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On the 2‐part of the Birch and Swinnerton‐Dyer conjecture for quadratic twists of elliptic curves
- Source :
- Journal of the London Mathematical Society. 101:714-734
- Publication Year :
- 2019
- Publisher :
- Wiley, 2019.
-
Abstract
- In the present paper, we prove, for a large class of elliptic curves defined over $\mathbb{Q}$, 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 $\mathbb{Q}$ without complex multiplication.
- Subjects :
- Large class
Conjecture
Rank (linear algebra)
Mathematics::Number Theory
General Mathematics
010102 general mathematics
Complex multiplication
Birch and Swinnerton-Dyer conjecture
01 natural sciences
Combinatorics
Elliptic curve
Quadratic equation
0103 physical sciences
010307 mathematical physics
0101 mathematics
Mathematics
Subjects
Details
- ISSN :
- 14697750 and 00246107
- Volume :
- 101
- Database :
- OpenAIRE
- Journal :
- Journal of the London Mathematical Society
- Accession number :
- edsair.doi...........df248b248e1ee2c294e9295519a9891d