The congruence criterion for power operations in Morava E-theory

We prove a congruence criterion for the algebraic theory of power operations in Morava E-theory, analogous to Wilkerson's congruence criterion for torsion free lambda-rings. In addition, we provide a geometric description of this congruence criterion, in terms of sheaves on the moduli problem of deformations of formal groups and Frobenius isogenies.Comment: Revised version for publicatio

Modular Isogeny Complexes

We describe a vanishing result on the cohomology of a cochain complex associated to the moduli of chains of finite subgroup schemes on elliptic curves. These results have applications to algebraic topology, in particular to the study of power operations for Morava E-theory at height 2.Comment: Minor revisions for publicatio

A streamlined proof of Goodwillie's n-excisive approximation

We give a shorter proof of Lemma 1.9 from Goodwillie, "Calculus III", which is the key step in proving that the construction P_nF gives an n-excisive functor.Comment: Expanded introductio