Search

In 2012, Chan proved a number of congruences for the coefficients of Ramanujan’s ϕ \phi function. In this paper, we prove some new congruences modulo powers of 2 and 3 for Ramanujan’s ϕ \phi function by employing Newman’s identities and theta function identities.

Let Q(x )= Q(x1 ,x 2, ..., xn) be a quadratic form over Z, p be an odd prime, and Δ = � (−1) n/2 det AQ/p � . A solution of the congruence Q(x) ≡ 0 (modp m ) is said to be a primitive solution if pxi for some i .W e prove that if this congruence has a primitive solution then it has a primitive

We prove that the action of the Hecke operators on the cohomology of a finite index non-congruence subgroup Γ \Gamma of a Bianchi group is essentially the same as the action of Hecke operators on the cohomology groups of Γ ^ \hat {\Gamma } , the congruence closure of Γ \Gamma . This is a generalization of Atkin’s conjecture, first confirmed in a special case by Serre in 1987 1987 and proved in general by Berger in 1994 1994 .

Let G G be a universal Chevalley-Demazure group scheme associated to a reduced irreducible root system of rank > 1. >1. For a commutative ring R R , we let Γ = E ( R ) \Gamma = E(R) denote the elementary subgroup of the group of R R -points G ( R ) . G(R). The congruence kernel C ( Γ ) C(\Gamma ) is then defined to be the kernel of the natural homomorphism Γ ^ → Γ ¯ , \widehat {\Gamma } \to \overline {\Gamma }, where Γ ^ \widehat {\Gamma } is the profinite completion of Γ \Gamma and Γ ¯ \overline {\Gamma } is the congruence completion defined by ideals of finite index. The purpose of this paper is to show that for an arbitrary noetherian ring R R (with some minor restrictions if G G is of type C n C_n or G 2 G_2 ), the congruence kernel C ( Γ ) C(\Gamma ) is central in Γ ^ . \widehat {\Gamma }.

We study the existence of a modular form satisfying a certain congruence relation. The existence of such modular forms plays an important role in the determination of the structure of a ring of modular forms modulo p p . We give a criterion for the existence of such a modular form in the case of Hermitian modular forms.

We generalize the methods of Pierce for counting solutions to the congruence X a ≡ Y b mod D X^a \equiv Y^b \bmod D and the square sieve method for counting squares in the sequence f ( X ) + g ( Y ) f(X) + g(Y) to the function field setting.

We show that if G G is a finite group, C C a conjugacy class of G G and d = | C | d=\left \vert C\right \vert , d 2 , d 3 , … , d m d_{2},d_{3},\ldots ,d_{m} are the distinct elements in the multiset { | C | χ ( C ) χ ( 1 ) | χ ∈ I r r ( G ) } \left \{ \frac {\left \vert C\right \vert \chi (C)} {\chi (1)}\ |\ \chi \in \mathrm {Irr}(G)\right \} (here χ ( C ) \chi (C) is the value of χ \chi on any element of C C ), then \[ | G / ⟨ C ⟩ | ⋅ ( d − d 2 ) ( d − d 3 ) ⋯ ( d − d m ) ≡ 0 mod ⁡ | G | . \left \vert G/\left \langle C\right \rangle \right \vert \cdot \left ( d-d_{2}\right ) \left ( d-d_{3}\right ) \cdots \left ( d-d_{m}\right ) \equiv 0\ \operatorname {mod}\ \left \vert G\right \vert . \] This is a dual to a generalization of a theorem of Blichfeldt stating that if G G is a finite group, θ \theta a generalized character and d = θ ( 1 ) , d 2 , d 3 , … , d m d=\theta (1),d_{2},d_{3},\ldots ,d_{m} are the distinct values of θ \theta , then \[ | ker ⁡ ( θ ) | ( d − d 2 ) ( d − d 3 ) ⋯ ( d − d m ) ≡ 0 mod ⁡ | G | . \left \vert \ker (\theta )\right \vert \left ( d-d_{2}\right ) \left ( d-d_{3}\right ) \cdots \left ( d-d_{m}\right ) \equiv 0\ \operatorname {mod} \ \left \vert G\right \vert . \] We also observe that d = θ ( 1 ) d=\theta (1) in Blichfeldt’s congruence can be replaced, with a minor adjustment, by any rational value of θ \theta . A similar change can be done to the first congruence above.

The phenomenon of U ( p ) U(p) -congruences was recently studied by Ahlgren and Ono (2005) and by Elkies, Ono and Yang (2005). We provide a necessary and sufficient condition which improves their general results.

