Article Frontmatter was published on July 1, 2021 in the journal Forum Mathematicum (volume 33, issue 4).

In the theory of harmonic Maaß forms and mock modular forms, mock theta functions are distinguished examples which arose from q -hypergeometric examples of Ramanujan. Recently, there has been a body of work on higher depth mock modular forms. Here, we introduce distinguished examples of these forms, which we call higher depth mock theta functions, and develop q -hypergeometric expressions for them

We study universal groups for right-angled buildings. Inspired by Simon Smith’s work on universal groups for trees, we explicitly allow local groups that are not necessarily finite nor transitive. We discuss various topological and algebraic properties in this extended setting. In particular, we characterise when these groups are locally compact, when they are abstractly simple, when they act primitively

We define the notion of a characteristically inert socle-regular Abelian p -group and explore such groups by focussing on their socles, thereby relating them to previously studied notions of socle-regularity. We show that large classes of p -groups, including all divisible, totally projective and torsion-complete p -groups, share this property when the prime p is odd. The present work generalizes notions

We provide a new perspective on extended Priestley duality for a large class of distributive lattices equipped with binary double quasioperators. Under this approach, non-lattice binary operations are each presented as a pair of partial binary operations on dual spaces. In this enriched environment, equational conditions on the algebraic side of the duality may more often be rendered as first-order

In this work, we study the non-spectrality of the self-affine measure μM,D{\mu_{M,D}} generated by an expanding integer matrix M∈M2⁢(ℤ){M\in M_{2}(\mathbb{Z})} with det⁡(M)∉2⁢ℤ{\det(M)\notin 2\mathbb{Z}} and the integer digit set D={(0,0)t,(α1,α2)t,(β1,β2)t,(-α1-β1,-α2-β2)t}D=\bigl{\{}(0,0)^{t},(\alpha_{1},\alpha_{2})^{t},(\beta_{1},\beta_{2})^{t},(-% \alpha_{1}-\beta_{1},-\alpha_{2}-\beta_{2})^{t}\bigr{\}}

Let θ:G↪GL⁢(n,F)\theta:G\hookrightarrow\mathrm{GL}(n,\mathbb{F}) be a representation of a finite group 𝐺 over the field 𝔽 and F⁢[V]⊗F⁢[V]GF⁢[V]\mathbb{F}[V]\otimes_{{\mathbb{F}[V]^{G}}}\mathbb{F}[V] the associated equivariant coinvariant algebra. The purpose of this manuscript is to determine the associated prime ideals of F⁢[V]⊗F⁢[V]GF⁢[V]\mathbb{F}[V]\otimes_{{\mathbb{F}[V]^{G}}}\mathbb{F}[V] for

In this paper, we undertake a systematic model- and valuation-theoretic study of the class of ordered fields which are dense in their real closure. We apply this study to determine definable henselian valuations on ordered fields, in the language of ordered rings. In light of our results, we re-examine the Shelah–Hasson Conjecture (specialized to ordered fields) and provide an example limiting its

In recent years, the problem of estimating Hankel determinants has attracted the attention of many mathematicians. Their research have been focused mainly on deriving the bounds of H2,2{H_{2,2}} or H3,1{H_{3,1}} over different subclasses of 𝒮{\mathcal{S}}. Only in a few papers third Hankel determinants for non-univalent functions were considered. In this paper, we consider two classes of analytic

We refine a result of [J. E. McCarthy, Common range of co-analytic Toeplitz operators, J. Amer. Math. Soc. 3 1990, 4, 793–799] and explore the common range of the co-analytic Toeplitz operators on a model space. The tools used to do this also yield information about when one can interpolate with an outer function.

