Principal Congruence Research Articles

Subresiduated Nelson algebras

In this paper we generalize the well known relation between Heyting algebras and Nelson algebras in the framework of subresiduated lattices. In order to make it possible, we introduce the variety of subresiduated Nelson algebras. The main tool for its study is the construction provided by Vakarelov. Using it, we characterize the lattice of congruences of a subresiduated Nelson algebra through some of its implicative filters. We use this characterization to describe simple and subdirectly irreducible algebras, as well as principal congruences. Moreover, we prove that the variety of subresiduated Nelson algebras has equationally definable principal congruences and also the congruence extension property. Additionally, we present an equational base for the variety generated by the totally ordered subresiduated Nelson algebras. Finally, we show that there exists an equivalence between the algebraic category of subresiduated lattices and the algebraic category of centedred subresiduated Nelson algebras.

Principal right congruences on completely 0-simple semigroups

Principal right congruences on completely 0-simple semigroups

Mixmaster chaos in an AdS black hole interior

We derive gravitational backgrounds that are asymptotically Anti-de Sitter, have a regular black hole horizon and which deep in the interior exhibit mixmaster chaotic dynamics. The solutions are obtained by coupling gravity with a negative cosmological constant to three massive vector fields, within an Ansatz that reduces to ordinary differential equations. At late interior times the equations are identical to those analysed in depth by Misner and by Belinskii-Khalatnikov-Lifshitz fifty years ago. We review and extend known classical and semiclassical results on the interior chaos, formulated as both a dynamical system of ‘Kasner eras’ and as a hyperbolic billiards problem. The volume of the universe collapses doubly-exponentially over each Kasner era. A remarkable feature is the emergence of a conserved energy, and hence a ‘time-independent’ Hamiltonian, at asymptotically late interior times. A quantisation of this Hamiltonian exhibits arithmetic chaos associated with the principal congruence subgroup Γ(2) of the modular group. We compute a large number of eigenvalues numerically to obtain the spectral form factor. While the spectral statistics is anomalous for a chaotic system, the eigenfunctions themselves display random matrix behaviour.

The density conjecture for principal congruence subgroups

The density conjecture for principal congruence subgroups

Wedge product matrices and orbits of principal congruence subgroups

Following Bump and Hoffstein in [2], the orbits in Γ∞(3)﹨Γ(3) are in bijection with sets of invariants satisfying certain relations. We explain how wedge product matrices give an alternative definition of the invariants of matrix orbits. This new method provides the possibility of performing similar computations with other congruence subgroups and arbitrary n×n matrices. Using Steinberg's refined version of the Bruhat decomposition, we construct an explicit choice of coset representative for each orbit in the orbit space Γ∞(3)﹨Γ(3) of 3×3 matrices over the PID of Eisenstein integers.

Factor principal congruences and Boolean products in filtral varieties

Motivated by Haviar and Ploščica’s 2021 characterisation of Boolean products of simple De Morgan algebras, we investigate Boolean products of simple algebras in filtral varieties. We provide two main theorems. The first yields Werner’s Boolean-product representation of algebras in a discriminator variety as an immediate application. The second, which applies to algebras in which the top congruence is compact, yields a generalisation of the Haviar–Ploščica result to semisimple varieties of Ockham algebras. The property of having factor principal congruences is fundamental to both theorems. While major parts of our general theorems can be derived from results in the literature, we offer new, self-contained and essentially elementary proofs.

On homology torsion growth

We prove new vanishing results on the growth of higher torsion homologies for suitable arithmetic lattices, Artin groups and mapping class groups. The growth is understood along Farber sequences, in particular, along residual chains. For principal congruence subgroups, we also obtain strong asymptotic bounds for the torsion growth. As a central tool, we introduce a quantitative homotopical method called effective rebuilding. This constructs small classifying spaces of finite index subgroups, at the same time controlling the complexity of the homotopy. The method easily applies to free abelian groups and then extends recursively to a wide class of residually finite groups.

Principal Equatorial Null Geodesic Congruences in the Kerr Metric, and Their Quantum Propagators

Principal Equatorial Null Geodesic Congruences in the Kerr Metric, and Their Quantum Propagators

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This paper establishes power-saving bounds for Kloosterman sums associated with the long Weyl element for GL(n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{GL}(n)$$\\end{document} for arbitrary n⩾3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n \\geqslant 3$$\\end{document}, as well as for another type of Weyl element of order 2. These bounds are obtained by establishing an explicit representation as exponential sums. As an application we go beyond Sarnak’s density conjecture for the principal congruence subgroup of prime level. We also obtain power-saving bounds for all Kloosterman sums on GL(4)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{GL}(4)$$\\end{document}.

Congruence subgroups and crystallographic quotients of small Coxeter groups

Abstract Small Coxeter groups are precisely the ones for which the Tits representation is integral, which makes the study of their congruence subgroups relevant. The symmetric group S n {S_{n}} has three natural extensions, namely the braid group B n {B_{n}} , the twin group T n {T_{n}} and the triplet group L n {L_{n}} . The latter two groups are small Coxeter groups, and play the role of braid groups under the Alexander–Markov correspondence for appropriate knot theories, with their pure subgroups admitting suitable hyperplane arrangements as Eilenberg-MacLane spaces. In this paper, we prove that the congruence subgroup property fails for infinite small Coxeter groups which are not virtually abelian. As an application, we deduce that the congruence subgroup property fails for both T n {T_{n}} and L n {L_{n}} when n ≥ 4 {n\geq 4} . We also determine subquotients of principal congruence subgroups of T n {T_{n}} , and identify the pure twin group P ⁢ T n {PT_{n}} and the pure triplet group P ⁢ L n {PL_{n}} with suitable principal congruence subgroups. Further, we investigate crystallographic quotients of these two families of small Coxeter groups, and prove that T n / P ⁢ T n ′ {T_{n}/PT_{n}^{\prime}} , T n / T n ′′ {T_{n}/T_{n}^{\prime\prime}} and L n / P ⁢ L n ′ {L_{n}/PL_{n}^{\prime}} are crystallographic groups. We also determine crystallographic dimensions of these groups and identify the holonomy representation of T n / T n ′′ {T_{n}/T_{n}^{\prime\prime}} .

