Principal Congruence Subgroup Research Articles

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.

Bounds for Kloosterman sums on $$\textrm{GL}(n)$$

AbstractThis paper establishes power-saving bounds for Kloosterman sums associated with the long Weyl element for $$\textrm{GL}(n)$$ GL ( n ) for arbitrary $$n \geqslant 3$$ n ⩾ 3 , 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 $$\textrm{GL}(4)$$ GL ( 4 ) .

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}} .

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.

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.

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.

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.

Modular flavor symmetry and vector-valued modular forms

We revisit the modular flavor symmetry from a more general perspective. The scalar modular forms of principal congruence subgroups are extended to the vector-valued modular forms, then we have more possible finite modular groups including ΓN and {Gamma}_N^{prime } as the flavor symmetry. The theory of vector-valued modular forms provides a method of differential equation to construct the modular multiplets, and it also reveals the simple structure of the modular invariant mass models. We review the theory of vector-valued modular forms and give general results for the lower dimensional vector-valued modular forms. The general finite modular groups are listed up to order 72. We apply the formalism to construct two new lepton mass models based on the finite modular groups A4× Z2 and GL(2, 3).

Automorphic Lie Algebras and Modular Forms

AbstractWe introduce and study certain hyperbolic versions of automorphic Lie algebras related to the modular group. Let $\Gamma $ be a finite index subgroup of $\textrm {SL}(2,\mathbb Z)$ with an action on a complex simple Lie algebra $\mathfrak g$, which can be extended to $\textrm {SL}(2,{\mathbb {C}})$. We show that the Lie algebra of the corresponding $\mathfrak {g}$-valued modular forms is isomorphic to the extension of $\mathfrak {g}$ over the usual modular forms. This establishes a modular analogue of a well-known result by Kac on twisted loop algebras. The case of principal congruence subgroups $\Gamma (N), \, N\leq 6$, is considered in more detail in relation to the classical results of Klein and Fricke and the celebrated Markov Diophantine equation. We finish with a brief discussion of the extensions and representations of these Lie algebras.

SL$ _n(\mathbb{Z}) $-normalizer of a principal congruence subgroup

<abstract><p>Let SL$ _n(\mathbb{Q}) $ be the set of matrices of order $ n $ over the rational numbers with determinant equal to 1. We study in this paper a subset $ \Lambda $ of SL$ _n(\mathbb{Q}) $, where a matrix $ B $ belongs to $ \Lambda $ if and only if the conjugate subgroup $ B\Gamma_q(n)B^{-1} $ of principal congruence subgroup $ \Gamma_q(n) $ of lever $ q $ is contained in modular group SL$ _n(\mathbb{Z}) $. The notion of least common denominator (LCD for convenience) of a rational matrix plays a key role in determining whether <italic>B</italic> belongs to $ \Lambda $. We show that LCD can be described by the prime decomposition of $ q $. Generally $ \Lambda $ is not a group, and not even a subsemigroup of SL$ _n(\mathbb{Q}) $. Nevertheless, for the case $ n = 2 $, we present two families of subgroups that are maximal in $ \Lambda $ in this paper.</p></abstract>

A remark on conductor, depth and principal congruence subgroups

In this paper we study a relation between conductor, depth, and the level of principal congruence subgroups for irreducible admissible representations of GLn(F) for a non-archimedean local field F of characteristic zero. As a global application, we estimate conductors of unitary irreducible automorphic cuspidal representations of GLn(AQ) in terms of principal congruence subgroups. We also give an explicit formula for the dimension of fixed vectors with respect to principal congruence subgroups for irreducible admissible representations of GL2(F) as a local application.

Modular symmetry at level 6 and a new route towards finite modular groups

We propose to construct the finite modular groups from the quotient of two principal congruence subgroups as Γ(N′)/Γ(N″), and the modular group SL(2, ℤ) is ex- tended to a principal congruence subgroup Γ(N′). The original modular invariant theory is reproduced when N′ = 1. We perform a comprehensive study of {Gamma}_6^{prime } modular symmetry corresponding to N′ = 1 and N″ = 6, five types of models for lepton masses and mixing with {Gamma}_6^{prime } modular symmetry are discussed and some example models are studied numerically. The case of N′ = 2 and N″ = 6 is considered, the finite modular group is Γ(2)/Γ(6) ≅ T′, and a benchmark model is constructed.

