{"title":"局部伽罗瓦表示的驯服多重性和导体","authors":"C. Bushnell, G. Henniart","doi":"10.2140/tunis.2020.2.337","DOIUrl":null,"url":null,"abstract":"Let $F$ be a non-Archimedean locally compact field of residual characteristic $p$. Let $\\sigma$ be an irreducible smooth representation of the absolute Weil group $\\Cal W_F$ of $F$ and $\\sw(\\sigma)$ the Swan exponent of $\\sigma$. Assume $\\sw(\\sigma) \\ge1$. Let $\\Cal I_F$ be the inertia subgroup of $\\Cal W_F$ and $\\Cal P_F$ the wild inertia subgroup. There is an essentially unique, finite, cyclic group $\\varSigma$, of order prime to $p$, so that $\\sigma(\\Cal I_F) = \\sigma(\\Cal P_F)\\varSigma$. In response to a query of Mark Reeder, we show that the multiplicity in $\\sigma$ of any character of $\\varSigma$ is bounded by $\\sw(\\sigma)$.","PeriodicalId":36030,"journal":{"name":"Tunisian Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8000,"publicationDate":"2018-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2140/tunis.2020.2.337","citationCount":"1","resultStr":"{\"title\":\"Tame multiplicity and conductor for local Galois representations\",\"authors\":\"C. Bushnell, G. Henniart\",\"doi\":\"10.2140/tunis.2020.2.337\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Let $F$ be a non-Archimedean locally compact field of residual characteristic $p$. Let $\\\\sigma$ be an irreducible smooth representation of the absolute Weil group $\\\\Cal W_F$ of $F$ and $\\\\sw(\\\\sigma)$ the Swan exponent of $\\\\sigma$. Assume $\\\\sw(\\\\sigma) \\\\ge1$. Let $\\\\Cal I_F$ be the inertia subgroup of $\\\\Cal W_F$ and $\\\\Cal P_F$ the wild inertia subgroup. There is an essentially unique, finite, cyclic group $\\\\varSigma$, of order prime to $p$, so that $\\\\sigma(\\\\Cal I_F) = \\\\sigma(\\\\Cal P_F)\\\\varSigma$. In response to a query of Mark Reeder, we show that the multiplicity in $\\\\sigma$ of any character of $\\\\varSigma$ is bounded by $\\\\sw(\\\\sigma)$.\",\"PeriodicalId\":36030,\"journal\":{\"name\":\"Tunisian Journal of Mathematics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.8000,\"publicationDate\":\"2018-09-15\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://sci-hub-pdf.com/10.2140/tunis.2020.2.337\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Tunisian Journal of Mathematics\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.2140/tunis.2020.2.337\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Tunisian Journal of Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.2140/tunis.2020.2.337","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
Tame multiplicity and conductor for local Galois representations
Let $F$ be a non-Archimedean locally compact field of residual characteristic $p$. Let $\sigma$ be an irreducible smooth representation of the absolute Weil group $\Cal W_F$ of $F$ and $\sw(\sigma)$ the Swan exponent of $\sigma$. Assume $\sw(\sigma) \ge1$. Let $\Cal I_F$ be the inertia subgroup of $\Cal W_F$ and $\Cal P_F$ the wild inertia subgroup. There is an essentially unique, finite, cyclic group $\varSigma$, of order prime to $p$, so that $\sigma(\Cal I_F) = \sigma(\Cal P_F)\varSigma$. In response to a query of Mark Reeder, we show that the multiplicity in $\sigma$ of any character of $\varSigma$ is bounded by $\sw(\sigma)$.