Related papers: Fundamental domains for quaternionic S-arithmetic …

Let D be a quaternion division algebra over a totally real number field F which splits exactly at one infinite place. We assume that there is a p-adic place where D doesn't split. Then the associated Shimura curve has a Cherednik…

Let $K=\mathbb{F}_q(C)$ be the global function field of rational functions over a smooth and projective curve $C$ defined over a finite field $\mathbb{F}_q$. The ring of regular functions on $C-S$ where $S \neq \emptyset$ is any finite set…

The main result of this paper is a construction of fundamental domains for certain group actions on Lorentz manifolds of constant curvature. We consider the simply connected Lie group G~, the universal cover of the group SU(1,1) of…

Heisenberg groups over algebras with central involution and their automorphism groups are constructed. The complex quaternion group algebra over a prime field is used as an example. Its subspaces provide finite models for each of the real…

We give an algorithm to determine finitely many generators for a subgroup of finite index in the unit group of an integral group ring $\mathbb{Z} G$ of a finite nilpotent group $G$, this provided the rational group algebra $\mathbb{Q} G$…

For a compact real form $U$ of a complex simple Lie group $G$, and an irreducible representation $\rho:\Gamma \to U$ of a Fuchsian group of the first kind $\Gamma$, it is shown that the classical isomorphism of Shimura, for the periods of a…

We consider quantum Schubert cells in the quantum grassmannian and give a cell decomposition of the prime spectrum via the Schubert cells. As a consequence, we show that all primes are completely prime in the generic case where the…

In this paper we determine the number of endomorphism rings of superspecial abelian surfaces over a field $\mathbb{F}_q$ of odd degree over $\mathbb{F}_p$ in the isogeny class corresponding to the Weil $q$-number $\pm\sqrt{q}$. This extends…

We give a strong fundamental domain for the quotient of $PGL_2\!\left(\mathbb{F}_q(\!(t^{-1})\!)\!\right)$ by $PGL_2\!\left(\mathbb{F}_q[t]\right)$ as a subset of distinct ordered triple points of $\mathbb{P}^1(\mathbb{F}_q(\!(t^{-1})\!))$.

Let $R$ be a Dedekind domain with field of fractions $K$ and $\operatorname{char}(R)\neq3$. In this paper, we generalize Bhargava's parametrization of $3$-torsion ideal classes by binary cubic forms to work over $R$. Specifically, we…

For $R$ a Euclidean number ring, and let $\Gamma_n(p)$ be the level-$p$ principal congruence subgroup of $\text{SL}_n(R)$. Borel--Serre showed that the cohomology of $\Gamma_n(p)$ vanishes above a degree $\nu$ that is quadratic in $n$. Let…

We apply the theory of Bruhat-Tits trees to the study of optimal embeddings of two and three dimensional commutative orders into quaternion algebras. Specifically, we determine how many conjugacy classes of global Eichler orders in a…

Let $H$ be a finite dimensional hereditary algebra over an algebraically closed field, and let $\mathcal{C}_{H}$ be the corresponding cluster category. We give a description of the (standard) fundamental domain of $\mathcal{C}_{H} $ in the…

The main aim of this paper is to give two infinite series of examples of Lorentz space forms that can be obtained from Lorentz polyhedra by identification of faces. These Lorentz space forms are bi-quotients of the form $\Gamma_1\backslash…

For $G$ a connected reductive group, $\gamma\in \kg(F)$ semisimple regular integral, we introduce a fundamental domain $F_{\gamma}$ for the affine Springer fibers $\xx_{\gamma}$. There is a beautiful way to reduce the purity conjecture of…

Let $\mathbb{K}$ be an unramified quadratic extension of $\mathbb{Q}_{p}$ for a fixed $p>2$. Projective general linear groups $G=\operatorname{PGL}_{2}(\mathbb{K})$ and $H=\operatorname{PGL}_{2}(\mathbb{Q}_{p})$ act transitively on…

We consider discrete subgroups Gamma of the simply connected Lie group SU~(1,1), the universal cover of SU(1,1), of finite level, i.e. the subgroup intersects the centre of SU~(1,1) in a subgroup of finite index, this index is called the…

We show by construction that every rhombic lattice $\Gamma$ in $\mathbb{R}^{2}$ has a fundamental domain whose symmetry group contains the point group of $\Gamma$ as a subgroup of index $2$. This solves the last open case of a question…

Let $F=\mathbb{F}_q(T)$ be the field of rational functions with $\mathbb{F}_q$-coefficients, and $A=\mathbb{F}_q[T]$ be the subring of polynomials. Let $D$ be a division quaternion algebra over $F$ which is split at $1/T$. Given an…

There are some distinguished ensembles of non-Hermitian random matrices for which the joint PDF can be written down explicitly, is unchanged by rotations, and furthermore which have the property that the eigenvalues form a Pfaffian point…