Zhao established a curious harmonic congruence for prime p > 3 p>3 : \[ ∑ i + j + k = p i , j , k > 0 1 i j k ≡ − 2 B p − 3 ( mod ⁡ p ) . \sum _{\substack {i+j+k=pi,j,k>0}} \frac {1}{ijk} \equiv -2B_{p-3}(\operatorname {mod} p). \] In this note the authors extend it to the following congruence for any prime p > 3 p > 3 and positive integer n ≤ p − 2 n \le p-2 : \[ ∑ l 1 + l 2 + ⋯ + l n = p l 1 , ⋯ , l n > 0 1 l 1 l 2 ⋯ l n ≡ { − ( n − 1 ) ! B p − n ( mod p ) a m p ; if 2 ∤ n , − n 2 ( n + 1 ) n ! B p − n − 1 p ( mod ⁡ p 2 ) a m p ; if 2 | n . \sum _{\substack {l_{1}+l_{2}+\cdots +l_{n}=pl_{1}, \cdots ,l_{n}>0}} \frac {1}{l_{1}l_{2}\cdots l_{n}}\equiv \begin {cases} -(n-1)!\ B_{p-n} (\textrm {mod}\; p) & \text {if $2\nmid n$},\\ -\frac {n}{2(n+1)}\ n!\ B_{p-n-1}\ p\ (\operatorname {mod} p^2) &\text {if $2|n$}. \end {cases} \] Other improvements on congruences of harmonic sums are also obtained.

The author gives a simple proof of the following curious congruence for odd prime p > 3 p>3 which was established by Jianqiang Zhao: ∑ i + j + k = p i , j , k > 0 1 i j k ≡ − 2 B p − 3 ( mod p ) . \begin{equation*}\sum _{\substack {{i+j+k=p} {i,\ j,\ k>0}}}\frac {1}{ijk}\equiv -2B_{p-3}(\text {mod}\ p).\end{equation*}

In this note, we show that if n n is a positive integer satisfying the congruence n 2 ≡ 1 ( m o d ϕ 2 ( n ) ) n^{2}\equiv 1~ (mod~\phi ^{2}(n)) , then n ≤ 3 n\le 3 .

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.

Let λ = 2 cos ⁡ ( π / 5 ) \lambda = 2\cos (\pi /5) . Let ( τ ) (\tau ) be an ideal of Z [ λ ] \mathbb {Z}[\lambda ] and let ( τ 0 ) (\tau _0) be the maximal ideal of Z [ λ ] \mathbb {Z}[\lambda ] such that ( τ 0 2 ) ⊆ ( τ ) (\tau _0^2)\subseteq (\tau ) . Then N ( G 0 ( τ ) ) ≤ G 0 ( τ 0 ) N(G_0(\tau ))\le G_0(\tau _0) . In particular, if τ \tau is square free, then G 0 ( τ ) G_{0}(\tau ) is self-normalized in P S L 2 ( R ) PSL_{2}(\mathbb {R}) .

Several formulas for Ramanujan's function r, defined by 00 00 xHI(, = EZr(n)xn (lxI < 1), 1 1 are presented. We also present a congruence modulo 3 for some of the function values.

We show that a residually finite, congruence meet-semidistributive variety of finite type is residually > N > N for some finite N N . This solves Pixley’s problem and a special case of the restricted Quackenbush problem.

For a cube B \mathcal {B} of size B B , we obtain a lower bound on B B so that B ∩ V \mathcal {B}\cap V is nonempty, where V V is the algebraic subset of F p n \mathbb {F}_{p}^{n} defined by x 1 x 2 x 3 … x n ≡ c ( mod p ) , \begin{equation*}x_{1}x_{2}x_{3}\dots x_{n}\equiv c\pmod p ,\end{equation*} n n a positive integer and c c an integer not divisible by p p . For n = 3 n=3 we obtain that B ∩ V \mathcal {B}\cap V is nonempty if B ≫ p 2 3 ( log ⁡ p ) 2 3 B\gg p^{\frac {2}{3}}(\log p)^{\frac {2}{3}} , for n = 4 n=4 we obtain that B ∩ V \mathcal {B}\cap V is nonempty if B ≫ p log ⁡ p B\gg \sqrt {p}\log p , and for n ≥ 5 n\ge 5 we obtain that B ∩ V \mathcal {B}\cap V is nonempty if B ≫ p 1 4 + 1 2 ( n + 4 ) ( log ⁡ p ) 3 2 B\gg p^{\frac {1}{4}+\frac {1}{\sqrt {2(n+4)}}}(\log p)^{\frac {3}{2}} . Using the assumption of the Grand Riemann Hypothesis we obtain B ∩ V \mathcal {B}\cap V is nonempty if B ≫ ϵ p 2 n + ϵ B\gg _{\epsilon }p^{\frac {2}{n}+\epsilon } .

