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On Euler systems and Nekovář-Selmer complexes

Dominik Bullach, David Burns

Abstract

We develop a theory of Euler and Kolyvagin systems relative to the Nekovář--Selmer complexes of $p$-adic representations over local complete Gorenstein rings. This theory is both finer and requires fewer hypotheses than those of Mazur and Rubin over discrete valuation rings and of Sakamoto et al. over Gorenstein rings. In particular, given appropriate Euler systems, it allows one to study Selmer groups defined relative to Greenberg local conditions. As initial applications, we prove new cases of Kato's generalised Iwasawa main conjecture for both $\mathbb{Z}_p(a)$ and the $p$-adic Tate modules of rational elliptic curves, new cases of the Quillen--Lichtenbaum Conjecture, and a strengthening of existing results on the Birch--Swinnerton-Dyer Conjecture for CM elliptic curves.

On Euler systems and Nekovář-Selmer complexes

Abstract

We develop a theory of Euler and Kolyvagin systems relative to the Nekovář--Selmer complexes of -adic representations over local complete Gorenstein rings. This theory is both finer and requires fewer hypotheses than those of Mazur and Rubin over discrete valuation rings and of Sakamoto et al. over Gorenstein rings. In particular, given appropriate Euler systems, it allows one to study Selmer groups defined relative to Greenberg local conditions. As initial applications, we prove new cases of Kato's generalised Iwasawa main conjecture for both and the -adic Tate modules of rational elliptic curves, new cases of the Quillen--Lichtenbaum Conjecture, and a strengthening of existing results on the Birch--Swinnerton-Dyer Conjecture for CM elliptic curves.

Paper Structure

This paper contains 6 sections, 6 theorems, 5 equations.

Table of Contents

  1. Introduction
  2. The general theory
  3. Algebraic preliminaries
  4. Hypotheses on rings
  5. $G_n$-rings
  6. Complete Gorenstein rings

Key Result

Theorem 1

Let $E$ be a rational elliptic curve and $K$ a finite abelian extension of $\mathds{Q}$ for which the Hasse--Weil $L$-function $L(E / K, s)$ does not vanish at $s=1$. Fix a prime $p > 3$ such that Then the pair $(h^1 (E / K) (1), \mathds{Z}_p [\mathrm{Gal}( K / \mathds{Q})])$ validates one inclusion in Kato's generalised Iwasawa main conjecture. Further, this case of Kato's conjecture is fully va

Theorems & Definitions (12)

  • Theorem
  • Theorem
  • Theorem
  • Definition 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4: Sakamoto
  • proof
  • Remark 2.7
  • ...and 2 more