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arXiv · 1511.07512

On 2-Selmer ranks of quadratic twists of elliptic curves

Myungjun Yu
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Abstract

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 $\Z_{\ge t_E}$ 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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Myungjun Yu. 2016-01-27. On 2-Selmer ranks of quadratic twists of elliptic curves. https://arxiv.org/abs/1511.07512

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