Principal Congruence Subgroup Research Articles

Twisted Conjugacy Classes in Abelian Extensions of Certain Linear Groups

AbstractGiven a group automorphismϕ: Γ → Γ, one has an action of Γ on itself byϕ-twisted conjugacy, namely,g.x = gxϕ(g-1). The orbits of this action are calledϕ-twisted conjugacy classes. One says that Γ has theR∞-property if there are infinitely manyϕ-twisted conjugacy classes for every automorphismϕof Γ. In this paper we show that SL(n; Z) and its congruence subgroups have the R8-property. Further we show that any (countable) abelian extension of Γ has the R8-property where Γ is a torsion free non-elementary hyperbolic group, or SL(n; Z); Sp(2n; Z) or a principal congruence subgroup of SL(n; Z) or the fundamental group of a complete Riemannian manifold of constant negative curvature.

COHOMOLOGY OF BRAIDS, PRINCIPAL CONGRUENCE SUBGROUPS AND GEOMETRIC REPRESENTATIONS

The main purpose of this article is to give the integral cohomology of classical principal congruence subgroups in SL(2, ℤ) as well as their analogues in the 3-strand braid group with local coefficients in symmetric powers of the natural symplectic representation. The resulting answers (1) correspond to certain modular forms in characteristic zero and (2) the cohomology of certain spaces in homotopy theory in characteristic p. The torsion is given in terms of the structure of a ‘p-divided power algebra’. The work is an extension of the work in Callegaro et al. [The cohomology of the braid group B3 and of SL2(ℤ) with coefficients in a geometric representation, Quart. J. Math. 64 (2013), 847–889] as well as extensions of a classical computation of Shimura to integral coefficients. The results here contrast the local coefficients such as that in Looijenga [Stable cohomology of the mapping class group with symplectic coefficients and of the universal Abel–Jacobi map, J. Algebraic Geom.5 (1996), 135–150] and Tillmann [The representation of the mapping class group of a surface on its fundamental group in stable homology, Quart. J. Math.61 (2010), 373–380].

Local-global principle for congruence subgroups of Chevalley groups

AbstractSuslin’s local-global principle asserts that if a matrix over a polynomial ring vanishes modulo the independent variable and is locally elementary then it is elementary. In this article we prove Suslin’s local-global principle for principal congruence subgroups of Chevalley groups. This result is a common generalization of the result of Abe for the absolute case and Apte, Chattopadhyay and Rao for classical groups. For the absolute case the localglobal principle was recently obtained by Petrov and Stavrova in the more general settings of isotropic reductive groups.

On locally analytic Beilinson–Bernstein localization and the canonical dimension

Let $$\mathbf{G}$$ be a connected split reductive group over a $$p$$ -adic field. In the first part of the paper we prove, under certain assumptions on $$\mathbf{G}$$ and the prime $$p$$ , a localization theorem of Beilinson–Bernstein type for admissible locally analytic representations of principal congruence subgroups in the rational points of $$\mathbf{G}$$ . In doing so we take up and extend some recent methods and results of Ardakov–Wadsley on completed universal enveloping algebras (Ardakov and Wadsley, Ann. Math., 2013) to a locally analytic setting. As an application we prove, in the second part of the paper, a locally analytic version of Smith’s theorem on the canonical dimension.

Representation zeta functions of compact p-adic analytic groups and arithmetic groups

We introduce new methods from p-adic integration into the study of representation zeta functions associated to compact p-adic analytic groups and arithmetic groups. They allow us to establish that the representation zeta functions of generic members of families of p-adic analytic pro-p groups obtained from a global, “perfect” Lie lattice satisfy functional equations. In the case of “semisimple” compact p-adic analytic groups, we exhibit a link between the relevant p-adic integrals and a natural filtration of the locus of irregular elements in the associated semisimple Lie algebra, defined by the centralizer dimension. Based on this algebro-geometric description, we compute explicit formulas for the representation zeta functions of principal congruence subgroups of the groups SL3(o), where o is a compact discrete valuation ring of characteristic 0, and of the groups SU3(O,o), where O is an unramified quadratic extension of o. These formulas, combined with approximative Clifford theory, allow us to determine the abscissae of convergence of representation zeta functions associated to arithmetic subgroups of algebraic groups of type A2. Assuming a conjecture of Serre on the congruence subgroup problem, we thereby prove a conjecture of Larsen and Lubotzky on lattices in higher-rank semisimple groups for algebraic groups of type A2 defined over number fields.

