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Let E/F be a cyclic extension of number fields of degree ℓn, where ℓ is a prime. It is proved that the ℓ-rank of the class group of E is bounded by ℓn(t-1+rkℓClF), where t is the number of primes of F, including infinite primes, which ramify in E and ClF is the class group of F. This generalizes a result of G. Cornell which applies when n = 1 and ℓ is odd. A similar result is shown to hold when the Galois group is an abelian ℓ-group and the Hasse norm theorem is valid for E/F. [ABSTRACT FROM AUTHOR]

We continue and complete our previous paper "Lifts of projective congruence groups" concerning the question of whether there exist noncongruence subgroups of SL2 (Z) that are projectively equivalent to one of the groups Γ0(N) or Γ1(N). A complete answer to this question is obtained: In case of Γ0(N) such noncongruence subgroups exist precisely if N ∉ {3, 4, 8} and we additionally have either that 4 | N or that N is divisible by an odd prime congruent to 3 modulo 4. In case of Γ1(N) these noncongruence subgroups exist precisely if N > 4. As in our previous paper the main motivation for this question is the fact that the above noncongruence subgroups represent a fairly accessible and explicitly constructible reservoir of examples of noncongruence subgroups of SL2(Z) that can serve as the basis for experimentation with modular forms on noncongruence subgroups. [ABSTRACT FROM AUTHOR]

Let n ≥ 1 and let p be any prime. Also, let En be the Lubin- Tate spectrum, Gn the extended Morava stabilizer group, and K(n) the nth Morava K-theory spectrum. Then work of Devinatz and Hopkins and some results due to Behrens and the first author of this paper show that if X is a finite spectrum, then the localization LK(n)(X) is equivalent to the homotopy fixed point spectrum (LK(n)(EnΛ X))hGn, which is formed with respect to the continuous action of Gn on LK(n)(EnΛ X). In this paper, we show that this equivalence holds for any (S-cofibrant) spectrum X. Also, we show that for all such X, the strongly convergent Adams-type spectral sequence abutting to π*(LK(n)(X)) is isomorphic to the descent spectral sequence that abuts to π*((LK(n)(EnΛ X))hGn). [ABSTRACT FROM AUTHOR]

Let m be an integer greater than three and l be an odd prime. In this paper we prove that at least one of the following groups: PΩ±2m(Fls), PSO±2m(Fls), PO2m±(Fls), or PGO±2m(Fls) is a Galois group of Q for infinitely many integers s > 0. This is achieved by making use of a slight modification of a group theory result of Khare, Larsen, and Savin, and previous results of the author on the images of the Galois representations attached to cuspidal automorphic representations of GL2m(AQ). [ABSTRACT FROM AUTHOR]

We prove a Wiener-Wintner ergodic type theorem for a Markov representation S = {Sg : g ∊ G} of a right amenable semitopological semigroup G. We assume that S is mean ergodic as a semigroup of linear Markov operators acting on (C(K),sup), where K is a fixed Hausdorff, compact space. The main result of the paper is necessary and sufficient conditions for mean ergodicity of a distorted semigroup {X(g)Sg : g ∊ G}, where X is a semigroup character. Such conditions were obtained before under the additional assumption that S is uniquely ergodic. [ABSTRACT FROM AUTHOR]

We recall that a finitely presented group G is properly 3-realizable if for some finite 2-dimensional CW-complex X with π1(X) ≅ G, the universal cover X has the proper homotopy type of a 3-manifold. This purely topological property is closely related to the asymptotic behavior of the group G. We show that proper 3-realizability is also a geometric property meaning that it is a quasi-isometry invariant for finitely presented groups. In fact, in this paper we prove that (after taking wedge with a single n-sphere) any two infinite quasiisometric groups of type Fn (n ≥ 2) have universal covers whose n-skeleta are proper homotopy equivalent. Recall that a group G is of type Fn if it admits a K(G, 1)-complex with finite n-skeleton. [ABSTRACT FROM AUTHOR]

The purpose of the present work is to answer an open problem which was raised by G. Costakis and A. Manoussos in their paper "J-class operators and hypercyclicity", J. Operator Theory 67 (2012), 101-119. More precisely, we give the spectral description of the closure of the set of J-class operators acting on a separable Hilbert space. [ABSTRACT FROM AUTHOR]

In this paper, we develop a difference Galois theory in the setting of real fields. After proving the existence and uniqueness of the real Picard- Vessiot extension, we define the real difference Galois group and prove a Galois correspondence. [ABSTRACT FROM AUTHOR]

