Principal Congruence Subgroup Research Articles

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].

Constructing certain special analytic Galois extensions

For every prime p≥5 for which a certain condition on the class group Cl(Q(μp)) is satisfied, we construct a p-adic analytic Galois extension of the infinite cyclotomic extension Q(μp∞) with some special ramification properties. In greater detail, this extension is unramified at primes above p and tamely ramified above finitely many rational primes and its Galois group over Q(μp∞) is isomorphic to a finite index subgroup of SL2(Zp) which contains the principal congruence subgroup. For the primes 107, 139, 271 and 379 such extensions were first constructed by Ohtani and Blondeau. The strategy for producing these special extensions at an abundant number of primes is through lifting two-dimensional reducible Galois representations which are diagonal when restricted to p.

The torsion in symmetric powers on congruence subgroups of Bianchi groups

In this paper we prove that for a fixed neat principal congruence subgroup of a Bianchi group the order of the torsion part of its second cohomology group with coefficients in an integral lattice associated with the m m th symmetric power of the standard representation of SL 2 ⁡ ( C ) \operatorname {SL}_2(\mathbb {C}) grows exponentially in m 2 m^2 . We give upper and lower bounds for the growth rate. Our result extends a result of W. Müller and S. Marshall, who proved the corresponding statement for closed arithmetic 3 3 -manifolds, to the finite-volume case. We also prove a limit multiplicity formula for combinatorial Reidemeister torsions on higher-dimensional hyperbolic manifolds.

Principal congruence subgroups in the infinite rank case

Principal congruence subgroups in the infinite rank case

A generalization of a theorem of Hecke for [formula omitted] to fundamental discriminants

Let p>3 be an odd prime, p≡3mod4 and let π+,π− be the pair of cuspidal representations of SL2(Fp). It is well known by Hecke that the difference mπ+−mπ− in the multiplicities of these two irreducible representations occurring in the space of weight 2 cusps forms with respect to the principal congruence subgroup Γ(p), equals the class number h(−p) of the imaginary quadratic field Q(−p). We extend this result to all fundamental discriminants −D of imaginary quadratic fields Q(−D) and prove that an alternating sum of multiplicities of certain irreducibles of SL2(Z/DZ) is an explicit multiple, up to a sign and a power of 2, of either the class number h(−D) or of the sums h(−D)+h(−D/2), h(−D)+2h(−D/2); the last two possibilities occur in some of the cases when D≡0mod8. The proof uses the holomorphic Lefschetz number.

Regular characters of classical groups over complete discrete valuation rings

Let o be a complete discrete valuation ring with finite residue field k of odd characteristic, and let G be a symplectic or special orthogonal group scheme over o. For any ℓ∈N let Gℓ denote the ℓ-th principal congruence subgroup of G(o). An irreducible character of the group G(o) is said to be regular if it is trivial on a subgroup Gℓ+1 for some ℓ, and if its restriction to Gℓ/Gℓ+1≃Lie(G)(k) consists of characters of minimal G(kalg)-stabilizer dimension. In the present paper we consider the regular characters of such classical groups over o, and construct and enumerate all regular characters of G(o), when the characteristic of k is greater than two. As a result, we compute the regular part of their representation zeta function.

