2-Selmer Groups over Multiquadratic Extensions

2024/01/16 by Ross Paterson, Paterson, Ross
Arts and Humanities · Mathematics · #11G05 (Primary) 11G07 #11N36 #11N45 #11R45 (Secondary) #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.2401.08828

openalex publication_date 2024/01/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let K be a multiquadratic number field. We investigate the average dimension of 2-Selmer groups over K for the family of all elliptic curves over the rational numbers (ordered by height). We give upper and lower bounds for this average. In the special case of quadratic fields, these bounds are arbitrarily close for a positive proportion of K. Our bounds are achieved by studying the genus theory invariant for 2-Selmer groups over such fields, whose average we similarly bound and, in many cases, determine. We make use of a variant of the Ekedahl sieve for local sums, which we present in appropriate generality for further applications.

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