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Eisenstein cocycles for imaginary quadratic fields

Emmanuel Lecouturier, Romyar Sharifi, Sheng-Chi Shih, Jun Wang

TL;DR

The paper develops Eisenstein cocycles for imaginary quadratic fields by constructing a CM-elliptic-curve framework that yields maps from the first homology of Bianchi spaces to second $K$-groups of ray class fields. The approach relies on motivic complexes of products of CM elliptic curves, a refined Δ-module system for Hecke actions, and specialization to $ rak N$-torsion to produce Eisenstein maps on $H_1(Y_1( rak N))$. Key contributions include integrality results, parabolicity analyses via Borel–Serre boundary comparisons, and the demonstration that these maps become Eisenstein away from the level, with an explicit $p$-adic decomposition into χ-components. The construction generalizes earlier cyclotomic cases, providing a CM-elliptic-curve analogue of the Fukaya–Kato–Sharifi program in the Bianchi setting and offering a path toward a deeper understanding of how CM theory and $K$-theory interact with automorphic and arithmetic geometry.

Abstract

We construct Eisenstein cocycles for arithmetic subgroups of GL_2 of imaginary quadratic fields valued in second K-groups of products of two CM elliptic curves. We use these to construct maps from the first homology groups of Bianchi spaces to corresponding second K-groups of ray class fields and to verify the Eisenstein property of these maps for prime-to-level Hecke operators.

Eisenstein cocycles for imaginary quadratic fields

TL;DR

The paper develops Eisenstein cocycles for imaginary quadratic fields by constructing a CM-elliptic-curve framework that yields maps from the first homology of Bianchi spaces to second -groups of ray class fields. The approach relies on motivic complexes of products of CM elliptic curves, a refined Δ-module system for Hecke actions, and specialization to -torsion to produce Eisenstein maps on . Key contributions include integrality results, parabolicity analyses via Borel–Serre boundary comparisons, and the demonstration that these maps become Eisenstein away from the level, with an explicit -adic decomposition into χ-components. The construction generalizes earlier cyclotomic cases, providing a CM-elliptic-curve analogue of the Fukaya–Kato–Sharifi program in the Bianchi setting and offering a path toward a deeper understanding of how CM theory and -theory interact with automorphic and arithmetic geometry.

Abstract

We construct Eisenstein cocycles for arithmetic subgroups of GL_2 of imaginary quadratic fields valued in second K-groups of products of two CM elliptic curves. We use these to construct maps from the first homology groups of Bianchi spaces to corresponding second K-groups of ray class fields and to verify the Eisenstein property of these maps for prime-to-level Hecke operators.
Paper Structure (20 sections, 56 theorems, 137 equations)

This paper contains 20 sections, 56 theorems, 137 equations.

Table of Contents

  1. Introduction
  2. Background and the main theorem
  3. Outline of the construction
  4. Hecke actions on cohomology
  5. Module systems
  6. Hecke operators as correspondences
  7. Hecke operators attached to ideals
  8. $\Delta$-module systems with pushforwards
  9. $\mathrm{GL}_2$-cocycles for CM elliptic curves
  10. CM elliptic curves
  11. Motivic complexes of products of elliptic curves
  12. Eisenstein cocycles for products of CM curves
  13. A representative cocycle
  14. Specialized cocycles for ray class fields
  15. Pullback by $\mathcal{N}$-torsion
  16. ...and 5 more sections

Key Result

Theorem 1

Let $F$ be an imaginary quadratic field with integer ring $\mathcal{O}$, and let $\mathcal{N}$ be an ideal of $\mathcal{O}$ such that the canonical map $\mathcal{O}^{\times} \to (\mathcal{O}/\mathcal{N})^{\times}$ is injective. Let $Y_1(\mathcal{N})$ be the Bianchi space for $F$ with $\Gamma_1(\math that is Eisenstein away from $\mathcal{N}$, which is a specialization at $\mathcal{N}$-torsion of a

Theorems & Definitions (113)

  • Theorem
  • Definition 2.1.1
  • Definition 2.1.2
  • Proposition 2.1.3
  • proof
  • Proposition 2.1.4
  • proof
  • Proposition 2.1.5
  • proof
  • Definition 2.2.1
  • ...and 103 more