Related papers: The size of $2$-Selmer groups for the $\frac{\pi}{…

We study the parity of rank of $2$-${\rm Selmer}$ groups associated to $\pi/3$ and $2\pi/3$-congruent numbers. Our second result gives some positive densities about $\pi/3$ and $2\pi/3$ non-congruent numbers which can support the even part…

This article presents a comprehensive data-scientific investigation into the arithmetic statistics of congruent number elliptic curves, leveraging a dataset of square-free integers up to $3$ million. We analyze the Mordell-Weil ranks,…

We prove an asymptotic formula for the number of ${\rm SL}_3({\mathbb Z})$-equivalence classes of integral ternary cubic forms having bounded invariants. We use this result to show that the average size of the 3-Selmer group of all elliptic…

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…

Employing a geometric setting inspired by the proof of the Fundamental Lemma, we study some counting problems related to the average size of 2-Selmer groups and hence obtain an estimate for it.

The elliptic curve $E_k \colon y^2 = x^3 + k$ admits a natural 3-isogeny $\phi_k \colon E_k \to E_{-27k}$. We compute the average size of the $\phi_k$-Selmer group as $k$ varies over the integers. Unlike previous results of Bhargava and…

In this article, we produce infinite families of non-congruent numbers in the residue class of $1,2,$ and $3$ modulo $8$ with arbitrarily many triples or quadruples prime factors. In short, we use Monsky matrix to show that the $2$-Selmer…

Several discrete geometry problems are closely related to the arithmetic theory of elliptic curves defined on the rational fields $\mathbb{Q}$. In this paper we consider the $\theta$-congruent number for $\theta=\frac{\pi}{3}$ and…

We consider elliptic curves over global fields of positive characteristic with two distinct marked non-trivial rational points. Restricting to a certain subfamily of the universal one, we show that the average size of the 2-Selmer groups of…

Recently, Alp\"oge-Bhargava-Shnidman determined the average size of the $2$-Selmer group in the cubic twist family of any elliptic curve over $\mathbb{Q}$ with $j$-invariant $0$. We obtain the distribution of the $3$-Selmer groups in the…

We consider the family of elliptic curves $E_{a,b}:y^2=x^3+a(x-b)^2$ with $a,b \in \mathbb{Z}$. These elliptic curves have a rational $3$-isogeny, say $\varphi$. We give an upper and a lower bound on the rank of the $\varphi$-Selmer group…

We determine average sizes/bounds for the $2$- and $3$-Selmer groups in various families of elliptic curves with marked points, thus confirming several cases of the Poonen--Rains heuristics. As a consequence, we deduce that the average…

In 2016, Balakrishnan-Ho-Kaplan-Spicer-Stein-Weigandt produced a database of elliptic curves over $\mathbb{Q}$ ordered by height in which they computed the rank, the size of the $2$-Selmer group, and other arithmetic invariants. They…

We take an approach toward counting the number of n for which the curves E_n: y^2=x^3-n^2x have 2-Selmer groups of a given size. This question was also discussed in a pair of Invent. Math. papers by Roger Heath-Brown. We discuss the…

In this article, we prove that the average rank of elliptic curves over $\mathbb{Q}$, when ordered by height, is less than $1$ (in fact, less than $.885$). As a consequence of our methods, we also prove that at least four fifths of all…

We study the parity of 2-Selmer ranks in the family of quadratic twists of an arbitrary elliptic curve E over an arbitrary number field K. We prove that the fraction of twists (of a given elliptic curve over a fixed number field) having…

Under suitable hypotheses, we construct a probability measure on the set of closed maximal isotropic subspaces of a locally compact quadratic space over F_p. A random subspace chosen with respect to this measure is discrete with probability…

Rank computation of elliptic curves has deep relations with various unsolved questions in number theory, most notably in the congruent number problem for right-angled triangles. Similar relations between elliptic curves and Heron triangles…

Consider the family of elliptic curves $E_n:y^2=x^3+n^2$, where $n$ varies over positive cubefree integers. There is a rational $3$-isogeny $\phi$ from $E_n$ to $\hat{E}_n:y^2=x^3-27n^2$ and a dual isogeny $\hat{\phi}:\hat{E}_n\rightarrow…

Given a sixth power free integer $a$, let $E_a$ be the elliptic curve defined by $y^2=x^3+a$. We prove explicit results for the lower density of sixth power free integers $a$ for which the $3$-isogeny induced Selmer group of $E_a$ over…