arXiv subjects
Publications and source records attributed to Pratiksha Shingavekar.
Let $p \in \{3, 5\}$ and consider a cyclic $p$-extension $L/\mathbb{Q}$. We show that there exists an effective positive density of elliptic curves $ E $ defined over $ \mathbb{Q} $, ordered by height, that are diophantine stable in $ L $.
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, 2-Selmer ranks, and 3-Selmer ranks of the corresponding elliptic curves $E_D: y^2 = x^3 - D^2x$, where $D$ is a square-free number. Our study empirically examines the Heath-Brown heuristics, which predict the distribution of $2$-Selmer ranks as well as congruent numbers based on their residue modulo $8$. In particular, offering statistical insights into the proportion of numbers whose associated elliptic curves have positive rank. We provide a rigorous verification of Goldfeld's Conjecture in this context, analyzing the distribution of analytic ranks and demonstrating their alignment with the conjectured $50/50$ split for ranks $0$ and $1$. Furthermore, we explore the conjectural asymptotic distribution of $2-$ and $3$-torsion part of the Tate-Shafarevich group of these curves. Based on empirical evidence, we also suggest potential statistical distribution of $3$-Selmer and Mordell-Weil ranks. We also examine the averages of Frobenius traces and observe that they tend to zero without exhibiting any murmuration-like patterns. In addition to these number-theoretic analyses, we apply machine learning techniques to classify and predict congruent numbers, exploring the efficacy of computational methods in distinguishing congruent from non-congruent numbers based on the arithmetic properties of elliptic curves. This interdisciplinary approach blends advanced number theory with modern data science, providing empirical support for conjectures as well as discovery of new patterns.
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 $φ$. We give an upper and a lower bound on the rank of the $φ$-Selmer group of $E_{a,b}$ over $K:=\mathbb{Q}(ζ_3)$ in terms of the $3$-part of the ideal class group of certain quadratic extension of $K$. Using our bounds on the Selmer groups, we construct infinitely many curves in this family with arbitrary large $3$-Selmer rank over $K$ and no non-trivial $K$-rational point of order $3$. We also show that for a positive proportion of natural numbers $n$, the curve $E_{n,n}/\mathbb{Q}$ has root number $-1$ and $3$-Selmer rank $=1$.
Consider a Mordell curve $E_a:y^2=x^3+a$ with $a \in \mathbb Z$. These curves have a rational $3$-isogeny, say $φ$. We give an upper and a lower bound on the rank of the $φ$-Selmer group of $E_a$ over $\mathbb Q(ζ_3)$ in terms of the $3$-part of the ideal class group of certain quadratic extension of $\mathbb Q(ζ_3)$. Using our bounds on the Selmer groups, we prove some cases of the rational cube sum problem. Further, using these bounds, we give explicit families of the Mordell curves to show that for a positive proportion of $E_a$, ${\rm Sel}^3(E_{a}/\mathbb Q)=0$ (respectively ${\rm Sel}^3(E_{a}/\mathbb Q)$ has $\mathbb F_3$-rank $1$).
Let $a$ be an integer which is not of the form $n^2$ or $-3 n^2$ for $n\in \mathbb{Z}$. Let $E_a$ be the elliptic curve with rational $3$-isogeny defined by $E_a:y^2=x^3+a$, and $K:=\mathbb{Q}(μ_3)$. Assume that the $3$-Selmer group of $E_a$ over $K$ vanishes. It is shown that there is an explicit infinite set of cubefree integers $m$ such that the $3$-Selmer groups over $K$ of $E_{m^2 a}$ and $E_{m^4 a}$ both vanish. In particular, the ranks of these cubic twists are seen to be $0$ over $K$. Our results are proven by studying stability properties of $3$-Selmer groups in cyclic cubic extensions of $K$, via local and global Galois cohomological techniques.
We study the Iwasawa theory of $p$-primary Selmer groups of elliptic curves $E$ over a number field $K$. Assume that $E$ has additive reduction at the primes of $K$ above $p$. In this context, we prove that the Iwasawa invariants satisfy an analogue of the Riemann--Hurwitz formula. This generalizes a result of Hachimori and Matsuno. We apply our results to study rank stability questions for elliptic curves in prime cyclic extensions of $\mathbb{Q}$. These extensions are ordered by their absolute discriminant and we prove an asymptotic lower bound for the density of extensions in which the Iwasawa invariants as well as the rank of the elliptic curve is stable.
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 $\mathbb{Q}(μ_3)$ has dimension $\leq 1$. The results are proven by refining the strategy of Davenport--Heilbronn, by relating the statistics for integral binary cubic forms to the statistics for $3$-isogeny induced Selmer groups.
Given an integer n>1, it is a classical Diophantine problem that whether n can be written as a sum of two rational cubes. The study of this problem, considering several special cases of n, has a copious history that can be traced back to the works of Sylvester, Satgé, Selmer etc. and up to the recent works of Alpöge-Bhargava-Shnidman. In this article, we consider the cube sum problem for cube-free integers n which has two distinct prime factors none of which is 3.