Volume 37, Issue 1

We prove that the extended frame spectral measures are of pure type and the Beurling dimension of any frame measure for an extended frame spectral measure is in its Fourier dimension and upper entropy dimension.

We construct all orthogonal separating coordinates in constant curvature spaces of arbitrary signature. Further, we construct explicit transformation between orthogonal separating and flat or generalised flat coordinates, as well as explicit formulas for the corresponding Killing tensors and Stäckel matrices.

We investigate the Cauchy–Szegő projection for quaternionic Siegel upper half space to obtain the pointwise (higher order) regularity estimates for Cauchy–Szegő kernel and prove that the Cauchy–Szegő kernel is nonzero everywhere, which further yields a non-degenerated pointwise lower bound. As applications, we prove the uniform boundedness of Cauchy–Szegő projection on every atom on the quaternionic Heisenberg group, which is used to give an atomic decomposition of regular Hardy space H p {H^{p}} on quaternionic Siegel upper half space for 2 3 < p ≤ 1 {\frac{2}{3}<p\leq 1} . Moreover, we establish the characterisation of singular values of the commutator of Cauchy–Szegő projection based on the kernel estimates. The quaternionic structure (lack of commutativity) is encoded in the symmetry groups of regular functions and the associated partial differential equations.

Sums of Kloosterman sums have deep connections with the theory of modular forms, and their estimation has many important consequences. Kuznetsov used his famous trace formula and got a power-saving estimate with respect to x with implied constants depending on m and n . Recently, in 2009, Sarnak and Tsimerman obtained a bound uniformly in x , m and n . The generalized Kloosterman sums are defined with multiplier systems and on congruence subgroups. Goldfeld and Sarnak bounded sums of them with main terms corresponding to exceptional eigenvalues of the hyperbolic Laplacian. Their error term is a power of x with implied constants depending on all the other factors. In this paper, for a wide class of half-integral weight multiplier systems, we get the same bound with the error term uniformly in x , m and n . Such uniform bounds have great applications. For the eta-multiplier, Ahlgren and Andersen obtained a uniform and power-saving bound with respect to m and n , which resulted in a convergent error estimate on the Rademacher exact formula of the partition function p ⁢ ( n ) {p(n)} . We also establish a Rademacher-type exact formula for the difference of partitions of rank modulo 3, which allows us to apply our power-saving estimate to the tail of the formula for a convergent error bound.

Employing Watson’s ϕ 7 8 {{}_{8}\phi_{7}} transformation formula, we unearth several q -supercongruences with a parameter s . Particularly, one of our results is an extension of a q -analogue of Van Hamme’s (G.2) supercongruence. In addition, we obtain a q -supercongruence modulo the fifth power of a cyclotomic polynomial and propose two related conjectures.

In this paper, we investigate certain submodules in C*-algebras associated to effective étale groupoids. First, we show that a submodule generated by normalizers is a closure of the set of compactly supported continuous functions on some open set. As a corollary, we show that discrete group coactions on groupoid C*-algebras are induced by cocycles of étale groupoids if the fixed point algebras contain C*-subalgebras of continuous functions vanishing at infinity on the unit spaces. In the latter part, we prove the Galois correspondence result for discrete group coactions on groupoid C*-algebras.

Let f be a polynomial with coefficients in the ring O S {O_{S}} of S -integers of a number field K , b a non-zero S -integer, and m an integer ≥ 2 {\geq 2} . We consider the following equation ( ⋆ ) {(\star)} : f ⁢ ( x ) = b ⁢ y m {f(x)=by^{m}} in x , y ∈ O S {x,y\in O_{S}} . Under the well-known LeVeque condition, we give fully explicit upper bounds in terms of K , S , f , m {K,S,f,m} and the S -norm of b for the heights of the solutions x of equation ( ⋆ ) {(\star)} . Further, we give an explicit bound C in terms of K , S , f {K,S,f} and the S -norm of b such that if m > C {m>C} equation ( ⋆ ) {(\star)} has only solutions with y = 0 {y=0} or a root of unity. Our results are more detailed versions of work of Trelina, Brindza, Shorey and Tijdeman, Voutier and Bugeaud, and extend earlier results of Bérczes, Evertse, and Győry to polynomials with multiple roots. In contrast with the previous results, our bounds depend on the S -norm of b instead of its height.

