On $2$-Selmer ranks of quadratic twists of elliptic curves

We study the $2$-Selmer ranks of elliptic curves. We prove that for an arbitrary elliptic curve $E$ over an arbitrary number field $K$, if the set $A_E$ of $2$-Selmer ranks of quadratic twists of $E$ contains an integer $c$, it contains all integers larger than $c$ and having the same parity as $c$. We also find sufficient conditions on $A_E$ such that $A_E$ is equal to $\mathbf{Z}_{\geq t_E}$ for some number $t_E$. When all points in $E[2]$ are rational, we give an upper bound for $t_E$.

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The author is very grateful to his advisor Karl Rubin for helpful suggestions and discussion. The author also thanks to the referee for careful reading the manuscript and many comments.

Received 23 November 2015

Accepted 10 May 2016

Published 29 December 2017