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

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.Comment: 37 pages. New title. Spelling mistake in bibliography corrected. To appear in Duke Math.

Non-stationarity of isomorphism between AF algebras defined by stationary Bratteli diagrams

We first study situations where the stable AF-algebras defined by two square primitive nonsingular incidence matrices with nonnegative integer matrix elements are isomorphic even though no powers of the associated automorphisms of the corresponding dimension groups are isomorphic. More generally we consider neccessary and sufficient conditions for two such matrices to determine isomorphic dimension groups. We give several examples.Comment: 16 page

Elliptic divisibility sequences and undecidable problems about rational points

Julia Robinson has given a first-order definition of the rational integers Z in the rational numbers Q by a formula (\forall \exists \forall \exists)(F=0) where the \forall-quantifiers run over a total of 8 variables, and where F is a polynomial. This implies that the \Sigma_5-theory of Q is undecidable. We prove that a conjecture about elliptic curves provides an interpretation of Z in Q with quantifier complexity \forall \exists, involving only one universally quantified variable. This improves the complexity of defining Z in Q in two ways, and implies that the \Sigma_3-theory, and even the \Pi_2-theory, of Q is undecidable (recall that Hilbert's Tenth Problem for Q is the question whether the \Sigma_1-theory of Q is undecidable). In short, granting the conjecture, there is a one-parameter family of hypersurfaces over Q for which one cannot decide whether or not they all have a rational point. The conjecture is related to properties of elliptic divisibility sequences on an elliptic curve and its image under rational 2-descent, namely existence of primitive divisors in suitable residue classes, and we discuss how to prove weaker-in-density versions of the conjecture and present some heuristics.Comment: 39 pages, uses calrsfs. 3rd version: many small changes, change of titl