On 2-Selmer ranks of quadratic twists of elliptic curves

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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 AE 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 suficient conditions on AE such that AE is equal to Z≥tE for some number tE. When all points in E[2] are rational, we give an upper bound for tE.

Original languageEnglish
Pages (from-to)1565-1583
Number of pages19
JournalMathematical Research Letters
Volume24
Issue number5
DOIs
Publication statusPublished - 2017

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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