Congruence subgroups, elliptic cohomology, and the Eichler—Shimura map

Abstract

In 1986 Landweber [7] introduced the connective and periodic elliptic cohomology theories whose coefficient rings can be interpreted as a ring of modular functions for certain congruence subgroups of SL 2( Z ). One of the open questions in the subject has been to produce a geometric definition of these theories. Nishida [8] defines a spectrum X Γ based on the congruence subgroup Γ, which is related to the connective elliptic cohomology theory when Γ = Γ 0(2). X Γ has a stable summand X Γ − , and he proposes that the Eichler-Shimura map gives a real vector space isomorphism from the modular forms of Γ of weight 2 k + 2 to the real cohomology of X Γ − in dimension 4 k + 1 for Γ = Γ 0(2). One of our main results is a proof of this claim when k > 0 for Γ = Γ 0( p) and when k ≥ 0 for Γ = Γ 0(2) or Γ = SL 2( Z ). Using obstruction theory, we are able to construct a non-trivial geometric map from ∑ 3 X Γ − to the 3-connected cover of the spectrum representing the connective theory which is an equivalence through dimension 4. We also produce a stable splitting of X Γ and of the spectrum representing the periodic theory introducd by Baker [2].