arXiv subjects

Debopam Chakraborty

Publications and source records attributed to Debopam Chakraborty.

10 recordsLinked to original sources

arXiv2026

Two infinite families of elliptic curves with Mordell-Weil rank at least $3$

In this paper, we consider two infinite parametric families of elliptic curves defined over $\mathbb{Q}$ given by the equations $E_{a,b} : y^{2} = x^{3} - a^{2}x + b^{2}$ and $E^{\prime}_{a,b} : y^{2} = x^{3} - a^{2}x + b^{6}$, where $a,b \in \mathbb{N}$ satisfy certain mild conditions. We prove that the torsion group of $E_{a,b}(\mathbb{Q})$ is trivial and the Mordell-Weil ranks of both $E_{a,b}(\mathbb{Q})$ and $E^{\prime}_{a,b}(\mathbb{Q})$ are at least $3$ for infinitely many choices of $a$ and $b$ by using the Néron-Tate height of a rational point and by exploiting the unit group of the ring of integers of $\mathbb{Q}(\sqrt{3})$. This is an extension of the results of Brown-Myres and Fujita-Nara where lower bounds of the ranks were provided under the assumption that $a = 1$ or $b = 1$. Also, our families of elliptic curves vastly generalize the curves recently investigated by Hatley and Stack.

math.NT↗

arXiv2026

On the Mordell-Weil rank and $2$-Selmer group of a family of elliptic curves

We consider the parametric family of elliptic curves over $\mathbb{Q}$ of the form $E_{m} : y^{2} = x(x - n_{1})(x - n_{2}) + t^{2}$, where $n_{1}$, $n_{2}$ and $t$ are particular polynomial expressions in an integral variable $m$. In this paper, we investigate the torsion group $E_{m}(\mathbb{Q})_{\rm{tors}}$, a lower bound for the Mordell-Weil rank $r({E_{m}})$ and the $2$-Selmer group ${\rm{Sel}}_{2}(E_{m})$ under certain conditions on $m$. This extends the previous works done in this direction, which are mostly concerned with the Mordell-Weil ranks of various parametric families of elliptic curves.

math.NT↗

arXiv2025

Some bi-quadratic Pólya fields and large Pólya groups of compositum of simplest cubic and quintic fields

The Pólya group $Po(K)$ of an algebraic number field $K$ is the subgroup of the ideal class group $Cl_{K}$ generated by the ideal classes of the products of prime ideals of the same norm. If $Po(K)$ is trivial, then the number field $K$ is said to be a Pólya field. In this article, we furnish three families $\mathbb{Q}(\sqrt{p},\sqrt{qrs})$, $\mathbb{Q}(\sqrt{2p},\sqrt{qrs})$ and $\mathbb{Q}(\sqrt{2p},\sqrt{2qrs})$ of bi-quadratic Pólya fields $K$ involving prime numbers $p,q,r$ and $s$ that satisfy certain quadratic residue conditions. It is worthwhile to note that in each of the fields, exactly five primes ramify in $K/\mathbb{Q}$ and this is the maximum possible number of ramified primes in a Pólya field over $\mathbb{Q}$. Towards the end of the paper, we discuss about large Pólya groups of the compositums of Shank's cubic fields and Lehmer's quintic fields and prove that there are infinitely many such fields with index $1$.

math.NT↗

arXiv2024

Bi-quadratic Pólya fields with five distinct ramified primes

For an algebraic number field $K$, the Pólya group of $K$, denoted by $Po(K),$ is the subgroup of the ideal class group $Cl_{K}$ generated by the ideal classes of the products of prime ideals of same norm. The number field $K$ is said to be Pólya if $Po(K)$ is trivial. Motivated by several recent studies on the group $Po(K)$ when $K$ is a totally real bi-quadratic field, we investigate the same with five distinct odd primes ramifying in $K/\mathbb{Q}$. This extends the previous results on this problem, where the number of distinct ramified primes was at most four.

math.NT↗

arXiv2023

Large Pólya groups in simplest cubic fields and consecutive bi-qaudratic fields

The Pólya group of an algebraic number field is the subgroup generated by the ideal classes of the products of prime ideals of equal norm inside the ideal class group. Inspired by a recent work on consecutive quadratic fields with large class numbers by Cherubini et al., we extend the notion of {\it consecutiveness} of number fields to certain parametric families of cyclic cubic fields and bi-quadratic fields and address the question of the existence of infinitely many such consecutive fields with large Pólya groups. This extends a recent result of the second author and Saikia for totally real bi-quadratic fields.

math.NT↗

arXiv2023

Heronian elliptic curves and the size of the $2$-Selmer group

A generalization of the congruent number problem is to find positive integers $n$ that appear as the areas of Heron triangles. Selmer group of a congruent number elliptic curve has been studied quite extensively. Here, we look into the $2$-Selmer group structure for Heronian elliptic curves associated with Heron triangles of area $n$ and one of the angle $θ$ such that $\tan \fracθ{2} = n^{-1}$ and $n^{2}+1=2q$ for some prime $q$.

math.NT↗

arXiv2022

Class number divisibility of $\mathbb{Q}(\sqrt{3p}, \sqrt{m-21pn^{2}})$ constructed from elliptic curves of $2$-Selmer rank exactly $1$

The class number divisibility problem for number fields is one of the classical problems in algebraic number theory, which originated from Gauss' class number conjectures. The relation between the points on an elliptic curve and class number divisibility of a number field has been explored through the works of various mathematicians. Here, we explicitly construct an unramified abelian extension of a bi-quadratic field generated from points of a certain type of elliptic curve. Moreover, showing the $2$-Selmer rank of the said elliptic curve as $1$, we also construct an infinite family of bi-quadratic fields of even class number.

math.NT↗

arXiv2022

Construction of an infinite family of elliptic curves of 2-selmer rank 1 from heron triangles

Given any positive integer n, it is well known that there always exist triangles with rational sides a, b and c such that the area of the triangle is n. Assuming finiteness of the Shafarevich-Tate group, we first construct a family of infinitely many Heronian elliptic curves of rank exactly 1 from Heron triangles of a certain type. We also explicitly produce a separate family of infinitely many Heronian elliptic curves with 2-Selmer rank lying between 1 and 3.

math.NT↗

arXiv2021

Heron triangles and a family of elliptic curves with rank zero

Given any positive integer $n$, it is well-known that there always exists a triangle with rational sides $a,b$ and $c$ such that the area of the triangle is $n$. For a given prime $p \not \equiv 1$ modulo $8$ such that $p^{2}+1=2q$ for a prime $q$, we look into the possibility of the existence of the triangles with rational sides with $p$ as the area and $\frac{1}{p}$ as $\tan \fracθ{2}$ for one of the angles $θ$. We also discuss the relation of such triangles with the solutions of certain Diophantine equations.

math.NT↗

arXiv2019

On a conjecture Mordell

A conjecture of Mordell states that if $p$ is a prime and $p$ is congruent to $3$ mod $4$, then $p$ does not divide $y$ where $(x,y)$ is the fundamental solution to $x^{2}-py^{2}=1$. The conjecture has been verified for primes not exceeding $10^{7}$. In this article, we show that Mordell's conjecture holds for four conjecturally infinite families of primes.

math.NT↗