Congruence subgroups, elliptic cohomology, and the Eichler—Shimura map Original Research Article

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 SL2(image). One of the open questions in the subject has been to produce a geometric definition of these theories. Nishida [8] defines a spectrum XΓ based on the congruence subgroup Γ, which is related to the connective elliptic cohomology theory when Γ = Γ0(2). XΓ has a stable summand XΓ−, and he proposes that the Eichler-Shimura map gives a real vector space isomorphism from the modular forms of Γ of weight 2k + 2 to the real cohomology of XΓ− in dimension 4k + 1 for Γ = Γ0(2). One of our main results is a proof of this claim when k> 0 for Γ = Γ0(p) and when k ≥ 0 for Γ = Γ0(2) or Γ = SL2(image). Using obstruction theory, we are able to construct a non-trivial geometric map from ∑3XΓ− 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 XΓ and of the spectrum representing the periodic theory introducd by Baker [2].