2025/07/07 by Andrew Graham, Graham, Andrew, Pol van Hoften +3
Mathematics · #11F85 #11S31 #14G22 #14G45 #Advanced Algebra and Geometry #Advanced Mathematical Identities #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT) #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2507.05374
openalex publication_date 2025/07/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We construct Fourier transforms relating functions and distributions on finite height p-divisible rigid analytic groups and objects in a dual category of ℤp-local systems with analyticity conditions. Our Fourier transforms are formulated as isomorphisms of solid Hopf algebras over arbitrary small v-stacks, and generalize earlier constructions of Amice and Schneider--Teitelbaum. We also construct compatible integral Fourier transforms for p-divisible groups and their dual Tate modules. As an application, we use the Weierstrass \wp-function to construct a global Eisenstein measure over the p-adic modular curve, extending previous constructions of Katz over the ordinary locus and at CM points, and show its generic fiber, the global Eisenstein distribution, gives rise to new families of quaternionic modular forms that overconverge from profinite sets in the rigid analytic supersingular locus.