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Effective bounds for Vinberg’s algorithm for arithmetic hyperbolic lattices
A group of isometries of a hyperbolic n -space is called a reflection group if it is generated by reflections in hyperbolic hyperplanes. Vinberg gave...
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Arithmetic purity of strong approximation for complete toric varieties
In this article, we establish the arithmetic purity of strong approximation for smooth loci of weighted projective spaces. By using this result and...
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Hyperbolic Punctured Spheres Without Arithmetic Systole Maximizers
We find bounds for the length of the systole—the shortest essential, non-peripheral closed curve—for arithmetic punctured spheres with n cusps, for
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Problems of Field Arithmetic
The first section of this chapter lists those problems of the first three editions of “Field Arithmetic”F that have been solved or partially solved....
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Translates of S-arithmetic periodic orbits and applications
We prove that certain sequences of periodic orbits of the diagonal group in the space of lattices equidistribute. As an application we obtain new...
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Arithmetic statistics for the fine Selmer group in Iwasawa theory
We study arithmetic statistics for Iwasawa invariants for fine Selmer groups associated to elliptic curves.
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Arithmetic fundamental lemma for the spherical Hecke algebra
We define Hecke correspondences and Hecke operators on unitary RZ spaces and study their basic geometric properties, including a commutativity...
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Reidemeister–Schreier rewriting process for matching uniform signal constellations to quotient groups of arithmetic Fuchsian groups
In this paper, we construct signal constellations from lattices in complex hyperbolic spaces. To construct a hyperbolic lattice, we identify an...
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FROM GEOMETRY TO ARITHMETIC OF COMPACT HYPERBOLIC COXETER POLYTOPES
We establish some geometric constraints on compact Coxeter polytopes in hyperbolic spaces and show that these constraints can be a very useful tool...
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Arithmetic progressions of squares and multiple Dirichlet series
We study a Dirichlet series in two variables which counts primitive three-term arithmetic progressions of squares. We show that this multiple...
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Commensurations of Aut(FN) and Its Torelli Subgroup
For N ≥3, the abstract commensurators of both Aut( F N ) and its Torelli subgroup IA N are isomorphic to Aut( F N ) itself.
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Poincaré and Arithmetic Revisited
Henri Poincaré’s forays into number theory have often been reduced to his pioneering use of automorphic forms or his contribution to the arithmetic...
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Charmenability of arithmetic groups of product type
We discuss special properties of the spaces of characters and positive definite functions, as well as their associated dynamics, for arithmetic...
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Polygons and Modular Arithmetic
There are connections between algebra and geometry that go well beyond the function–graph–analytic-geometry connections studied in high school. We...
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Topological characteristic factors and independence along arithmetic progressions
In 1994 Glasner studied the topological characteristic factor for minimal systems. It is shown that up to canonically defined proximal extensions, a...