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2-Selmer Groups over Multiquadratic Extensions

Ross Paterson

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.

2-Selmer Groups over Multiquadratic Extensions

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.
Paper Structure (27 sections, 37 theorems, 124 equations, 7 tables)

This paper contains 27 sections, 37 theorems, 124 equations, 7 tables.

Table of Contents

  1. Introduction
  2. Main Results
  3. The Genus Theory Invariant
  4. Local sums over Ekedahl-type sets
  5. Outline
  6. Acknowledgements
  7. Notation and Conventions
  8. Reduction to Genus Theory Invariant
  9. Proof of Explicit \ref{['thm:INTRO MQ Selmer Avg']} (Assuming \ref{['thm:INTRO MQGenusTheoryAverage']})
  10. Averages of Local Sums on Ekedahl-type Sets
  11. The Ekedahl Sieve
  12. Average at a Fixed Prime
  13. Tail Estimate
  14. Local Sums
  15. Local Norm Indices
  16. ...and 12 more sections

Key Result

Theorem 2

Let $K/\QQ$ be a multiquadratic extension. Then

Theorems & Definitions (83)

  • Definition 1
  • Theorem 2: \ref{['cor:MQ Selmer Avg']}
  • Remark 3
  • Remark 4
  • Definition 5: Paterson2021*Definition 4.10
  • Remark 6
  • Theorem 7: \ref{['thm:MQGenusTheoryAverage']}
  • Corollary 8
  • Theorem 9: \ref{['thm:average of acceptable function over all lattice points']}
  • Definition 10
  • ...and 73 more