Congruence Groups Research Articles

A Trace Map on Higher Scissors Congruence Groups

Abstract Cut-and-paste $K$-theory has recently emerged as an important variant of higher algebraic $K$-theory. However, many of the powerful tools used to study classical higher algebraic $K$-theory do not yet have analogues in the cut-and-paste setting. In particular, there does not yet exist a sensible notion of the Dennis trace for cut-and-paste $K$-theory. In this paper we address the particular case of the $K$-theory of polyhedra, also called scissors congruence $K$-theory. We introduce an explicit, computable trace map from the higher scissors congruence groups to group homology, and use this trace to prove the existence of some nonzero classes in the higher scissors congruence groups. We also show that the $K$-theory of polyhedra is a homotopy orbit spectrum. This fits into Thomason’s general framework of $K$-theory commuting with homotopy colimits, but we give a self-contained proof. We then use this result to re-interpret the trace map as a partial inverse to the map that commutes homotopy orbits with algebraic $K$-theory.

A refined scissors congruence group and the third homology of SL2

There is a natural connection between the third homology of SL2(A) and the refined Bloch group RB(A) of a commutative ring A. In this article we investigate this connection and as the main result we show that if A is a universal GE2-domain such that −1∈A×2, then we have the exact sequenceH3(SM2(A),Z)→H3(SL2(A),Z)→RB(A)→0, where SM2(A) is the group of monomial matrices in SL2(A). Moreover, we show that RP1(A), the refined scissors congruence group of A, is naturally isomorphic to the relative homology group H3(SL2(A),SM2(A);Z).

Third homology of SL2 over number fields: The norm-Euclidean quadratic imaginary case

In the article The third homology ofSL2(Q) ([7]), Hutchinson determined the structure of H3(SL2(Q),Z[12]) by expressing it in terms of K3ind(Q)≅Z/24 and the scissor congruence group of the residue field Fp with p a prime number. In this paper, we develop further the properties of the refined scissors congruence group in order to extend this result to the case of imaginary quadratic number fields whose ring of integers is a Euclidean domain with respect to the norm.

Groups of convex bodies

In this paper we introduce and study a topological abelian group of convex bodies, analogous to the scissors congruence group and McMullen’s polytope algebra, with the universal property that continuous valuations on convex bodies correspond to continuous homomorphisms on the group of convex bodies. To study this group, we first obtain a version of McMullen polynomiality for valuations that take values not in fields or vector spaces, but in abelian groups. Using this, we are able to equip the group of convex bodies with a grading that consists of real vector spaces in all positive degrees, mirroring one of the main structural properties of the polytope algebra. It is hoped that this work can serve as the starting point for a K-theoretic interpretation of valuations on convex bodies.

CRT-Based Homomorphic Encryption over the Fraction

Homomorphic encryption technology is the holy grail of cryptography and has a wide range of applications in practice. This paper proposes a homomorphic encryption scheme over the fraction based on the Chinese remainder theorem (CRT) Dayan qiuyi rule. This homomorphic scheme performs encryption and decryption operations by forming congruence groups and has homomorphism. The solution in this paper first combines the traditional CRT algorithm with the Dayan qiuyi rule to obtain the CRTF algorithm that can be operated on the fraction field. Finally, in the decryption process, modulo arithmetic is used twice to obtain the correct plaintext components, restored to plaintext by CRTF. The scheme’s security is related to a decisional version of an approximate GCD problem. The proof of theoretical derivation shows that this paper’s homomorphic encryption scheme can realize the homomorphic addition operation in the fraction field. Compared with the CKKS scheme, efficiency is improved.

Jutila's circle method and [formula omitted] shifted convolution sums

Let λf(n) be the normalized Fourier coefficients of a Hecke-Maass or Hecke holomorphic cusp form f for congruence group Γ0(N) with level N and nebentypus χN. Let Q(x) be a positive definite integral quadratic form, and r(n,Q) denote the number of representations of n by the quadratic form Q. In this paper, we apply Jutila's circle method to derive a sharp bound for the shifted convolution sum of Fourier coefficients λf(n) and r(n,Q). We generalize and improve previous results without the Ramanujan conjecture.