In this paper, we prove the Eichler cohomology theorem of weakly parabolic generalized modular forms of real weights on subgroups of finite index in the full modular group. We explicitly establish the isomorphism for large weights by constructing the map from the space of cusp forms to the cohomology group. [ABSTRACT FROM AUTHOR]

In this paper, we study the bilinear Littlewood-Paley square function introduced by M. Lacey. We give an easy proof of its boundedness from Lp(…d) × Lq(…d) into Lr(…d), d ≥ 1, for all possible values of exponents p, q, r, i.e. for 2 ≤ p,q ≤∞, 1 ≤ r ≤ ∞ satisfying 1/p + 1/q = 1/r. We also prove analogous results for bilinear square functions on the torus group …d. [ABSTRACT FROM AUTHOR]

A k-involution is an involution with a fixed point set of codimension k. The conjugacy class of such an involution, denoted Sk, generates Möb(n)(the group of isometries of hyperbolic n-space) if k is odd and its orientation-preserving subgroup if k is even. In this paper, we supply effective lower and upper bounds for the Sk word length of Möb(n) if k is odd and the Sk word length of Möb+(n) if k is even. As a consequence, for a fixed codimension k, the length of Möb+(n) with respect to Sk, k even, grows linearly with n, with the same statement holding for Möb(n) in the odd case. Moreover, the percentage of involution conjugacy classes for which Möb+(n) has length two approaches zero as n approaches infinity. [ABSTRACT FROM AUTHOR]

A twisted generalized Weyl algebra A of degree n depends on a base algebra R, n commuting automorphisms σi of R, n central elements ti of R and on some additional scalar parameters. In a paper by Mazorchuk and Turowska, it is claimed that certain consistency conditions for σi and ti are sufficient for the algebra to be nontrivial. However, in this paper we give an example which shows that this is false. We also correct the statement by finding a new set of consistency conditions and prove that the old and new conditions together are necessary and sufficient for the base algebra R to map injectively into A. In particular they are sufficient for the algebra A to be nontrivial. We speculate that these consistency relations may play a role in other areas of mathematics, analogous to the role played by the Yang-Baxter equation in the theory of integrable systems. [ABSTRACT FROM AUTHOR]

Let K be a field, and let σ be an automorphism of K of finite order. Let K(X; σ) be the quotient ring of the skew polynomial ring K[X; σ]. Then any order in K(X; σ) which contains K and its center is a valuation ring of the center of K(X; σ) is a crossed-product algebra Af, where f is some normalized 2-cocycle. Associated to f is a subgroup H of ‹σ› and a graph. In this paper, we determine the connections between hereditary-ness and maximal order properties of Af and the properties of H, f and the graph of f. [ABSTRACT FROM AUTHOR]

H. Furusho proved the beautiful result that of the three defining equations for associators, the pentagon implies the two hexagons. In this paper we present a simpler proof for this theorem (although our paper is less dense and hence only slightly shorter). In particular, we package the use of algebraic geometry and Groethendieck-Teichmüller groups into a useful and previously known principle, and, less significantly, we eliminate the use of spherical braids. [ABSTRACT FROM AUTHOR]

In our earlier paper it was proved that the singular locus of Ag (coarse moduli space of principally polarized abelian varieties over C) is expressed as the union of irreducible varieties Ag(p, a) representing abelian varieties with an order p automorphism of fixed entire representation. In this paper we prove that Ag(p, α) is an irreducible component of SingAg if and only if for a general element of this variety its automorphism group modulo {±1}, G+, satisfies the equivalent conditions: G+ = _a_ or NG+(_a_) = _a_. We illustrate how these results can be used by studying the case g = 4 and p = 5. [ABSTRACT FROM AUTHOR]

The main result in this paper is that there is an example of a conjugacy between two expanding Blaschke products on the circle which preserve the Lebesgue measure such that this conjugacy is symmetric at one point but not symmetric on the whole unit circle. Since the proof uses a symmetric rigidity result in a work by Y. Jiang, we use martingale sequences for uniformly quasisymmetric circle endomorphisms developed in an earlier work of the author to give a simple proof. Furthermore, we give a detailed proof of the result in that prior work of the author that the limiting martingale is invariant under symmetric conjugacy. [ABSTRACT FROM AUTHOR]

In 2000, Keating and Snaith suggested that the value distribution of the Riemann zeta function ζ(1/2+it) is related to that of the characteristic polynomials of random unitary matrices, ΛU(θ) = ∏n=1N(1 - ei(θn+θ)), with respect to the circular unitary ensemble. They derived the conjecture for the moment of the Riemann zeta function through computing the exact formula for the moments of the characteristic polynomials. In this paper, we compute the shifted moments of the characteristic polynomials of random unitary matrices and express them in a determinant form. When shifts are the roots of unity, we can obtain a precise formula, and this also leads to a new formula analogous to the Selberg's identity applied in Keating and Snaith's computation. [ABSTRACT FROM AUTHOR]

