Congruence Groups Research Articles

Certain congruences on E-inversive E-semigroups

A semigroup S is called E-inversive if for every a ∈ S there exists x ∈ S such that ax is idempotent. S is called E-semigroup if the set of idempotents of S forms a subsemigroup. In this paper some special congruences on E-inversive E-semigroups are investigated, such as the least group congruence, a certain semilattice congruence, some regular congruences and a certain idempotent-separating congruence.

Elliptic functions and equations of modular curves

Let $p\ge 5$ be a prime. We show that the space of weight one Eisenstein series defines an embedding into ${\mathbb P}^{(p-3)/2} $ of the modular curve $ X_1(p)$ for the congruence group $\Gamma_1(p)$ that is scheme-theoretically cut out by explicit quadratic equations.

Environmental Congruence, Group Importance, and Well-Being Among Paratroopers

The study examined the relation between variables of environmental congruence (or Person-environment fit; P-E fit) and group importance on the one hand and variables of satisfaction and somatic complaints on the other. Participants were 267 paratroopers who belonged to 35 units (environments). Both the diadic and the census methods were used to assign congruence scores. Findings showed (a) group importance correlated .55 and -.25 (p < .05) with satisfaction and somatic complaints, respectively, while congruence did not show significant correlations; (b) The nonlinear examination of the relationships showed a significant (p < .05) main effect for group importance on satisfaction and for the interaction of group importance and congruence both on satisfaction and on somatic complaints. It is suggested on the basis of these and earlier findings that group importance is a sufficient condition for satisfaction and that congruence interacts with it to enhance satisfaction. The superiority of the diadic method was not statistically confirmed.

Classification of torsion-free genus zero congruence groups

We study and classify all torsion-free genus zero congruence subgroups of the modular group.

Toric modular forms and nonvanishing of L-functions

In a previous paper [1], we defined the space of toric forms T (l), and showed that it is a finitely generated subring of the holomorphic modular forms of integral weight on the congruence group Γ1(l). In this article we prove the following theorem: modulo Eisenstein series, the weight two toric forms coincide exactly with the vector space generated by all cusp eigenforms f such that L(f, 1) 6= 0. The proof uses work of Merel, and involves an explicit computation of the intersection pairing on Manin symbols.

Spectral theory of Laplacians for Hecke groups with primitive character

It was proved by [Sel] tha t the Laplacian A(F) for congruence subgroups F of the modular group Fz has an infinite sequence of embedded eigenvalues {Ai} satisfying a Weyl law #{~i~<A}~(rFf/47r)A for A--+cx~. Here I l l is the area of the fundamental domain F of the group F, and the eigenvalues Ai are counted according to multiplicity. The same holds true for the Laplacian A(F; X), where X is a character on F and A(F; X) is associated with a congruence subgroup F1 of F. It is an important question whether this is a characteristic of congruence groups or it may hold also for some non-congruence subgroups of Fz. To investigate this problem Phillips and Sarnak studied per turbat ion theory for Laplacians A(F) with regular per turbat ions derived from modular forms of

Some remarks on a spectral correspondence for maass waveforms

We prove a detailed result in classical language on a special case of the Jacquet-Langlands correspondence: Let O1 be the (cocompact) Fuchsian group consisting of the units of norm one in a maximal order O in an indefinite quaternion algebra over ℚ. Let d=d(O) be the discriminant of O⁠. Our result involves a certain integral transform Θ constructed by Dennis Hejhal (1985). We prove that when chosen appropriately, the map Θ gives a linear bijection between each discrete eigenspace of the Laplacian on O1\ℋ and the space of newforms of the same eigenvalue on the congruence group Γ0(d). The proof uses trace formulas for the Hecke operators on O1 and Γ0(d)

Hecke theory of series formed with modular symbols and relations among convolution L -functions

In [G1], [G2], the distribution of modular symbols is studied and a new class of functions which satisfy a transformation law involving these objects is introduced. The goal of Goldfeld’s program is to prove Szpiro’s conjecture which states that for elliptic curves with minimal discriminant D and conductor N there is an absolute constant κ such that D ? N. To do this, an equivalent conjecture involving modular symbols is established in [G4] using the, now proven, conjecture of Shimura, Taniyama and Weil. It seems that a sufficiently good understanding of the new series proposed by Goldfeld should yield a resolution of these conjectures. We repeat here their definition in a somewhat more general form to include period polynomials rather than modular symbols only. For positive integers M,N such that M |N, let m, k be non-negative integers such that m ≥ k − 2 ≥ 0 and let χ be a Dirichlet character modulo N. First, for each f ∈ Sk(M) = {holomorphic cusp forms of weight k and level M} we denote by rf the map which sends γ ∈ Γ0(M), the Hecke congruence group of level M , to the polynomial function:

