Congruence subgroups, elliptic cohomology, and the Eichler-Shimura map
type: Item
instance of: scholarly article
title: Congruence subgroups, elliptic cohomology, and the Eichler-Shimura map
zbMATH Open document ID: 0871.55013
DOI: 10.1016/0022-4049(95)00088-7
author: Thomas C. Ratliff
published in: Journal of Pure and Applied Algebra
publication date: 1996-07-03T00:00:00Z
review text: For \(\Gamma\subset \text{SL}_2 (\mathbb{Z})\) a congruence subgroup one can look at the Borel construction \(X_\Gamma= E\Gamma \times_\Gamma B^2 (\mathbb{Z}\times \mathbb{Z})\). Using a Serre-spectral-sequence argument one sees that there is interesting cohomology only in odd dimensions -- away from 2 and 3 torsion. Group cohomology then gives a purely algebraic description. Complex conjugation induces an action on \(X_\Gamma\) which produces a (stable) splitting of (the cohomology) of \(X_\Gamma\). Using results of [\textit{G. Shimura}, Introduction to the arithmetic theory of automorphic functions, Publ. Math. Soc. Japan 11 (1971; Zbl 0221.10029)] and putting the focus to real coefficients -- so that analysis becomes available -- for certain congruence subgroups the (--1) eigenspace of the cohomology can be calculated by means modular forms (Theorem 19, Proposition 27). The author also discusses interesting relations to elliptic cohomology [\textit{P. S. Landweber}, Elliptic cohomology and modular forms, Lect. Notes Math. 1326, 55-68 (1988; Zbl 0649.57022)].
Mathematics Subject Classification ID: 55R40
Mathematics Subject Classification ID: 11F11
Mathematics Subject Classification ID: 55P42
zbMATH DE Number: 896559
zbMATH Keywords: Borel construction
zbMATH Keywords: cohomology
zbMATH Keywords: splitting
zbMATH Keywords: modular forms
zbMATH Keywords: elliptic cohomology
MaRDI profile type: Publication
full work available at URL: 0022-4049(95)00088-7
OpenAlex ID: W2002701584
cites work: Group representations, \(\lambda\)-rings and the \(J\)-homomorphism
cites work: Hecke operators as operations in elliptic cohomology
cites work: Q4143495
cites work: On the Cohomology of Classifying Spaces of Torus Bundles and Automorphic Forms
cites work: Q3346401
cites work: Homological Properties of Comodules Over MU ∗ (MU) and BP ∗ (BP)
cites work: Q3794903
cites work: Q3979928
cites work: Q4875376
description: scientific article; zbMATH DE number 896559