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A NOTE ON HIGHER TWISTS OF ELLIPTIC CURVES

Published online by Cambridge University Press:  29 March 2010

MACIEJ ULAS
MACIEJ ULAS*
Affiliation:
Institute of Mathematics of the Polish Academy of Sciences, Śniadeckich 8, 00-956 Warszawa, Poland Jagiellonian University, Institute of Mathematics, Łojasiewicza 6, 30-348 Kraków, Poland e-mail: maciej.ulas@uj.edu.pl
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Abstract

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We show that for any pair of elliptic curves E1, E2 over ℚ with j-invariant equal to 0, we can find a polynomial D ∈ ℤ[u, v] such that the cubic twists of the curves E1, E2 by D(u, v) have positive rank over ℚ(u, v). We also prove that for any quadruple of pairwise distinct elliptic curves Ei, i = 1, 2, 3, 4, with j-invariant j = 0, there exists a polynomial D ∈ ℤ[u] such that the sextic twists of Ei, i = 1, 2, 3, 4, by D(u) have positive rank. A similar result is proved for quadruplets of elliptic curves with j-invariant j = 1, 728.

Keywords

11G05
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 52 , Issue 2 , May 2010 , pp. 371 - 381
Copyright
Copyright © Glasgow Mathematical Journal Trust 2010

References

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