Let $H$ denote a finite index subgroup of the modular group $\Gamma$ and let $\rho$ denote a finite-dimensional complex representation of $H.$ Let $M\left(\rho \right)$ denote the collection of holomorphic vector-valued modular forms for $\rho$ and let $M\left(H\right)$ denote the collection of modular forms on $H$. Then $M\left(\rho \right)$ is a $\mathbb{Z}$-graded $M\left(H\right)$-module. It has been proven that $M\left(\rho \right)$ may not be projective as a $M\left(H\right)$-module. We prove that $M\left(\rho \right)$ is Cohen-Macaulay as a $M\left(H\right)$-module. We also explain how to apply this result to prove that if $M\left(H\right)$ is a polynomial ring, then...

In this paper some results on direct summands of Goldie extending elements are studied in a modular lattice. An element $a$ of a lattice $L$ with $0$ is said to be a Goldie extending element if and only if for every $b\le a$ there exists a direct summand $c$ of $a$ such that $b\wedge c$ is essential in both $b$ and $c$. Some characterizations of decomposition of a Goldie extending element in a modular lattice are obtained.

Suppose that $f$ is a primitive Hecke eigenform or a Mass cusp form for ${\Gamma }_0\left(N\right)$ with normalized eigenvalues ${\lambda }_f\left(n\right)$ and let $X>1$ be a real number. We consider the sum $${𝒮}_k\left(X\right):=\sum _{n<X}\sum _{n=n_1,n_2,...,n_k}{\lambda }_f\left(n_1\right){\lambda }_f\left(n_2\right)...{\lambda }_f\left(n_k\right)$$ and show that ${𝒮}_k\left(X\right){\ll }_{f,ϵ}{X}^{1-3/\left(2\right(k+3\left)\right)+ϵ}$ for every $k\ge 1$ and $ϵ>0$. The same problem was considered for the case $N=1$, that is for the full modular group in Lü (2012) and Kanemitsu et al. (2002). We consider the problem in a more general setting and obtain bounds which are better than those obtained by the classical result of Landau (1915) for $k\ge 5$. Since the result is valid...

Let $H\subseteq Z\subseteq 2^{\omega }$. For n ≥ 2, we prove that if Selivanovski measurable functions from $2^{\omega }$ to Z give as preimages of H all Σₙ¹ subsets of $2^{\omega }$, then so do continuous injections.

We give a full description of locally finite $2$-groups $G$ such that the normalized group of units of the group algebra $FG$ over a field $F$ of characteristic $2$ has exponent $4$.

Let $N$ be an odd and squarefree positive integer divisible by at least two relative prime integers bigger or equal than $4$. Our main theorem is an asymptotic formula solely in terms of $N$ for the stable arithmetic self-intersection number of the relative dualizing sheaf for modular curves $X_1\left(N\right)/\mathbb{Q}$. From our main theorem we obtain an asymptotic formula for the stable Faltings height of the Jacobian $J_1\left(N\right)/\mathbb{Q}$ of $X_1\left(N\right)/\mathbb{Q}$, and, for sufficiently large $N$, an effective version of Bogomolov’s conjecture for $X_1\left(N\right)/\mathbb{Q}$. ...

Let $F$ be an imaginary quadratic field and $𝒪$ its ring of integers. Let $𝔫\subset 𝒪$ be a non-zero ideal and let $p\&gt;5$ be a rational inert prime in $F$ and coprime with $𝔫$. Let $V$ be an irreducible finite dimensional representation of ${\overline{𝔽}}_p\left[{\mathrm{GL}}_2\left({𝔽}_{p^2}\right)\right]$. We establish that a system of Hecke eigenvalues appearing in the cohomology with coefficients in $V$ already lives in the cohomology with coefficients in ${\overline{𝔽}}_p\otimes de{t}^e$ for some $e\ge 0$; except possibly in some few cases.

Let $M(X,𝒜)$ ($M^{*}(X,𝒜)$) be the $f$-ring of all (bounded) real-measurable functions on a $T$-measurable space $(X,𝒜)$, let $M_K(X,𝒜)$ be the family of all $f\in M(X,𝒜)$ such that $\mathrm{coz}\left(f\right)$ is compact, and let $M_{\infty }(X,𝒜)$ be all $f\in M(X,𝒜)$ that $\{x\in X:|f\left(x\right)|\ge \frac{1}{n}\}$ is compact for any $n\in \mathbb{N}$. We introduce realcompact subrings of $M(X,𝒜)$, we show that $M^{*}(X,𝒜)$ is a realcompact subring of $M(X,𝒜)$, and also $M(X,𝒜)$ is a realcompact if and only if $(X,𝒜)$ is a compact measurable space. For every nonzero real Riesz map $\varphi :M(X,𝒜)\to ℝ$, we prove that there is an element $x_0\in X$ such that $\varphi \left(f\right)=f\left(x_0\right)$ for every $f\in M(X,𝒜)$ if $(X,𝒜)$ is a compact measurable space....

