Israel Journal of Mathematics最新文献

We twist Zagier’s function f_k,D by a sign function and a genus character. Assuming weight 0 < k ≡ 2 (mod 4), and letting D be a positive non-square discriminant, we prove that the obstruction to modularity caused by the sign function can be corrected obtaining a locally harmonic Maaß form or a local cusp form of the same weight. In addition, we provide an alternative representation of our new function in terms of a twisted trace of modular cycle integrals of a Poincaré series due to Petersson.

We consider the horocyclic flow corresponding to a (topologically mixing) Anosov flow or diffeomorphism, and establish the uniqueness of transverse quasi-invariant measures with Hölder Jacobians. In the same setting, we give a precise characterization of the equilibrium states of the hyperbolic system, showing that existence of a family of Radon measures on the horocyclic foliation such that any probability (invariant or not) having conditionals given by this family, necessarily is the unique equilibrium state of the system.

Let k ≥ 2 and s be positive integers. Let θ ∈ (0, 1) be a real number. In this paper, we establish that if s > k(k + 1) and θ > 0.55, then every sufficiently large natural number n, subject to certain congruence conditions, can be written as

, where p_i (1 ≤ i ≤ s) are primes in the interval (({({n over s})^{{1 over k}}} - {n^{{theta over k}}},{({n over s})^{{1 over k}}} + {n^{{theta over k}}}]). The second result of this paper is to show that if (s > {{k(k + 1)} over 2}) and θ > 0.55, then almost all integers n, subject to certain congruence conditions, have the above representation.

Given a hyperelliptic hyperbolic surface S of genus g ≥ 2, we find bounds on the lengths of homologically independent loops on S. As a consequence, we show that for any λ ∈ (0, 1) there exists a constant N(λ) such that every such surface has at least (leftlceil {lambda cdot {2 over 3}g} rightrceil ) homologically independent loops of length at most N(λ), extending the result in [Mu] and [BPS]. This allows us to extend the constant upper bound obtained in [Mu] on the minimal length of non-zero period lattice vectors of hyperelliptic Riemann surfaces to almost ({2 over 3}g) linearly independent vectors.

Given an Artin group A_Γ, a common strategy in the study of A_Γ is the reduction to parabolic subgroups whose defining graphs have small diameter, i.e., showing that A_Γ has a specific property if and only if all “small” parabolic subgroups of A_Γ have this property. Since “small” parabolic subgroups are the building blocks of A_Γ one needs to study their behavior, in particular their intersections. The conjecture we address here says that the class of parabolic subgroups of A_Γ is closed under intersection. Under the assumption that intersections of parabolic subgroups in complete Artin groups are parabolic, we show that the intersection of a complete parabolic subgroup with an arbitrary parabolic subgroup is parabolic. Further, we connect the intersection behavior of complete parabolic subgroups of A_Γ to fixed point properties and to automatic continuity of A_Γ using Bass–Serre theory and a generalization of the Deligne complex.

P. Hall constructed a universal countable locally finite group U, determined up to isomorphism by two properties: every finite group C is a subgroup of U, and every embedding of C into U is conjugate in U. Every countable locally finite group is a subgroup of U. We prove that U is a subgroup of the abstract commensurator of a finite-rank nonabelian free group.

Let (K: = {rm{G}}{{rm{L}}_n}({cal O})) denote the maximal compact subgroup of GL_n(F), where F is a nonarchimedean local field with ring of integers ({cal O}). We study the decomposition of the space of locally constant functions on the unit sphere in Fⁿ into irreducible K-modules; for F = ℚ_p, these are the p-adic analogues of spherical harmonics. As an application, we characterise the newform and conductor exponent of a generic irreducible admissible smooth representation of GL_n(F) in terms of distinguished K-types. Finally, we compare our results to analogous results in the archimedean setting.

We show that a version of the cube axiom holds in cosimplicial unstable coalgebras and cosimplicial spaces equipped with a resolution model structure. As an application, classical theorems in unstable homotopy theory are extended to this context.

For a non-cyclic free group F, the second homology of its pronilpotent completion ({H_2}(widehat F)) is not a cotorsion group.

We study the mean (sumnolimits_{x in {cal X}} {|sumnolimits_{p le N} {{u_p}e(xp){|^ell}}} ) when ℓ covers the full range [2, ∞) and ({cal X} subset mathbb{R}/mathbb{Z}) is a well-spaced set, providing a smooth transition from the case ℓ = 2 to the case ℓ > 2 and improving on the results of J. Bourgain and of B. Green and T. Tao. A uniform Hardy–Littlewood property for the set of primes is established as well as a sharp upper bound for (sumnolimits_{x in {cal X}} {|sumnolimits_{p le N} {{u_p}e(xp){|^ell}}}) when ({cal X}) is small. These results are extended to primes in any interval in a last section, provided the primes are numerous enough therein.