On zeta functions associated to symmetric matrices, II: Functional equations and special values

AbstractNew simple functional equations of zeta functions of the prehomogeneous vector spaces consisting of symmetric matrices are obtained, using explicit forms of zeta functions in the previous paper, Part I, and real analytic Eisenstein series of half-integral weight. When the matrix size is 2, our functional equations are identical with the ones by Shintani, but we give here an alternative proof. The special values of the zeta functions at nonpositive integers and the residues are also explicitly obtained. These special values, written by products of Bernoulli numbers, are used to give the contribution of “central” unipotent elements in the dimension formula of Siegel cusp forms of any degree. These results lead us to a conjecture on explicit values of dimensions of Siegel cusp forms of any torsion-free principal congruence subgroups of the symplectic groups of general degree.

On Hilbert modular threefolds of discriminant 49

Let $$K$$ be the totally real cubic field of discriminant $$49$$ , let $${\fancyscript{O}}$$ be its ring of integers, and let $$p\subset {\fancyscript{O}}$$ be the prime over $$7$$ . Let $$\Gamma (p)\subset \Gamma = SL_{2} ({\fancyscript{O}})$$ be the principal congruence subgroup of level $$p$$ . This paper investigates the geometry of the Hilbert modular threefold attached to $$\Gamma (p) $$ and some related varieties. In particular, we discover an octic in $$\mathbb{P }^3$$ with $$84$$ isolated singular points of type $$A_2$$ .

Multiple commutator formulas

Let A be a quasi-finite R-algebra (i.e., a direct limit of module finite algebras) with identity. Let Ii, i = 0, …,m, be two-sided ideals of A, GLn(A, Ii) be the principal congruence subgroup of level Ii in GLn(A) and En(A, Ii) be the relative elementary subgroup of level Ii. We prove the multiple commutator formula $$\left[ {{E_n}(A,{I_0}),{\rm{G}}{{\rm{L}}_n}(A,{I_1}),{\rm{G}}{{\rm{L}}_n}(A,{I_2}), \ldots ,{\rm{G}}{{\rm{L}}_n}(A,{I_m})} \right] = \left[ {{E_n}(A,{I_0}),{E_n}(A,{I_1}),{E_n}(A,{I_2}), \ldots ,{E_n}(A,{I_m})} \right],$$ , which is a broad generalization of the standard commutator formulas. This result contains all the published results on commutator formulas over commutative rings and answers a problem posed by A. Stepanov and N. Vavilov.

Equidistribution of Zeros of Holomorphic Sections in the Non-compact Setting

We consider N-tensor powers of a positive Hermitian line bundle L over a non-compact complex manifold X. In the compact case, B. Shiffman and S. Zelditch proved that the zeros of random sections become asymptotically uniformly distributed with respect to the natural measure coming from the curvature of L, as N tends to infinity. Under certain boundedness assumptions on the curvature of the canonical line bundle of X and on the Chern form of L we prove a non-compact version of this result. We give various applications, including the limiting distribution of zeros of cusp forms with respect to the principal congruence subgroups of SL2(Z) and to the hyperbolic measure, the higher dimensional case of arithmetic quotients and the case of orthogonal polynomials with weights at infinity. We also give estimates for the speed of convergence of the currents of integration on the zero-divisors.

