{"title":"Tate Cohomology of Whittaker Lattices and Base Change of Generic Representations of GLn","authors":"Santosh Nadimpalli, Sabyasachi Dhar","doi":"10.1093/imrn/rnae183","DOIUrl":null,"url":null,"abstract":"Let $p$ and $l$ be two distinct odd primes, and let $n\\geq 2$ be a positive integer. Let $E$ be a finite Galois extension of degree $l$ of a $p$-adic field $F$. Let $q$ be the cardinality of the residue field of $F$. Let $\\pi _{F}$ be an integral $l$-adic generic representation of $\\mathrm{GL}_{n}(F)$, and let $\\pi _{E}$ be the base change of $\\pi _{F}$. Let $J_{l}(\\pi _{F})$ (resp. $J_{l}(\\pi _{E})$) be the unique generic component of the mod-$l$ reduction $r_{l}(\\pi _{F})$ (resp. $r_{l}(\\pi _{E})$). Assuming that $l$ does not divide $|\\mathrm{GL}_{n-1}(\\mathbb{F}_{q})|$, we prove that the Frobenius twist of $J_{l}(\\pi _{F})$ is the unique generic subquotient of the Tate cohomology group $\\widehat{H}^{0}(\\mathrm{Gal}(E/F), J_{l}(\\pi _{E}))$—considered as a representation of $\\mathrm{GL}_{n}(F)$.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":"168 1","pages":""},"PeriodicalIF":0.9000,"publicationDate":"2024-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"International Mathematics Research Notices","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1093/imrn/rnae183","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Let $p$ and $l$ be two distinct odd primes, and let $n\geq 2$ be a positive integer. Let $E$ be a finite Galois extension of degree $l$ of a $p$-adic field $F$. Let $q$ be the cardinality of the residue field of $F$. Let $\pi _{F}$ be an integral $l$-adic generic representation of $\mathrm{GL}_{n}(F)$, and let $\pi _{E}$ be the base change of $\pi _{F}$. Let $J_{l}(\pi _{F})$ (resp. $J_{l}(\pi _{E})$) be the unique generic component of the mod-$l$ reduction $r_{l}(\pi _{F})$ (resp. $r_{l}(\pi _{E})$). Assuming that $l$ does not divide $|\mathrm{GL}_{n-1}(\mathbb{F}_{q})|$, we prove that the Frobenius twist of $J_{l}(\pi _{F})$ is the unique generic subquotient of the Tate cohomology group $\widehat{H}^{0}(\mathrm{Gal}(E/F), J_{l}(\pi _{E}))$—considered as a representation of $\mathrm{GL}_{n}(F)$.
期刊介绍:
International Mathematics Research Notices provides very fast publication of research articles of high current interest in all areas of mathematics. All articles are fully refereed and are judged by their contribution to advancing the state of the science of mathematics.