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On products of quadratic twists and ranks of elliptic curves over large fields
Author(s) -
Im Bo-Hae,
Lozano-Robledo Álvaro
Publication year - 2009
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/jdn048
Subject(s) - elliptic curve , rank (graph theory) , mathematics , twists of curves , field (mathematics) , abelian group , quadratic equation , extension (predicate logic) , pure mathematics , property (philosophy) , combinatorics , schoof's algorithm , geometry , quarter period , computer science , philosophy , epistemology , programming language
In this paper, we give examples of elliptic curves E / K over a number field K satisfying the property that there exist P 1 , P 2 ∈ K [ t ] such that the twistsE P 1, E P 2andE P 1 P 2are of positive rank over K ( t ). As a consequence of this result on twists, we show that for those elliptic curves E / K , and for each σ ∈ G a l ( K ¯ / K ) , the rank of E over the fixed field ( K ab ) σ under σ is infinite, where K ab is the maximal abelian extension of K .