We define solitons for the generalized Ricci flow on an exact Courant algebroid. We then define a family of flows for left-invariant Dorfman brackets on an exact Courant algebroid over a simply connected nilpotent Lie group, generalizing the bracket flows for nilpotent Lie brackets in a way that might make this new family of flows useful for the study of generalized geometric flows such as the generalized

In this paper, we establish the endpoint estimate (0

In recent works [G. A. Bagheri-Bardi, A. Elyaspour and G. H. Esslamzadeh, Wold-type decompositions in Baer ∗\ast-rings, Linear Algebra Appl. 539 2018, 117–133] and [G. A. Bagheri-Bardi, A. Elyaspour and G. H. Esslamzadeh, The role of algebraic structure in the invariant subspace theory, Linear Algebra Appl. 583 2019, 102–118], the algebraic analogues of the three major decomposition theorems of Wold

Let 𝔽q{\mathbb{F}_{q}} be the finite field of order q , where q is an odd prime power. Then a k -dimensional quadratic subspace (W,Q){(W,Q)} of (𝔽qn,x12+x22+⋯+xn2){(\mathbb{F}_{q}^{n},x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2})} is called dot𝐤{\operatorname{dot}_{\mathbf{k}}}-subspace if Q is isometrically isomorphic to x12+x22+⋯+xk2{x_{1}^{2}+x_{2}^{2}+\cdots+x_{k}^{2}}. In this paper, we obtain bounds

Let π be an automorphic irreducible cuspidal representation of GLm{\operatorname{GL}_{m}} over ℚ{\mathbb{Q}} with unitary central character, and let λπ⁢(n){\lambda_{\pi}(n)} be its n -th Dirichlet series coefficient. We study short sums of isotypic trace functions associated to some sheaves modulo primes q of bounded conductor, twisted by multiplicative functions λπ⁢(n){\lambda_{\pi}(n)} and μ⁢(n)

Article Frontmatter was published on May 1, 2021 in the journal Forum Mathematicum (volume 33, issue 3).

We prove a central limit theorem for log⁡|ζ⁢(12+i⁢t)|{\log\lvert\zeta(\frac{1}{2}+it)\rvert} with respect to the measure |ζ(m)⁢(12+i⁢t)|2⁢k⁢d⁢t{\lvert\zeta^{(m)}(\frac{1}{2}+it)\rvert^{2k}\,dt} (k,m∈ℕ{k,m\in\mathbb{N}}), assuming RH and the asymptotic formula for twisted and shifted integral moments of zeta. Under the same hypotheses, we also study a shifted case, looking at the measure |ζ⁢(12+i⁢t

If ω is a closed G -invariant 2-form and μ is a moment map, we obtain necessary and sufficient conditions for equivariant prequantizability that can be computed in terms of the moment map μ. Our main result is that G -equivariant prequantizability is related to the fact that the moment map μ should be quantized for certain vectors on the Lie algebra of G . We also compute the obstructions to lift the

We are interested in characterising the commutative rings for which a 1-tilting cotorsion pair (𝒜,𝒯){(\mathcal{A},\mathcal{T})} provides for covers, that is when the class 𝒜{\mathcal{A}} is a covering class. We use Hrbek’s bijective correspondence between the 1-tilting cotorsion pairs over a commutative ring R and the faithful finitely generated Gabriel topologies on R . Moreover, we use results

We apply differential operators to modular forms on orthogonal groups O⁢(2,ℓ){\mathrm{O}(2,\ell)} to construct infinite families of modular forms on special cycles. These operators generalize the quasi-pullback. The subspaces of theta lifts are preserved; in particular, the higher pullbacks of the lift of a (lattice-index) Jacobi form ϕ are theta lifts of partial development coefficients of ϕ. For

In this paper, we show that an action on the set of elliptic curves with j=1728j=1728 preserves a certain kind of symmetry on the local root number of Hecke characters attached to such elliptic curves. As a consequence, we give results on the distribution of the root numbers and their average of the aforementioned Hecke characters.

