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Quadratic twists of elliptic curves
Author(s) -
Coates John,
Li Yongxiong,
Tian Ye,
Zhai Shuai
Publication year - 2015
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/pdu059
Subject(s) - mathematics , elliptic curve , quadratic equation , conjecture , twists of curves , lemma (botany) , prime (order theory) , pure mathematics , supersingular elliptic curve , combinatorics , schoof's algorithm , quarter period , geometry , ecology , poaceae , biology
The paper generalizes, for a wide class of elliptic curves defined over Q , the celebrated classical lemma of Birch and Heegner about quadratic twists with prime discriminants, to quadratic twists by discriminants having any prescribed number of prime factors. In addition, it proves stronger results for the family of quadratic twists of the modular elliptic curveX 0 ( 49 ) , including showing that there is a large class of explicit quadratic twists whose complex L ‐series does not vanish at s = 1 , and for which the full Birch–Swinnerton‐Dyer conjecture is valid.