In this paper, we investigate nonlocal partial systems that incorporate the fractional Laplace operator. Our primary focus is to establish a theorem concerning the existence of optimal solutions for these equations. To achieve this, we utilize two fundamental tools: information obtained from an iterative reconstruction algorithm and a variant of the Phragmén–Lindelöf principle of concentration and compactness tailored for fractional systems. By employing these tools, we provide valuable insights into the nature of nonlocal partial systems and their optimal solutions.

Given a proper holomorphic surjective morphism f : X → Y {f:X\rightarrow Y} between compact Kähler manifolds, and a Nakano semipositive holomorphic vector bundle E on X , we prove Kollár-type vanishing theorems on cohomologies with coefficients in R q ⁢ f ∗ ⁢ ( ω X ⁢ ( E ) ) ⊗ F {R^{q}f_{\ast}(\omega_{X}(E))\otimes F} , where F is a k -positive vector bundle on Y . The main inputs in the proof are the deep results on the Nakano semipositivity of the higher direct images due to Berndtsson and Mourougane–Takayama, and an L 2 {L^{2}} -Dolbeault resolution of the higher direct image sheaf R q ⁢ f ∗ ⁢ ( ω X ⁢ ( E ) ) {R^{q}f_{\ast}(\omega_{X}(E))} , which is of interest in itself.

Define an infinite matrix D α = ( d n , v α ) \mathfrak{D}^{\alpha}=(d^{\alpha}_{n,v}) by d n , v α = { v α σ ( α ) ⁢ ( n ) , v ∣ n , 0 , v ∤ n , d^{\alpha}_{n,v}=\begin{cases}\dfrac{v^{\alpha}}{\sigma^{(\alpha)}(n)},&v\mid n,\\ 0,&v\nmid n,\end{cases} where σ ( α ) ⁢ ( n ) \sigma^{(\alpha)}(n) is defined to be the sum of the 𝛼-th power of the positive divisors of n ∈ N n\in\mathbb{N} , and construct the matrix domains ℓ p ⁢ ( D α ) \ell_{p}(\mathfrak{D}^{\alpha}) ( 0 < p < ∞ 0<p<\infty ), c 0 ⁢ ( D α ) c_{0}(\mathfrak{D}^{\alpha}) , c ⁢ ( D α ) c(\mathfrak{D}^{\alpha}) and ℓ ∞ ⁢ ( D α ) \ell_{\infty}(\mathfrak{D}^{\alpha}) defined by the matrix D α \mathfrak{D}^{\alpha} . We develop Schauder bases and determine 𝛼-, 𝛽- and 𝛾-duals of these new spaces. We characterize some matrix transformation from ℓ p ⁢ ( D α ) \ell_{p}(\mathfrak{D}^{\alpha}) , c 0 ⁢ ( D α ) c_{0}(\mathfrak{D}^{\alpha}) , c ⁢ ( D α ) c(\mathfrak{D}^{\alpha}) and ℓ ∞ ⁢ ( D α ) \ell_{\infty}(\mathfrak{D}^{\alpha}) to ℓ ∞ \ell_{\infty} , 𝑐, c 0 c_{0} and ℓ 1 \ell_{1} . Furthermore, we determine some criteria for compactness of an operator (or matrix) from X ∈ { ℓ p ⁢ ( D α ) , c 0 ⁢ ( D α ) , c ⁢ ( D α ) , ℓ ∞ ⁢ ( D α ) } X\in\{\ell_{p}(\mathfrak{D}^{\alpha}),c_{0}(\mathfrak{D}^{\alpha}),c(\mathfrak{D}^{\alpha}),\ell_{\infty}(\mathfrak{D}^{\alpha})\} to ℓ ∞ \ell_{\infty} , 𝑐, c 0 c_{0} or ℓ 1 \ell_{1} .

