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.

中文翻译：

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关于 $$p$$ -Adic 局部泛函方程的注释

摘要

给定素数\(\ell\ne p\)，我们在此记录一个在\(\mathbf{Q}_\ell\)上的局部场上的\(p\) -adic 值傅立叶理论，它是在透视下开发的的团体计划。作为一种应用，通过用刚性分析代替复分析，它自然会导致\(\ell\)处的\(p\) -adic 局部泛函方程，这与 Tate 论文中的复数非常相似。