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Statistics for Anticyclotomic Iwasawa Invariants of Elliptic Curves

Jeffrey Hatley, Debanjana Kundu, Anwesh Ray

Abstract

We study the average behaviour of the Iwasawa invariants for Selmer groups of elliptic curves, considered over anticyclotomic $\mathbb{Z}_p$-extensions in both the definite and indefinite settings. The results in this paper lie at the intersection of arithmetic statistics and Iwasawa theory.

Statistics for Anticyclotomic Iwasawa Invariants of Elliptic Curves

Abstract

We study the average behaviour of the Iwasawa invariants for Selmer groups of elliptic curves, considered over anticyclotomic -extensions in both the definite and indefinite settings. The results in this paper lie at the intersection of arithmetic statistics and Iwasawa theory.

Paper Structure

This paper contains 18 sections, 36 theorems, 107 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Background and Preliminaries
  3. The Euler characteristic
  4. Results for a fixed elliptic curve and varying prime
  5. Ridgdill curves
  6. Results for varying imaginary quadratic number field : Vanishing of the -invariant
  7. Results for varying imaginary quadratic number field : the rank zero case
  8. Results for varying elliptic curves
  9. Cohen--Lenstra Heuristics for Tate--Shafarevich Groups
  10. Indefinite Case
  11. Indefinite Case: Assumptions
  12. Auxiliary Selmer Groups
  13. A criterion for the triviality of
  14. Statistics
  15. Fix $E$ and $K$, vary $p$.
  16. ...and 3 more sections

Key Result

Lemma \oldthetheorem

Let $E_{/K}$ be an elliptic curve and assume that the Selmer group $\mathrm{Sel}_{p^{\infty}}(E/K_{\infty})$ is cotorsion as a $\Lambda$-module. Then $\lambda_p(E/K_{\infty})\geq \mathrm{rank}_{\mathbb{Z}} E(K)$.

Theorems & Definitions (85)

  • Conjecture \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • proof
  • Definition \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • proof
  • ...and 75 more