Integral structures on $p$-adic Fourier theory

TL;DR

This paper constructs an explicit $p$-adic Fourier transform, enabling the development of an integral basis for locally analytic functions on $ ext{O}_K$, and applies it to prove congruences of Bernoulli-Hurwitz numbers at supersingular primes.

Contribution

It provides a new explicit construction of the $p$-adic Fourier transform and an integral basis for locally analytic functions on $ ext{O}_K$, extending previous work.

Findings

Explicit $p$-adic Fourier transform construction.

Integral basis for $K$-locally analytic functions.

Proof of Bernoulli-Hurwitz number congruences at supersingular primes.

Abstract

In this article, we give an explicit construction of the $p$-adic Fourier transform by Schneider and Teitelbaum, which allows for the investigation of the integral property. As an application, we give a certain integral basis of the space of $K$-locally analytic functions on the ring of integers $\mathcal{O}_K$ for any finite extension $K$ of $\mathbb{Q}_p$, generalizing the basis constructed by Amice for locally analytic functions on $\mathbb{Z}_p$. We also use our result to prove congruences of Bernoulli-Hurwitz numbers at non-ordinary (i.e. supersingular) primes originally investigated by Katz and Chellali.