Principal Congruence Research Articles

Representing some families of monotone maps by principal lattice congruences

For a lattice L with 0 and 1, let Princ(L) denote the set of principal congruences of L. Ordered by set inclusion, it is a bounded ordered set. In 2013, G. Gratzer proved that every bounded ordered set is representable as Princ(L); in fact, he constructed L as a lattice of length 5. For {0, 1}-sublattices $${A \subseteq B}$$ of L, congruence generation defines a natural map Princ(A) $${\longrightarrow}$$ Princ(B). In this way, every family of {0, 1}-sublattices of L yields a small category of bounded ordered sets as objects and certain 0-separating {0, 1}-preserving monotone maps as morphisms such that every hom-set consists of at most one morphism. We prove the converse: every small category of bounded ordered sets with these properties is representable by principal congruences of selfdual lattices of length 5 in the above sense. As a corollary, we can construct a selfdual lattice L in G. Gratzer's above-mentioned result.

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 .

Homomorphisms and principal congruences of bounded lattices I. Isotone maps of principal congruences

Homomorphisms and principal congruences of bounded lattices I. Isotone maps of principal congruences

Semi-Nelson Algebras

Generalizing the well known and exploited relation between Heyting and Nelson algebras to semi-Heyting algebras, we introduce the variety of semi-Nelson algebras. The main tool for its study is the construction given by Vakarelov. Using it, we characterize the lattice of congruences of a semi-Nelson algebra through some of its deductive systems, use this to find the subdirectly irreducible algebras, prove that the variety is arithmetical, has equationally definable principal congruences, has the congruence extension property and describe the semisimple subvarieties.

Invariants of some compactified Picard modular surfaces and applications

The paper investigates invariants of compactified Picard modular surfaces by principal congruence subgroups of Picard modular groups. The applications to the surface classification and modular forms are also discussed.

Semantical conditions for the definability of functions and relations

Let \({\mathcal{L}\subseteq \mathcal{L}^\prime}\) be first order languages, let \({R \in \mathcal{L}^\prime- \mathcal{L}}\) be a relation symbol, and let \({\mathcal{K}}\) be a class of \({\mathcal{L}^\prime}\)-structures. In this paper, we present semantical conditions equivalent to the existence of an \({\mathcal{L}}\)-formula \({\varphi(\vec{x})}\) such that \({\mathcal{K}\vDash \varphi(\vec{x}) \leftrightarrow R(\vec{x})}\), where \({\varphi}\) has a specific syntactical form (e.g., quantifier free, positive and quantifier free, existential Horn, etc.). For each of these definability results for relations, we also present an analogous version for the definability of functions. Several applications to natural definability questions in universal algebra have been included; most notably definability of principal congruences. The paper concludes with a look at term-interpolation in classes of structures with the same techniques used for definability. Here we obtain generalizations of two classical term-interpolation results: Pixley’s theorem for quasiprimal algebras, and the Baker–Pixley Theorem for finite algebras with a majority term.

An independence theorem for ordered sets of principal congruences and automorphism groups of bounded lattices

An independence theorem for ordered sets of principal congruences and automorphism groups of bounded lattices

Arithmetic of Eisenstein series of level T for the function field modular group [formula omitted

In this paper we study the structure of the algebra of Drinfeld modular forms for the principal congruence subgroup Γ(T) of the full modular group GL(2,Fq[T]).To this end and based on work of Cornelissen, we introduce a new class of Eisenstein series and describe their fundamental properties. These so-called modified Eisenstein series allow for a simplification of prior results and are an indispensable tool for future work on Drinfeld modular forms of level T.

Principal congruences in weak Heyting algebras

Let A be a weak Heyting algebra and let $${a, b \in A}$$ . We give a description for the congruence generated by the pair (a, b), and we use it in order to give a necessary and sufficient condition for a function $${f : A^{k} \rightarrow A}$$ to be compatible with every congruence of A. We also find conditions on a not necessarily polynomial function g(a, b) in A that imply that the function $${a \mapsto {\rm min}\{b \in A : g(a, b) \leq b}\}$$ is compatible when defined.

Bolza Quaternion Order and Asymptotics of Systoles Along Congruence Subgroups

We give a detailed description of the arithmetic Fuchsian group of the Bolza surface and the associated quaternion order. This description enables us to show that the corresponding principal congruence covers satisfy the bound on the systole, where g is the genus. We also exhibit the Bolza group as a congruence subgroup, and calculate out a few examples of “Bolza twins” (using magma). Like the Hurwitz triplets, these correspond to the factoring of certain rational primes in the ring of integers of the invariant trace field of the surface. We exploit random sampling combined with the Reidemeister–Schreier algorithm as implemented in magma to generate these surfaces.

