BSD Invariants and Murmurations of Elliptic Curves
The University of Arizona（亚利桑那大学）
Dane Wachs
备注：
Corrected version, 21 pages, 5 figures, 6 tables. The main claim of v1-v2 (that the Tate-Shafarevich group carries information on Frobenius traces beyond the other BSD invariants taken together) is withdrawn; corrections are listed in Section 7.3 and the ancillary file CORRECTIONS_V1.md
We study how the murmurations of elliptic curves over $\mathbb{Q}$ depend on the invariants in the Birch-Swinnerton-Dyer formula, using the Frobenius traces at all primes up to $10^5$ for one curve in each isogeny class of conductor below $10^5$. In these data, at the murmuration scale the rank-$0$ profile depends on the central value $L(E,1)$, in its shape and in the position of its sign changes. Through the series for $L(E,1)$, the covariance of $a_p$ with $L(E,1)$ is exactly a sum of covariances of $a_p$ with the coefficients $a_n$ weighted by a factor that depends on $n$ and the conductor; numerically, in this conductor range, the terms with $n\le50$ reproduce its shape (correlation $0.998$, largest deviation $0.29$ against a largest absolute covariance of $3.03$). Empirically, for $q=3$, $5$, $7$ and $11$ the unweighted covariance of $a_p$ with $a_q$ has nearly the shape of the rank-$0$ profile dilated by a factor near $q$; for $q=2$ the agreement is weaker. Weighting by the Tamagawa product changes the rank-$0$ profile substantially on $0.025<y\le0.125$; after standardisation to the deciles of $\log L(E,1)$, the change summed over these bins is 13% of the unstandardised one (17% for sums of absolute values), and in the lowest bin the standardised change is larger than the unstandardised one and of opposite sign. This version replaces the earlier versions (v1 and v2); their claim that the Tate-Shafarevich group carries information about Frobenius traces beyond the other invariants taken together cannot hold, because its analytic order is defined by the BSD formula, and it is withdrawn.
Murmurations, Mestre--Nagao sums, and Convolutional Neural Networks for elliptic curves
University of Redlands（雷德兰兹大学）；Massachusetts Institute of Technology（麻省理工学院）；Center for Communications Research（通信研究中心）；University of Connecticut（康涅狄格大学）；Korea Institute for Advanced Study（韩国高等研究院）；Brown University（布朗大学）；University of Westminster（威斯敏斯特大学）；Northeastern University（东北大学）；Harvard University（哈佛大学）
Shanghai Institute for Mathematics and Interdisciplinary Sciences (SIMIS)（上海数学与交叉学科研究院）；Research Institute of Intelligent Complex Systems, Fudan University（复旦大学智能复杂系统研究院）
Constructing non-trivial elements of the Shafarevich-Tate group of an Abelian Variety over a Number Field
Birch and Swinnerton-Dyer（BSD）猜想的第二部分根据曲线的其他可计算不变量给出了椭圆曲线的Shafarevich-Tate群的阶的猜想公式。Cremona和Mazur提出了一个理论，该理论通常可以用来验证BSD猜想，通过构造一条椭圆曲线的Shafarevich-Tate群的非平凡元素，借助于一条环境曲线的Mordell-Weil群。本文推广了Cremona和Mazur的工作，给出了数域上阿贝尔簇的Shafarevich-Tate群可构造的精确条件.然后，我们给出了我们的一般结果的一个扩展，提供了新的理论证据的BSD猜想。
Main conjectures for non-CM elliptic curves at good ordinary primes
Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；Beijing Institute of Mathematical Sciences and Applications（北京应用数学与计算科学研究所）；Yau Mathematical Sciences Center, Tsinghua University（清华大学丘成桐数学科学中心）
Machine Learning meets Number Theory: The Data Science of Birch-Swinnerton-Dyer
University of Copenhagen（哥本哈根大学）；City, University of London（伦敦大学城市学院）；The Alan Turing Institute（艾伦·图灵研究所）；University of Oxford（牛津大学）；NanKai University（南开大学）
