{"title":"Note on $$p$$ -Adic Local Functional Equation","authors":"Luochen Zhao","doi":"10.1134/s2070046622030037","DOIUrl":null,"url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> Given primes <span>\\(\\ell\\ne p\\)</span>, we record here a <span>\\(p\\)</span>-adic valued Fourier theory on a local field over <span>\\(\\mathbf{Q}_\\ell\\)</span>, which is developed under the perspective of group schemes. As an application, by substituting rigid analysis for complex analysis, it leads naturally to the <span>\\(p\\)</span>-adic local functional equation at <span>\\(\\ell\\)</span>, which strongly resembles the complex one in Tate’s thesis. </p>","PeriodicalId":44654,"journal":{"name":"P-Adic Numbers Ultrametric Analysis and Applications","volume":"17 1","pages":""},"PeriodicalIF":0.5000,"publicationDate":"2022-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"P-Adic Numbers Ultrametric Analysis and Applications","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1134/s2070046622030037","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS, INTERDISCIPLINARY APPLICATIONS","Score":null,"Total":0}
引用次数: 0
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.
期刊介绍：
This is a new international interdisciplinary journal which contains original articles, short communications, and reviews on progress in various areas of pure and applied mathematics related with p-adic, adelic and ultrametric methods, including: mathematical physics, quantum theory, string theory, cosmology, nanoscience, life sciences; mathematical analysis, number theory, algebraic geometry, non-Archimedean and non-commutative geometry, theory of finite fields and rings, representation theory, functional analysis and graph theory; classical and quantum information, computer science, cryptography, image analysis, cognitive models, neural networks and bioinformatics; complex systems, dynamical systems, stochastic processes, hierarchy structures, modeling, control theory, economics and sociology; mesoscopic and nano systems, disordered and chaotic systems, spin glasses, macromolecules, molecular dynamics, biopolymers, genomics and biology; and other related fields.
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