Twists of elliptic curves |
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Authors: | K ONO |
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Institution: | (1) School of Mathematics, Institute for Advanced Study, Princeton, New Jersey, 08540;(2) Department of Mathematics, Penn State University, University Park, Pennsylvania, 16802 |
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Abstract: | If E is an elliptic curve over
, then let E(D) denote theD-quadratic twist of E. It is conjectured that there are infinitely many primesp for which E(p) has rank 0, and that there are infinitely many primes
for which
has positive rank. For some special curvesE we show that there is a set S of primes p with density
for which if
is a squarefree integer where
, then E(D) has rank 0. In particular E(p) has rank 0 for every
. As an example let E1 denote the curve
.Then its associated set of primes S1 consists of the prime11 and the primes p for which the order of the reduction ofX0(11) modulo p is odd. To obtain the general result we show for primes
that the rational factor of L(E(p),1) is nonzero which implies thatE(p) has rank 0. These special values are related to surjective
Galois representations that are attached to modularforms. Another example of this result is given, and we conclude with someremarks regarding the existence of positive rank prime twists via polynomialidentities. |
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Keywords: | elliptic curves modular forms |
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