In this paper, we introduce splitting numbers of subvarieties in a smooth complex variety for a Galois cover, and prove that the splitting numbers are invariant under certain homeomorphisms. In particular cases, we show that splitting numbers enable us to distinguish the topology of complex plane curves even if the fundamental groups of the complements of plane curves are isomorphic. Consequently, we prove that there are π1-equivalent Zariski k-plets for any integer k ≥ 2. [ABSTRACT FROM AUTHOR]

In this paper, we study an analytic curve ϕ : I = [a, b] → M(n × n,R) in the space of n by n real matrices. There is a natural map u : M(n × n,R) → H = SL(2n,R). Let G be a Lie group containing H and Γ < G be a lattice of G. Let X = G/Γ. Then given a dense H-orbit in X, one could embed u(ϕ(I)) into X. We consider the expanding translates of the curve by some diagonal subgroup A = {a(t) : t ∈ R} ⊂ H. We will prove that if ϕ satisfies certain geometric conditions, then the expanding translates will tend to be equidistributed in G/Γ, as t → +∞. As an application, we show that for almost every point on ϕ(I), the Diophantine approximation given by Dirichlet's Theorem is not improvable. [ABSTRACT FROM AUTHOR]

Dominated and Pointwise Ergodic Theorems for bounded representations of second countable locally compact amenable groups by Lamperti operators in Lp(O,F,m), p > 1 fixed, are proved; we restrict ourselves to Cesàro averages in this paper. These theorems generalize or are closely related to well-known theorems for powers of power bounded Lamperti operators. [ABSTRACT FROM AUTHOR]

Let m be an integer greater than three and l be an odd prime. In this paper we prove that at least one of the following groups: PΩ±2m(Fls), PSO±2m(Fls), PO2m±(Fls), or PGO±2m(Fls) is a Galois group of Q for infinitely many integers s > 0. This is achieved by making use of a slight modification of a group theory result of Khare, Larsen, and Savin, and previous results of the author on the images of the Galois representations attached to cuspidal automorphic representations of GL2m(AQ). [ABSTRACT FROM AUTHOR]

If a linear algebraic group G acts on M(m, n), then it also acts on a rank variety M(r)(m, n) = {X ∈, M(m, n)| rankX = r}. In this paper, we give the necessary and sufficient condition that this variety has a Zariski-dense G-orbit. We consider everything over the complex number field …. [ABSTRACT FROM AUTHOR]

In this paper, we consider various extension problems associated with elements in the closure with respect to either the multiplier norm or the completely bounded multiplier norm of the Fourier algebra of a locally compact group. In particular, we show that it is not always possible to extend an element in the closure with respect to the multiplier norm of the Fourier algebra of the free group on two generators to a multiplier of the Fourier algebra of SL(2,R...). [ABSTRACT FROM AUTHOR]

For m ∈ N, let Sm be the Suzuki curve defined over F22m+1. It is well-known that Sm is supersingular, but the p-torsion group scheme of its Jacobian is not known. The α-number is an invariant of the isomorphism class of the p-torsion group scheme. In this paper, we compute a closed formula for the α-number of Sm using the action of the Cartier operator on H0(Sm,Ω¹). [ABSTRACT FROM AUTHOR]

The main results of this paper provide a lower bound on the p-rank of the finite p-groups. [ABSTRACT FROM AUTHOR]

We study the arithmetic of division fields of semistable abelian varieties A/Q. The Galois group of Q(A[2])/Q is analyzed when the conductor of A is odd and squarefree. The irreducible semistable mod 2 representations of small conductor are determined under GRH. These results are used in our paper Paramodular abelian varieties of odd conductor. [ABSTRACT FROM AUTHOR]

This paper discusses the twisted conjugacy classes in loop groups. We restrict to classical groups and give some explicit classifications. [ABSTRACT FROM AUTHOR]

Let p be a prime and r, e positive integers. In this paper we prove that the full affine group AGL(r, pe) of dimension r over the field with pe elements can be realized as the geometric monodromy group of a polynomial in characteristic p. We also determine the arithmetic monodromy group of this polynomial and the ramification structure it induces. [ABSTRACT FROM AUTHOR]

