Related papers: Equidistribution in S-arithmetic and adelic spaces

In previous work, the authors introduced "soft" methods to prove the effective (i.e. with power savings error) equidistribution of "shears" in cusped hyperbolic surfaces. In this paper, we study the same problem but now allow full use of…

We prove a joint value equidistribution statement for Hecke-Maa{\ss} cusp forms on the hyperbolic three-space $\mathbb{H}^3$. This supports the conjectural statistical independence of orthogonal cusp forms.

We study the dynamics of stochastic families of rational maps on the projective line. As such families can be infinite and may not typically be defined over a single number field, we introduce the concept of generalized adelic measures,…

We study the growth of the number of conjugacy classes of infinite dihedral subgroups of lattices in PSL(2,R), generalizing earlier work of Sarnak and Bourgain-Kontorovich on the growth of the number of reciprocal geodesics on the modular…

We introduce variational approximations for curve evolutions in two-dimensional Riemannian manifolds that are conformally flat, i.e.\ conformally equivalent to the Euclidean space. Examples include the hyperbolic plane, the hyperbolic disk,…

Dynamical systems on an infinite translation surface with the lattice property are studied. The geodesic flow on this surface is found to be recurrent in all but countably many rational directions. Hyperbolic elements of the affine…

We define the spherical Hecke algebra for an (untwisted) affine Kac-Moody group over a local non-archimedian field. We prove a generalization of the Satake isomorphism for these algebras, relating it to integrable representations of the…

We address the prime geodesic theorem in arithmetic progressions, and resolve conjectures of Golovchanski\u{\i}-Smotrov (1999). In particular, we prove that the traces of closed geodesics on the modular surface do not equidistribute in the…

In this article, we introduce the notion of global adelic space of an arithmetic variety over an adelic curve and prove an equidistribution theorem for a generic sequence of subvarieties. As an application, we prove a Bogomolov type theorem…

We consider the the intersections of the complex nodal set of the analytic continuation of an eigenfunction of the Laplacian on a real analytic surface with the complexification of a geodesic. We prove that if the geodesic flow is ergodic…

Periodic geodesics on the modular surface correspond to periodic orbits of the geodesic flow in its unit tangent bundle $\mathrm{PSL}_2(\mathbb{Z})\backslash\mathrm{PSL}_2(\mathbb{R})$. The complement of any finite number of orbits is a…

We investigate how often geodesics have homology in a fixed set of the homology lattice of a compact Riemann surface. We prove that closed geodesics are equidistributed on any sets with asymptotic density with respect to a specific norm. We…

For a fixed prime p, we consider the (finite) set of supersingular elliptic curves over $\bar{\mathbb{F}}$. Hecke operators act on this set. We compute the asymptotic frequence with which a given supersingular elliptic curve visits another…

We extend Teichmueller dynamics to a flow on the total space of a flat bundle of deformation spaces of representations of the fundamental group of a fixed surface S in a Lie group G. The resulting dynamical system is a continuous version of…

We calculate explicitly an adelic quotient group for an excellent Noetherian normal integral two-dimensional separated scheme. An application to an irreducible normal projective algebraic surface over a field is given.

This paper is a continuation of a previous paper in which the first two authors have introduced the spherical Hecke algebra and the Satake isomorphism for an untwisted affine Kac-Moody group over a non-archimedian local field. In this paper…

Three types of reciprocity laws for arithmetic surfaces are established. For these around a point or along a vertical curve, we first construct $K_2$ type central extensions, then introduce reciprocity symbols, and finally prove the law as…

This article is a first step towards the understanding of the dynamics of the horocycle flow on foliated manifolds by hyperbolic surfaces. This is motivated by a question formulated by M. Martinez and A. Verjovsky on the minimality of this…

We consider Abelian covers of compact hyperbolic surfaces. We establish an asymptotic expansion of the correlations for the horocycle flow on $\mathbb{Z}^d$-covers, thus proving a strong form of Krickeberg mixing. We also prove that the…

We give a formula for the topological pressure of the geodesic flow of a compact rank 1 manifold in terms of the growth of the number of closed hyperbolic (rank 1) geodesics. We derive an equidistribution result for these geodesics with…