The aim of this paper is to characterize the $L^p-L^q$ boundedness of two classes of integral operators from $L^p(𝒰,\mathrm{d}{V}_{\alpha })$ to $L^q(𝒰,\mathrm{d}{V}_{\beta })$ in terms of the parameters $a$, $b$, $c$, $p$, $q$ and $\alpha$, $\beta$, where $𝒰$ is the Siegel upper half-space. The results in the presented paper generalize a corresponding result given in C. Liu, Y. Liu, P. Hu, L. Zhou (2019).

The concept of a Goldie extending module is generalized to a Goldie extending element in a lattice. An element $a$ of a lattice $L$ with $0$ is said to be a Goldie extending element if and only if for every $b\le a$ there exists a direct summand $c$ of $a$ such that $b\wedge c$ is essential in both $b$ and $c$. Some properties of such elements are obtained in the context of modular lattices. We give a necessary condition for the direct sum of Goldie extending elements to be Goldie extending. Some characterizations...

Let ⟨X,Y⟩ be a duality pair of M-spaces X,Y of measurable functions from Ω ⊂ ℝ ⁿ into ${ℝ}^d$. The paper deals with Y-weak cluster points ϕ̅ of the sequence $\varphi (·,z_j\left(·\right))$ in X, where $z_j:\Omega \to {ℝ}^m$ is measurable for j ∈ ℕ and $\varphi :\Omega \times {ℝ}^m\to {ℝ}^d$ is a Carathéodory function. We obtain general sufficient conditions, under which, for some negligible set $A_{\varphi }$, the integral $I(\varphi ,{\nu }_x):={\int }_{{ℝ}^m}\varphi (x,\lambda )d{\nu }_x\left(\lambda \right)$ exists for $x\in \Omega \setminus A_{\varphi }$ and $\varphi ̅\left(x\right)=I(\varphi ,{\nu }_x)$ on $\Omega \setminus A_{\varphi }$, where $\nu ={{\nu }_x}_{x\in \Omega }$ is a measurable-dependent family of Radon probability measures on ${ℝ}^m$.

We prove an interpolatory estimate linking the directional Haar projection $P^{\left(\epsilon \right)}$ to the Riesz transform in the context of Bochner-Lebesgue spaces $L^p(ℝⁿ;X)$, 1 < p < ∞, provided X is a UMD-space. If ${\epsilon }_{i₀}=1$, the result is the inequality $\left|\right|P^{\left(\epsilon \right)}{u\left|\right|}_{L^p(ℝⁿ;X)}{\le C\left|\right|u\left|\right|}_{L^p(ℝⁿ;X)}^{1/}\left|\right|R_{i₀}u{\left|\right|}_{L^p(ℝⁿ;X)}^{1-1/}$, (1) where the constant C depends only on n, p, the UMD-constant of X and the Rademacher type of $L^p(ℝⁿ;X)$. In order to obtain the interpolatory result (1) we analyze stripe operators $S_{\lambda }$, λ ≥ 0, which are used as basic building blocks to dominate the directional Haar projection....

Let $f$, $g$ and $h$ be three distinct primitive holomorphic cusp forms of even integral weights $k_1$, $k_2$ and $k_3$ for the full modular group $\Gamma =\mathrm{SL}(2,\mathbb{Z})$, respectively, and let ${\lambda }_f\left(n\right)$, ${\lambda }_g\left(n\right)$ and ${\lambda }_h\left(n\right)$ denote the $n$th normalized Fourier coefficients of $f$, $g$ and $h$, respectively. We consider the cancellations of sums related to arithmetic functions ${\lambda }_g\left(n\right)$, ${\lambda }_h\left(n\right)$ twisted by ${\lambda }_f\left(n\right)$ and establish the following results: $$\sum _{n\le x}{\lambda }_f\left(n\right){\lambda }_g{\left(n\right)}^i{\lambda }_h{\left(n\right)}^j{\ll }_{f,g,h,\epsilon }{x}^{1-1/2^{i+j}+\epsilon }$$ for any $\epsilon >0$, where $1\le i\le 2$, $j\ge 5$ are any fixed positive integers.