Related papers: Geometric configurations in the ring of integers m…

In the present paper, we study various Erd\H{o}s type geometric problems in the setting of the integers modulo $q$, where $q=p^l$ is an odd prime power. More precisely, we prove certain results about the distribution of triangles and…

The Erd\H os-Falconer distance problem in $\mathbb{Z}_q^d$ asks one to show that if $E \subset \mathbb{Z}_q^d$ is of sufficiently large cardinality, then $\Delta(E) := \{(x_1 - y_1)^2 + \dots + (x_d - y_d)^2 : x, y \in E\}$ satisfies…

We explore variants of Erd\H os' unit distance problem concerning dot products between successive pairs of points chosen from a large finite subset of either $\mathbb F_q^d$ or $\mathbb Z_q^d,$ where $q$ is a power of an odd prime.…

We construct polynomial solutions modulo $p^s$ of the differential KZ and dynamical equations where $p$ is an odd prime number.

For the integer $ D=pq$ of the product of two distinct odd primes, we construct an elliptic curve $E_{2rD}:y^2=x^3-2rDx$ over $\mathbb Q$, where $r$ is a parameter dependent on the classes of $p$ and $q$ modulo 8, and show, under the parity…

We study a family of variants of Erd\H os' unit distance problem, concerning distances and dot products between pairs of points chosen from a large finite point set. Specifically, given a large finite set of $n$ points $E$, we look for…

Let $\ell\ge5$ be an odd prime and $j, s$ be positive integers. We study the distribution of the coefficients of integer and half-integral weight modular forms modulo odd positive integer $M$. As a consequence, we prove that for each…

This paper is a contribution to the description of some congruences on the odd prime factors of the class number of the number fields. An example of results obtained is: Let L/Q be a finite Galois solvable extension with [L:Q]=N, where N >…

We consider polynomial equations, or systems of polynomial equations, with integer coefficients, modulo prime numbers $p$. We offer an elementary approach based on a counting method. The outcome is a weak form of the Lang-Weil lower bound…

We show that substantially more than a quarter of the odd integers of the form $pq$ up to $x$, with $p,q$ both prime, satisfy $p\equiv q\equiv 3 \pmod 4$.

Fix m >= 1 and let E be an elliptic curve over Q with complex multiplication. We formulate conjectures on the density of primes p (congruent to one modulo m) for which the pth Fourier coefficient of E is an mth power modulo p; often these…

Let $p$ be an odd prime. For field extensions $L/\mathbb{Q}_p$ with Galois group isomorphic to the dihedral group $D_{2p}$ of order $2p$, we consider the problem of computing a basis of the associated order in each Hopf Galois structure and…

Let $q$ be an odd prime and $k$ be a natural number. We show that a finite subset of integers $S$ that does not contain any perfect $q^{th}$ power, contains a $q^{th}$ power residue modulo almost every natural numbers $N$ with at most $k$…

Let $q$ be a fixed odd prime. We show that a finite subset $B$ of integers, not containing any perfect $q^{th}$ power, contains a $q^{th}$ power modulo almost every prime if and only if $B$ corresponds to a blocking set (with respect to…

We study the following variant of the Erd\H{o}s distance problem. Given $E$ and $F$ a point sets in $\mathbb{R}^d$ and $p = (p_1, \ldots, p_q)$ with $p_1+ \cdots + p_q = d$ is an increasing partition of $d$ define $$ B_p(E,F)=\{(|x_1-y_1|,…

Let $q$ be a prime. We classify the odd primes $p\neq q$ such that the equation $x^2\equiv q\pmod{p}$ has a solution, concretely, we find a subgroup $\mathbb{L}_{4q}$ of the multiplicative group $\mathbb{U}_{4q}$ of integers relatively…

We study Erd\H os's distinct distances problem under $\ell_p$ metrics with integer $p$. We improve the current best bound for this problem from $\Omega(n^{4/5})$ to $\Omega(n^{6/7-\epsilon})$, for any $\epsilon>0$. We also characterize the…

Consider a $q$-Weil polynomial $f$ of degree $2g$. Using an equidistribution assumption that is too strong to be true, we define and compute a product of local relative densities of matrices in $\rm{GSp}_{2g}(\mathbb{F}_\ell)$ with…

Let F/k be a Galois extension of number fields with dihedral Galois group of order 2q, where q is an odd integer. We express a certain quotient of S-class numbers of intermediate fields, arising from Brauer-Kuroda relations, as a unit…

Let $q=p^{e}$ be a prime power, $\ell$ be a prime number different from $p$, and $n$ be a positive integer divisible by neither $p$ nor $\ell$. In this paper we define the $\ell$-adic $q$-cyclotomic system $\mathcal{PC}(\ell,q,n)$ with base…