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

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(Gamma) based on the congruence subgroup Gamma, which is related to the connective elliptic cohomology theory when Gamma = Gamma(0)(2) X(Gamma) has a stable summand X(Gamma-), and he proposes that the Eichler-Shimura map gives a real vector space isomorphism from the modular forms of Gamma of weight 2k + 2 to the real cohomology of X(Gamma-) in dimension 4k + 1 for Gamma = Gamma(0)(2). One of our main results is a proof of this claim when k > 0 for Gamma = Gamma(0)(p) and when k greater than or equal to 0 for Gamma = Gamma(0)(2) or Gamma = SL(2)(Z). Using obstruction theory, we are able to construct a non-trivial geometric map from Sigma(3)X(Gamma-) 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(Gamma) and of the spectrum representing the periodic theory introducd by Baker [2].