$p$-adic Fourier theory in families

TL;DR

This paper develops a comprehensive $p$-adic Fourier theory for $p$-divisible groups, generalizing classical constructions, and applies it to create new families of quaternionic modular forms with overconvergence properties.

Contribution

It introduces a novel framework for $p$-adic Fourier transforms as isomorphisms of solid Hopf algebras, extending previous theories and constructing a global Eisenstein measure on the $p$-adic modular curve.

Findings

Constructed Fourier transforms as isomorphisms of solid Hopf algebras.

Extended Eisenstein measure to the $p$-adic modular curve.

Produced new families of quaternionic modular forms with overconvergence.

Abstract

We construct Fourier transforms relating functions and distributions on finite height $p$-divisible rigid analytic groups and objects in a dual category of $\mathbb{Z}_p$-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.