The power operation structure on Morava

Abstract

The power operation structure on Morava E-theory of height 2 at the prime 3 YIFEI ZHUWe give explicit calculations of the algebraic theory of power operations for a specific Morava E -theory spectrum and its K(1)-localization. These power operations arise from the universal degree-3 isogeny of elliptic curves associated to the E -theory.55S12; 55N20, 55N34

Algebraic theories of power operations

“…At $$p=3$$, for Lubin–Tate spectra $$E$$ associated to certain elliptic curves, $$E$$‐power operations have been computed by Nendorf [59] and by Zhu [77]. The latter also discusses the power operation structure on $$L_{K(1)} E$$ for the height $$h=2$$ Lubin–Tate spectrum $$E$$ in question.…”

Elliptic Cohomology Is Unique Up to Homotopy

“…(1) From our choice of A , we have , where represents and represents . Moreover, both and are integral domains; see [Zhu14, Proposition 2.1] and [BO16, Theorem 1.1.1], respectively, where the following isomorphisms are constructed: It then follows that can be written in the following commutative diagram of graded rings: The ring A is Noetherian as it is finitely presented over , so both and are Noetherian integral domains. In particular, the completion maps are flat for .…”