Related papers: Fundamental domains for quaternionic S-arithmetic …

Let $\Lambda$ be a maximal $\mathbb{F}_q[T]$-order in a division quaternion algebra over $\mathbb{F}_q(T)$ which is split at the place $\infty$. The present article gives an algorithm to compute a fundamental domain for the action of the…

In this note we compute the fundamental domain of the action on the corresponding Bruhat-Tits tree of the arithmetic subgroup of PGL2 with respect to an arbitrary place of the rational function field.

We describe an algorithm for computing certain quaternionic quotients of the Bruhat-Tits tree for GL2(Qp). As an application, we describe an algorithm to obtain (conjectural) equations for the canonical embedding of Shimura curves.

In this work, we construct fundamental domains for congruence subgroups of $SL_2(F_q[t])$ and $PGL_2(F_q[t])$. Our method uses Gekeler's description of the fundamental domains on the Bruhat- Tits tree $X = X_{q+1}$ in terms of cosets of…

A practical algorithm to compute the fundamental domain of an arithmetic Fuchsian group was given by Voight, and implemented in Magma. It was later expanded by Page to the case of arithmetic Kleinian groups. We combine and improve on parts…

We initiate a study of the spectral theory of the locally symmetric space $X=\Gamma\backslash G/K$, where $G=SO(3,Complex)$, $\Gamma=SO(3,Z[i])$, $K=SO{3}$. We write down explicit equations defining a fundamental domain for the action of…

We prove a higher dimensional generalization of Gross and Zagier's theorem on the factorization of differences of singular moduli. Their result is proved by giving a counting formula for the number of isomorphisms between elliptic curves…

We exhibit an algorithm to compute a Dirichlet domain for a cofinite Fuchsian group Gamma. As a consequence, we compute the invariants of Gamma, including an explicit finite presentation for Gamma.

We give an explicit description of fundamental domains associated to the $p$-adic uniformisation of families of Shimura curves of discriminant $Dp$ and level $N\geq 1$, for which the one-sided ideal class number $h(D,N)$ is $1$. The…

Let $p_{1}, p_{2}$ be two distinct prime integers, let $n$ be a positive integer, $n$$\geq 3$ and let $\xi_{n} $ be a primitive root of order $n$ of the unity. In this paper we obtain a complete characterization for a quaternion algebra…

We study the action on the Bruhat-Tits tree of unit groups of maximal orders in certain quaternion algebras over $\mathbb{F}_q(T)$ and discuss applications to arithmetic geometry and group theory.

Given the spherical subalgebra $B$ of a rational Cherednik algebra, we aim to classify all finite groups $\Gamma$ for which there exists a domain $R$ on which $\Gamma$ acts by ring automorphisms, such that $B=R^{\Gamma}.$ We describe such…

We will consider a totally real Galois field $K$ of degree 4 as the linear coordinate space $\mathbb{Q}^4\subset\mathbb{R}^4$. An element $k\in K$ is called strictly positive, if all its conjugates are positive. The set of strictly positive…

Let $G=SO(3,C)$, $\Gamma=SO(3,Z[i])$, $K=SO(3)$, and let $X$ be the locally symmetric space $\Gamma\backslash G/K$. In this paper, we write down explicit equations defining a fundamental domain for the action of $\Gamma$ on $G/K$. The…

Let $K$ be a global function field of characteristic $p$, and let $\Gamma$ be a finite-index subgroup of an arithmetic group defined with respect to $K$ and such that any torsion element of $\Gamma$ is a $p$-torsion element. We define…

It is shown that most lattices $\Gamma$ in $\mathbb{R}^2$ and $\mathbb{R}^3$ possess a fundamental domain $F$ for the action of $\Gamma$ on $\mathbb{R}^2$, respectively $\mathbb{R}^3$, having more symmetries than the point group…

In this paper we construct explicit LPS-type Ramanujan graphs from each definite quaternion algebra over $\mathbb Q$ of class number 1, extending the constructions of Lubotzky, Phillips, Sarnak, and later Chiu, and answering in the…

For a totally real number field $F$ and a nonarchimedean prime $\mathfrak{p}$ of $F$ lying above a prime number $p$ we introduce certain sheaf cohomology groups that intertwine the $\mathfrak{p}^{\infty}$-tower of a quaternionic Hilbert…

Let K be a function field with constant field k and let "infinity" be a fixed place of K. Let C be the Dedekind domain consisting of all those elements of K which are integral outside "infinity". The group G=GL_2(C) is important for a…

Building on our previous work on rigid analytic uniformizations, we introduce Darmon points on Jacobians of Shimura curves attached to quaternion algebras over Q and formulate conjectures about their rationality properties. Moreover, if K…