Volume 30, Issue 3

Given an ample, Hausdorff groupoid 𝒢{\mathcal{G}}, and a unital commutative ring R , we consider the Steinberg algebra AR⁢(𝒢){A_{R}(\mathcal{G})}. First we prove a uniqueness theorem for this algebra and then, when 𝒢{\mathcal{G}} is graded by a cocycle, we study graded ideals in AR⁢(𝒢){A_{R}(\mathcal{G})}. Applications are given for two classes of ample groupoids, namely those coming from actions of groups on graphs, and also to groupoids defined in terms of Boolean dynamical systems.

We study a parametric Robin problem driven by a nonlinear nonhomogeneous differential operator and with a superlinear Carathéodory reaction term. We prove a bifurcation-type theorem for small values of the parameter. Also, we show that as the parameter λ>0{\lambda>0} approaches zero, we can find positive solutions with arbitrarily big and arbitrarily small Sobolev norm. Finally, we show that for every admissible parameter value, there is a smallest positive solution uλ*{u^{*}_{\lambda}} of the problem, and we investigate the properties of the map λ↦uλ*{\lambda\mapsto u^{*}_{\lambda}}.

Let F be a totally real number field and let π be a cuspidal automorphic representation of GSp4⁢(𝔸F){\mathrm{GSp_{4}}(\mathbb{A}_{F})}, which contributes irreducibly to coherent cohomology. If π has a Bessel model, we may attach a period p⁢(π){p(\pi)} to this datum. In the present paper, which is Part I in a series of two, we establish a relation of these Bessel periods p⁢(π){p(\pi)} and all of their twists p⁢(π⊗ξ){p(\pi\otimes\xi)} under arbitrary algebraic Hecke characters ξ. In the appendix, we show that (𝔤,K){(\mathfrak{g},K)}-cohomological cusp forms of GSp4⁢(𝔸F){\mathrm{GSp_{4}}(\mathbb{A}_{F})} all qualify to be of the above type – providing a large source of examples. We expect that these period relations for GSp4⁢(𝔸F){\mathrm{GSp_{4}}(\mathbb{A}_{F})} will allow a conceptual, fine treatment of rationality relations of special values of the spin L -function, which we hope to report on in Part II of this paper.

In 1874, Franz Mertens proved the so-called Mertens’ theorem, and in 1974, Kenneth S. Williams showed Mertens’ theorem associated with a character. In a previous paper, we presented a graph-theoretic analogue to Williams’ theorem. In this paper, we generalize our previous work in the sense that a character is extended to a representation. To our knowledge, a number-theoretic analogue to our result is not yet known. So, we expect that, by using our methods, it can be proven in the future.

Let L=-div⁡(A⁢∇){L=-\operatorname{div}(A\nabla)} be a second-order divergence form elliptic operator and let A be an accretive, n×n{n\times n} matrix with bounded measurable complex coefficients in ℝn{{\mathbb{R}}^{n}}. Let L-α2{L^{-\frac{\alpha}{2}}} be the fractional integral associated to L for 0<α<n{0<\alpha<n}. For b∈Lloc⁢(ℝn){b\in L_{\mathrm{loc}}({\mathbb{R}}^{n})} and k∈ℕ{k\in{\mathbb{N}}}, the k -th order commutator of b and L-α2{L^{-\frac{\alpha}{2}}} is given by (L-α2)b,k⁢f⁢(x)=L-α2⁢((b⁢(x)-b)k⁢f)⁢(x).(L^{-\frac{\alpha}{2}})_{b,k}f(x)=L^{-\frac{\alpha}{2}}((b(x)-b)^{k}f)(x). In the paper, we mainly show that if b∈BMO⁢(ℝn){b\in\mathrm{BMO}({\mathbb{R}}^{n})}, 0<λ<n{0<\lambda<n} and 0<α<n-λ{0<\alpha<n-\lambda}, then (L-α2)b,k{(L^{-\frac{\alpha}{2}})_{b,k}} is bounded from Lp,λ{L^{p,\lambda}} to Lq,λ{L^{q,\lambda}} for p-⁢(L)<p<q<p+⁢(L)⁢n-λn{p_{-}(L)<p<q<p_{+}(L)\frac{n-\lambda}{n}} and 1q=1p-αn-λ{\frac{1}{q}=\frac{1}{p}-\frac{\alpha}{n-\lambda}}, where p-⁢(L){p_{-}(L)} and p+⁢(L){p_{+}(L)} are the two critical exponents for the Lp{L^{p}} uniform boundedness of the semigroup {e-t⁢L}t>0{\{e^{-tL}\}_{t>0}}. Also, we establish the boundedness of the commutator of the fractional integral with Lipschitz function on Morrey spaces. The results encompass what is known for the classical Riesz potentials and elliptic operators with Gaussian domination by the classical heat operator.

We prove Vojta’s conjecture for some rational surfaces. Moreover, for similar but different rational surfaces, we show that their Vojta’s conjecture is related to the abc conjecture. More specifically, we prove that Vojta’s conjecture on these surfaces implies a special case of the abc conjecture, while the abc conjecture implies Vojta’s conjecture on these surfaces. The argument carries over to the holomorphic case, so we unconditionally obtain Griffiths’ conjecture for the same situation. To prove these results, we prove and use some properties of Farey sequences.

Let K/F{K/F} be a finite extension of number fields of degree n≥2{n\geq 2}. We establish effective field-uniform unconditional upper bounds for the least norm of a prime ideal 𝔭{\mathfrak{p}} of F which is degree 1 over ℚ{\mathbb{Q}} and does not ramify or split completely in K . We improve upon the previous best known general estimates due to Li [7] when F=ℚ{F=\mathbb{Q}}, and Murty and Patankar [9] when K/F{K/F} is Galois. Our bounds are the first when K/F{K/F} is not assumed to be Galois and F≠ℚ{F\neq\mathbb{Q}}.

