Selmer ranks of quadratic twists of elliptic curves with partial rational two-torsion Article Swipe
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· 2016
· Open Access
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· DOI: https://doi.org/10.1090/tran/6744
· OA: W1534446261
This paper investigates which integers can appear as 2-Selmer ranks within the quadratic twist family of an elliptic curve $E$ defined over a number field $K$ with $E(K)[2] \simeq \mathbb {Z}/2\mathbb {Z}$. We show that if $E$ does not have a cyclic 4-isogeny defined over $K(E[2])$ with kernel containing $E(K)[2]$, then subject only to constant 2-Selmer parity, each non-negative integer appears infinitely often as the 2-Selmer rank of a quadratic twist of $E$. If $E$ has a cyclic 4-isogeny with kernel containing $E(K)[2]$ defined over $K(E[2])$ but not over $K$, then we prove the same result for 2-Selmer ranks greater than or equal to $r_2$, the number of complex places of $K$. We also obtain results about the minimum number of twists of $E$ with rank $0$ and, subject to standard conjectures, the number of twists with rank $1$, provided $E$ does not have a cyclic 4-isogeny defined over $K$.