The homology of SL2 of discrete valuation rings

Let A be a discrete valuation ring with field of fractions F and residue field k such that |k|≠2,3,4,5,7,8,9,16,27,32,64. We prove that there is a natural exact sequenceH3(SL2(A),Z[12])→H3(SL2(F),Z[12])→RP1(k)[12]→0, where RP1(k) is the refined scissors congruence group of k. Let Γ0(mA) denote the congruence subgroup consisting of matrices in SL2(A) whose lower off-diagonal entry lies in the maximal ideal mA. We also prove that there is an exact sequence0→P‾(k)[12]→H2(Γ0(mA),Z[12])→H2(SL2(A),Z[12])→I2(k)[12]→0, where I2(k) is the second power of the fundamental ideal of the Grothendieck-Witt ring GW(k) and P‾(k) is a certain quotient of the scissors congruence group (in the sense of Dupont-Sah) P(k) of k.

On group congruences on the semigroup $\boldsymbol{B}_{\omega}^{\mathscr{F}}$ and its homomorphic retracts in the case when the family $F$ consists of inductive non-empty subsets of $\omega$

On group congruences on the semigroup $\boldsymbol{B}_{\omega}^{\mathscr{F}}$ and its homomorphic retracts in the case when the family $F$ consists of inductive non-empty subsets of $\omega$

On the space of theta functions whose levels are square-free

Hecke conjectured that an explicit set of theta series obtained from a quaternion algebra defined over ℚ ramified at a prime N is a basis of a space of holomorphicmodular forms of weight 2 for the Hecke congruence group Γ0(N). However, Eichler noticed that Hecke’s conjecture is not true in general. Hence it is natural to ask the dimension of the subspace of M2(Γ0(N)) spanned by the theta series, and this question is called Hecke’s basis problem, which we have shown an answer in [K. Sugiyama, On the space of theta functions for a prime level, Comment. Math. Univ. St. Pauli, 67(1):66–81, 2019]. In this paper, we generalize the results for a square-free positive integer N.

The third homology of [formula omitted

We calculate the structure of H3(SL2(Q),Z[12]). Let H3(SL2(Q),Z)0 denote the kernel of the (split) surjective homomorphism H3(SL2(Q),Z)→K3ind(Q). Each prime number p determines an operator 〈p〉 on H3(SL2(Q),Z) with square the identity. We prove that H3(SL2(Q),Z[12])0 is the direct sum of the (−1)-eigenspaces of these operators. The (−1)-eigenspace of 〈p〉 is the scissors congruence group, over Z[12], of the field Fp, which is a cyclic group whose order is the odd part of p+1.

A step toward measuring children’s college-bound identity in children’s savings accounts programs: The case of promise scholars

Research indicates that children’s college expectations are an important predictor of children’s school outcomes and Children’s Savings Accounts (CSAs) are a predictor of children’s educational expectations. Typically measured with a single question, children’s college expectations have been used as s proxy for college-bound identity. Building on Identity-Based Motivation theory, we take a step toward creating a scale for measuring college-bound identity. This scale consists of 22items with three first-order identity factors: salience, group congruence, and interpretation of difficulty as normal. Findings indicate that a second-order identity latent named college-bound identity may be reflected in the data. We then test the association between Promise Early Award Scholarship (Promise Scholars), a CSA and early award scholarship program, and college-bound identity. Promise Scholars participants’ college-bound identity is predicted to be 0.02 higher than that of non-Promise Scholars participants. The authors offer potential next steps in the attempt to better measure children’s college-bound identity.

Cusps, congruence groups and Monstrous dessins

We study general properties of the dessins d’enfants associated with the Hecke congruence subgroups Γ0(N) of the modular group PSL2(Z). The definition of the Γ0(N) as the stabilisers of couples of projective lattices in a two-dimensional vector space gives an interpretation of the quotient set Γ0(N)∖PSL2(Z) as the projective lattices N-hyperdistant from a reference one, and hence as the projective line over the ring Z∕NZ. The natural action of PSL2(Z) on the lattices defines a dessin d’enfant structure, allowing for a combinatorial approach to features of the classical modular curves, such as the torsion points and the cusps. We tabulate the dessins d’enfants associated with the 15 Hecke congruence subgroups of genus zero, which arise in Moonshine for the Monster sporadic group.