The problem considered in this paper is whether an inequality of ω-terms is valid in a given level of a concatenation hierarchy of star-free languages. The main result shows that this problem is decidable for all (integer and half) levels of the Straubing–Thérien hierarchy.

The purpose of this paper is to collect and make explicit the results of Langlands [16], Bump [3], Miyazaki [18] and Manabe, Ishii and Oda [17] for the GL⁢(3){\mathrm{GL}(3)} Eisenstein series and Whittaker functions which are non-trivial on SO⁢(3,ℝ){\mathrm{SO}(3,\mathbb{R})}. The final goal for the series of papers is a complete and completely explicit spectral expansion for L2⁢(SL⁢(3,ℤ)∖SL⁢(3,ℝ)){L^{2}(\mathrm{SL}(3,\mathbb{Z})\setminus\mathrm{SL}(3,\mathbb{R}))} in the style of Duke, Friedlander and Iwaniec’s paper [8]. We derive a number of new results on the Whittaker functions and Eisenstein series, and give new, concrete proofs of the functional equations and spectral expansion in place of the general constructions of Langlands.

This paper is a companion paper to [6], where sharp estimates are proven for Fourier transforms of compactly supported functions built out of two-dimensional real-analytic functions. The theorems of [6] are stated in a rather general form. In this paper, we expand on the results of [6] and show that there is a class of “well-behaved” functions that contains a number of relevant examples for which such estimates can be explicitly described in terms of the Newton polygon of the function. We will further see that for a subclass of these functions, one can prove noticeably more precise estimates, again in an explicitly describable way.

We introduce a class of first order G -structures, each of which has an underlying almost conformally symplectic structure. There is one such structure for each real simple Lie algebra which is not of type Cn{C_{n}} and admits a contact grading. We show that a structure of each of these types on a smooth manifold M determines a canonical compatible linear connection on the tangent bundle TM{\mathrm{TM}}. This connection is characterized by a normalization condition on its torsion. The algebraic background for this result is proved using Kostant’s theorem on Lie algebra cohomology. For each type, we give an explicit description of both the geometric structure and the normalization condition. In particular, the torsion of the canonical connection naturally splits into two components, one of which is exactly the obstruction to the underlying structure being conformally symplectic. This article is the first in a series aiming at a construction of differential complexes naturally associated to these geometric structures.

Since positive definite Hermitian matrices have become fundamental objects in many areas, a variety of theoretical and computational research topics have been arisen. Especially, the average of positive definite matrices is a very important notion to see the central tendency of objects. There are many different kinds of averages for a finite number of positive definite matrices such as quasi-arithmetic means, power means and Cartan barycenters. We generalize these averages to the setting of positive definite matrices equipped with probability measures of compact support, and show the monotonicity of quasi-arithmetic means for parameters ≥1{\geq 1}, and connections with inequalities between quasi-arithmetic means and power means, and between quasi-arithmetic means and Cartan barycenters.

We estimate the number of relatively r -prime lattice points in Km{K^{m}} with their components having a norm less than x , where K is a number field. The error terms are estimated in terms of x and the discriminant D of the field K , as both x and D grow. The proof uses the bounds of Dedekind zeta functions. We obtain uniform upper bounds as K runs through number fields of any degree under assuming the Lindelöf hypothesis. We also show unconditional results for abelian extensions with a degree less than or equal to 6.

In this article, we prove an Omega result for the Hecke eigenvalues λF⁢(n){\lambda_{F}(n)} of Maass forms F which are Hecke eigenforms in the space of Siegel modular forms of weight k , genus two for the Siegel modular group S⁢p2⁢(ℤ){Sp_{2}({\mathbb{Z}})}. In particular, we prove λF⁢(n)=Ω⁢(nk-1⁢exp⁡(c⁢log⁡nlog⁡log⁡n)),\lambda_{F}(n)=\Omega\biggl{(}n^{k-1}\exp\biggl{(}c\frac{\sqrt{\log n}}{\log% \log n}\biggr{)}\biggr{)}, when c>0{c>0} is an absolute constant. This improves the earlier result λF⁢(n)=Ω⁢(nk-1⁢(log⁡nlog⁡log⁡n))\lambda_{F}(n)=\Omega\biggl{(}n^{k-1}\biggl{(}\frac{\sqrt{\log n}}{\log\log n}% \biggr{)}\biggr{)} of Das and the third author. We also show that for any n≥3{n\geq 3}, one has λF⁢(n)≤nk-1⁢exp⁡(c1⁢log⁡nlog⁡log⁡n),\lambda_{F}(n)\leq n^{k-1}\exp\biggl{(}c_{1}\sqrt{\frac{\log n}{\log\log n}}% \biggr{)}, where c1>0{c_{1}>0} is an absolute constant. This improves an earlier result of Pitale and Schmidt. Further, we investigate the limit points of the sequence {λF⁢(n)/nk-1}n∈ℕ{\{\lambda_{F}(n)/n^{k-1}\}_{n\in{\mathbb{N}}}} and show that it has infinitely many limit points. Finally, we show that λF⁢(n)>0{\lambda_{F}(n)>0} for all n , a result proved earlier by Breulmann by a different technique.

We construct special classes of totally geodesic almost regular foliations, namely, complex radial foliations in Hermitian manifolds and quaternionic radial foliations in quaternionic Kähler manifolds, and we give criteria for their harmonicity and minimality. Then examples of these foliations on complex and quaternionic space forms, which are harmonic and minimal, are presented.