Selmer Ranks of Quadratic Twists of Elliptic Curves with Partial Rational Two-Torsion

2012/01/25 by Zev Klagsbrun, Klagsbrun, Zev · 1 citation
Mathematics · #11G05 #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1201.5408

openalex publication_date 2012/01/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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] = Z/2Z. We show that if E does not have a cyclic 4-isogeny defined over K(E[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 defined over K(E[2]) but not over K, then we prove the same result for 2-Selmer ranks greater than or equal to r₂, 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.

Cited by

• Selmer ranks under quadratic twists satisfying the Heegner hypothesis

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