Related papers: Counting and equidistribution in Heisenberg groups

We compute the group homology, the topological K-theory of the reduced C^*-algebra, the algebraic K-theory and the algebraic L-theory of the group ring of the semi-direct product of the three-dimensional discrete Heisenberg group by Z/4.…

We prove asymptotic equidistribution results for pieces of large closed horospheres in cofinite hyperbolic manifolds of arbitrary dimension. This extends earlier results by Hejhal and Str\"ombergsson in dimension 2. Our proofs use spectral…

Let group generators having finite-dimensional representation be realized as Hermitian linear differential operators without nhomogeneous terms as takes place, for example, for the SO(n) group. Then orresponding group Hamiltonians…

We show that each connected component of the moduli space of smooth real binary quintics is isomorphic to an open subset of an arithmetic quotient of the real hyperbolic plane. Moreover, our main result says that the induced metric on this…

We prove that infinite orbits of Zariski dense hyperbolic groups equidistribute in homogeneous spaces, in the sense that the family of measures obtained by averaging along spheres in the Cayley graph converges to Haar measure.

We explicitly compute the ellitpic points and isotropy groups for the action of the Picard modular group over the Gaussian integers on 2-dimensional complex hyperbolic space.

We construct \'etale generalized Heisenberg group covers of hyperelliptic curves over number fields. We use these to produce infinite families of quadratic extensions of cyclotomic fields that admit everywhere unramified generalized…

We derive explicit isomorphisms between certain congruence subgroups of the Siegel modular group, the Hermitian modular group over an arbitrary imaginary-quadratic number field and the modular group over the Hurwitz quaternions of degree 2…

We prove that for a metric space $X$ and a finite group $G$ acting on $X$ by isometries, if $X$ coarsely embeds into a Hilbert space, then so does the quotient $X/G$. A crucial step towards our main result is to show that for any integer $k…

We present an alternative relatively easy way to understand and determine the zeros of a quintic whose Galois group is isomorphic to the group of rotational symmetries of a regular icosahedron. The extensive algebraic procedures of Klein in…

We use Kor\'anyi--Reimann complex cross--ratios to prove the Ptolemaean inequality and the Theorem of Ptolemaeus in the setting of the boundary of complex hyperbolic space and the first Heisenberg group.

We provide a general framework for proving asymptotic equidistribution, convexity, and log concavity of coefficients of generating functions on arithmetic progressions. Our central tool is a variant of Wright's Circle Method proved by two…

We develop a heuristic for the density of integer points on affine cubic surfaces. Our heuristic applies to smooth surfaces defined by cubic polynomials that are log K3, but it can also be adjusted to handle singular cubic surfaces. We…

Duke, Imamo\=glu, and T\'oth have recently constructed a new geometric invariant, a hyperbolic orbifold, associated to each narrow ideal class of a real quadratic field. Furthermore, they have shown that the projection of these hyperbolic…

Given a fixed quadratic extension K of Q, we consider the distribution of elements in K of norm 1 (denoted N). When K is an imaginary quadratic extension, N is naturally embedded in the unit circle in C and we show that it is…

Let $k$ be a positive integer and let $X_k$ be the cubic hypersurface defined by the equation $x^3-(y_1^2+\cdots+y_{4k}^2)z=0$. In this paper, we give an asymptotic formula for the counting function of semi-integral points on $X_k$. We also…

We show the existence of group-theoretic sections of the "etale-by-geometrically abelian" quotient of the arithmetic fundamental group of hyperbolic curves over $p$-adic local fields relative to a proper and flat model which are…

Let $E$ be an elliptic curve defined over a number field $K$ without complex multiplication. If $\Gamma \subset E(\overline{K})$ is a subgroup of finite rank, a very special case of a conjecture of R\'emond predicts that points of small…

We Classify the rational quadratic extensions K and the finite groups G for which the group ring R[G] of G over the ring R of integers of K has the property that the group of units of augmentation 1 of R[G] is hyperbolic. We also construct…

We prove quantitative equidistribution results for actions of Abelian subgroups of the $2g+1$ dimensional Heisenberg group acting on compact $2g+1$-dimensional homogeneous nilmanifolds. The results are based on the study of the…