Disparity in Selmer ranks of quadratic twists of elliptic curves

2011/11/30 by Zev Klagsbrun, Barry Mazur, Karl Rubin · 27 citations
Mathematics · #Absolute Galois group #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Algebraic number field #Analytic Number Theory Research #Elliptic curve #Hessian form of an elliptic curve #Order (exchange) #Product (mathematics) #Quadratic equation #Rank (graph theory) #Twists of curves #math.NT #msc:11G05 #msc:11G40 #msc:14G25

paper · pdf · open access · doi:10.4007/annals.2013.178.1.5

published in Annals of Mathematics 178(1), 287-320 (Princeton University) · The proof of Example 7.11 in the published version of this paper was incorrect, because we applied [5, Theorem 1.3] incorrectly. The statement and proof here have been corrected. We thank Lilybelle Cowland Kellock for pointing out the error

openalex publication_date 2013/04/04 · openalex created_date 2016/06/24 · arxiv created 2022/10/09 · arxiv updated 2022/10/11 · openalex updated_date 2026/07/28

Abstract

We study the parity of 2-Selmer ranks in the family of quadratic twists of an arbitrary elliptic curve E over an arbitrary number field K. We prove that the fraction of twists (of a given elliptic curve over a fixed number field) having even 2-Selmer rank exists as a stable limit over the family of twists, and we compute this fraction as an explicit product of local factors. We give an example of an elliptic curve E such that as K varies, these fractions are dense in [0, 1]. More generally, our results also apply to p-Selmer ranks of twists of 2-dimensional self-dual F_p-representations of the absolute Galois group of K by characters of order p.

Citations

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

Cited by

• A Markov model for Selmer ranks in families of twists
• Strong Selmer companion elliptic curves
• p-twisted Selmer near-companion curves
• A rank zero p-converse to a theorem of Gross--Zagier, Kolyvagin and Rubin
• Elliptic curves of rank one over number fields
• Mordell--Lang and disparate Selmer ranks of odd twists of some superelliptic curves over global function fields
• Second descent and rational points on Kummer varieties

Related

• On 2-Selmer ranks of quadratic twists of elliptic curves
• Selmer Ranks of Quadratic Twists of Elliptic Curves with Partial Rational Two-Torsion
• On the Distribution of 2-Selmer Ranks within Quadratic Twist Families of Elliptic Curves with Partial Rational Two-Torsion
• A Markov model for Selmer ranks in families of twists
• Increasing the p-Selmer rank by twisting