Congruence Groups Research Articles

Zero-Semidistributive Inverse Semigroups

An inverse semigroup S is said to be 0-semidistributive if its lattice ℒF (S) of full inverse subsemigroups is 0-semidistributive. We show that it is sufficient to study simple inverse semigroups which are not groups. Our main theorem states that such a simple inverse semigroup S is 0-semidistributive if and only if (1) S is E-unitary, (2) S is aperiodic, (3) for any a,b ∈ S/σ with ab ≠ 1, there exist nonzero integers n and m such that (ab) m = a n or (ab) m = b n , where σ is the minimum group congruence on S.

R-Unipotent Congruences on Eventually Regular Semigroups

A semigroup S is called an eventually regular semigroup if for every a ∈ S, there exists a positive integer n such that anis regular. In this paper, the R-unipotent, inverse semigroup and group congruences on an eventually regular semigroup S are described by means of certain congruence pairs (ξ, K), where ξ is a normal congruence on the subsemigroup 〈E(S)〉 generated by E(S), and K is a normal subsemigroup of S.

Compactness of the congruence group of measurable functions in several variables

We solve a problem, which appears in functional analysis and geometry, on the group of symmetries of functions of several arguments. Let \(f:\prod\limits_{i = 1}^n {X_i \to Z} \) be a measurable function defined on the product of finitely many standard probability spaces (Xi, \(\mathfrak{B}_i \), μi), 1 ≤ i ≤ n, that takes values in any standard Borel space Z. We consider the Borel group of all n-tuples (g1, ..., gn) of measure preserving automorphisms of the respective spaces (Xi, \(\mathfrak{B}_i \), μi) such that f(g1 x 1, ..., gnxn) = f(x1, ..., xn) almost everywhere and prove that this group is compact, provided that its “trivial” symmetries are factored out. As a consequence, we are able to characterize all groups that result in such a way. This problem appears with the question of classifying measurable functions in several variables, which was solved by the first author but is interesting in itself. Bibliography: 5 titles.

Numerical computations with the trace formula and the Selberg eigenvalue conjecture

We verify the Selberg eigenvalue conjecture for congruence groups of small squarefree conductor, improving on a result of Huxley [M. N. Huxley, Introduction to Kloostermania, in: Elementary and analytic theory of numbers, Banach Center Publ. 17, Warsaw (1985), 217–306.]. The main tool is the Selberg trace formula which, unlike previous geometric methods, allows for treatment of cases where the eigenvalue 1/4 is present. We present a few other sample applications, including the classification of even 2-dimensional Galois representations of small squarefree conductor.

TRACES OF SINGULAR VALUES AND BORCHERDS PRODUCTS

Let p be a prime for which the congruence group Γ0(p)* is of genus zero, and let j p * be the corresponding Hauptmodul. Let f be a nearly holomorphic modular form of weight 1/2 on Γ0(4p) which satisfies some congruence condition on its Fourier coefficients. We interpret f as a vector valued modular form. Applying Borcherds lifting of vector valued modular forms we construct infinite products associated to j p * and relate them to Zagier's trace formula for the singular values of j p * . 2000 Mathematics Subject Classification 11F03, 11F30 (primary); 11F22, 11F37, 11F50 (secondary).

A converse theorem for [formula omitted

We prove that a Dirichlet series with a functional equation and Euler product of a particular form can only arise from a holomorphic cusp form on the Hecke congruence group Γ 0 ( 13 ) . The proof does not assume a functional equation for the twists of the Dirichlet series. The main new ingredient is a generalization of the familiar Weil's lemma that played a prominent role in previous converse theorems.

SOME INTUITIONISTIC FUZZY CONGRUENCES

First, we introduce the concept of intuitionistic fuzzy group con- gruence and we obtain the characterizations of intuitionistic fuzzy group con- gruences on an inverse semigroup and a T -pure semigroup, respectively. Also, we study some properties of intuitionistic fuzzy group congruence. Next, we introduce the notion of intuitionistic fuzzy semilattice congruence and we give the characterization of intuitionistic fuzzy semilattice congruence on a T -pure semigroup. Finally, we introduce the concept of intuitionistic fuzzy normal congruence and we prove that (IFNC(ES),\,_) is a complete lattice. And we find the greatest intuitionistic fuzzy normal congruence containing an in- tuitionistic fuzzy congruence on ES.

H. Weyl’s Asymptotics and Rankin Convolutions

A well-known H. Weyl’s asymptotics for eigenvalues is obtained by an arithmetic method. For congruence groups, the remainder in this asymptotics is a square root of the principal term. Bibliography: 14 titles.

More on super-replication formulae

We extend Norton–Borcherds–Koike's replication formulae to super-replicable ones by working with the congruence groups Γ 1 ( N ) and find the product identities which characterize super-replicable functions. These will provide a clue for constructing certain new infinite-dimensional Lie superalgebras whose denominator identities coincide with the above product identities. Therefore it could be one way to find a connection between modular functions and infinite-dimensional Lie algebras.

A Representation for $F$-Regular Semigroups

A regular (inverse) semigroup S is called F-regular (F-inverse), if each class of the least group congruence σS contains a greatest element with respect to the natural partial order on S. Such a semigroup is necessarily an E-unitary regular (hence orthodox) monoid. We show that each F-regular semigroup S is isomorphic to a well determined subsemigroup of a semidirect product of a band X by S/σS, where X belongs to the band variety, generated by the band of idempotents ES of S. Our main result, Theorem 4, is the regular version of the corresponding fact for inverse semigroups, and might be useful to generalize further features of the theory of F-inverse semigroups to the F-regular case.