We establish some higher differentiability results of integer and fractional order for solutions to non-autonomous obstacle problems of the form min⁡{∫Ωf⁢(x,D⁢v⁢(x)):v∈Kψ⁢(Ω)},\min\biggl{\{}\int_{\Omega}f(x,Dv(x)):v\in\mathcal{K}_{\psi}(\Omega)\biggr{\}}, where the function 𝑓 satisfies 𝑝-growth conditions with respect to the gradient variable, for 1

Let 𝔽q{\mathbb{F}_{q}} be the finite field of q=pk{q=p^{k}} elements with p being a prime and let k be a positive integer. For any y,z∈𝔽q{y,z\in\mathbb{F}_{q}}, let Ns⁢(z){N_{s}(z)} and Ts⁢(y){T_{s}(y)} denote the numbers of zeros of x13+⋯+xs3=z{x_{1}^{3}+\cdots+x_{s}^{3}=z} and x13+⋯+xs-13+y⁢xs3=0{x_{1}^{3}+\cdots+x_{s-1}^{3}+yx_{s}^{3}=0}, respectively. Gauss proved that if q=p{q=p}, p≡1(mod3){p\equiv

Let 𝐹 be a non-Archimedean local field, and let 𝜎 be a non-trivial Galois involution with fixed field F0F_{0}. When the residue characteristic of F0F_{0} is odd, using the construction of cuspidal representations of classical groups by Stevens, we classify generic cuspidal representations of U⁢(2,1)⁢(F/F0)U(2,1)(F/F_{0}).

In this paper, we construct a class of new modules for the quantum group Uq⁢(s⁢l2)U_{q}(\mathfrak{sl}_{2}) which are free of rank 1 when restricted to C⁢[K±1]\mathbb{C}[K^{\pm 1}]. The irreducibility of these modules and submodule structure for reducible ones are determined. It is proved that any C⁢[K±1]\mathbb{C}[K^{\pm 1}]-free Uq⁢(s⁢l2)U_{q}(\mathfrak{sl}_{2})-module of rank 1 is isomorphic to one

This paper aims to introduce a construction technique of set-theoretic solutions of the Yang–Baxter equation, called strong semilattice of solutions . This technique, inspired by the strong semilattice of semigroups, allows one to obtain new solutions. In particular, this method turns out to be useful to provide non-bijective solutions of finite order. It is well-known that braces, skew braces and

The simplicial volume of oriented closed connected smooth manifolds that admit a non-trivial smooth S1{S^{1}}-action vanishes. In the present work, we prove a version of this result for the integral foliated simplicial volume of aspherical manifolds: The integral foliated simplicial volume of aspherical oriented closed connected smooth manifolds that admit a non-trivial smooth S1{S^{1}}-action vanishes

We show that, on compact Riemannian surfaces of nonpositive curvature, the generalized periods, i.e. the 𝜈-th order Fourier coefficients of eigenfunctions eλe_{\lambda} over a closed smooth curve 𝛾 which satisfies a natural curvature condition, go to 0 at the rate of O⁢((log⁡λ)-12)O((\log\lambda)^{-\frac{1}{2}}) in the high energy limit λ→∞\lambda\to\infty if 0<|ν|λ<1-δ0<\frac{\lvert\nu\rvert}{\lambda}<1-\delta

We find the complete integer solutions of the equation X2+Y2+Z2-4⁢X⁢Y-4⁢Y⁢Z+10⁢X⁢Z=1X^{2}+Y^{2}+Z^{2}-4XY-4YZ+10XZ=1. As an application, we prove that, for each solution (a,b,c)(a,b,c) such that a>0a>0, b-2⁢a>0b-2a>0 and (b-2⁢a)2≥4⁢a(b-2a)^{2}\geq 4a, there is a vector bundle 𝐸 on P3\mathbb{P}^{3} defined by a minimal linear resolution 0→OP3⁢(-2)a→OP3⁢(-1)b→OP3c→E→00\to\mathcal{O}_{\mathbb{P}^{3}