We establish a direct correspondence between two congruence poroperties for finite algebras. The first property is that minimal sets of type i omit tails. The second property is that congruence lattices omit pentagons of type i.

In this note we compute the integral cohomology groups of the subgroups ro(n) of SL(2, Z) and the corresponding subgroups Pro(n) of its quotient, the classical modular group, PSL(2, Z).

We prove there is no infinite set of congruences with: every integer satisfying exactly one congruence, distinct moduli, the sum of the reciprocals of the moduli equal to 1, the lcm of the moduli divisible by only finitely many primes, and a prime greater than 3 dividing any of the moduli.

With the help of the Pell sequence we obtain the following new congruence for odd primes: \[ ∑ k = 1 ( p − 1 ) / 2 1 k ⋅ 2 k ≡ ∑ k = 1 [ 3 p / 4 ] ( − 1 ) k − 1 k ( mod p ) . \sum \limits _{k = 1}^{(p - 1)/2} {\frac {1}{{k \cdot {2^k}}} \equiv \sum \limits _{k = 1}^{[3p/4]} {\;\frac {{{{( - 1)}^{k - 1}}}}{k}} \quad \pmod p.} \]

The elementary Chevalley-Warning congruence method is applied to obtain several p-adic estimates of character sums over finite fields.

We show that there is a uniform bound for the numbers of generators for all principal congruence subgroups of SL ( n , Z ) {\text {SL}}(n,Z) for n ≥ 3 n \geq 3 . On the other hand, we show that the numbers are unbounded if we work with all arithmetic subgroups of SL ( n , Z ) {\text {SL}}(n,Z) .

Let M 2 n {M^{2n}} be a smooth, closed, orientable 2 n 2n -manifold and suppose that K x 2 n − 2 K_x^{2n - 2} is an orientable submanifold of M 2 n {M^{2n}} dual to a cohomology class x x . If d d is a positive integer, the signatures of K d x 2 n − 2 K_{dx}^{2n - 2} and K x 2 n − 2 K_x^{2n - 2} are related by a numerical congruence. If n n is odd, then any codimension 2 submanifold of C P n {\mathbf {C}}{P^n} fixed by a diffeomorphism of odd prime order is dual to the generator of the cohomology algebra.

In this note, we give a simple and elementary proof of the following curious congruence which was established by Zhi-Wei Sun: \[ ∑ k = 1 ( p − 1 ) / 2 1 k ⋅ 2 k ≡ ∑ k = 1 [ 3 p / 4 ] ( − 1 ) k − 1 k ( m o d p ) . \sum ^{(p-1)/2}_{k=1}\frac {1}{k\cdot 2^k}\equiv \sum ^{[3p/4]}_{k=1} \frac {(-1)^{k-1}}{k}\quad (\mathrm {mod}\,p). \]

Let p p be an odd prime, and let k k be a positive integer with i ⩽ k ⩽ ( p − 1 ) / 2 i \leqslant k \leqslant (p - 1)/2 . In this note we give a formula for the number of solutions ( x 1 , … , x k ) ({x_1}, \ldots ,{x_k}) of the congruence x 1 2 + ⋯ + x k 2 ≡ 0 ( mod p ) x_1^2 + \cdots + x_k^2 \equiv 0\;(\bmod p) , 1 ⩽ x 1 > ⋯ > x k ⩽ ( p − 1 ) / 2 1 \leqslant {x_1} > \cdots > {x_k} \leqslant (p - 1)/2 .

In this paper, we prove that, if a metrizable space X {\mathbf {X}} has an admissible metric such that X {\mathbf {X}} has no two distinct congruent subsets of cardinality 3, then ind X ≤ 1 {\text {ind}}{\mathbf {X}} \leq 1 . We also show that if a non-empty metrizable space X {\mathbf {X}} has an admissible star-rigid metric, then ind X = 0 {\text {ind}}{\mathbf {X}} = 0 . The latter answers a question of L. Janos and H. Martin [3].

We give an explicit construction for non-congruence subgroups in the fundamental group of a flexible hyperbolic 3-manifold.

