A Fourier transform for Banach–Colmez spaces

Abstract

We define and study an $\ell$-adic Fourier transform for a relative version of Banach–Colmez spaces (over a perfectoid space which is not necessarily a geometric point), which can be thought of as some analytic analogue of the $\ell$-adic Fourier transform for unipotent perfect group schemes. We explicitly describe it on some examples.