We prove one divisibility relation of the anticyclotomic Iwasawa Main Conjecture for a higher weight ordinary modular form f and an imaginary quadratic field satisfying a “relaxed” Heegner hypothesis. Let Λ be the anticyclotomic Iwasawa algebra. Following the approach of Howard and Longo–Vigni, we construct the Λ-adic Kolyvagin system of generalized Heegner classes coming from Heegner points on a suitable Shimura curve. As its application, we also prove one divisibility relation in the Iwasawa–Greenberg main conjecture for the p -adic L -function defined by Magrone.

In this paper we prove some generalizations of the classical Hasse–Davenport product relation for certain arithmetic factors defined on a p -adic field F , among them one finds the ϵ-factors appearing in Tate’s thesis. We then show that these generalizations are equivalent to some representation theoretic identities relating the determinant of ramified local coefficients matrices defined for coverings of SL 2 ⁢ ( F ) {\mathrm{SL}_{2}(F)} to Plancherel measures and γ-factors.

Given a totally real algebraic number field k of degree s , we consider locally symmetric spaces X G / Γ {X_{G}/\Gamma} associated with arithmetic subgroups Γ of the special linear algebraic k -group G = SL M 2 ⁢ ( D ) {G=\mathrm{SL}_{M_{2}(D)}} , attached to a quaternion division k -algebra D . The group G is k -simple, of k -rank one, and non-split over k . Using reduction theory, one can construct an open subset Y Γ ⊂ X G / Γ {Y_{\Gamma}\subset X_{G}/\Gamma} such that its closure Y ¯ Γ {\overline{Y}_{\Gamma}} is a compact manifold with boundary ∂ ⁡ Y ¯ Γ {\partial\overline{Y}_{\Gamma}} , and the inclusion Y ¯ Γ → X G / Γ {\overline{Y}_{\Gamma}\rightarrow X_{G}/\Gamma} is a homotopy equivalence. The connected components Y [ P ] {Y^{[P]}} of the boundary ∂ ⁡ Y ¯ Γ {\partial\overline{Y}_{\Gamma}} are in one-to-one correspondence with the finite set of Γ-conjugacy classes of minimal parabolic k -subgroups of G . We show that each boundary component carries the natural structure of a torus bundle. Firstly, if the quaternion division k -algebra D is totally definite, that is, D ramifies at all archimedean places of k , we prove that the basis of this bundle is homeomorphic to the torus T s - 1 {T^{s-1}} of dimension s - 1 {s-1} , has the compact fibre T 4 ⁢ s {T^{4s}} , and its structure group is SL 4 ⁢ s ⁢ ( ℤ ) {\mathrm{SL}_{4s}(\mathbb{Z})} . We determine the cohomology of Y [ P ] {Y^{[P]}} . Secondly, if the quaternion division k -algebra D is indefinite, thus, there exists at least one archimedean place v ∈ V k , ∞ {v\in V_{k,\infty}} at which D v {D_{v}} splits over ℝ {\mathbb{R}} , that is, D v ≅ M 2 ⁢ ( ℝ ) {D_{v}\cong M_{2}(\mathbb{R})} , the fibre is homeomorphic to T 4 ⁢ s {T^{4s}} , but the base space of the bundle is more complicated.

