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I am new to field of Elliptic curves. When I was seeing some papers related to this area I have come across elliptic curves having quadratic(some times cubic ) twists with zero rank. What is the consequence by this? Any help is great. Thank you in advance.

Bruno Joyal
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MKJ
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1 Answers1

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If $E/\mathbb Q$ is an elliptic curve, it is conjectured that one-half of the quadratic twists of $E$ have rank $0$, and one-half have rank $1$ ("one-half" being understood asymptotically, with twists ordered by conductor).

If $E/\mathbb Q$ is a CM curve admitting cubic (resp. quartic) twists, then it is conjectured that the proportion of cubic (resp. quartic) twists of $E$ with positive rank is $0$.

Conjecturally, what you have said should be true of any elliptic curve over $\mathbb Q$.

For reference, see this paper.

Bruno Joyal
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