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The structure of Selmer groups and the Iwasawa main conjecture for elliptic curves

Chan-Ho Kim

TL;DR

This paper develops a refined Iwasawa-theoretic framework for elliptic curves by linking Kato’s Euler systems to a discrete family of Kurihara numbers derived from modular symbols. A central technical advance is proving that the non-triviality of Kato’s Kolyvagin system modulo height-one primes is equivalent to a localized Iwasawa main conjecture, enabling a structural description of Selmer groups for curves of arbitrary rank without requiring the full IMC. The authors then transfer the Mazur–Rubin Kolyvagin-system structure results to classical Selmer groups, yielding a refined Birch–Swinnerton-Dyer type picture: the corank and Tate–Shafarevich group structure are determined by the Kurihara numbers, and, under finiteness assumptions, explicit rank and $p$-part information follows. They also confirm Kurihara’s mod $p$ semi-local conjecture in their setting and develop a $p$-adic BSD perspective for Kato’s zeta elements, linking Iwasawa-module structure to $p$-adic invariants. The work provides both conceptual and computational gains by making the Selmer-group structure explicit via modular-symbol data, with numerical examples illustrating the effectiveness of the approach and potential applications to refined BSD-type conjectures and Iwasawa theory.

Abstract

We reveal a new and refined application of (a weaker statement than) the Iwasawa main conjecture for elliptic curves to the structure of Selmer groups of elliptic curves of arbitrary rank. For a large class of elliptic curves, we obtain the following arithmetic consequences. 1. Kato's Kolyvagin systems is non-trivial. It is the cyclotomic analogue of the Kolyvagin conjecture. 2. The structure of Selmer groups of elliptic curves over the rationals is completely determined in terms of certain modular symbols. It is a structural refinement of Birch and Swinnerton-Dyer conjecture. 3. The rank zero $p$-converse, the $p$-parity conjecture, and a new upper bound of the ranks of elliptic curves are obtained. 4. The conjecture of Kurihara on the semi-local description of mod $p$ Selmer groups is confirmed. 5. An application of the $p$-adic Birch and Swinnerton-Dyer conjecture to the structure of Iwasawa modules is discussed.

The structure of Selmer groups and the Iwasawa main conjecture for elliptic curves

TL;DR

This paper develops a refined Iwasawa-theoretic framework for elliptic curves by linking Kato’s Euler systems to a discrete family of Kurihara numbers derived from modular symbols. A central technical advance is proving that the non-triviality of Kato’s Kolyvagin system modulo height-one primes is equivalent to a localized Iwasawa main conjecture, enabling a structural description of Selmer groups for curves of arbitrary rank without requiring the full IMC. The authors then transfer the Mazur–Rubin Kolyvagin-system structure results to classical Selmer groups, yielding a refined Birch–Swinnerton-Dyer type picture: the corank and Tate–Shafarevich group structure are determined by the Kurihara numbers, and, under finiteness assumptions, explicit rank and -part information follows. They also confirm Kurihara’s mod semi-local conjecture in their setting and develop a -adic BSD perspective for Kato’s zeta elements, linking Iwasawa-module structure to -adic invariants. The work provides both conceptual and computational gains by making the Selmer-group structure explicit via modular-symbol data, with numerical examples illustrating the effectiveness of the approach and potential applications to refined BSD-type conjectures and Iwasawa theory.

Abstract

We reveal a new and refined application of (a weaker statement than) the Iwasawa main conjecture for elliptic curves to the structure of Selmer groups of elliptic curves of arbitrary rank. For a large class of elliptic curves, we obtain the following arithmetic consequences. 1. Kato's Kolyvagin systems is non-trivial. It is the cyclotomic analogue of the Kolyvagin conjecture. 2. The structure of Selmer groups of elliptic curves over the rationals is completely determined in terms of certain modular symbols. It is a structural refinement of Birch and Swinnerton-Dyer conjecture. 3. The rank zero -converse, the -parity conjecture, and a new upper bound of the ranks of elliptic curves are obtained. 4. The conjecture of Kurihara on the semi-local description of mod Selmer groups is confirmed. 5. An application of the -adic Birch and Swinnerton-Dyer conjecture to the structure of Iwasawa modules is discussed.
Paper Structure (42 sections, 52 theorems, 109 equations)

This paper contains 42 sections, 52 theorems, 109 equations.

Table of Contents

  1. Introduction
  2. Overview
  3. Kato's Kolyvagin systems and the Iwasawa main conjecture
  4. The non-triviality of Kato's Kolyvagin systems
  5. Modular symbols and Kurihara numbers
  6. The structure of Selmer groups and Kurihara numbers
  7. The conjecture of Kurihara
  8. Applications to Birch and Swinnerton-Dyer conjecture
  9. Iwasawa modules and a $p$-adic Birch and Swinnerton-Dyer conjecture
  10. Comparison with related conjectures and results
  11. BSD
  12. $p$-adic BSD
  13. Refined BSD type conjectures
  14. Refined Iwasawa theory
  15. Kolyvagin systems for elliptic curves
  16. ...and 27 more sections

Key Result

Theorem 1.1

Let $E$ be an elliptic curve over $\mathbb{Q}$ and $p \geq 5$ a semi-stable reduction prime for $E$ such that the mod $p$ representation $\overline{\rho} : \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \to \mathrm{Aut}_{\mathbb{F}_p}(E[p])$ is surjective. If the Iwasawa main conjecture (Conjecture If we assume the finiteness of $\textrm{\normalfont SH}(E/\mathbb{Q})[p^\infty]$ as well, then the

Theorems & Definitions (109)

  • Theorem 1.1: Corollaries \ref{['cor:main-ks']} and \ref{['cor:main-kn']}
  • Theorem 1.2: Corollary \ref{['cor:rank-zero-converse']}
  • Conjecture 1.3: IMC
  • Theorem 1.4
  • proof
  • Corollary 1.5
  • proof
  • Corollary 1.6
  • proof
  • Conjecture 1.7
  • ...and 99 more