ON THE LIMITING BEHAVIOUR OF ARITHMETIC TORAL EIGENFUNCTIONS

We consider a wide class of families (Fm)m is an element of N of Gaussian fields on 1Cd = Rd/Zd defined by Fm : x-* 1 _ X p|Am| l lambda ei2 pi lambda is an element of Lambda m where the l lambda's are independent standard normals and Am is the set of solutions lambda is an element of Zd to the equation p(lambda) = m for some fixed elliptic polynomial p with integer coefficients. The case p(x) = x21 ++x2d amounts to considering a random Laplace eigenfunction whose law is sometimes called the arithmetic random wave and has been studied in the past by many authors. In contrast, we consider three classes of polynomials p: a certain family of positive definite quadratic forms in two variables, all positive definite quadratic forms in three variables except the multiples of x21 + x22 + x23, and a wide family of polynomials in many variables. For these three classes of polynomials, we study the (d - 1)-dimensional volume Vm of the zero set of Fm. We compute the asymptotics, as m * +infinity along certain well chosen subsequences of integers, of the expectation and variance of Vm. Moreover, we prove that in the same limit, Vm-E[Vm] root Var(Vm) converges to a standard normal. As in previous analogous works on this topic for the arithmetic random wave, a very general method reduces the problem of these asymptotics to the study of certain arithmetic properties of the sets of solutions to p(lambda) = m. More precisely, we need to study the number of such solutions for a fixed m, as well as the number of quadruples of solutions (lambda, mu, nu, iota) satisfying lambda+mu+nu+iota = 0, a.k.a. 4-correlations, and the rate of convergence of the (rescaled) counting measure of Am towards a certain limiting measure on the hypersurface {p(x) = 1}. To this end, we use many previous results on this topic but also prove a new estimate on correlations which may be of independent interest.