On fundamental domains of arithmetic Fuchsian groups

Let be a totally real algebraic number field and an order in a quaternion algebra over . Assume that the group of units in with reduced norm equal to is embedded into as an arithmetic Fuchsian group. It is shown how Ford's algorithm can be effectively applied in order to determine a fundamental domain of as well as a complete system of generators of .

     

Dirichlet fundamental domains and topology of projective varieties

We prove that for every finitely-presented group G there exists a 2-dimensional irreducible complex-projective variety W with the fundamental group G, so that all singularities of W are normal crossings and Whitney umbrellas.     

Ford fundamental domains in symmetric spaces of rank one

We show the existence of isometric (or Ford) fundamental regions for a large class of subgroups of the isometry group of any rank one Riemannian symmetric space of noncompact type. The proof does not use the classification of symmetric spaces. All hitherto known existence results of isometric fundamental regions and domains are essentially subsumed by our work.     

The fundamental group of period domains over finite fields

We determine the fundamental group of period domains over finite fields. This answers a question of M. Rapoport raised in [M. Rapoport, Period domains over finite and local fields, in: Proc. Sympos. Pure Math., vol. 62, part 1, 1997, pp. 361-381].     

On fundamental skew distributions

A new class of multivariate skew-normal distributions, fundamental skew-normal distributions and their canonical version, is developed. It contains the product of independent univariate skew-normal distributions as a special case. Stochastic representations and other main properties of the associated distribution theory of linear and quadratic forms are considered. A unified procedure for extending this class to other families of skew distributions such as the fundamental skew-symmetric, fundamental skew-elliptical, and fundamental skew-spherical class of distributions is also discussed.     

On fundamental Heronian triangles

Translated from Matematicheskie Zametki, Vol. 55, No. 2, pp. 72–79, February, 1994.     

On the fundamental group-scheme

We show that the fundamental group-scheme of a separably rationally connected variety defined over an algebraically closed field is trivial. Let X be a geometrically irreducible smooth projective variety defined over a finite field k admitting a k-rational point. Let {En,σn}_n?0 be a flat principal G-bundle over X, where G is a reductive linear algebraic group defined over k. We show that there is a positive integer a such that the principal G-bundle is isomorphic to E₀, where FX is the absolute Frobenius morphism of X. From this it follows that E₀ is given by a representation of the fundamental group-scheme of X in G.     

On Noether's fundamental theorem

Ohne Zusammenfassung     

On arithmetic fundamental lemmas

We present a relative trace formula approach to the Gross–Zagier formula and its generalization to higher-dimensional unitary Shimura varieties. As a crucial ingredient, we formulate a conjectural arithmetic fundamental lemma for unitary Rapoport–Zink spaces. We prove the conjecture when the Rapoport–Zink space is associated to a unitary group in two or three variables.     

On euclidean domains

We consider Euclidean domains and their groups of units. Let K(a,b) be the set of remainders in the division of a by b. If Card K(a,b) = 1 for any a and b from a Euclidean domain R, then R is known to be isomorphic to the ring of polynomials over some field, see [4], [5]. On the other hand, the condition Card K(a,b) = 2 for any a and b implies that R is isomorphic to the ring Z of integers, see [2]. We give characterization of Euclidean domains and their groups of units under some other conditions on K(a,b).     

On comb domains

A result for comb domains is proved which is stronger than but in particular implies a conjecture of Rodin and Warschawski.

     

On almost-divided domains

Let B be a domain, Q a maximal ideal of B, π: B → B/Q the canonical surjection, D a subring of B/Q, and A:=π ⁻¹(D). If both B and D are almost-divided domains (resp., n-divided domains), then A = B × _B/Q D is an almost-divided domain (resp., an n-divided domain); the converse holds if B is quasilocal. If 2 ≤ d ≤ ∞, an example is given of an almost-divided domain of Krull dimension d which is not a divided domain.      

On regularly perturbed fundamental matrices

In this note we characterize regular perturbations of finite state Markov chains in terms of continuity properties of its fundamental matrix. A perturbation turns out to be regular if and only if the fundamental matrix can be approximated by the discounted deviation from stationarity for small perturbation parameters. We also give bounds to asses the quality of the approximation.