Almost structural completeness; an algebraic approach

A deductive system is structurally complete if all of its admissible inference rules are derivable. For several important systems, like the modal logic S5, failure of structural completeness is caused only by the underivability of a passive rule, i.e., a rule whose premise is not unifiable by any substitution. Neglecting passive rules leads to the notion of almost structural completeness, that means, to the derivability of admissible non-passive rules. We investigate almost structural completeness for quasivarieties and varieties of general algebras. The results apply to all algebraizable deductive systems.Firstly, various characterizations of almost structurally complete quasivarieties are presented. Two of them are general: the one expressed with finitely presented algebras, and the one expressed with subdirectly irreducible algebras. The next one is restricted to quasivarieties with the finite model property and equationally definable principal relative congruences, where the condition is verifiable on finite subdirectly irreducible algebras. Some connections with exact and projective unification are included.Secondly, examples of almost structurally complete varieties are provided. Particular emphasis is put on varieties of closure algebras, that are known to constitute adequate semantics for normal extensions of the modal logic S4. A certain infinite family of such almost structurally complete, but not structurally complete, varieties is constructed. Every variety from this family has a finitely presented unifiable algebra which does not embed into any free algebra for this variety. Hence unification is not unitary there. This shows that almost structural completeness is strictly weaker than projective unification for varieties of closure algebras.

The ordered set of principal congruences of a countable lattice

For a lattice L, let Princ(L) denote the ordered set of principal congruences of L. In a pioneering paper, G. Gratzer characterized the ordered set Princ(L) of a finite lattice L; here we do the same for a countable lattice. He also showed that every bounded ordered set H is isomorphic to Princ(L) of a bounded lattice L. We prove a related statement: if an ordered set H with a least element is the union of a chain of principal ideals (equivalently, if 0 \({\in}\) H and H has a cofinal chain), then H is isomorphic to Princ(L) of some lattice L.

Petrie polygons, Fibonacci sequences and Farey maps

We consider the regular triangular maps corresponding to the principal congruence subgroups Γ ( n ) of the classical modular group. We relate the sizes of the Petrie polygons on these maps to the periods of reduced Fibonacci sequences.

Congruence Subgroups of Lattices in Rank 2 Kac–Moody Groups over Finite Fields

Let G be a rank 2 complete affine or hyperbolic simply-connected Kac–Moody group over a finite field k. Then G is locally compact and totally disconnected. Let B− = HU− be the negative minimal parabolic subgroup of G, where H is the analog of a diagonal subgroup and U− is generated by all negative real root groups. Let w1 and w2 be the generators of the Weyl group. Let be the negative standard parabolic subgroup of G corresponding to w1. It is known that the subgroups U−, B− and , are nonuniform lattice subgroups of G. Here we construct an infinite sequence of congruence subgroups of as natural generalizations of the corresponding notions for lattices in Lie groups. We also show that the group U− contains analogous congruence subgroups. Our technique involves determining graphs of groups presentations for U−, B−, and with the fundamental apartment of the Bruhat–Tits tree X a quotient graph for U− and for B− on X. When k = 𝔽q and q = 2s, the graph of groups for has the the positive half of the fundamental apartment as quotient graph. We explicitly construct the graphs of groups for the principal (level 1) congruence subgroup of and the analogous subgroups of U− giving generalized amalgam presentations for them.

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

New 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.

Action de Hopf sur les opérateurs de Hecke modulaires tordus

Let \Gamma \subseteq SL_2(\mathbb Z) be a principal congruence subgroup. For each \sigma \in SL_2(\mathbb Z) , we introduce the collection \mathcal A_\sigma (\Gamma) of modular Hecke operators twisted by \sigma . Then, \mathcal A_\sigma(\Gamma) is a right \mathcal A (\Gamma) -module, where \mathcal A (\Gamma) is the modular Hecke algebra introduced by Connes and Moscovici. Using the action of a Hopf algebra \mathfrak{h}_0 on \mathcal A_\sigma (\Gamma) , we define reduced Rankin–Cohen brackets on \mathcal A_\sigma (\Gamma) . Moreover \mathcal A_\sigma (\Gamma) carries an action of \mathcal H_1 , where \mathcal H_1 is the Hopf algebra of foliations of codimension 1. Finally, we consider operators between the levels \mathcal A_\sigma (\Gamma) , {\sigma\in SL_2(\mathbb Z)} . We show that the action of these operators can be expressed in terms of a Hopf algebra \mathfrak{h}_{\mathbb Z} .

Conformal scattering on the Schwarzschild metric

We show that existing decay results for scalar fields on the Schwarzschild metric are sufficient to obtain a conformal scattering theory. Then we re-interpret this as an analytic scattering theory defined in terms of wave operators, with an explicit comparison dynamics associated with the principal null geodesic congruences. The case of the Kerr metric is also discussed.

Lower Bounds on the Dimension of the Cohomology of Bianchi Groups via Sczech Cocyles

Using Sczech cocyles, we compute the traces of certain involutions on the Eisenstein cohomology of principal congruence subgroups of Bianchi groups. These traces, combined with results of [13, 14, 2], give explicit lower bounds for the cuspidal cohomology of these groups. The asymptotic lower bounds that follow from our results complement the recent asymptotic upper bounds found in [4, 5, 12].

On the cusp forms of congruence subgroups of an almost simple Lie group

In this paper we address the issue of existence of newforms among the cusp forms for almost simple Lie groups using the approach of the second author combined with local information on supercuspidal representations for p-adic groups known by the first author. We pay special attention to the case of $$SL_M({\mathbb {R}})$$ where we prove various existence results for principal congruence subgroups.

Operator properties of congruence permutable varieties with strongly definable principal congruences

In an attempt to describe the partially ordered monoid of operators generated by the operators H (homomorphic images), S (subalgebras), \({P_{\rm f}}\) (filtered products) for the variety \({\mathcal{R}_{\rm c}}\) of commutative rings, several results about congruence permutable varieties have been discovered.