Related papers: Lines generate the Picard groups of certain Fermat…

Let $\epsilon>0$. In this article we will present a deterministic algorithm which does the following. The input is a hyperelliptic curve $C$ of genus $g$ over a finite field $k$ of cardinality $q$ given by $y^2+h(x)y=f(x)$ such that the…

In this paper, we develop the python code for generating unit graph $G(\mathbb{Z}_{n}\oplus\mathbb{Z}_{m})$, for any integers $m\ \& \ n$. For any prime $r$, we construct $r$-ary linear codes from the incidence matrix of the unit graph…

Explicit bounds are given on the norms of prime ideals generating arbitrary subgroups of ray class groups of number fields, assuming the Extended Riemann Hypothesis. These are the first explicit bounds for this problem, and are…

We give a new general technique for constructing and counting number fields with an ideal class group of nontrivial m-rank. Our results can be viewed as providing a way of specializing the Picard group of a variety V over $\mathbb{Q}$ to…

We suggest a short proof of O.Benoist and O.Wittenberg theorem (arXiv:1907.10859) which states that for each real non-singular cubic hypersurface $X$ of dimension $\ge 2$ the real lines on $X$ generate the whole group $H_1(X(\Bbb R);\Bbb…

Let $E$ be a quadratic imaginary field and let $p$ be a prime which is inert in $E.$ We study three types of Picard modular surfaces in positive characteristic $p$ and the morphisms between them. The first Picard surface, denoted $S$,…

In this paper, it is proved that there is an arithmetic progression of positive integers such that each of which is expressible neither as $p+F_m$ nor as $q+L_n$, where $ p,q $ are primes, $ F_m $ denotes the $ m $-th Fibonacci number and $…

The dynamical system generated by the iterated calculation of the high order gaps between neighboring terms of a sequence of natural numbers is remarkable and only incidentally characterized at the boundary by the notable Proth-Glibreath…

Let $F[X]$ be the polynomial ring over a finite field $F$. It is shown that, for $n\geq 3$, the special linear group $SL_n(F[X])$ is boundedly generated by the elementary matrices.

We prove an analogue of the Brauer-Siegel theorem for Fermat surfaces over a finite field. Namely, letting $F_d$ be the Fermat surface of degree $d$ over $\mathbb{F}_q$ and $p_g(F_d)$ be its geometric genus, we consider the product of the…

We study lines on smooth cubic surfaces over the field of $p$-adic numbers, from a theoretical and computational point of view. Segre showed that the possible counts of such lines are $0,1,2,3,5,7,9,15$ or $27$. We show that each of these…

Let $K$ be an imaginary quadratic field and $\mathcal{O}_K$ be its ring of integers. Let $h_E$ be the Weber function on certain elliptic curve $E$ with complex multiplication by $\mathcal{O}_K$. We show that if $N$ ($>1$) is an integer…

We give an explicit set of generators for various natural subgroups of the real Cremona group Bir_R(P^2). This completes and unifies former results by several authors.

We construct a class of positive linear maps on matrix algebras. We find conditions when these maps are atomic, decomposable and completely positive. We obtain a large class of atomic positive linear maps. As applications in quantum…

Throughout this paper, all groups are finite. Let $\sigma =\{\sigma_{i} | i\in I \}$ be some partition of the set of all primes $\Bbb{P}$. If $n$ is an integer, the symbol $\sigma (n)$ denotes the set $\{\sigma_{i} |\sigma_{i}\cap \pi…

In the first part of this paper we give a survey of classical results on Kummer surfaces with Picard number 17 from the point of view of lattice theory. We prove ampleness properties for certain divisors on Kummer surfaces and we use them…

This article is the first in a series devoted to computing the class groups of real quadratic fields. We present a new relation between the class number and the index of unit groups. This relation generalizes Hilbert class field theory for…

It is known that infinitely many number fields and function fields of any degree $m$ have class number divisible by a given integer $n$. However, significantly less is known about the indivisibility of class numbers of such fields. While…

In this article, we prove that any complex smooth rational surface $X$ which has no automorphism of positive entropy has a finite number of real forms (this is especially the case if $X$ cannot be obtained by blowing up $\mathbb…

We construct two series of linear codes over finite ring $\mathbb{F}_{q}[x]/(x^2)$ and Galois ring $GR(p^2,m)$ respectively reaching the Griesmer bound. They derive two series of codes over finite field $\mathbb{F}_{q}$ by Gray map. The…