For a prime p > 2, an integer a with gcd(a,p) = 1 and real 1 ≤ X,Y < p, we consider the set of points on the modular hyperbola ${}_{a,p}(X,Y)=(x,y):xy\equiv a\left(modp\right),1\le x\le X,1\le y\le Y$. We give asymptotic formulas for the average values ${\sum }_{\begin{array}{c}(\end{array}x,y){\in }_{a,p}(X,Y)x\ne y*}\phi \left(\right|x-y\left|\right)/|x-y|$ and ${\sum }_{\begin{array}{c}(\end{array}x,y){\in }_{a,p}(X,X)x\ne y*}\phi \left(\right|x-y\left|\right)$ with the Euler function φ(k) on the differences between the components of points of ${}_{a,p}(X,Y)$.

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 $X_1\left(N\right)/\mathbb{Q}$. ...

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 $4$.

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...

The purpose of this paper is to investigate efficient representations of the residue classes modulo $q$, by performing sum and product set operations starting from a given subset $A$ of ${\mathbb{Z}}_q$. We consider the case of very small sets $A$ and composite $q$ for which not much seemed known (nontrivial results were recently obtained when $q$ is prime or when log $\left|A\right|\sim logq$). Roughly speaking we show that all residue classes are obtained from a $k$-fold sum of an $r$-fold product set of $A$, where $r\ll logq$ and $logk\ll logq$, provided the...

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.

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 ${𝒫}_2$ denote a positive integer with at most $2$ prime factors, counted according to multiplicity. For integers $a$, $q$ such that $(a,q)=1$, let ${𝒫}_2(q,a)$ denote the least ${𝒫}_2$ in the arithmetic progression ${\{nq+a\}}_{n=1}^{\infty }$. It is proved that for sufficiently large $q$, we have $${𝒫}_2(q,a)\ll q^{1.825}.$$ This result constitutes an improvement upon that of J. Li, M. Zhang and Y. Cai (2023), who obtained ${𝒫}_2(q,a)\ll q^{1.8345}.$

Let ${𝒫}_r$ denote an almost-prime with at most $r$ prime factors, counted according to multiplicity. Suppose that $a$ and $q$ are positive integers satisfying $(a,q)=1$. Denote by ${𝒫}_2(a,q)$ the least almost-prime ${𝒫}_2$ which satisfies ${𝒫}_2\equiv a(modq)$. It is proved that for sufficiently large $q$, there holds $${𝒫}_2(a,q)\ll q^{1.8345}.$$ This result constitutes an improvement upon that of Iwaniec (1982), who obtained the same conclusion, but for the range $1.845$ in place of $1.8345$.