Commutators, commensurators, and PSL2(Z)

Let $H<\mathrm{PSL}_2(\mathbb{Z})$ be a finite index normal subgroup which is contained in a principal congruence subgroup, and let $\Phi(H)\neq H$ denote a term of the lower central series or the derived series of $H$. In this paper, we prove that the commensurator of $\Phi(H)$ in $\mathrm{PSL}_2(\mathbb{R})$ is discrete. We thus obtain a natural family of thin subgroups of $\mathrm{PSL}_2(\mathbb{R})$ whose commensurators are discrete, establishing some cases of a conjecture of Shalom.

On the top-dimensional cohomology groups of congruence subgroups of SL(n, ℤ)

Let $\Gamma_n(p)$ be the level-$p$ principal congruence subgroup of $\text{SL}_n(\mathbb{Z})$. Borel-Serre proved that the cohomology of $\Gamma_n(p)$ vanishes above degree $\binom{n}{2}$. We study the cohomology in this top degree $\binom{n}{2}$. Let $\mathcal{T}_n(\mathbb{Q})$ denote the Tits building of $\text{SL}_n(\mathbb{Q})$. Lee-Szczarba conjectured that $H^{\binom{n}{2}}(\Gamma_n(p))$ is isomorphic to $\widetilde{H}_{n-2}(\mathcal{T}_n(\mathbb{Q})/\Gamma_n(p))$ and proved that this holds for $p=3$. We partially prove and partially disprove this conjecture by showing that a natural map $H^{\binom{n}{2}}(\Gamma_n(p)) \rightarrow \widetilde{H}_{n-2}(\mathcal{T}_n(\mathbb{Q})/\Gamma_n(p))$ is always surjective, but is only injective for $p \leq 5$. In particular, we completely calculate $H^{\binom{n}{2}}(\Gamma_n(5))$ and improve known lower bounds for the ranks of $H^{\binom{n}{2}}(\Gamma_n(p))$ for $p \geq 5$.

Joint universality theorem of Selberg zeta functions for principal congruence subgroups

In this paper, we investigate the joint functional distribution of Selberg zeta functions for principal congruence subgroups. We prove that the joint universality theorem for these zeta functions holds in the strip 0.85<σ<1. As a corollary, we obtain the functional independence for the zeta functions.

Arithmetic properties of septic partition functions

Congruences and related identities are derived for a set of colored and weighted partition functions whose generating functions generate the graded algebra of integer weight modular forms of level seven. The work determines a general strategy for identifying and proving identities and associated congruences for modular forms on the principal congruence subgroup of level [Formula: see text]. Ramanujan’s partition congruence modulo [Formula: see text] serves as a prototype for the process used to prove new congruences for modular forms of level [Formula: see text].

Modular symmetry by orbifolding magnetized T2\u2009\xd7\u2009T2: realization of double cover of \u0393N

We study the modular symmetry of zero-modes on {T}_1^2times {T}_2^2 and orbifold compactifications with magnetic fluxes, M1, M2, where modulus parameters are identified. This identification breaks the modular symmetry of {T}_1^2times {T}_2^2 , SL(2, ℤ)1× SL(2, ℤ)2 to SL(2, ℤ) ≡ Γ. Each of the wavefunctions on {T}_1^2times {T}_2^2 and orbifolds behaves as the modular forms of weight 1 for the principal congruence subgroup Γ(N), N being 2 times the least common multiple of M1 and M2. Then, zero-modes transform each other under the modular symmetry as multiplets of double covering groups of ΓN such as the double cover of S4.

Representations of the modular group arising from Drinfeld doubles of finite abelian groups

We show that the image of the representation of the modular group [Formula: see text] arising from the representation category [Formula: see text] of the Drinfeld double [Formula: see text] of a finite abelian group [Formula: see text] of exponent [Formula: see text] is isomorphic to the special linear group [Formula: see text], where [Formula: see text] denotes the ring of integers modulo [Formula: see text]. As a consequence, we establish that the kernel of the representation in question is the principal congruence subgroup of level [Formula: see text].