Generalized $F$-semigroups

A semigroup $S$ is called a generalized $F$-semigroup if there exists a group congruence on $S$ such that the identity class contains a greatest element with respect to the natural partial order $\le _{S}$ of $S$. Using the concept of an anticone, all partially ordered groups which are epimorphic images of a semigroup $(S,\cdot ,\le _{S})$ are determined. It is shown that a semigroup $S$ is a generalized $F$-semigroup if and only if $S$ contains an anticone, which is a principal order ideal of $(S,\le _{S})$. Also a characterization by means of the structure of the set of idempotents or by the existence of a particular element in $S$ is given. The generalized $F$-semigroups in the following classes are described: monoids, semigroups with zero, trivially ordered semigroups, regular semigroups, bands, inverse semigroups, Clifford semigroups, inflations of semigroups, and strong semilattices of monoids.

Asymptotic densities of Maass newforms

We define the counting function for Maass newforms of Hecke congruence groups and calculate the three main terms of this counting function. We then give necessary and sufficient conditions for this expansion to have the same shape as if it were counting eigenvalues related to cocompact surfaces. We relate the result to classical instances of the Jacquet–Langlands correspondence.

The role of health congruence in functional status and depression.

Few studies have attempted to examine the meaning of health congruence, particularly in the oldest old. Participants were drawn from a longitudinal study of the oldest old (N = 151; M = 90 years). Dichotomized objective health was cross classified with dichotomized subjective health, producing four health congruence groups: good health realists, poor health realists, optimists, and pessimists. Both good health realists and pessimists had good objective health, yet pessimists had the highest depression, lowest functional status, and frequent reports of hospitalization. By contrast, the poor health realists and optimist groups had poor objective health, but the optimists had better outcomes on depression. This suggests that discrepancies between objective and subjective health may have significant implications for health outcomes.

Motivic Cohomology of Pairs of Simplices

For an arbitrary field $F$ we study the double scissors congruence groups $A_n(F)$ generated by admissible pairs of simplices in the projective $n$-space ${\mathbb P}_F^n$ over $F$. Let $(L, M)$ be an admissible pair of simplices whose faces are defined as hyperplanes in ${\mathbb P}_F^n$ such that they do not have common faces of the same dimension. When $L$ and $M$ are in general positions we define a linearly constructible motivic perverse sheaf producing a motivic cohomology whose Betti realization is the relative cohomology $H^{\ast}({\mathbb P}_F^n\setminus L, M\setminus L; {\mathbb Q})$. The motivic cohomology provides a simple explanation of why the generic part of $A_\bullet$ forms a Hopf algebra with well-defined coproduct. When $n = 2$ we define explicitly the coproduct on $A_2$ which simplifies the approach of Beilinson et al. We also complete the same task for $A_3$ and $A_4$ which enables us to develop further results in another paper connecting Aomoto trilogarithms to the classical ones.

F-regular semigroups

A semigroup S is called F-regular if S is regular and if there exists a group congruence ρ on S such that every ρ-class contains a greatest element with respect to the natural partial order of S (see [K.S. Nambooripad, Proc. Edinburgh Math. Soc. 23 (1980) 249–260]). These semigroups were investigated in [C.C. Edwards, Semigroup Forum 19 (1980) 331–345] where a description similar to the F-inverse case (see [R. McFadden, L. O'Carroll, Proc. London Math. Soc. 22 (1971) 652–666]) is given. Further characterizations of F-regular semigroups, including an axiomatic one, are provided. The main objective is to give a new representation of such semigroups by means of Szendrei triples (see [M. Szendrei, Acta Sci. Math. 51 (1987) 229–249]). The particular case of F-regular semigroups S satisfying the identity (xy) ∗=y ∗x ∗ , where x ∗∈S denotes the greatest element of the ρ-class containing x∈ S, is considered. Also the F-inversive semigroups, for which this identity holds, are characterized.

Euclidean scissor congruence groups and mixed Tate motives over dual numbers

We define Euclidean scissor congruence groups for an arbitrary algebraically closed field F and propose their conjectural description. We suggest how they should be related to mixed Tate motives over dual numbers for F.

A generalization of Cohen- Eisenstein series and Shimura liftings and some congruences between Cusp forms and Eisenstein series

In this paper we introduce some modular forms of half-integral weight on congruence group Гo(4N) withN an odd positive integer which can be viewed as a natural generalization of Cohen-Eisenstein series. Using these series, we can prove that the restriction of Shimura lifting on Eisenstein spaceEk+1/2+(4N,χl) gives an isomorphism fromEk+1/2+(4N,χl) toE2k(N). We consider some congruence relationships between modular forms in use of Shimura lifting.

GALOIS ORBITS OF PRINCIPAL CONGRUENCE HECKE CURVES

It is shown that the natural action of the absolute Galois group on the ideals defining principal congruence subgroups of certain nonarithmetic Fuchsian triangle groups is compatible with its action on the algebraic curves that these congruence groups uniformize.

Congruence Subgroups of PSL(2, Z) of Genus Less than or Equal to 24

In this paper, we report the computation and tabulation, using MAGMA, of all congruence subgroups of PSL(2, Z) of genus less than or equal to 24. We include full tables of the congruence groups of genus 0, 1, 2, and 3 and a summary of the remaining cases.

Modular Subgroups, Forms, Curves and Surfaces

AbstractWe study a class of subgroups of PSL2(ℤ) which can be characterized in different ways, such as congruence groups, modular forms, modular curves, elliptic surfaces, lattices and graphs.