ON KAZHDAN CONSTANTS OF FINITE INDEX SUBGROUPS IN SLn(ℤ)

We prove that for any finite index subgroup Γ in SL n(ℤ), there exists k = k(n) ∈ ℕ, ϵ = ϵ(Γ) > 0, and an infinite family of finite index subgroups in Γ with a Kazhdan constant greater than ϵ with respect to a generating set of order k. On the other hand, we prove that for any finite index subgroup Γ of SL n(ℤ), and for any ϵ > 0 and k ∈ ℕ, there exists a finite index subgroup Γ′ ≤ Γ such that the Kazhdan constant of any finite index subgroup in Γ′ is less than ϵ, with respect to any generating set of order k. In addition, we prove that the Kazhdan constant of the principal congruence subgroup Γn(m), with respect to a generating set consisting of elementary matrices (and their conjugates), is greater than [Formula: see text], where c > 0 depends only on n. For a fixed n, this bound is asymptotically best possible.

Hecke Operators and the Stable Homology of GL(n)

Let R be a field of any characteristic and A a principal ideal domain. We make a conjecture that asserts that any Hecke operator T acts punctually on any Hecke eigenclass in the stable homology with trivial coefficients R of a principal congruence subgroup Γ in GL(n, A), i.e., as multiplication by the number of single cosets contained in T. In the case where A = ℤ, this conjecture implies that the Galois representation consisting of the sum of the 0th, 1st, ⋅, n − 1st powers of the cyclotomic character is attached to any stable Hecke eigenclass. When A = ℤ and Γ =GL(n, ℤ), this conjecture was already made by Calegari and Venkatesh. We obtain partial results giving evidence for the conjecture. These results imply that some part of the Hecke algebra acts punctually. If the characteristic of R is 0, they show that the entire Hecke algebra acts punctually, which was already known in a completely different way using a result of A. Borel.

Subgroups of algebraic groups which are clopen in the S-congruence topology

AbstractLetBy applying the lemma to

Hecke operators on line bundles over modular curves

Given a principal congruence subgroup Γ = Γ ( N ) ⊆ SL 2 ( Z ) , Connes and Moscovici have introduced a modular Hecke algebra A ( Γ ) that incorporates both the pointwise multiplicative structure of modular forms and the action of the classical Hecke operators. It is well known that a Γ -modular form g of weight k may be described as a global section of the k -th tensor power of a certain line bundle p ( Γ ) : L ( Γ ) → Γ \ H . The purpose of this paper is to develop a theory of modular Hecke algebras for Hecke correspondences between the line bundles L ( Γ ) that lift the classical Hecke correspondences between modular curves Γ \ H .

Zeroes of Eisenstein series for principal congruence subgroups over rational function fields

We determine the zeroes of Drinfeld–Goss Eisenstein series for the principal congruence subgroups Γ ( N ) of Γ = GL ( 2 , F q [ T ] ) on the Drinfeld modular curve X ( N ) .

On the some normal subgroups of the extended modular group

Abstract In this paper, we give the group structures and the signatures of some normal subgroups of the extended modular group Π containing the principal congruence subgroup Γ (12).

Generalized Commutator Formulas

Let A be an algebra which is a direct limit of module finite algebras over a commutative ring R with 1. Let I, J be two-sided ideals of A, GL n (A, I) the principal congruence subgroup of level I in GL n (A), and E n (A, I) the relative elementary subgroup of level I. Using Bak's localization-patching method, we prove the commutator formula which is a generalization of the standard commutator formular. This answers a problem posed by Stepanov and Vavilov.

Lifts of projective congruence groups

We show that noncongruence subgroups of SL2(ℤ) that are projectively equivalent to congruence subgroups are ubiquitous. More precisely, they always exist if the congruence subgroup in question is a principal congruence subgroup Γ(N) of level N>2, and they also exist in many cases for Γ0(N). The motivation for asking this question is related to modular forms: projectively equivalent groups have the same spaces of cusp forms for all even weights, whereas the spaces of cusp forms of odd weights are distinct in general. We make some initial observations on this phenomenon for weight 3 via geometric considerations of the attached elliptic modular surfaces. We also develop algorithms that construct all subgroups that are projectively equivalent to a given congruence subgroup and decide which of them are congruence. A crucial tool in this is the generalized level concept of Wohlfahrt.

