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>

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 $$ {\Gamma}_6^{\prime } $$ Γ 6 ′ modular symmetry corresponding to N′ = 1 and N″ = 6, five types of models for lepton masses and mixing with $$ {\Gamma}_6^{\prime } $$ Γ 6 ′ 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.

Modular symmetry by orbifolding magnetized T2 × T2: realization of double cover of ΓN

We study the modular symmetry of zero-modes on $$ {T}_1^2\times {T}_2^2 $$ T 1 2 × T 2 2 and orbifold compactifications with magnetic fluxes, M1, M2, where modulus parameters are identified. This identification breaks the modular symmetry of $$ {T}_1^2\times {T}_2^2 $$ T 1 2 × T 2 2 , SL(2, ℤ)1× SL(2, ℤ)2 to SL(2, ℤ) ≡ Γ. Each of the wavefunctions on $$ {T}_1^2\times {T}_2^2 $$ T 1 2 × 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.

A note on cusp forms and representations of SL2(𝔽𝑝)

Cusp forms are certain holomorphic functions defined on the upper half-plane, and the space of cusp forms for the principal congruence subgroup \Gamma(p), 𝑝 a prime, is acted on by \mathrm{SL}_{2}(\mathbb{F}_{p}). Meanwhile, there is a finite field incarnation of the upper half-plane, the Deligne–Lusztig (or Drinfeld) curve, whose cohomology space is also acted on by \mathrm{SL}_{2}(\mathbb{F}_{p}). In this note, we compute the relation between these two spaces in the weight 2 case.

Explicit presentation of an Iwasawa algebra: The case of pro-p Iwahori subgroup of SLn(ℤp)

Iwasawa algebras of compact p-adic Lie groups are completed group algebras with applications in number theory in studying class numbers of towers of number fields and representation theory of p-adic Lie groups. We previously determined an explicit presentation of the Iwasawa algebra for the first principal congruence kernel of Chevalley groups over {\mathbb{Z}_{p}} which were uniform pro-p groups in the sense of Dixon, du Sautoy, Mann and Segal. In this paper, for prime {p>n+1}, we determine the explicit presentation, in the form of generators and relations, of the Iwasawa algebra of the pro-p Iwahori subgroup of {\mathrm{GL}_{n}(\mathbb{Z}_{p})} which is not, in general, a uniform pro-p group.

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].