Note on $p$-adic Local Functional Equation

TL;DR

This paper develops a $p$-adic Fourier theory on local fields over $ extbf{Q}_ extbf{l}$ using group schemes, leading to a $p$-adic local functional equation analogous to Tate's complex functional equation.

Contribution

It introduces a $p$-adic Fourier theory on local fields over $ extbf{Q}_ extbf{l}$ and derives a $p$-adic local functional equation using rigid analysis.

Findings

Establishment of a $p$-adic Fourier theory on local fields.

Derivation of a $p$-adic local functional equation similar to Tate's thesis.

Connection between rigid analysis and $p$-adic functional equations.

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.