On 2-Selmer ranks of quadratic twists of elliptic curves

2015/11/24 by Myungjun Yu, Yu, Myungjun
Arts and Humanities · Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Historical Studies and Socio-cultural Analysis #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1511.07512

openalex publication_date 2015/11/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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_{≥ tE} for some number t_E. When all points in E[2] are rational, we give an upper bound for t_E.

Citations

• Finding large Selmer rank via an arithmetic theory of local constants

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