In 1769, Euler proved the following result: ∫ 0 π 2 log ⁡ ( sin ⁡ θ ) ⁢ 𝑑 θ = - π 2 ⁢ log ⁡ 2 . \int_{0}^{\frac{\pi}{2}}\log(\sin\theta)\,d\theta=-\frac{\pi}{2}\log 2. In this paper, as a generalization, we evaluate the definite integrals ∫ 0 x θ r - 2 ⁢ log ⁡ ( cos ⁡ θ 2 ) ⁢ 𝑑 θ \int_{0}^{x}\theta^{r-2}\log\biggl{(}\cos\frac{\theta}{2}\biggr{)}\,d\theta for r = 2 , 3 , 4 , … r=2,3,4,\dots  . We show that it can be expressed by the special values of Kurokawa and Koyama’s multiple cosine functions 𝒞 r ⁢ ( x ) {\mathcal{C}_{r}(x)} or by the special values of alternating zeta and Dirichlet lambda functions. In particular, we get the following explicit expression of the zeta value: ζ ⁢ ( 3 ) = 4 ⁢ π 2 21 ⁢ log ⁡ ( e 4 ⁢ G π ⁢ 𝒞 3 ⁢ ( 1 4 ) 16 2 ) , \zeta(3)=\frac{4\pi^{2}}{21}\log\Biggl{(}\frac{e^{\frac{4G}{\pi}}\mathcal{C}_{% 3}\bigl{(}\frac{1}{4}\bigr{)}^{16}}{\sqrt{2}}\Biggr{)}, where G is Catalan’s constant and 𝒞 3 ⁢ ( 1 4 ) {\mathcal{C}_{3}(\frac{1}{4})} is the special value of Kurokawa and Koyama’s multiple cosine function 𝒞 3 ⁢ ( x ) {\mathcal{C}_{3}(x)} at 1 4 {\frac{1}{4}} . Furthermore, we prove several series representations for the logarithm of multiple cosine functions log ⁡ 𝒞 r ⁢ ( x 2 ) {\log\mathcal{C}_{r}(\frac{x}{2})} by zeta functions, L -functions or polylogarithms. One of them leads to another expression of ζ ⁢ ( 3 ) {\zeta(3)} : ζ ⁢ ( 3 ) = 72 ⁢ π 2 11 ⁢ log ⁡ ( 3 1 72 ⁢ 𝒞 3 ⁢ ( 1 6 ) 𝒞 2 ⁢ ( 1 6 ) 1 3 ) . \zeta(3)=\frac{72\pi^{2}}{11}\log\Biggl{(}\frac{3^{\frac{1}{72}}\mathcal{C}_{3% }\bigl{(}\frac{1}{6}\bigr{)}}{\mathcal{C}_{2}\bigl{(}\frac{1}{6}\bigr{)}^{% \frac{1}{3}}}\Biggr{)}.

In this paper, we prove several versions of the classical Paley inequality for the Weyl transform. As for some applications, we prove a version of the Hörmander’s multiplier theorem to discuss L p {L^{p}} - L q {L^{q}} boundedness of the Weyl multipliers and prove the Hardy–Littlewood inequality. We also consider the vector-valued version of the inequalities of Paley, Hausdorff–Young, and Hardy–Littlewood and their relations. Finally, we also prove Pitt’s inequality for the Weyl transform.

In this paper, we characterize homogeneous arithmetically Cohen–Macaulay (ACM) bundles and Ulrich bundles on rational homogeneous spaces. From this result, we see that there are only finitely many irreducible homogeneous ACM bundles (up to twist) and Ulrich bundles on these varieties. Moreover, we give numerical criteria for some special irreducible homogeneous bundles to be ACM bundles.

Given a prime power q and a positive integer n , let 𝔽 q n {\mathbb{F}_{q^{n}}} represent a finite extension of degree n of the finite field 𝔽 q {{\mathbb{F}_{q}}} . In this article, we investigate the existence of m elements in arithmetic progression, where every element is primitive and at least one is normal with prescribed norms. Moreover, for n ≥ 6 {n\geq 6} , q = 3 k {q=3^{k}} , m = 2 {m=2} we establish that there are only 10 possible exceptions.