Since (D2 > (PI, there is a smallest i satisfying E3(xi, yi) 2, it follows that IDl = e(u, v). Hence we can choose e2 = xi-,, h = z, and el = Yi-1. If i is even, then we proceed dually. [1

For a finite distributive lattice D with n join-irreducible elements, we construct a finite (planar) lattice L with O ( n 2 ) O({n^2}) elements such that the congruence lattice of L is isomorphic to D. This improves on an early result of R. P. Dilworth (around 1940) and G. Grätzer and E. T. Schmidt (1962) constructing such a (nonplanar) lattice L with O ( 2 2 n ) O({2^{2n}}) elements, and on a recent construction of G. Grätzer and H. Lakser which yields a finite (planar) lattice L with O ( n 3 ) O({n^3}) elements.

We show that there is a uniform bound for the numbers of generators for all principal congruence subgroups of SL(n, Z) for n ≥ 3. On the other hand, we show that the numbers are unbounded if we work with all arithmetic subgroups of SL(n, Z).

Let K _ \underline K be a concrete category and ∼ \sim a congruence on K _ \underline K . Let ∼ \sim be generated by a class M = M 1 ∪ M 2 M = {M_1} \cup {M_2} of Pairs of K _ \underline K -morphisms such that { dom ⁡ f ; ( ∃ g ) ( ( f , g ) ∈ M 1 ) } \{ \operatorname {dom} f;(\exists g)((f,g) \in {M_1})\} and { rng ⁡ f ; ( ∃ g ) ( ( f , g ) ∈ M 2 ) } \left \{ {\operatorname {rng} f;(\exists g)((f,g) \in {M_2})} \right \} are small sets. Then K _ / ∼ \underline K / \sim is concrete. Consequently, if ∼ \sim is generated by a small set of pairs of morphisms, then K _ / ∼ \underline K / \sim is concrete.

Let p be an odd prime. In this paper, we prove that if p≢3 (mod 4) and x, y, z are integers satisfying x p +y p =z p , p/xyz, 0 2 −1/p p 6p−2 and z-x>1/2p 6p−3

A subsemigroup T T of a commutative semigroup S S is called a mild ideal if for any a ∈ S , a T ∩ T ≠ ϕ a \in S,aT \cap T \ne \phi . It is shown here that any maximal cancellative subsemigroup T T of a commutative, idempotent-free, archimedean semigroup S S must be a mild ideal of S S . Maximal cancellative subsemigroups exist in abundance due to Zorn’s lemma. It is also shown that if T T is mild ideal of a commutative semigroup S S , then every cancellative congruence of T T has a unique extension to a cancellative congruence of S S .

Classifying space techniques are used to solve a smooth version of the classical scissors congruence problem.

In this paper, simple algebraic proofs are given for the completeness theorems for the implicational and universal logics of algebras. The proofs are obtained by examining congruences,θ\theta, on the algebra of terms,F(ω)F(\omega ), such thatF(ω)/θF(\omega )/\thetabelongs to the given class of algebras. Thus, they are direct analogs of G. Birkhoff’s proof of the completeness theorem for equational logic.

A formula for Ramanujan’s tau function τ \tau , defined by ∑ 1 ∞ τ ( n ) x n = x ∏ 1 ∞ ( 1 − x n ) 24 ( | x | > 1 ) \sum \nolimits _1^\infty {\tau (n){x^n} = } x\prod _1^\infty {(1 - {x^n})^{24}}(\left | x \right | > 1) , is presented. The author then observes that some of the known congruence properties of τ \tau are immediate consequences of this formula representation.

0. Introduction. In [2] R. Freese and B. Jonsson have shown that the congruence lattices of algebras in modular varieties are in fact arguesian. Since the congruence lattice of an algebra W may be thought of as the (abstract) projective geometry of W it is natural to ask whether there might be an affine analogue to this result. An affine geometry, Kongruenzklassengeometrie by name, was introduced and investigated by R. Wille in [7]. Simply take the elements of W as points and every congruence class as a subspace. In this general setting there is no transition nor even a connection known between these two geometries. Thus it may be surprising that there is an affine analogue to the above-mentioned result of Freese and Jonsson. Namely, as we are about to prove, the "Little Desarguesian Theorem" is true in the Kongruenzklassengeometrie of algebras in modular varieties.

For every n ≥ 3 n \geq 3 there exists a finite nonabelian algebra whose congruence lattice has M n {M_n} as a 0 , 1 0, 1 -sublattice. This answers a question of R. McKenzie and D. Hobby.