Relations between cusp forms on congruence and noncongruence groups

We construct a quadratic relation between cusp forms of weight two on four genus 1 subgroups of SL2 (Z). Two of the subgroups are congruence and two are noncongruence. We then generalize this to subgroups of F(N) of index 2. Let A be a finite-index subgroup of SL2(Z), and let XA be the associated Riemann surface (*/A. If K is a number field and C/K is a projective nonsingular curve, we say that C/K is a model for XA if there exists a morphism : C * JPl defined over K and a complex analytic isomorphism p: XA 0(QC) such that p(oo) is rational over K and XA P 0 (C) (1) 1 jjc * /SL2 (Z) i PI1 (C) commutes. (Here, the left hand arrow is the natural projection map.) It is known that for any A one can find a C/K as above, and, in fact, Belyi's theorem [B] tells us that the converse is true as well. In light of this, one may wish to study the usual arithmetic objects associated with congruence groups (i.e., Hecke operators, Fourier coefficients, and so forth) for arbitrary subgroups of finite index in SL2(Z). While it is true that on a general noncongruence subgroup, Hecke operators and Fourier coefficients are not related, there are nevertheless results concerning both. (See [Be] for Hecke operators on noncongruence subgroups and [S] for Fourier coefficients. From [Be], we see that Hecke operators on noncongruence subgroups give essentially no new information, but the Atkin-Swinnerton-Dyer congruences of [S] are highly nontrivial.) What has not been previously studied, however, is the relationship between cusp forms on related congruence and noncongruence groups. Since cusp forms on congruence groups are to some extent understood, any explicit information we obtain may be viewed as shedding light on the noncongruence forms. In particular, an actual polynomial relationship between forms would lead to an infinite number of relations between Fourier coefficients. One may view this as a Received by the editors November 16, 1998. 1991 Mathematics Subject Classification. Primary lIF1I; Secondary 1F30.

Fuzzy complete inner-unitary subsemigroups and fuzzy group congruences on a regular semigroup

In this paper, we introduce the concepts of the fuzzy complete inner-unitary subsemigroups and the fuzzy group congruences on a semigroup. Some of their properties are investigated. Also, we show that there is a one to one correspondence between the fuzzy complete inner-unitary subsemigroups and the fuzzy group congruences on a regular semigroup.

Graph expansions of unipotent monoids

Margolis and Meakin use the Cayley graph of a group presentation to construct E-unitary inverse monoids [11]. This is the technique we refer to as graph expansion. In this paper we consider graph expansions of unipotent monoids, where a monoid is unipotent if it contains a unique idempotent. The monoids arising in this way are E-unitary and belong to the quasivariety of weakly left ample monoids. We give a number of examples of such monoids. We show that the least unipotent congruence on a weakly left ample monoid is given by the same formula as that for the least group congruence on an inverse monoid and we investigate the notion of proper for weakly left ample monoids. Using graph expansions we construct a functor Fe from the category U of unipotent monoids to the category PWLA of proper weakly left ample monoids. The functor Fe is an expansion in the sense of Birget and Rhodes [2]. If we equip proper weakly left ample monoids with an extra unary operation and denote the corresponding category by PWLA 0 then regarded as a functor U→PWLA 0 Fe is a left adjoint of the functor Fσ : PWLA 0 → U that takes a proper weakly left ample monoid to its greatest unipotent image. Our main result uses the covering theorem of [8] to construct free weakly left ample monoids.

A Spectral Correspondence for Maa� Waveforms

Let \( {\cal O}^1 \) be a (cocompact) Fuchsian group, given as the group of units of norm one in a maximal order \( {\cal O} \) in an indefinite quaternion division algebra over \( {\Bbb Q} \). Using the (classical) Selberg trace formula, we show that the eigenvalues of the automorphic Laplacian for \( {\cal O}^1 \) and their multiplicities coincide with the eigenvalues and multiplicities of the Laplacian defined on the Maas newforms for the Hecke congruence group \( \Gamma_0(d) \), when d is the discriminant of the maximal order \( {\cal O} \). We also show the equality of the traces of certain Hecke operators defined on the Laplace eigenspaces for \( {\cal O}^1 \) and the newforms of level d, respectively.

