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Arkabrata Ghosh

Publications and source records attributed to Arkabrata Ghosh.

11 recordsLinked to original sources
arXiv2026

Classification of the rank of a certain family of elliptic curves

In this article, we study the family of elliptic curves $E_{-2pq}: y^2=x^3-2pqx$, where $p$ and $q$ are distinct odd primes. Using a $2$-isogeny methods and some elementary techniques, we obtain explicit possibilities for the Mordell--Weil ranks, conditional on the Parity Conjecture. Moreover, in the rank-one case, we are also able to derive explicit conditions that are independent of the parity conjecture. Moreover, the main results depend only on the residue classes of $(p,q)$ modulo $8$ and the Legendre symbols $\legendre{p}{q}$.

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arXiv2026

Exact classification of elliptic curves $y^{2}=x^{3}-pqx$ with rank $0$ and trivial $\Sha[2]$

For the elliptic curves $E_{p,q}: y^{2}=x^{3}-pqx$ where $p$ and $q$ are distinct odd primes, we establish necessary and sufficient conditions under which rank$\,E_{p,q}(\mathbb{Q})$ and $\dim_{\mathbb{F}_{2}} \Sha \left( E_{p,q}/\bbQ \right)[2]$ are both $0$. We do so via a similar characterisation of when the Selmer groups associated with the degree-$2$ isogeny $\phi$ and its dual $\widehat{\phi}$ are both of minimal size, along with results about a cokernel that arises from a related exact sequence.

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arXiv2025

Class groups of imaginary biquadratic fields

We present two distinct families of imaginary biquadratic fields, each of which contains infinitely many members, with each member having large class groups. Construction of the first family involves elliptic curves and their quadratic twists, whereas to find the other family, we use a combination of elliptic and hyperelliptic curves. Two main results are used, one from Soleng and the other from Banerjee and Hoque.

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arXiv2025

Rank of the family of elliptic curves $y^2 = x^3- 5px$

This article considers the family of elliptic curves given by $E_{p}: y^2=x^3-5px$ and certain conditions on an odd prime $p$. More specifically, we have shown that if $p \equiv 7, 23 \pmod {40}$, then the rank of $E_{p}$ is zero for both $ \mathbb{Q} $ and $ \mathbb{Q}(i) $. Furthermore, if the prime $ p $ is of the form $ 40k_1 + 3 $ or $ 40k_2 + 27$, where $k_1, k_2 \in \mathbb{Z}$ such that $(5k_1+1)$ or $(5k_2 +4)$ are perfect squares, then the given family of elliptic curves has rank one over $\mathbb{Q}$ and rank two over $\mathbb{Q}(i)$. Moreover, if the prime $ p $ is of the form $ 40k_3 + 11 $ or $ 40k_4 + 19$ where $k_3 ~\text{and}~ k_4 \in \mathbb{Z}$ such that $(160k_3+49)$ or $(160k_4 + 81) $ are perfect squares, then the given family of elliptic curves has rank at least one over $\mathbb{Q}$ and rank at least two over $\mathbb{Q}(i)$.

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arXiv2025

On the Family of Elliptic Curves $y^2=x^3-5pqx$

This article considers the family of elliptic curves given by $E_{pq}: y^2=x^3-5pqx$ and certain conditions on odd primed $p$ and $q$. More specifically, we have proved that if $p \equiv 33 \pmod {40}$ and $ q \equiv 7 \pmod {40}$, then the rank of $E_{pq}$ is zero over both $ \mathbb{Q} $ and $ \mathbb{Q}(i) $. Furthermore, if the primes $ p $ and $q$ are of the form $ 40k + 33 $ and $ 40l + 27$, where $k,l \in \mathbb{Z}$ such that $(25k+ 5l +21)$ is a perfect square, then the given family of elliptic curves has rank one over $\mathbb{Q}$ and rank two over $\mathbb{Q}(i)$. Finally, we have shown that torsion of $E_{pq}$ over $\mathbb{Q}$ is isomorphic to $ \mathbb{Z}/ 2\mathbb{Z}$.

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arXiv2024

The interplay between additive and symmetric large sets and their combinatorial applications

The study of symmetric structures is a new trend in Ramsey theory. Recently in [7], Di Nasso initiated a systematic study of symmetrization of classical Ramsey theoretical results, and proved a symmetric version of several Ramsey theoretic results. In this paper Di Nasso asked if his method could be adapted to find new non-linear Diophantine equations that are partition regular [7,Final remarks (4)]. By analyzing additive, multiplicative, and symmetric large sets, we construct new partition regular equations that give a first affirmative answer to this question. A special case of our result shows that if $P$ is a polynomial with no constant term then the equation $x+P(y-x)=z+w+zw$, where $y\neq x$ is partition regular. Also we prove several new monochromatic patterns involving additive, multiplicative, and symmetric structures. Throughout our work, we use tools from the Algebra of the Stone-\v{C}ech Compactifications of discrete semigroups.

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arXiv2024

Solution of the Diophantine equation $x^2 + p^k=y^n$

The main aim of this article is to find all solutions of the Diophantine equation $x^2 + p^k=y^n$ where $p \equiv 1 \pmod 4$, $\frac{p-1}{3}$ is a perfect square and the class number of $\mathbb{Z}[\sqrt{-p}]$ is $2$. In this article, I used a method involving prime factorization and class numbers which is different from using congruent number argument which is widely used in this type of problem.

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arXiv2024

On the family of elliptic curves $y^2=x^3-m^2x + (pqr)^2$

In this article, we consider a family of elliptic curves defined by $E_{m}: y^2= x^3 -m^2 x + (pqr)^2 $ where $m $ is a positive integer and $p, q, ~\text{and}~ r$ are distinct odd primes and study the torsion as well the rank of $E_{m}(\mathbb{Q})$. More specifically, we proved that if $m \not \equiv 0 \pmod{3}, m \not \equiv 0 \pmod{4} ~\text{and}~ m \equiv 2 \pmod {2^{k}}$ where $k \geq 5$, then the torsion subgroup of $E_{m}(\mathbb{Q})$ is trivial and lower bound of the $\mathbb{Q}$ rank of this family of elliptic curves is $2$.

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arXiv2023

General Solution of the Diophantine equation involving Mersenne Prime

In this article, I study and solve the exponential Diophantine equation $M_p^{x} + (M_q + 1)^{y}= (lz)^2$ where $M_p$ and $M_q$ are Mersenne primes, $l$ is a prime number, and $x,y$, and $z$ are non-negative integers. Several illustrations are presented as well as cases where no solution of the given Diophantine equation is present.

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arXiv2023

The Prym variety of a dilated double cover of metric graphs

We calculate the volume of the tropical Prym variety of a harmonic double cover of metric graphs having non-trivial dilation. We show that the tropical Prym variety behaves discontinuously under deformations of the double cover that change the number of connected components of the dilation subgraph.

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