For a DOL-scheme ( X , φ ) (X,\varphi ) , where X X is a finite alphabet and φ \varphi is an endomorphism of X ∗ {X^*} , we study the properties of the congruence φ ¯ \bar \varphi induced by φ \varphi in terms of the properties of X ∗ φ {X^*}\varphi . We prove that every submonoid of X ∗ {X^*} has a disjunctive subset (for any X X ) and deduce that φ ¯ \bar \varphi is a syntactic congruence. As special cases, we consider the conditions on φ \varphi which are equivalent to φ ¯ \bar \varphi being perfect or uniquely perfect or linear. The latter is introduced in the paper together with a ramification.

Let A \mathfrak {A} be a finitely generated algebra of finite type. If θ \theta is a congruence relation of A \mathfrak {A} such that A / θ \mathfrak {A}/\theta is finite then θ \theta is compact in the lattice Con ( A ) {\text {Con}}(\mathfrak {A}) of all congruence relations of A \mathfrak {A} . Moreover, if A \mathfrak {A} is infinite then there is a congruence relation θ \theta such that A / θ \mathfrak {A}/\theta is infinite and A / θ ′ \mathfrak {A}/\theta ’ is finite for every θ ′ > θ \theta ’ > \theta in Con ( A ) {\text {Con}}(\mathfrak {A}) .

It is shown that the classical nonassociative algebras which have an identity element can be defined in terms of congruence relations modulo the base field.

In a recent paper, D. B. McAlister gave several characterizations of proper inverse semigroups. In this paper, the concept of proper is extended to the class of regular semigroups. This is done by requiring that the set of idempotents of the semigroup coincides with the kernel of the minimum group congruence on the semigroup. A theorem is presented which contains several characterizations of proper regular semigroups, and the related result of McAlister then follows as a corollary. The paper concludes with some open questions and examples.

A semigroup S S is called (left, right) absolutely flat if all of its (left, right) S S -sets are flat. Let S = ∪ { S γ : γ ∈ Γ } S = \cup \{ {S_\gamma }:\gamma \in \Gamma \} be the least semilattice decomposition of a band S S . It is known that if S S is left absolutely flat then S S is right regular (that is, each S γ {S_\gamma } is right zero). In this paper it is shown that, in addition, whenever α , β ∈ Γ , α > β \alpha ,\beta \in \Gamma ,\alpha > \beta , and F F is a finite subset of S β × S β {S_\beta } \times {S_\beta } , there exists w ∈ S α w \in {S_\alpha } such that ( w u , w v ) ∈ θ R ( F ) (wu,wv) \in {\theta _R}(F) for all ( u , v ) ∈ F ( θ R ( F ) (u,v) \in F({\theta _R}(F) denotes the smallest right congruence on S S containing F F ). This condition in fact affords a characterization of left absolute flatness in certain classes of right regular bands (e.g. if Γ \Gamma is a chain, if all chains contained in Γ \Gamma have at most two elements, or if S S is right normal).

For each fixed prime p ≠ 5 p \ne 5 , we prove Leopoldt’s conjecture in two infinite families of fields of degree five whose normal closure has Galois group over the rationals isomorphic to S 5 {S_5} . The units of these fields were determined by Maus [4]; we develop and apply a simple reformulation of Leopoldt’s conjecture to obtain the result. We also observe that Leopoldt’s conjecture in one field can imply the same in a second field related by congruence conditions.

This paper studies conditions under which every congruence relation θ \theta in a direct product A × B A \times B of algebras is of the form θ 1 × θ 2 {\theta _1} \times {\theta _2} , where θ 1 {\theta _1} and θ 2 {\theta _2} are congruence relations in A A and B B respectively. It is shown that for any equational class K K , every such θ \theta in every A × B A \times B in K K has this property if and only if K K satisfies certain identities.

If A \mathfrak {A} and B \mathfrak {B} are algebras, where A ⊆ B \mathfrak {A} \subseteq \mathfrak {B} , and θ \theta is a congruence relation on A \mathfrak {A} , let θ B {\theta ^\mathfrak {B}} be the smallest congruence relation on B \mathfrak {B} containing θ \theta . A \mathfrak {A} is called a congruence subalgebra of B \mathfrak {B} if A ⊆ B \mathfrak {A} \subseteq \mathfrak {B} and, for every congruence relation θ \theta on A , θ B ∩ | A | 2 = θ \mathfrak {A},{\theta ^\mathfrak {B}} \cap |\mathfrak {A}{|^2} = \theta . Elementary subalgebras are congruence subalgebras, and there are Directed Union and Loewenheim-Skolem Theorems for congruence subalgebras analogous to those for elementary subalgebras. Consequently we obtain full analogues of the Jónsson-Morley-Vaught results concerning homogeneous-universal algebras, where the notion of “subalgebra” is everywhere replaced by “congruence subalgebra".