A pair (G, K) of a group and its subgroup is called a Gelfand pair if the induced trivial representation of K on G is multiplicity free. Let (aj) be a sequence of positive integers of length n, and let (bi) be its non-decreasing rearrangement. The sequence (ai) is called a parking function of length n if bi ≤ i for all i = 1, …, n. In this paper we study certain Gelfand pairs in relation with parking functions. In particular, we find explicit descriptions of the decomposition of the associated induced trivial representations into irreducibles. We obtain and study a new q-analogue of the Catalan numbers 1/n+1 (n2n), n ≤ 1. [ABSTRACT FROM AUTHOR]

In this paper, we study the properties of the positive solutions of a nonlinear integral system involving Wolff potentials: {u1 = Wβ,γ(f1(u)) um = Wβ,γ(fm(u)), where u = (u1, … , um) and Wβ,γ(f)(x) = ∫ ∞ 0 [ ∫Bt(x) f(y)dy/tn-βγ ] 1/γ-1 dt/t with 1 < γ < 2 and n > βγ. First, we estimate the decay rate of the positive solutions at infinity. Based on this, we prove radial symmetry and monotonicity for those solutions by the refined method of moving planes in integral forms, which was established by Chen, Li and Ou. Since the Kelvin transform cannot be used in such a Wolff type system, we have to find a new technique to study the asymptotic estimate, which is essential when we move the planes. [ABSTRACT FROM AUTHOR]

The Ore conjecture, proved by the authors, states that every element of every finite non-abelian simple group is a commutator. In this paper we use similar methods to prove that every element of every finite simple group is a product of two squares. This can be viewed as a non-commutative analogue of Lagrange's four squares theorem. Results for higher powers are also obtained. [ABSTRACT FROM AUTHOR]

It was proved by M. Bonk, J. Heinonen and P. Koskela that the quasihyperbolic metric hyperbolizes (in the sense of Gromov) uniform metric spaces. In this paper we introduce a new metric that hyperbolizes all locally compact noncomplete metric spaces. The metric is generic in the sense that (1) it can be defined on any metric space; (2) it preserves the quasiconformal geometry of the space; (3) it generalizes the j-metric, the hyperbolic cone metric and the hyperbolic metric of hyperspaces; and (4) it is quasi-isometric to the quasihyperbolic metric of uniform metric spaces. In particular, the Gromov hyperbolicity of these metrics also follows from that of our metric. [ABSTRACT FROM AUTHOR]

In Chermak and Delgado's paper ``A measuring argument for finite groups'', a certain ``measuring lemma'' was shown to hold. This lemma has been successfully applied in many recent papers. We generalize this lemma by expanding the discussion from groups acting on groups to groups acting on sets. As applications, we obtain the main results of several earlier papers. [ABSTRACT FROM AUTHOR]

It is well known that for irreducible, square-integrable representations of a locally compact group, there exist so-called admissible vectors which allow the construction of generalized continuous wavelet transforms. In this paper we discuss when the irreducibility requirement can be dropped, using a connection between generalized wavelet transforms and Plancherel theory. For unimodular groups with type I regular representation, the existence of admissible vectors is equivalent to a finite measure condition. The main result of this paper states that this restriction disappears in the nonunimodular case: Given a nondiscrete, second countable group $G$ with type I regular representation $\lambda_G$, we show that $\lambda_G$ itself (and hence every subrepresentation thereof) has an admissible vector in the sense of wavelet theory iff $G$ is nonunimodular. [ABSTRACT FROM AUTHOR]

In previous papers, nice trinomial equations were given for unramified coverings of the once punctured affine line in nonzero characteristic $p$ with the projective general group $\mathrm{PGL}(m,q)$ and the general linear group $\mathrm{GL}(m,q)$ as Galois groups where $m>1$ is any integer and $q>1$ is any power of $p$. These Galois groups were calculated over an algebraically closed ground field. Here we show that, when calculated over the prime field, as Galois groups we get the projective general semilinear group $\mathrm{P}\Gamma \mathrm{L}(m,q)$ and the general semilinear group $\Gamma \mathrm{L}(m,q)$. We also obtain the semilinear versions of the local coverings considered in previous papers. [ABSTRACT FROM AUTHOR]

The supersingular polynomial бℓ(x) ∈ Fℓ[x] has many wellstudied lifts to Q[x]. Among these is one introduced by Kaneko and Zagier, which, when interpreted as a specialized Jacobi polynomial, is seen to coincide with a lift discovered by Brillhart and Morton a few years later. The algebraic properties of this family of lifts of бℓ(x) are not well-understood. We focus on a conjecture of Mahlburg and Ono regarding the maximality of their Galois groups (when shorn of their trivial linear factors) and also establish their irreducibility in some previously unknown cases. [ABSTRACT FROM AUTHOR]