We find the number sk⁢(p,Ω)s_{k}(p,\Omega) of cuspidal automorphic representations of GSp⁢(4,AQ)\mathrm{GSp}(4,\mathbb{A}_{\mathbb{Q}}) with trivial central character such that the archimedean component is a holomorphic discrete series representation of weight k≥3k\geq 3, and the non-archimedean component at 𝑝 is an Iwahori-spherical representation of type Ω and unramified otherwise. Using the automorphic

Let Y be a smooth complete intersection of a quadric and a cubic in ℙn{\mathbb{P}^{n}}, with n even. We show that Y has a multiplicative Chow–Künneth decomposition, in the sense of Shen–Vial. As a consequence, the Chow ring of (powers of) Y displays K3-like behavior. As a by-product of the argument, we also establish a multiplicative Chow–Künneth decomposition for the resolution of singularities of

Article Frontmatter was published on March 1, 2021 in the journal Forum Mathematicum (volume 33, issue 2).

This paper studies the Prasad conjecture for the special orthogonal group SO3,3\mathrm{SO}_{3,3}. Then we use the local theta correspondence between Sp4\mathrm{Sp}_{4} and O⁢(V)\mathrm{O}(V) to study the Sp4\mathrm{Sp}_{4}-distinction problems over a quadratic field extension E/FE/F and dim⁡V=4\dim V=4 or 6. Thus we can verify the Prasad conjecture for a square-integrable representation of Sp4⁢(E)\mathrm{Sp}_{4}(E)

In this paper, we consider a finite-dimensional vector space 𝒫{{\mathcal{P}}} over the Galois field GF⁡(p){\operatorname{GF}(p)}, with p being an odd prime, and the family ℬkx{{\mathcal{B}}_{k}^{x}} of all k -sets of elements of 𝒫{\mathcal{P}} summing up to a given element x . The main result of the paper is the characterization, for x=0{x=0}, of the permutations of 𝒫{\mathcal{P}} inducing permutations

In this paper, our first goal is to rigorously define a Lévy process pinned at random time. Our second task is to establish the Markov property with respect to its completed natural filtration and thus with respect to the usual augmentation of the latter. The resulting conclusion is the right-continuity of completed natural filtration. Certain examples of such process are considered.

Let X and Y be pathwise connected and paracompact Hausdorff spaces equipped with free involutions T:X→X{T:X\to X} and S:Y→Y{S:Y\to Y}, respectively. Suppose that there exists a sequence (Xi,Ti)⁢⟶hi⁢(Xi+1,Ti+1) for ⁢1≤i≤k,(X_{i},T_{i})\overset{h_{i}}{\longrightarrow}(X_{i+1},T_{i+1})\quad\text{for }% 1\leq i\leq k, where, for each i , Xi{X_{i}} is a pathwise connected and paracompact Hausdorff space

The theory of R. Crowell on derived modules is approached within the theory of non-commutative differential modules. We also seek analogies to the theory of cotangent complex from differentials in the commutative ring setting. Finally, we give examples motivated from the theory of Galois coverings of curves.

We explicitly prove the quantum modularity of partial theta series with even or odd periodic coefficients. As an application, we show that the Kontsevich–Zagier series ℱt⁢(q){\mathscr{F}_{t}(q)} which matches (at a root of unity) the colored Jones polynomial for the family of torus knots T⁢(3,2t){T(3,2^{t})}, t≥2{t\geq 2}, is a weight 32{\frac{3}{2}} quantum modular form. This generalizes Zagier’s

The aim of this paper is to extend results from [A. Cañada, J. A. Montero and S. Villegas, Lyapunov inequalities for partial differential equations, J. Funct. Anal. 237 (2006), 1, 176–193] about Lyapunov-type inequalities for linear partial differential equations to nonlinear partial differential equations with 𝑝-Laplacian with zero Neumann or Dirichlet boundary conditions.

