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Mazur's main conjecture at Eisenstein primes

Francesc Castella, Giada Grossi, Christopher Skinner

Abstract

Let $E/\mathbb{Q}$ be an elliptic curve, let $p>2$ be a prime of good reduction for $E$, and assume that $E$ admits a rational $p$-isogeny with kernel $\mathbb{F}_p(φ)$. In this paper we prove the cyclotomic Iwasawa main conjecture for $E$, as formulated by Mazur in 1972, when $φ\vert_{G_p}\neq 1,ω$, where $G_p$ is a decomposition group at $p$ and $ω$ is the Teichmüller character. Our proof is based on a study of the anticyclotomic Iwasawa theory of $E$ over an imaginary quadratic field $K$ in which $p$ splits, and a congruence argument exploiting the cyclotomic Euler system of Beilinson--Flach classes.

Mazur's main conjecture at Eisenstein primes

Abstract

Let be an elliptic curve, let be a prime of good reduction for , and assume that admits a rational -isogeny with kernel . In this paper we prove the cyclotomic Iwasawa main conjecture for , as formulated by Mazur in 1972, when , where is a decomposition group at and is the Teichmüller character. Our proof is based on a study of the anticyclotomic Iwasawa theory of over an imaginary quadratic field in which splits, and a congruence argument exploiting the cyclotomic Euler system of Beilinson--Flach classes.
Paper Structure (37 sections, 37 theorems, 224 equations)

This paper contains 37 sections, 37 theorems, 224 equations.

Table of Contents

  1. Introduction
  2. Some ideas from the proof
  3. Application to the $p$-part of the Birch--Swinnerton Dyer formula
  4. Further applications and relation to previous works
  5. Outline of the paper
  6. Acknowledgements
  7. $p$-adic $L$-functions
  8. Cyclotomic $p$-adic $L$-function
  9. Two-variable $p$-adic $L$-function, I
  10. Anticyclotomic $p$-adic $L$-function
  11. Two-variable $p$-adic $L$-function, II
  12. Twists and imprimitive $p$-adic $L$-functions
  13. Selmer groups
  14. Selmer structures
  15. Discrete coefficients
  16. ...and 22 more sections

Key Result

Theorem 1

Let $E/\mathbb{Q}$ be an elliptic curve, and let $p>2$ be a prime of good reduction for $E$. Suppose that $p$ is Eisenstein with $\phi\vert_{G_p}\neq 1,\omega$, where $G_p\subset G_\mathbb{Q}$ is a decomposition group at $p$. Then $\mathfrak{X}_{\rm ord}(E/\mathbb{Q}_\infty)$ is $\Lambda_\mathbb{Q}$ and hence Mazur's main conjecture holds.

Theorems & Definitions (81)

  • Theorem 1
  • Conjecture 2: Perrin-Riou
  • Theorem 3
  • Theorem 4
  • proof
  • Remark 1.2.1
  • Theorem 2.1.1
  • proof
  • Theorem 2.2.1
  • proof
  • ...and 71 more