On (bi) linked group congruences on (bi) linked semigroups

In this paper we introduce the notion of (bi) linked group congruence on a (bi) linked semigroup. Inclusion preserving bijection between the set of all (bi) linked group congruences and the set of all dense k-ideals has firstly been obtained. Subsequently these results have been refined to lattice isomorphisms. For a linked semigroup (S, T, f) (bi-linked semigroup (S, T, f, g)) with the left operator semiring L and the right operator semiring R there correspond six lattices viz., the lattice $${\mathcal {G}}{\mathcal {C}}(S)$$ ( $${\mathcal {B}}{\mathcal {G}}{\mathcal {C}}(S)$$ ) of all linked group congruences on (S, T, f) (respectively, bi-linked group congruences on (S, T, f, g)), the lattice $${\mathcal {I}}(S)$$ of all dense k-ideals of (S, T, f), the lattice $${\mathcal {R}}{\mathcal {C}}(L)$$ of ring congruences on L, the lattice $${\mathcal {I}}(L)$$ of dense k-ideals of L, the lattice $${\mathcal {R}}{\mathcal {C}}(R)$$ of ring congruences on R, the lattice $${\mathcal {I}}(R)$$ of dense k-ideals of R. Any two of these lattices have been shown to be isomorphic. Modularity, distributivity and completeness of these lattices have also been investigated. Finally the least (bi) linked group congruence on a (bi) linked semigroup has been identified under a suitable restriction.

Modular Dedekind symbols associated to Fuchsian groups and higher-order Eisenstein series

Let E(z, s) be the non-holomorphic Eisenstein series for the modular group $${\mathrm{SL}}(2,{{\mathbb {Z}}})$$. The classical Kronecker limit formula shows that the second term in the Laurent expansion at $$s=1$$ of E(z, s) is essentially the logarithm of the Dedekind eta function. This eta function is a weight 1/2 modular form and Dedekind expressed its multiplier system in terms of Dedekind sums. Building on work of Goldstein, we extend these results from the modular group to more general Fuchsian groups $${\Gamma }$$. The analogue of the eta function has a multiplier system that may be expressed in terms of a map $$S:{\Gamma }\rightarrow {{\mathbb {R}}}$$ which we call a modular Dedekind symbol. We obtain detailed properties of these symbols by means of the limit formula. Twisting the usual Eisenstein series with powers of additive homomorphisms from $${\Gamma }$$ to $${{\mathbb {C}}}$$ produces higher-order Eisenstein series. These series share many of the properties of E(z, s) though they have a more complicated automorphy condition. They satisfy a Kronecker limit formula and produce higher-order Dedekind symbols $$S^*:{\Gamma }\rightarrow {{\mathbb {R}}}$$. As an application of our general results, we prove that higher-order Dedekind symbols associated to genus one congruence groups $${\Gamma }_0(N)$$ are rational.

Experiencia Compartida: The effect of discrimination on Latino attitudes toward illegal immigration

Increasingly, research is being done on American attitudes toward immigration, particularly illegal immigration. Such research has focused on overall U.S. attitudes, predominantly the positions of Whites and African Americans, leaving a gap in the literature on Latino attitudes. In popular discussions on immigration, especially news accounts of “solidarity” movements, there is an underlying assumption of group congruence among Latinos. Latinos, however, are a diverse population, with varied opinions on political and social issues, including that of immigration. Using data from the 2010 National Survey of Latinos (https://www.pewresearch.org/hispanic/dataset/2010-national-survey-of-latinos/), this study explores the attitudes of Latinos in the U.S. toward illegal immigration by examining the effects of discrimination on the development of these attitudes. The outcomes of ordered logit (ordinal regression) show that having experienced discrimination increases favorability toward illegal immigration. This suggests that attitudes toward illegal immigration go beyond ethnic solidarity to include perceived shared experiences.

Transformation laws for generalized Dedekind sums associated to Fuchsian groups

We establish transformation laws for generalized Dedekind sums associated to the Kronecker limit function of non-holomorphic Eisenstein series and their higher-order variants. These results apply to general Fuchsian groups of the first kind, and examples are provided in the cases of the Hecke triangle groups, the Hecke congruence groups $\Gamma_0(N)$, and the non-congruence arithmetic groups $\Gamma_0(N)^+$.

