Spectral moduli problems for level structures and an integral Jacquet-Langlands dual of Morava E-theory

2025/08/31 by Xiao-Yan Ma, Yifei Zhu, Ma, Xuecai +1
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Number Theory (math.NT) #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2509.00690

openalex publication_date 2025/08/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Given an E-infinity ring spectrum R, with motivation from chromatic homotopy theory, we define relative effective Cartier divisors for a spectral Deligne-Mumford stack over Spet(R) and prove that, as a functor from connective R-algebras to topological spaces, it is representable. This enables us to solve various moduli problems of level structures on spectral abelian varieties, overcoming difficulty at primes dividing the level. In particular, we obtain higher-homotopical refinement for finite levels of a Lubin-Tate tower as E-infinity ring spectra, which generalizes Morava, Hopkins, Miller, Goerss, and Lurie's spectral realization of the deformation ring at the ground level. Moreover, passing to the infinite level and then descending along the equivariantly isomorphic Drinfeld tower, we obtain a Jacquet-Langlands dual to the Morava E-theory spectrum, along with homotopy fixed point spectral sequences dual to those studied by Devinatz and Hopkins. These serve as potential tools for computing higher-periodic homotopy types from pro-etale cohomology of p-adic general linear groups.

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Related

• ell-adic topological Jacquet-Langlands duality
• The Morava E-theory of Eilenberg-Mac Lane spaces
• The Tate spectrum of the higher real K-theories at height n=p-1
• There aren't that many Morava E-theories
• Level structures on p-divisible groups from the Morava E-theory of abelian groups