Quasimodular Hecke algebras and Hopf actions

Let \Gamma=\Gamma(N) be a principal congruence subgroup of SL_2(\mathbb Z) . In this paper, we extend the theory of modular Hecke algebras due to Connes and Moscovici to define the algebra \mathcal Q(\Gamma) of quasimodular Hecke operators of level \Gamma . Then, \mathcal Q(\Gamma) carries an action of “the Hopf algebra \mathcal H_1 of codimension 1 foliations” that also acts on the modular Hecke algebra \mathcal A(\Gamma) of Connes and Moscovici. However, in the case of quasimodular forms, we have several new operators acting on the quasimodular Hecke algebra \mathcal Q(\Gamma) . Further, for each \sigma\in SL_2(\mathbb Z) , we introduce the collection \mathcal Q_\sigma(\Gamma) of quasimodular Hecke operators of level \Gamma twisted by \sigma . Then, \mathcal Q_\sigma(\Gamma) is a right \mathcal Q(\Gamma) -module and is endowed with a pairing (\_\_,\_\_):\mathcal Q_\sigma(\Gamma)\otimes \mathcal Q_\sigma(\Gamma)\longrightarrow \mathcal Q_\sigma(\Gamma). We show that there is a "Hopf action" of a certain Hopf algebra \mathfrak{h}_1 on the pairing on \mathcal Q_\sigma(\Gamma) . Finally, for any \sigma\in SL_2(\mathbb Z) , we consider operators acting between the levels of the graded module \mathbb Q_\sigma(\Gamma)=\underset{m\in \mathbb Z}{\oplus}\mathcal Q_{\sigma(m)}(\Gamma) , where \sigma(m)=\begin{pmatrix} 1 & m \\ 0 & 1 \\ \end{pmatrix}\cdot \sigma for any m\in \mathbb Z . The pairing on \mathcal Q_\sigma(\Gamma) can be extended to a graded pairing on \mathbb Q_\sigma(\Gamma) and we show that there is a Hopf action of a larger Hopf algebra \mathfrak{h}_{\mathbb Z}\supseteq \mathfrak{h}_1 on the pairing on \mathbb Q_\sigma(\Gamma) .

The dimensions of spaces of Siegel cusp forms of general degree

In this paper, we give a dimension formula for spaces of Siegel cusp forms of general degree with respect to neat arithmetic subgroups. The formula was conjectured before by several researchers (cf. [40,43,45,75]). The dimensions are expressed by special values of Shintani zeta functions for spaces of symmetric matrices at non-positive integers. This formula was given by Shintani for only a small part of the geometric side of the trace formula (see [64]). To be precise, it is the contribution of unipotent elements corresponding to the partitions (2j,12n−2j), where n denotes the degree and 0≤j≤n. Hence, our work is to show that all the other contributions vanish. Therefore, one finds that Shintani's formula means the dimension itself. Combining our formula and an explicit formula of the Shintani zeta functions, which was discovered by Ibukiyama and Saito (cf. [44,42,61]), we can derive an explicit dimension formula for the principal congruence subgroups of level greater than two (cf. [43,45]). In this explicit dimension formula, the dimensions are described by degree n, weight k, level N, and the Bernoulli numbers Bm.

Decompositions of congruence subgroups of Chevalley groups

We formulate and prove relative versions of several decompositions known in the theory of Chevalley groups over commutative rings. These decompositions are used to obtain factorizations in terms of subsystem subgroups of type [Formula: see text] and upper estimates of the width of principal congruence subgroups with respect to Tits–Vaserstein generators.

Palindromic automorphisms of right-angled Artin groups

We introduce the palindromic automorphism group and the palindromic Torelli group of a right-angled Artin group A_\Gamma . The palindromic automorphism group \Pi A_\Gamma is related to the principal congruence subgroups of GL (n,\mathbb Z) and to the hyperelliptic mapping class group of an oriented surface, and sits inside the centraliser of a certain hyperelliptic involution in Aut (A_\Gamma) . We obtain finite generating sets for \Pi A_\Gamma and for this centraliser, and determine precisely when these two groups coincide. We also find generators for the palindromic Torelli group.

Modular Hecke algebras over Möbius categories

We extend the modular Hecke operators of Connes and Moscovici by taking values in the modular incidence algebra M [ C ] over a Möbius category C . Here M is the ‘modular tower’ consisting of all modular forms of all weights across all levels. We construct multiple product structures on the collection A ( Γ ) [ C ] of modular Hecke operators of level Γ over C , where Γ is a principal congruence subgroup. These product structures are then shown to be well behaved with respect to Hopf actions on A ( Γ ) [ C ] . While A ( Γ ) [ C ] already carries an action of the Hopf algebra H 1 of codimension 1-foliations, the noncommutativity of the modular incidence algebra M [ C ] allows us to construct additional operators on A ( Γ ) [ C ] . The use of Möbius categories provides a single framework for describing modular Hecke operators taking values in various rings: from formal power series rings over M to arithmetic functions over M and algebras of upper triangular matrices with entries in M . Moreover, we use functors between Möbius categories to study relations between various modular Hecke algebras.