We use Levinson’s method and the work of Blomer and Harcos on the GL2\mathrm{GL}_{2} shifted convolution problem to prove that at least 6.96 % of the nontrivial zeros of the 𝐿-function of a GL2\mathrm{GL}_{2} automorphic form lie on the critical line.

The main result of this paper is to give that if b∈Lip⁡(ℝn){b\in\operatorname{Lip}(\mathbb{R}^{n})}, hj∈BMO⁡(ℝn){h_{j}\in\operatorname{BMO}(\mathbb{R}^{n})}, j=1,…,k{j=1,\ldots,k}, k∈ℤ+{k\in\mathbb{Z}^{+}} and w∈Ap{w\in A_{p}}, 1

This article addresses two characterizations of BMO⁢(ℝn){\mathrm{BMO}(\mathbb{R}^{n})}-type space via the commutators of Hardy operators with homogeneous kernels on Lebesgue spaces: (i) characterization of the central BMO⁢(ℝn){\mathrm{BMO}(\mathbb{R}^{n})} space by the boundedness of the commutators; (ii) characterization of the central BMO⁢(ℝn){\mathrm{BMO}(\mathbb{R}^{n})}-closure of Cc∞⁢(ℝn){C_

We apply Weyl 𝑛-algebras to prove formality theorems for higher Hochschild cohomology. We present two approaches: via propagators and via the factorization complex. It is shown that the second approach is equivalent to the first one taken with a new family of propagators we introduce.

We develop a theory of syzygies in equivariant cohomology for tori as well as p -tori and coefficients in 𝔽p{\mathbb{F}_{p}}. A noteworthy feature is a new algebraic approach to the partial exactness of the Atiyah–Bredon sequence, which also covers all instances considered so far.

By work of John Tate we can associate an epsilon factor with every multiplicative character of a local field. In this paper, we determine the explicit signs of the epsilon factors for symplectic type characters of K×{K^{\times}}, where K/F{K/F} is a wildly ramified quadratic extension of a non-Archimedean local field F of characteristic zero.

In this paper we show that, for topological dynamical systems with a dense set (in the weak topology) of periodic measures, a typical (in Baire's sense) invariant measure has, for each $q>0$, zero lower $q$-generalized fractal dimension. This implies, in particular, that a typical invariant measure has zero upper Hausdorff dimension and zero lower rate of recurrence. Of special interest is the full-shift

Abstract We classify commutative algebraic monoid structures on normal affine surfaces over an algebraically closed field of characteristic zero. The answer is given in two languages: comultiplications and Cox coordinates. The result follows from a more general classification of commutative monoid structures of rank 0, n - 1 n-1 or 𝑛 on a normal affine variety of dimension 𝑛.

Abstract The spectrum of the Laplacian operator on the positive theta line bundle over the quasi-torus reduces to eigenvalues π ⁢ ℓ {\pi\ell} , ℓ = 0 , 1 , … {\ell=0,1,\ldots{}} , which are called Landau levels. This paper discusses the coherent state transform for each eigenspace associated with a Landau level. We construct a unitary transform valid for each eigenspace. A concrete form of the inverse

Abstract In this paper, we prove a genuine analogue of the Wiener Tauberian theorem for L p , 1 ⁢ ( G ) {L^{p,1}(G)} ( 1 ≤ p < 2 {1\leq p<2} ), with G = SL ⁢ ( 2 , ℝ ) {G=\mathrm{SL}(2,\mathbb{R})} .

