Zeév Rudnick

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.

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).

Hecke theory and equidistribution for the quantization of linear maps of the torus

1.1. Background. One of the key issues of “Quantum Chaos” is the nature of the semiclassical limit of eigenstates of classically chaotic systems. When the classical system is given by the geodesic flow on a compact Riemannian manifoldM (or rather, on its cotangent bundle), one can formulate the problem as follows: The quantum Hamiltonian is, in suitable units, represented by the positive Laplacian − on M . To measure the distribution of its eigenstates, we start with a (smooth) classical observable, that is, a (smooth) function on the unit cotangent bundle S∗M; via some choice of quantization from symbols to pseudodifferential operators, we form its quantization Op(f ). This is a zero-order pseudodifferential operator with principal symbol f . The expectation value of Op(f ) in the eigenstate ψ is 〈Op(f )ψ,ψ〉. Letψj be a sequence of normalized eigenfunctions: ψj+λjψj = 0, ∫ M |ψj |2= 1. The problem then is to understand the possible limits as λj →∞ of the distributions f ∈ C∞(S∗M) −→ 〈Op(f )ψj ,ψj 〉. (1.1)

On the asymptotic distribution of zeros of modular forms

We study the distribution of zeros of holomorphic modular forms. Assuming the Generalized Riemann Hypothesis we show that the zeros of Hecke eigenforms for the modular group become equidistributed with respect to the hyperbolic measure on the modular domain as the weight grows.

Lower bounds for moments of L-functions

The moments of central values of families of L-functions have recently attracted much attention and, with the work of Keating and Snaith [(2000) Commun. Math. Phys. 214, 57-89 and 91-110], there are now precise conjectures for their limiting values. We develop a simple method to establish lower bounds of the conjectured order of magnitude for several such families of L-functions. As an example we work out the case of the family of all Dirichlet L-functions to a prime modulus.

Eigenvalue Spacings for Regular Graphs

We carry out a numerical study of fluctuations in the spectra of regular graphs. Our experiments indicate that the level spacing distribution of a generic k-regular graph approaches that of the Gaussian Orthogonal Ensemble of random matrix theory as we increase the number of vertices. A review of the basic facts on graphs and their spectra is included.

The distribution of spacings between quadratic residues

We study the distribution of spacings between squares modulo q, where q is square-free and highly composite, in the limit as the number of prime factors of q goes to infinity. We show that all correlation functions are Poissonian, which among other things, implies that the spacings between nearest neighbors, normalized to have unit mean, have an exponential distribution. Date: Dec 14, 1998. Supported in part by a grant from the Israel Science Foundation. In addition, the first author was partially supported by the EC TMR network ”Algebraic Lie Representations”, EC-contract no ERB FMRX-CT97-0100. 1

On the Volume of Nodal Sets for Eigenfunctions of the Laplacian on the Torus

Abstract.We study the volume of nodal sets for eigenfunctions of the Laplacian on the standard torus in two or more dimensions. We consider a sequence of eigenvalues 4π2E with growing multiplicity

STATISTICS OF THE ZEROS OF ZETA FUNCTIONS IN FAMILIES OF HYPERELLIPTIC CURVES OVER A FINITE FIELD

{\mathcal{N}} \rightarrow \infty