arXiv subjects

Vinodkumar Ghale

Publications and source records attributed to Vinodkumar Ghale.

6 recordsLinked to original sources

arXiv2023

Quadratic Twists of Heronian Elliptic Curves with Arbitrarily Large 2-Selmer Rank

This paper explicitly determines the 2-Selmer group of the family of elliptic curves arising from Heron triangles of area 2mn. By analyzing the quadratic twists induced by powers of 2(varying m), we show precisely how the 2-Selmer group transforms under such twisting. For any squarefree odd integer n, the 2-Selmer rank is determined by the parity of m, the congruence class of q = (n2 + 1)/2 (mod 8), and the prime factorizations of the factors n1, n2 of n = n1n2. These results yield explicit formulas for the 2-Selmer rank and provide an explicit upper bound on the Mordell-Weil rank. In particular, this generalizes the family studied in [1] to arbitrary m, constructing explicit families of Heronian elliptic curves with prescribed and arbitrarily large 2-Selmer rank.

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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 $\theta$ such that $\tan \frac{\theta}{2} = n^{-1}$ and $n^{2}+1=2q$ for some prime $q$.

math.NT↗

arXiv2023

A Diophantine Criterion for the Shafarevich-Tate Groups of Elliptic Curves from Heron Triangles

The solvability of Diophantine quartic equations is a contemporary area of interest due to its connection with generalized Fermat's equation. In this work, we are interested in the integer solutions of a similar Diophantine equation p u^2 = v^2 + w^2. For a particular form of u, v, and w, we prove that the elliptic curves E_p: y^2 = x(x-1)(x+p^2), which arise from Heron triangles, for primes p = 1 (mod 8) where q = (p^2+1)/2 is also prime, exhibit a sharp dichotomy based on the solution of the aforementioned Diophantine equation: either rank(E_p(Q)) = 2 with trivial Shafarevich-Tate group or rank = 0 with III(E_p/Q)[2] = (Z/2Z)^2.

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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.

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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.

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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 $\tan \frac{\theta}{2}$ for one of the angles $\theta$. We also discuss the relation of such triangles with the solutions of certain Diophantine equations.

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