We show that if p is a prime and G is a finite p-solvable group satisfying the condition that a prime q divides the degree of no irreducible p-Brauer character of G, then the normalizer of some Sylow q-subgroup of G meets all the conjugacy classes of p-regular elements of G. [ABSTRACT FROM AUTHOR]

The maximal nilpotent and supersoluble automorphism groups of Riemann surfaces were given in earlier papers by this author. In this note the author wishes to correct the necessity of the condition given in Theorem (4.3) of \textit{Bounds for the order of supersoluble automorphism groups of Riemann surfaces} (Proc. Amer. Math. Soc. {\bf108} (1990), 587--600), which was left out at the time of writing the paper. The author also wishes to apologize to the readers for that. [ABSTRACT FROM AUTHOR]

We give a short axiomatic introduction to Carnot groups and their subRiemannian and subFinsler geometry. We explain how such spaces can be metrically described as exactly those proper geodesic spaces that admit dilations and are isometrically homogeneous. [ABSTRACT FROM AUTHOR]

R. Baer has proved that if the factor-group G/ζn(G) of a group G by the member ζn(G) of its upper central series is finite (here n is a positive integer), then the member γn+1(G) of the lower central series of G is also finite. In particular, in this case, the nilpotent residual of G is finite. This theorem admits the following simple generalization, which has been published very recently by M. de Falco, F. de Giovanni, C. Musella and Ya. P. Sysak: "If the factor-group G/Z of a group G modulo its upper hypercenter Z is finite, then G has a finite normal subgroup L such that G/L is hypercentral." In the current article we offer a new, simpler, very short proof of this theorem and specify it substantially. In fact, we prove that if |G/Z| = t, then |L| ≤ tk, where k = ¼(logpt + 1) and p is the least prime divisor of t. [ABSTRACT FROM AUTHOR]

We study twisted cohomology and homology groups on a onedimensional complex torus minus n distinct points with coefficients in a certain local system of rank one. This local system comes from the integrand of the Riemann-Wirtinger integral introduced by Mano. We construct bases of nonvanishing cohomology and homology groups, give an interpretation as a pairing of a cohomology class and a homology class to the Riemann-Wirtinger integral, and finally describe briefly the Gauss-Manin connection on the cohomology groups. [ABSTRACT FROM AUTHOR]

We give a detailed and clear exposition of the basic result describing the structure of a finite subset A of a nonabelian group G with a small doubling |A²| < 1.5|A|. [ABSTRACT FROM AUTHOR]

Assume that A is a purely infinite simple C*-algebra whose K0 is a torsion group, namely, contains no free element. Then a positive element a ∈ A can be written as a finite sum of projections in A if and only if either a is a projection or ||a|| > 1. [ABSTRACT FROM AUTHOR]

A product A = F1 … FN of invertible block-diagonal matrices will be banded with a banded inverse: Aij = 0 and also (A-1)ij = 0 for ¶i-j¶ > w. We establish this factorization with the number N controlled by the bandwidths w and not by the matrix size n. When A is an orthogonal matrix, or a permutation, or banded plus finite rank, the factors Fi have w = 1 and we find generators of that corresponding group. In the case of infinite matrices, the A = LPU factorization is now established but conjectures remain open. [ABSTRACT FROM AUTHOR]

For an odd integer N, we study the action of Atkin's U(2)-operator on the modular function x(t) associated to the Fermat curve: XN + YN = 1. The function x(t) is modular for the Fermat group φ(N), generically a noncongruence subgroup. If x(t) = q-1+Σ∞i=1 a(iN-1)qiN-1, we essentially prove that limn→0 a(n) = 0 in the 2-adic topology. [ABSTRACT FROM AUTHOR]

In this paper, we show that if $ \varphi(x; y)$-types $ p$-isolated $ \varphi$. Moreover, we can choose $ p'$elementary $ \varphi$ of $ p$ $ \vert\mathrm{dom}(p') - \mathrm{dom}(p)\vert \le 2 \cdot \mathrm{ID}(\varphi)$ being dependent. Finally, we give some corollaries of this theorem and draw some parallels to the stable setting. [ABSTRACT FROM AUTHOR]

A topological group is locally pseudocompact if it contains a non-empty open set with pseudocompact closure. In this paper, we prove that if $ G$ is locally pseudocompact, then $ G_0$, and $ G/G_0$ may fail to be zero-dimensional even for totally minimal pseudocompact groups. [ABSTRACT FROM AUTHOR]