相关论文: BSD Invariants and Murmurations of Elliptic Curves

We compute the first murmurations for elliptic curves over function fields F_q(t): oscillatory patterns in average Frobenius traces that separate rank-0 from rank-1 curves, with z-scores up to 256. For the family E_D: y^2 = x^3 + x + D(t)…

Empirical analysis is often the first step towards the birth of a conjecture. This is the case of the Birch-Swinnerton-Dyer (BSD) Conjecture describing the rational points on an elliptic curve, one of the most celebrated unsolved problems…

We prove that over function fields F_q(t), the Tate-Shafarevich group |Sha| is an invariant of the cyclotomic type of the L-polynomial, so that |Sha|-stratified murmuration densities reduce to type-weighted densities with no within-type…

The second part of the Birch and Swinnerton-Dyer (BSD) conjecture gives a conjectural formula for the order of the Shafarevich-Tate group of an elliptic curve in terms of other computable invariants of the curve. Cremona and Mazur initiated…

Recent work of Burungale-Skinner-Tian-Wan established the first infinite families of quadratic twists of non-CM elliptic curves over $\mathbb{Q}$ for which the strong Birch-Swinnerton-Dyer (BSD) conjecture holds. Building on their results,…

This is an investigation into the possible existence and consequences of a Birch-Swinnerton-Dyer-type formula for L-functions of elliptic curves twisted by Artin representations. We translate expected properties of L-functions into purely…

Let $ E $ be an elliptic curve defined over a number field, the conjecture of Birch and Swinnerton-Dyer (BSD, for short) asserts a deep relation between the group $ E(K) $ of rational points and the $ L-$function $ L(E/K, s)$ of $ E $ at $…

We prove that there exist infinitely many pairs of non-isomorphic elliptic curves over $\mathbb{Q}$ sharing the same BSD invariants -- including their Mordell--Weil groups, Tate--Shafarevich groups, Tamagawa numbers, regulators, and real…

Recently, all Birch and Swinnerton-Dyer invariants, except for the order of Sha, have been computed for all curves of genus 2 contained in the L-functions and Modular Forms Database. This report explains the improvements made to the…

We develop the theory and algorithms necessary to be able to verify the strong Birch--Swinnerton-Dyer Conjecture for absolutely simple modular abelian varieties over $\mathbf{Q}$. We apply our methods to all 28 Atkin--Lehner quotients of…

Let $E$ be an elliptic curve over $\mathbb{Q}$. Let $\underline{III}(E)$ be a certain group of equivalence classes of homogeneous spaces of $E$ called its Tate-Shafarevich group. We show in this paper that this group has finite cardinality…

He, Lee, Oliver, and Pozdnyakov~\cite{HLOP} have empirically observed that the average of the $p$th coefficients of the $L$-functions of elliptic curves of particular ranks in a given range of conductors $N$ appears to approximate a…

We study elliptic curves of the form $x^3+y^3=2p$ and $x^3+y^3=2p^2$ where $p$ is any odd prime satisfying $p\equiv 2\bmod 9$ or $p\equiv 5\bmod 9$. We first show that the $3$-part of the Birch-Swinnerton-Dyer conjecture holds for these…

Let $E/\mathbb{Q}$ be an optimal elliptic curve of analytic rank zero. It follows from the Birch and Swinnerton-Dyer conjecture for elliptic curves of analytic rank zero that the order of the torsion subgroup of $E/\mathbb{Q}$ divides the…

We propose a novel derived cohomological framework for the Birch and Swinnerton-Dyer (BSD) conjecture for elliptic curves. In our approach, local arithmetic data are encoded in derived sheaves which, when glued via a mapping cone…

Let $E/\mathbb{Q}$ be an elliptic curve with conductor $N=N_+N_-$, where $N_+$ and $N_-$ are coprime and $N_-$ is squarefree. Let $D$ be a positive fundamental discriminant satisfying the modified Heegner hypothesis with respect to…

Let $E/\mathbb{Q}$ be an elliptic curve. We study the behavior of the Tate--Shafarevich group of $E$ under quadratic extensions $\mathbb{Q}(\sqrt{D})/\mathbb{Q}$. By analyzing the cokernel of the restriction map, without assuming the…

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…

The structure of the Tate-Shafarevich groups of a class of elliptic curves over global function fields is determined. These are known to be finite abelian groups from the monograph [1] and hence they are direct sums of finite cyclic groups…

We review the recent proof of the N.Takahashi's conjecture on genus $0$ Gromov-Witten invariants of $(\mathbb{P}^2, E)$, where $E$ is a smooth cubic curve in the complex projective plane $\mathbb{P}^2$. The main idea is the use of the…