discussion below will convince the reader otherwise. Time and again it has turned out that the crux of a problem lies in the theory of these functions. At some level it is not entirely clear to us why L-functions should enter decisively, though in hindsight one can give reasons. Our plan is to introduce L-functions and describe the central problems connected with them. We give a sample (this is certainly not meant to be a survey) of results towards these conjectures as well as some problems that can be resolved by finessing these conjectures. We also mention briefly some of the successful present-day tools and the role they might play in the big picture.

On the period matrix of a Riemann surface of large genus (with an Appendix by J. H. Conway and N. J. A. Sloane).

SummaryRiemann showed that a period matrix of a compact Riemann surface of genusg≧1 satisfies certain relations. We give a further simple combinatorial property, related to the length of the shortest non-zero lattice vector, satisfied by such a period matrix, see (1.13). In particular, it is shown that for large genus the entire locus of Jacobians lies in a very small neighborhood of the boundary of the space of principally polarized abelian varieties.We apply this to the problem of congruence subgroups of arithmetic lattices in SL2(ℝ). We show that, with the exception of a finite number of arithmetic lattices in SL2(ℝ), every such lattice has a subgroup of index at most 2 which is noncongruence. A notable exception is the modular groupSL2(ℤ).