Note on $p$-adic Local Functional Equation

Abstract

Given primes $\ell \ne p$, we record here a $p$-adic valued Fourier theory on a local field over ${\mathbf{Q}}_{\ell }$, which is developed under the perspective of group schemes. As an application, by substituting rigid analysis for complex analysis, it leads naturally to the $p$-adic local functional equation at $\ell$, which strongly resembles the complex one in Tate's thesis.

arXiv comment26 pages; revised version with §B.1 rewritten; accepted by p-Adic Numbers, Ultrametric Analysis and Applications