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Rational points of order 7
Author(s) -
Dummigan Neil
Publication year - 2008
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/bdn093
Subject(s) - mathematics , rational number , isogeny , conjecture , order (exchange) , elliptic curve , rational point , pure mathematics , product (mathematics) , class (philosophy) , point (geometry) , combinatorics , algebra over a field , mathematical analysis , geometry , finance , artificial intelligence , computer science , economics , algebraic number
For an elliptic curve over the rationals, optimal in its isogeny class, with a rational point of order 7 but L E (1) ≠ 0, we prove that 7 divides the product of the Tamagawa factors with the order of the Shafarevich–Tate group. This is a small consequence of the Birch and Swinnerton‐Dyer conjecture.