On the 2‐part of the Birch and Swinnerton‐Dyer conjecture for quadratic twists of elliptic curves

Li Cai, Chao Li, Shuai Zhai

Type: Article

Publication Date: 2019-10-21

Citations: 4

DOI: https://doi.org/10.1112/jlms.12284

Abstract

In the present paper, we prove, for a large class of elliptic curves defined over Q, the existence of an explicit infinite family of quadratic twists with analytic rank 0. In addition, we establish the 2-part of the conjecture of Birch and Swinnerton-Dyer for many of these infinite families of quadratic twists. Recently, Xin Wan has used our results to prove for the first time the full Birch–Swinnerton-Dyer conjecture for some explicit infinite families of elliptic curves defined over Q without complex multiplication.

Locations

Journal of the London Mathematical Society - View - PDF
arXiv (Cornell University) - View - PDF

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Works Cited by This (19)

Action	Title	Year	Authors
+	How to do a 𝑝-descent on an elliptic curve	2003	Edward F. Schaefer Michael Stoll
+	Algorithms for Modular Elliptic Curves	1992	J. E. Cremona
+ PDF Chat	Selmer ranks of quadratic twists of elliptic curves with partial rational two-torsion	2016	Zev Klagsbrun
+	Iwasawa Main Conjecture for Supersingular Elliptic Curves	2014	Xin Wan
+	Kolyvagin's work for modular elliptic curves	1991	Benedict H. Gross
+	The arithmetic of elliptic curves	1974	John Tate
+	FINITENESS OF $ E(\mathbf{Q})$ AND $ \textrm{Ø}(E,\mathbf{Q})$ FOR A SUBCLASS OF WEIL CURVES	1989	V. A. Kolyvagin
+	PARABOLIC POINTS AND ZETA-FUNCTIONS OF MODULAR CURVES	1972	Ju I Manin
+ PDF Chat	Explicit Gross–Zagier and Waldspurger formulae	2014	Li Cai Jie Shu Ye Tian
+	The Conjecture of Birch and Swinnerton-Dyer	2016	J. Coates Andrew Wiles