On inverse submonoids of the monoid of almost monotone injective co-finite partial selfmaps of positive integers

In this paper we study submonoids of the monoid $\mathscr{I}_{\infty}^{\,\Rsh\!\!\nearrow}(\mathbb{N})$ of almost monotone injective co-finite partial selfmaps of positive integers $\mathbb{N}$. Let $\mathscr{I}_{\infty}^{\nearrow}(\mathbb{N})$ be a submonoid of $\mathscr{I}_{\infty}^{\,\Rsh\!\!\nearrow}(\mathbb{N})$ which consists of cofinite monotone partial bijections of $\mathbb{N}$ and $\mathscr{C}_{\mathbb{N}}$ be a subsemigroup of $\mathscr{I}_{\infty}^{\,\Rsh\!\!\nearrow}(\mathbb{N})$ which is generated by the partial shift $n\mapsto n+1$ and its inverse partial map. We show that every automorphism of a full inverse subsemigroup of $\mathscr{I}_{\infty}^{\!\nearrow}(\mathbb{N})$ which contains the semigroup $\mathscr{C}_{\mathbb{N}}$ is the identity map. We construct a submonoid $\mathbf{I}\mathbb{N}_{\infty}^{[\underline{1}]}$ of $\mathscr{I}_{\infty}^{\,\Rsh\!\!\nearrow}(\mathbb{N})$ with the following property: if $S$ is an inverse submonoid of $\mathscr{I}_{\infty}^{\,\Rsh\!\!\nearrow}(\mathbb{N})$ such that $S$ contains $\mathbf{I}\mathbb{N}_{\infty}^{[\underline{1}]}$ as a submonoid, then every non-identity congruence $\mathfrak{C}$ on $S$ is a group congruence. We show that if $S$ is an inverse submonoid of $\mathscr{I}_{\infty}^{\,\Rsh\!\!\nearrow}(\mathbb{N})$ such that $S$ contains $\mathscr{C}_{\mathbb{N}}$ as a submonoid then $S$ is simple and the quotient semigroup $S/\mathfrak{C}_{\mathbf{mg}}$, where $\mathfrak{C}_{\mathbf{mg}}$ is the minimum group congruence on $S$, is isomorphic to the additive group of integers. Also, we study topologizations of inverse submonoids of $\mathscr{I}_{\infty}^{\,\Rsh\!\!\nearrow}(\mathbb{N})$ which contain $\mathscr{C}_{\mathbb{N}}$ and embeddings of such semigroups into compact-like topological semigroups.

On the monoid of cofinite partial isometries of $\qq{N}^n$ with the usual metric

In this paper we study the structure of the monoid Iℕn ∞ of cofinite partial isometries of the n-th power of the set of positive integers ℕ with the usual metric for a positive integer n > 2. We describe the group of units and the subset of idempotents of the semigroup Iℕn ∞, the natural partial order and Green's relations on Iℕn ∞. In particular we show that the quotient semigroup Iℕn ∞/Cmg, where Cmg is the minimum group congruence on Iℕn ∞, is isomorphic to the symmetric group Sn and D = J in Iℕn ∞. Also, we prove that for any integer n ≥2 the semigroup Iℕn ∞ is isomorphic to the semidirect product Sn ×h(P∞(Nn); U) of the free semilattice with the unit (P∞(Nn); U) by the symmetric group Sn.

Weyl bound for p-power twist of GL(2)L-functions

Let f be a cuspidal eigenform (holomorphic or Maass) for the congruence group Γ 0 ( N ) with N square-free. Let p be a prime and let χ be a primitive character of modulus p 3 r . We shall prove the Weyl-type subconvex bound L ( 1 2 + i t , f ⊗ χ ) ≪ f , t , e p r + e , where e > 0 is any positive real number.

The transcendence of zeros of canonical basis elements of the space of weakly holomorphic modular forms for Γ0(2)

We consider the canonical basis elements fk,mε for the space of weakly holomorphic modular forms of weight k for the Hecke congruence group Γ0(2) and we prove that for all m≥c(k) for some constant c(k), if z0 in a fundamental domain for Γ0(2) is a zero of fk,mε, then either z0 is in {i2,−12+i2,12+i2,−1+i74,1+i74}, or z0 is transcendental.