Semisimplicity and Congruence 3-Permutabilty for Quasivarieties with Equationally Definable Principal Congruences

Semisimplicity and Congruence 3-Permutabilty for Quasivarieties with Equationally Definable Principal Congruences

Computing a Link Diagram From Its Exterior

A knot is a circle piecewise-linearly embedded into the 3-sphere. The topology of a knot is intimately related to that of its exterior, which is the complement of an open regular neighborhood of the knot. Knots are typically encoded by planar diagrams, whereas their exteriors, which are compact 3-manifolds with torus boundary, are encoded by triangulations. Here, we give the first practical algorithm for finding a diagram of a knot given a triangulation of its exterior. Our method applies to links as well as knots, and allows us to recover links with hundreds of crossings. We use it to find the first diagrams known for 23 principal congruence arithmetic link exteriors; the largest has over 2500 crossings. Other applications include finding pairs of knots with the same 0-surgery, which relates to questions about slice knots and the smooth 4D Poincaré conjecture.

Dimension formulas for Siegel modular forms of level 4

AbstractWe prove several dimension formulas for spaces of scalar‐valued Siegel modular forms of degree 2 with respect to certain congruence subgroups of level 4. In case of cusp forms, all modular forms considered originate from cuspidal automorphic representations of whose local component at admits nonzero fixed vectors under the principal congruence subgroup of level 2. Using known dimension formulas combined with dimensions of spaces of fixed vectors in local representations at , we obtain formulas for the number of relevant automorphic representations. These, in turn, lead to new dimension formulas, in particular for Siegel modular forms with respect to the Klingen congruence subgroup of level 4.

On classification of fermionic rational conformal field theories

We systematically study how the integrality of the conformal characters shapes the space of fermionic rational conformal field theories in two dimensions. The integrality suggests that conformal characters on torus with a given choice of spin structures should be invariant under a principal congruence subgroup of PSL(2, ℤ). The invariance strongly constrains the possible values of the central charge as well as the conformal weights in both Neveu-Schwarz and Ramond sectors, which improves the conventional holomorphic modular bootstrap method in a significant manner. This allows us to make much progress on the classification of fermionic rational conformal field theories with the number of independent characters less than five.

The stable representations of GL N over finite local principal ideal rings

Let be a discrete valuation ring with maximal ideal and with finite residue field , the field with q elements where q is a power of a prime p. For , we write for the reduction of modulo the ideal . An irreducible ordinary representation of the finite group is called stable if its restriction to the principal congruence kernel , where , consists of irreducible representations whose stabilizers modulo , where , are centralizers of certain matrices in , called stable matrices. The study of stable representations is motivated by constructions of strongly semisimple representations, introduced by Hill, which is a special case of stable representations. In this paper, we explore the construction of stable irreducible representations of the finite group for . Communicated by Scott Chapman

Analytic torsion for arithmetic locally symmetric manifolds and approximation of L2-torsion

In this paper we define a regularized version of the analytic torsion for quotients of a symmetric space of non-positive curvature by arithmetic lattices. The definition is based on the study of the renormalized trace of the corresponding heat operators, which is defined as the geometric side of the Arthur trace formula applied to the heat kernel. Then we study the limiting behavior of the analytic torsion as the lattices run through a sequence of congruence subgroups of a fixed arithmetic subgroup. Our main result states that for sequences of principal congruence subgroups, which converge to 1 at a fixed finite set of places and strongly acyclic flat bundles, the logarithm of the analytic torsion, divided by the index of the subgroup, converges to the L2-analytic torsion.

On Relative Principal Congruences in Term Quasivarieties

On Relative Principal Congruences in Term Quasivarieties

Bounded generation for congruence subgroups of Sp4(R)

For rings of algebraic integers [Formula: see text] in a number field [Formula: see text] called [Formula: see text]-pseudo-good, this paper describes a bounded generation result concerning the minimal number of conjugates of suitable elementary matrices (or more precisely root elements) in [Formula: see text] needed to write any element of the [Formula: see text]-principal congruence subgroup of [Formula: see text] as their product. Using this bounded generation result, we give explicit bounds for the diameter of word norms on [Formula: see text] given by conjugacy classes thereby continuing an investigation into such diameters by Kedra et al. Additionally, we present some examples of [Formula: see text]-pseudo-good rings and classify normally generating subsets of [Formula: see text] for [Formula: see text] the ring of algebraic integers in any number field.

On the systole growth in congruence quaternionic hyperbolic manifolds

We provide an explicit lower bound for the systole in principal congruence covers of compact quaternionic hyperbolic manifolds. We also prove the optimality of this lower bound.

Une description combinatoire du sous-groupe principal de congruence Γ(2) de SL(2, Z)

We characterize sequences of positive integers (c 1 , c 2 , ..., cn) for which the (2 × 2)-matrix c 1 −1 1 0 · · · cn −1 1 0 belongs to the principal congruence subgroup of level 2 in SL(2, Z). The answer is given in terms of dissections of a convex n-gon into a mixture of triangles and quadrilaterals.