Note on $p$-adic Local Functional Equation

Abstract

Given primes ℓ p, we record here a p-adic valued Fourier theory on a local field over Q ℓ , 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 ℓ, which strongly resembles the complex one in Tate's thesis.