An approximation principle for congruence subgroups

The motivating question of this paper is roughly the following: given a flat group scheme G over \mathbb Z_p , p prime, with semisimple generic fiber G_{\mathbb Q_p} , how far are open subgroups of G(\mathbb Z_p) from subgroups of the form X(\mathbb Z_p)\mathbf K_p(p^n) , where X is a subgroup scheme of G and \mathbf K_p(p^n) is the principal congruence subgroup Ker (G(\mathbb Z_p)\to G(\mathbb Z/p^n\mathbb Z)) ? More precisely, we will show that for G_{\mathbb Q_p} simply connected there exist constants J\ge1 and \varepsilon>0 , depending only on G , such that any open subgroup of G (\mathbb Z_p) of level p^n admits an open subgroup of index \le J which is contained in X(\mathbb Z_p)\mathbf K_p(p^{\lceil \varepsilon n\rceil}) for some proper, connected algebraic subgroup X of G defined over \mathbb Q_p . Moreover, if G is defined over \mathbb Z , then \varepsilon and J can be taken independently of p . We also give a correspondence between natural classes of \mathbb Z_p -Lie subalgebras of \mathfrak {g}_{\mathbb Z_p} and of closed subgroups of G(\mathbb Z_p) that can be regarded as a variant over \mathbb Z_p of Nori's results on the structure of finite subgroups of GL (N_0,\mathbb F_p) for large p [Nor 87]. As an application we give a bound for the volume of the intersection of a conjugacy class in the group G (\hat{\mathbb Z}) = \prod_p G (\mathbb Z_p) , for G defined over \mathbb Z , with an arbitrary open subgroup. In a companion paper, we apply this result to the limit multiplicity problem for arbitrary congruence subgroups of the arithmetic lattice G (\mathbb Z) .

Modular embeddings and automorphic Higgs bundles

We investigate modular embeddings for semi-arithmetic Fuchsian groups. First we prove some purely algebro-geometric or even topological criteria for a regular map from a smooth complex curve to a quaternionic Shimura variety to be covered by a modular embedding. Then we set up an adelic formalism for modular embeddings and apply our criteria to study the effect of abstract field automorphisms of $\mathbb{C}$ on modular embeddings. Finally we derive that the absolute Galois group of $\mathbb{Q}$ operates on the dessins d'enfants defined by principal congruence subgroups of (arithmetic or non-arithmetic) triangle groups by permuting the defining ideals in the tautological way.

APPROXIMATION OF -ANALYTIC TORSION FOR ARITHMETIC QUOTIENTS OF THE SYMMETRIC SPACE

In [31] we defined a regularized analytic torsion for quotients of the symmetric space$\operatorname{SL}(n,\mathbb{R})/\operatorname{SO}(n)$by arithmetic lattices. In this paper we study the limiting behavior of the analytic torsion as the lattices run through sequences of congruence subgroups of a fixed arithmetic subgroup. Our main result states that for principal congruence subgroups and strongly acyclic flat bundles, the logarithm of the analytic torsion, divided by the index of the subgroup, converges to the$L^{2}$-analytic torsion.