Affine models of the modular curves X(p) and its application

Farkas, Kra and Kopeliovich (Commun. Anal. Geom. 4(2):207–259, 1996) showed that the quotients F1 and F2 of modified theta functions generate the function field \(\mathcal{K}(X(p))\) of the modular curve X(p) for a principal congruence subgroup Γ(p) with prime p≥7. For such primes p we first find affine models of X(p) over ℚ represented by Φp(X,Y)=0, from which we are able to obtain the algebraic relations Ψp(X,Y)=0 of F1 and F2 presented by Farkas et al. As its application we construct the ray class field K(p) modulo p over an imaginary quadratic field K and then explicitly calculate its class polynomial by using the Shimura reciprocity law.

Increasing the number of fibered faces of arithmetic hyperbolic 3-manifolds

We exhibit a closed hyperbolic 3-manifold which satisfies a very strong form of Thurston's Virtual Fibration Conjecture. In particular, this manifold has finite covers which fiber over the circle in arbitrarily many ways. More precisely, it has a tower of finite covers where the number of fibered faces of the Thurston norm ball goes to infinity, in fact faster than any power of the logarithm of the degree of the cover, and we give a more precise quantitative lower bound. The example manifold $M$ is arithmetic, and the proof uses detailed number-theoretic information, at the level of the Hecke eigenvalues, to drive a geometric argument based on Fried's dynamical characterization of the fibered faces. The origin of the basic fibration $M\to S^1$ is the modular elliptic curve $E=X_0(49)$, which admits multiplication by the ring of integers of ${\Bbb Q}[\sqrt{-7}]$. We first base change the holomorphic differential on $E$ to a cusp form on ${{\rm GL}(2)}$ over $K={\Bbb Q}[\sqrt{-3}]$, and then transfer over to a quaternion algebra $D/K$ ramified only at the primes above $7$; the fundamental group of $M$ is a quotient of the principal congruence subgroup of ${\cal O}_D^\ast$ of level $7$. To analyze the topological properties of $M$, we use a new practical method for computing the Thurston norm, which is of independent interest. We also give a noncompact finite-volume hyperbolic 3-manifold with the same properties by using a direct topological argument.

A variant of Jacobi type formula for Picard curves

The classical Jacobi formula for the elliptic integrals (Gesammelte Werke I, p. 235) shows a relation between Jacobi theta constants and periods of ellptic curves E ( λ ) : w 2 = z ( z - 1 ) ( z - λ ) . In other words, this formula says that the modular form ϑ 00 4 ( τ ) with respect to the principal congruence subgroup Γ ( 2 ) of PSL ( 2 , Z ) has an expression by the Gauss hypergeometric function F ( 1 / 2 , 1 / 2 , 1 ; 1 - λ ) via the inverse of the period map for the family of elliptic curves E ( λ ) (see Theorem 1.1). In this article we show a variant of this formula for the family of Picard curves C ( λ 1 , λ 2 ) : w 3 = z ( z - 1 ) ( z - λ 1 ) ( z - λ 2 ) , those are of genus three with two complex parameters. Our result is a two dimensional analogy of this context. The inverse of the period map for C ( λ 1 , λ 2 ) is established in [S] and our modular form ϑ 0 3 ( u , v ) (for the definition, see (2.7)) is defined on a two dimensional complex ball D = { 2 Re v + | u | 2 < 0 } , that can be realized as a Shimura variety in the Siegel upper half space of degree 3 by a modular embedding. Our main theorem says that our theta constant is expressed in terms of the Appell hypergeometric function F 1 ( 1 / 3 , 1 / 3 , 1 / 3 , 1 ; 1 - λ 1 , 1 - λ 2 ) .