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On the cyclotomic Iwasawa invariants of elliptic curves of rank one

Foivos Chnaras

TL;DR

The paper develops explicit numerical criteria to determine when the Iwasawa invariants $\mu_p$ and $\lambda_p$ of a rank-one elliptic curve over $\mathbb{Q}$ attain their minimal possible value in the cyclotomic $\mathbb{Z}_p$-extension, distinguishing good ordinary and supersingular primes. It introduces a Fermat-quotient based test via a specific multiple $Q$ of a Mordell–Weil generator, linking $a(Q)^{p-1}$ modulo $p^2$ (and $d(Q)$ modulo $p^2$ in the supersingular case) to $\mu_p$ and $\lambda_p$ through the $p$-adic regulator and $p$-adic BSD. The framework relies on p-adic regulator constructions (Mazur–Tate sigma function, Dieudonné module, signed $p$-adic $L$-functions) and the main conjectures for ordinary and supersingular cases, providing concrete, computable criteria and illustrating them with explicit examples. Additionally, the work discusses model-minimality issues, Tamagawa-scale corrections, and a Tamagawa-theorem that clarifies how minimal models affect local indices, with practical implications for computing the $p$-adic height and Iwasawa data. Overall, the results offer a practical, theory-driven approach to assessing when Rank $1$ curves exhibit minimal Iwasawa invariants in the cyclotomic tower, supported by computational evidence and explicit algorithms.

Abstract

Fix an elliptic curve $E$ over $\mathbb Q$ of rank $1$. In this paper, we develop an explicit numerical criterion, comparable to Gold's criterion, that determines whether the Iwasawa invariants of the elliptic curve at a good (ordinary or supersingular) prime attain their smallest possible value, i.e. whether $μ_p^\pm(E) +λ^\pm_p(E)=1$ in the supersingular case or $μ_p(E) + λ_p(E)=1$ in the ordinary case.

On the cyclotomic Iwasawa invariants of elliptic curves of rank one

TL;DR

The paper develops explicit numerical criteria to determine when the Iwasawa invariants and of a rank-one elliptic curve over attain their minimal possible value in the cyclotomic -extension, distinguishing good ordinary and supersingular primes. It introduces a Fermat-quotient based test via a specific multiple of a Mordell–Weil generator, linking modulo (and modulo in the supersingular case) to and through the -adic regulator and -adic BSD. The framework relies on p-adic regulator constructions (Mazur–Tate sigma function, Dieudonné module, signed -adic -functions) and the main conjectures for ordinary and supersingular cases, providing concrete, computable criteria and illustrating them with explicit examples. Additionally, the work discusses model-minimality issues, Tamagawa-scale corrections, and a Tamagawa-theorem that clarifies how minimal models affect local indices, with practical implications for computing the -adic height and Iwasawa data. Overall, the results offer a practical, theory-driven approach to assessing when Rank curves exhibit minimal Iwasawa invariants in the cyclotomic tower, supported by computational evidence and explicit algorithms.

Abstract

Fix an elliptic curve over of rank . In this paper, we develop an explicit numerical criterion, comparable to Gold's criterion, that determines whether the Iwasawa invariants of the elliptic curve at a good (ordinary or supersingular) prime attain their smallest possible value, i.e. whether in the supersingular case or in the ordinary case.

Paper Structure

This paper contains 27 sections, 23 theorems, 93 equations, 1 table.

Table of Contents

  1. Introduction
  2. The good ordinary case
  3. The supersingular case
  4. Non-minimal models
  5. Overview
  6. Examples
  7. p-adic Regulator
  8. p-adic sigma function
  9. The ordinary case
  10. Dieudonne Module
  11. The supersingular case
  12. Supersingular p-adic Regulator
  13. p-adic L-series and the p-adic Birch and Swinnerton-Dyer Conjecture
  14. The ordinary case
  15. The supersingular case
  16. ...and 12 more sections

Key Result

Theorem 1

Assume $E$ is an elliptic curve over a number field $K$ with good ordinary reduction at all the primes above $p$. Assume also that $\mathop{\mathrm{Sel}}\nolimits_{p^\infty}(E/K)$ is finite. Then where $N_v:=\# \tilde{E}(\mathbb F_v)$ the number of points of the elliptic curve $\mod v$ and where $c_v$ are the Tamagawa numbers. Finally, $f_E(T)$ is the characteristic polynomial of the Pontryagin d

Theorems & Definitions (39)

  • Theorem : Greenberg
  • Theorem 1.1
  • Definition 1.1
  • Theorem 1.2
  • Theorem : Gold's Criterion
  • Conjecture 1.1: Bernardi--Perrin-Riou
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.1
  • Proposition 1.1
  • ...and 29 more