Comparison of spectra. II

A new version of the Selberg spectral conjecture is proved, i.e., for a congruence group with a nontrivial multiplier. Bibliography: 7titles.

Quotients of theta series as rational functions of j1,4

Let Q( n, 1) be the set of even unimodular positive definite integral quadratic forms in n-variables. Then n is divisible by 8. For A[ x] in Q( n, 1), the theta series 0 A(z): = ∑ XϵZ ne πizA¦X¦ (z ϵ h , the complex upper half plane) is a modular form of weight n 2 for the congruence group Γ 1(4) = {γ ϵ SL 2(Z)¦γ  1 ∗ 0 1 mod 4} . If n ≥ 24 and A[ X], B[ X] are two quadratic forms in Q( n, 1), then the quotient θ A(z) θ B(z) is a modular function for Γ 1(4). Since we can identify the field of modular functions for Γ 1(4) with the function field K( X 1(4)) over the modular curve x 1(4) = Γ 1(4) h∗ (the extended plane of h ) with genus 0, in this paper, we express it as a rational function j 1,4 which is a field generator over C of K( X 1(4)) and defined by j 1.4(z): = θ 2(2z) 4 (θ 3(2z) 4 . Here, θ 2 and θ 3 denote the classical Jacobi theta functions.

Comparison of spectra

The Selberg conjecture concerning the absence of small eigenvalues in the spectrum of the congruence group is discussed. An approach to the proof via comparison of two different spectral expansions for one and the same arithmetic shift-averaged convolution is suggested. Bibliography: 6 titles.

Volumes of hyperbolic manifolds and mixed Tate motives

Two different constructions of an invariant of an odd-dimensional hyperbolic manifold with values in K 2 n − 1 ( Q ¯ ) ⊗ Q K_{2n-1}(\overline {\mathbb Q})\otimes \mathbb Q are given. We prove that the volume of the manifold equals the value of the Borel regulator on this invariant. The scissors congruence groups in noneuclidean geometries are studied and related to mixed Tate motives and algebraic K-theory of C \mathbb C . We contribute to the general theory of mixed Hodge structures by introducing for Hodge-Tate structures the big period map with values in C ⊗ C ∗ ( n − 2 ) \mathbb C \otimes \mathbb C^*(n-2) .

Convolutions. Comparison of spectral expansions at different vertices

Two spectral expansions of one and the same convolution at two different vertices of the fundamental domain Fq for the congruence group Гo(q) are compared. As a result, a representation of the space of functions of a given eigenvalue λn in terms of the whole remaining spectrum is obtained. Bibliography:3 titles.

On the scattering matrix for the Eisenstein series for the Hecke congruence groups

On the scattering matrix for the Eisenstein series for the Hecke congruence groups

The Emergence of Orientation and the Geometry of Logic

In this paper it is said that cultural evolution consists of four phases which are denoted by the words nature, mythology I, mythology II and history. Pictograms, ideograms and logograms signify two phases fundamental for the later emergence of the linear writing and time as history and thought. The nowaday operational structures of cognition appear as rooted in prehistoric thought meaning vision, ideas – images seen with the inner eye – and logos as orientation of logograms. Thus it is assumed that logic in the form of the elementary first order propositional calculus should be derived from the orientation symmetry of space. The author gives an exact proof in terms of the Clifford algebra that this is indeed the case. Four basic representations in Cl2, Cl3, Cl3 and Cl3,1 of the 16 expressions of second order logic relations are given. The proposition of transformational self-reference is defined. The most basic self-referent representation of Peirce's box-X icons (the 16 binary logic connectives) is derivated from the spin-representation δD2d of the spatial congruence group of the square or the mandala of the sunwheel. Finally it is shown that the representation in terms of the idempotents of the Clifford algebra Cl3,1 is isomorphic with Zellweger's logic garnet.

Homology of O( n) and O1(1, n) made discrete: An application of edgewise subdivision

The group of isometrics of hyperbolic n-space contains the orthogonal group O( n) as a subgroup. We prove that this inclusion induces a stable isomorphism of discrete group homology. The unstable version of this result implies in particular that the scissors congruence group P ( S 3 ) in spherical 3-space is a rational vectorspace.