Related papers: Prime geodesic theorem for the Picard manifold

In Euclidean and Hyperbolic space, and the hemisphere in $S^n$, geodesic balls maximize the gap $\lambda_2 - \lambda_1$ of Dirichlet eigenvalues, amoung domains with fixed $\lambda_1$. We prove an upper bound on $\lambda_2 - \lambda_1$ for…

A geometric version of the Poincar\'e Lemma is established for the topological vector space of differential chains. In particular, every differential k-cycle with compact support in a contractible open subset U of a smooth n-manifold M is…

The aim of this article is to understand the geometry of limit sets in pseudo-Riemannian hyperbolic geometry. We focus on a class of subgroups of $\mathrm{PO}(p,q+1)$ introduced by Danciger, Gu\'eritaud and Kassel, called…

Soundararajan and Young (2013) proposed a new approach to improve the error term of the prime geodesic theorem for the modular group and actually obtained the exponent 25/36 of the error term. In the present paper, we sharpen it to 9/13.

We say that a collection Gamma of geodesics in the hyperbolic plane H^2 is a modular pattern if Gamma is invariant under the modular group PSL_2(Z), if there are only finitely many PSL_2(Z)-equivalence classes of geodesics in Gamma, and if…

Algebraic hyperbolicity serves as a bridge between differential geometry and algebraic geometry. Generally, it is difficult to show that a given projective variety is algebraically hyperbolic. However, it was established recently that a…

Suppose that $\Sigma$ is a hyperbolic surface and $f:\mathbb R_+\to\mathbb R_+$ a monotonic function. We study the closure in the projective tangent bundle $PT\Sigma$ of the set of all geodesics $\gamma$ satisfying $I(\gamma,\gamma)\leq…

The article is devoted to the question whether the orbit space of a compact linear group is a topological manifold and a homological manifold. In the paper, the case of a simple three-dimensional group is considered. An upper bound is…

In [FI09], Friedlander and Iwaniec studied the so-called Hyperbolic Prime Number Theorem, which asks for an infinitude of elements gamma in SL(2,Z) such that the norm squared |gamma|^2 = a^2 + b^2 + c^2 + d^2 = p, a prime. Under the…

Let K be the totally real cubic field of discriminant 49, let O be its ring of integers, and let p be the prime over 7. Let Gamma (p)\subset Gamma = SL_2(O) be the principal congruence subgroup of level p. This paper investigates the…

We present an algorithm to compute the torsion component $\mathrm{Pic}^\tau X$ of the Picard scheme of a smooth projective variety $X$ over a field $k$. Specifically, we describe $\mathrm{Pic}^\tau X$ as a closed subscheme of a projective…

For a Poisson manifold $M$ we develop systematic methods to compute its Picard group $Pic(M)$, i.e., its group of self Morita equivalences. We establish a precise relationship between $Pic(M)$ and the group of gauge transformations up to…

For the ternary quadratic form Q(x) = x^2 + y^2 - z^2 and a non-zero Pythagorean triple x_0 in Z^3 lying on the cone Q(x) = 0, we consider an orbit O = x_0 Gamma of a finitely generated subgroup Gamma < SO_Q(Z) with critical exponent…

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.

A homogeneous Riemannian manifold $(M=G/K, g)$ is called a space with homogeneous geodesics or a $G$-g.o. space if every geodesic $\gamma (t)$ of $M$ is an orbit of a one-parameter subgroup of $G$, that is $\gamma(t) = \exp(tX)\cdot o$, for…

Based on high precision computation of periods and lattice reduction techniques, we compute the Picard group of smooth surfaces. We also study the lattice reduction technique that is employed in order to quantify the possibility of…

A closed 3-form $H \in \Omega^3_0(M)$ defines an extension of $\Gamma(TM)$ by $\Omega^2_0(M)$. This fact leads to the definition of the group of $H$-twisted Hamiltonian symmetries $\Ham(M, \JJ; H)$ as well as Hamiltonian action of Lie group…

Pascal's triangle will give the number of geodesics from the identity to each point of ${\bf Z}^2$ if you write it in each of the quadrants. Given a group $G$ and generating set $\cal G$ we take the {\it Pascal's function} $p_{\cal G}: G…

We seek to characterize homology classes of Lagrangian projective spaces embedded in irreducible holomorphic-symplectic manifolds, up to the action of the monodromy group. This paper addresses the case of manifolds deformation-equivalent to…

Let $\pi$ be a fixed Hecke--Maass cusp form for $\mathrm{SL}(3,\mathbb{Z})$ and $\chi$ be a primitive Dirichlet character modulo $M$, which we assume to be a prime. Let $L(s,\pi\otimes \chi)$ be the $L$-function associated to $\pi\otimes…