Abstract In this paper we show that how the representation theory of subcategories (of the category of modules over an Artin algebra) can be connected to the representation theory of all modules over some algebra. The subcategories dealing with are some certain subcategories of the morphism categories (including submodule categories studied recently by Ringel and Schmidmeier) and of the Gorenstein

Abstract Let G n {G_{n}} denote either the group SO ⁢ ( 2 ⁢ n + 1 , F ) {\mathrm{SO}(2n+1,F)} or Sp ⁢ ( 2 ⁢ n , F ) {\mathrm{Sp}(2n,F)} over a non-archimedean local field of characteristic different than two. We study parabolically induced representations of the form 〈 Δ 〉 ⋊ σ {\langle\Delta\rangle\rtimes\sigma} , where 〈 Δ 〉 {\langle\Delta\rangle} denotes the Zelevinsky segment representation of the

Abstract In this paper, we study hybrid subconvexity bounds for class group L-functions associated to quadratic extensions K / ℚ {K/\mathbb{Q}} (real or imaginary). Our proof relies on relating the class group L-functions to Eisenstein series evaluated at Heegner points using formulas due to Hecke. The main technical contribution is the uniform sup norm bound for Eisenstein series E ⁢ ( z , 1 / 2 +

Abstract We establish explicit Ichino’s formulae for the central values of the triple product L-functions with emphasis on the calculations for the real place. The key ingredient for our computations is Proposition 8 which generalizes a result in [P. Michel and A. Venkatesh, The subconvexity problem for GL 2 {\rm GL}_{2} , Publ. Math. Inst. Hautes Études Sci. 111 2010, 171–271]. As an application we

Abstract We develop a version of Ekedahl’s geometric sieve for integral quadratic forms of rank at least five. As one ranges over the zeros of such quadratic forms, we use the sieve to compute the density of coprime values of polynomials, and furthermore, to address a question about local solubility in families of varieties parameterised by the zeros.

Abstract Let ℛ {\mathcal{R}} be a finite valuation ring of order q r {q^{r}} . In this paper, we prove that for any quadratic polynomial f ⁢ ( x , y , z ) ∈ ℛ ⁢ [ x , y , z ] {f(x,y,z)\in\mathcal{R}[x,y,z]} that is of the form a ⁢ x ⁢ y + R ⁢ ( x ) + S ⁢ ( y ) + T ⁢ ( z ) {axy+R(x)+S(y)+T(z)} for some one-variable polynomials R , S , T {R,S,T} , we have | f ⁢ ( A , B , C ) | ≫ min ⁡ { q r , | A | ⁢

Abstract We consider iterated integrals of log ⁡ ζ ⁢ ( s ) {\log\zeta(s)} on certain vertical and horizontal lines. Here, the function ζ ⁢ ( s ) {\zeta(s)} is the Riemann zeta-function. It is a well-known open problem whether or not the values of the Riemann zeta-function on the critical line are dense in the complex plane. In this paper, we give a result for the denseness of the values of the iterated

Abstract We study the generalised Iwasawa invariants of ℤ p d {\mathbb{Z}_{p}^{d}} -extensions of a fixed number field K. Based on an inequality between ranks of finitely generated torsion ℤ p ⁢ [ [ T 1 , … , T d ] ] {\mathbb{Z}_{p}[\kern-2.133957pt[T_{1},\dots,T_{d}]\kern-2.133957pt]} -modules and their corresponding elementary modules, we prove that these invariants are locally maximal with respect

Abstract Recently, the problem of bounding the sup norms of L 2 {L^{2}} -normalized cuspidal automorphic newforms ϕ on GL 2 {\mathrm{GL}_{2}} in the level aspect has received much attention. However at the moment strong upper bounds are only available if the central character χ of ϕ is not too highly ramified. In this paper, we establish a uniform upper bound in the level aspect for general χ. If the

Journal Name: Forum Mathematicum Volume: 33 Issue: 1 Pages: i-iv

We study the number of integer solutions $(x,y,l)$ of an equation $F(x,y)=\Pi_K(l)$, where $F(x,y)$ is a homogeneous polynomial with integer coefficients and $\Pi_K(l)$ is a generalized factorial function over number fields. We show a necessary condition for the existence of infinitely many solutions. As a corollary, we obtain the finiteness of solution for $P(x)=l!$, where $P$ is decomposed into irreducible