Congruence subgroups, elliptic cohomology, and the Eichler-Shimura map
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)].
scientific article; zbMATH DE number 896559