Adjoint orbits of matrix groups over finite quotients of compact discrete valuation rings and representation zeta functions

This paper gives methods to describe the adjoint orbits of $\mathbf{G}(\mathfrak{o}_r)$ on $\mathrm{Lie}(\mathbf{G})(\mathfrak{o}_r)$ where $\mathfrak{o}_r=\mathfrak{o}/\mathfrak{p}^r$ ($r\in\mathbb{N}$) is a finite quotient of the localization $\mathfrak{o}$ of the ring of integers of a number field at a prime ideal $\mathfrak{p}$ and $\mathbf{G}$ is a closed $\mathbb{Z}$-subgroup scheme of $\mathrm{GL}_{n}$ for an $n\in\mathbb{N}$ and such that the Lie ring $\mathrm{Lie}(\mathbf{G})(\mathfrak{o})$ is quadratic.. The main result is a classification of the adjoint orbits in $\mathrm{Lie}(\mathbf{G})(\mathfrak{o}_{r+1})$ whose reduction $\bmod\,\mathfrak{p}^{r}$ contains $a\in\mathrm{Lie}(\mathbf{G})(\mathfrak{o}_r)$ in terms of the reduction $\bmod\mathfrak{p}$ of the stabilizer of $a$ for the $\mathbf{G}(\mathfrak{o}_r)$-adjoint action. As an application, this result is then used to compute the representation zeta function of the principal congruence subgroups of $\mathrm{SL}_{3}(\mathfrak{o})$.

Regular characters of groups of type [formula omitted] over discrete valuation rings

Let o be a complete discrete valuation ring with finite residue field k of odd characteristic. Let G be a general or special linear group or a unitary group defined over o and let g denote its Lie algebra. For every positive integer ℓ, let Kℓ be the ℓ-th principal congruence subgroup of G(o). A continuous irreducible representation of G(o) is called regular of level ℓ if it is trivial on Kℓ+1 and its restriction to Kℓ/Kℓ+1≃g(k) consists of characters with G(k‾)-stabiliser of minimal dimension. In this paper we construct the regular characters of G(o), compute their degrees and show that the latter satisfy Ennola duality. We give explicit uniform formulae for the regular part of the representation zeta functions of these groups.

Zariski density and computing in arithmetic groups

For $n > 2$, let $\Gamma$ denote either $SL(n, Z)$ or $Sp(n, Z)$. We give a practical algorithm to compute the level of the maximal principal congruence subgroup in an arithmetic group $H\leq \Gamma$. This forms the main component of our methods for computing with such arithmetic groups $H$. More generally, we provide algorithms for computing with Zariski dense groups in $\Gamma$. We use our GAP implementation of the algorithms to solve problems that have emerged recently for important classes of linear groups.

Multiple commutator formulas for unitary groups

Let (A,Λ) be a formring such that A is quasi-finite R-algebra (i.e., a direct limit of module finite algebras) with identity. We consider the hyperbolic Bak’s unitary groups GU(2n, A, Λ), n ≥ 3. For a form ideal (I, Γ) of the form ring (A, Λ) we denote by EU(2n, I, Γ) and GU(2n, I, Γ) the relative elementary group and the principal congruence subgroup of level (I, Γ), respectively. Now, let (I i , Γ i ), i = 0,...,m, be form ideals of the form ring (A, Λ). The main result of the present paper is the following multiple commutator formula: [EU(2n, I 0, Γ 0),GU(2n, I 1, Γ 1), GU(2n, I 2, Γ 2),..., GU(2n, I m , Γ m )] =[EU(2n, I 0, Γ 0), EU(2n, I 1, Γ 1), EU(2n, I 2, Γ 2),..., EU(2n, I m , Γ m )], which is a broad generalization of the standard commutator formulas. This result contains all previous results on commutator formulas for classicallike groups over commutative and finite-dimensional rings.

On the automorphic atoms of length 2 of $\mathrm{GL}_2(\mathbb{Q}_p)$

Let p>3 be a prime. The aim of this paper is to give a description of the invariant space, under principal and Iwahori congruence subgroups of arbitrary level, of extensions of generic principal series representations appearing in the p -modular local Langlands correspondence for \bold{GL}_2(\bold{Q}_p) . As an application we describe Hecke isotypical components of the mod p cohomology of the modular curve over \bold{Q} with deeply ramified level at p .