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Kolyvagin systems of rank 0 and the structure of the Selmer group of elliptic curves over abelian extensions

Alberto Angurel

TL;DR

The paper advances Kolyvagin-system theory by treating core rank zero, introducing a minimal relaxation of a Selmer structure at a single prime to recover a primitive Kolyvagin system whose local data control the Fitting ideals of the Selmer group. It then specializes to elliptic curves over $\mathbb{Q}$ and their finite abelian extensions, expressing the Galois structure of the $p$-primary Selmer group in terms of twisted Kurihara numbers and Kato’s Euler system; a key linkage is made through the Bloch–Kato dual exponential map and Mazur–Tate/Stickelberger data. A central contribution is the equivalence between the Iwasawa main conjecture for twists $f_\chi$ and the primitivity of the twisted Kato KS, enabling explicit decompositions of Selmer groups into character components with precise Fitting ideals. The results combine local-global dualities, Euler-Kolyvagin machinery, and modular-symbol data to provide computable descriptions of Selmer groups over abelian extensions, along with concrete numerical examples validating the theory and suggesting practical algorithms for Sha-rank and torsion in these settings.

Abstract

With the motivation to study the Selmer group af an elliptic curve, we improve the theory of Kolyvagin systems to describe the Fitting ideals of a Selmer group in the core rank zero situation. By relaxing a Selmer structure of rank zero at certain prime, we can construct an auxiliary Kolyvagin system whose localisation determines most of the Fitting ideals of the Selmer group, and all of them when the Galois representation is not self-dual. With this new theory, one can describe, in terms of the modular symbols, the Galois structure of the Selmer group of an elliptic curve $E/\mathbb{Q}$ over a finite abelian extension whose degree is coprime to $p$.

Kolyvagin systems of rank 0 and the structure of the Selmer group of elliptic curves over abelian extensions

TL;DR

The paper advances Kolyvagin-system theory by treating core rank zero, introducing a minimal relaxation of a Selmer structure at a single prime to recover a primitive Kolyvagin system whose local data control the Fitting ideals of the Selmer group. It then specializes to elliptic curves over and their finite abelian extensions, expressing the Galois structure of the -primary Selmer group in terms of twisted Kurihara numbers and Kato’s Euler system; a key linkage is made through the Bloch–Kato dual exponential map and Mazur–Tate/Stickelberger data. A central contribution is the equivalence between the Iwasawa main conjecture for twists and the primitivity of the twisted Kato KS, enabling explicit decompositions of Selmer groups into character components with precise Fitting ideals. The results combine local-global dualities, Euler-Kolyvagin machinery, and modular-symbol data to provide computable descriptions of Selmer groups over abelian extensions, along with concrete numerical examples validating the theory and suggesting practical algorithms for Sha-rank and torsion in these settings.

Abstract

With the motivation to study the Selmer group af an elliptic curve, we improve the theory of Kolyvagin systems to describe the Fitting ideals of a Selmer group in the core rank zero situation. By relaxing a Selmer structure of rank zero at certain prime, we can construct an auxiliary Kolyvagin system whose localisation determines most of the Fitting ideals of the Selmer group, and all of them when the Galois representation is not self-dual. With this new theory, one can describe, in terms of the modular symbols, the Galois structure of the Selmer group of an elliptic curve over a finite abelian extension whose degree is coprime to .
Paper Structure (22 sections, 66 theorems, 301 equations)

This paper contains 22 sections, 66 theorems, 301 equations.

Table of Contents

  1. Introduction
  2. Acknowledgments
  3. Galois cohomology
  4. Local Galois cohomology
  5. Selmer structures
  6. Kolyvagin systems
  7. General theory
  8. Lambda-adic Kolyvagin systems
  9. Euler systems
  10. Localisation of Kolyvagin systems
  11. Fitting ideals of Selmer groups of core rank zero
  12. Non-self-dual case
  13. Selmer group of an elliptic curve
  14. Main results
  15. Dual exponential map
  16. ...and 7 more sections

Key Result

Theorem 1.1

(Theorem th:kur_par) Under the assumptions Hffr-Hprimes in §sec:global, for every $i\in \mathbb Z_{\geq 0}$ we have that The equality holds when $i$ satisfies one of the following conditions:

Theorems & Definitions (154)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Remark 1.6
  • Theorem 1.7
  • Definition 2.1.1
  • Proposition 2.1.2
  • Definition 2.1.3
  • ...and 144 more