Abstract. Using the spectral multiplicities of the standard torus, we endow the Laplace eigenspaces with Gaussian probability measures. This induces a notion of random Gaussian Laplace eigenfunctions on the torus ("arithmetic random waves"). We study the distribution of the nodal length of random eigenfunctions for large eigenvalues, and our primary result is that the asymptotics for the variance is non-universal, and is intimately related to the arithmetic of lattice points lying on a circle with radius corresponding to the energy.

Abstract. We study semi-classical limits of eigenfunctions of a quantized linear hyperbolic automorphism of the torus ("cat map"). For some values of Planck's constant, the spectrum of the quantized map has large degeneracies. Our first goal in this paper is to show that these degeneracies are coupled to the existence of quantum symmetries. There is a commutative group of unitary operators on the state-space which commute with the quantized map and therefore act on its eigenspaces. We call these "Hecke operators", in analogy with the setting of the modular surface.We call the eigenstates of both the quantized map and of all the Hecke operators "Hecke eigenfunctions". Our second goal is to study the semiclassical limit of the Hecke eigenfunctions. We will show that they become equidistributed with respect to Liouville measure, that is the expectation values of quantum observables in these eigenstates converge to the classical phase-space average of the observable.

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The number of points on a hyperelliptic curve over a field of q elements may be expressed as q + 1 + S where S is a certain character sum. We study fluctuations of S as the curve varies over a large family of hyperelliptic curves of genus g. For fixed genus and growing q, Katz and Sarnak showed that S/ √ q is distributed as the trace of a random 2g × 2g unitary symplectic matrix.When the finite field is fixed and the genus grows, we find that the limiting distribution of S is that of a sum of q independent trinomial random variables taking the values ±1 with probabilities 1/2(1 + q −1 ) and the value 0 with probability 1/(q + 1). When both the genus and the finite field grow, we find that S/ √ q has a standard Gaussian distribution.

We prove a strong version of quantum ergodicity for linear hyperbolic maps of the torus ("cat maps"). We show that there is a density one sequence of integers so that as N tends to infinity along this sequence, all eigenfunctions of the quantum propagator at inverse Planck constant N are uniformly distributed.A key step in the argument is to show that for a hyperbolic matrix in the modular group, there is a density one sequence of integers N for which its order (or period) modulo N is somewhat larger than √ N .