{"title":"p-adic 四元数代数的 Toric 周期","authors":"U. K. Anandavardhanan, Basudev Pattanayak","doi":"10.1007/s00209-024-03551-3","DOIUrl":null,"url":null,"abstract":"<p>Let <i>G</i> be a compact group with two given subgroups <i>H</i> and <i>K</i>. Let <span>\\(\\pi \\)</span> be an irreducible representation of <i>G</i> such that its space of <i>H</i>-invariant vectors as well as the space of <i>K</i>-invariant vectors are both one dimensional. Let <span>\\(v_H\\)</span> (resp. <span>\\(v_K\\)</span>) denote an <i>H</i>-invariant (resp. <i>K</i>-invariant) vector of unit norm in a given <i>G</i>-invariant inner product <span>\\(\\langle ~,~ \\rangle _\\pi \\)</span> on <span>\\(\\pi \\)</span>. We are interested in calculating the correlation coefficient </p><span>$$\\begin{aligned} c(\\pi \\text {;}\\,H,K) = |\\langle v_H,v_K \\rangle _\\pi |^2. \\end{aligned}$$</span><p>In this paper, we compute the correlation coefficient of an irreducible representation of the multiplicative group of the <i>p</i>-adic quaternion algebra with respect to any two tori. In particular, if <span>\\(\\pi \\)</span> is such an irreducible representation of odd minimal conductor with non-trivial invariant vectors for two tori <i>H</i> and <i>K</i>, then its root number <span>\\(\\varepsilon (\\pi )\\)</span> is <span>\\(\\pm 1\\)</span> and <span>\\(c(\\pi \\text {;}\\, H, K)\\)</span> is non-vanishing precisely when <span>\\(\\varepsilon (\\pi ) = 1\\)</span>.</p>","PeriodicalId":18278,"journal":{"name":"Mathematische Zeitschrift","volume":"41 1","pages":""},"PeriodicalIF":1.0000,"publicationDate":"2024-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Toric periods for a p-adic quaternion algebra\",\"authors\":\"U. K. Anandavardhanan, Basudev Pattanayak\",\"doi\":\"10.1007/s00209-024-03551-3\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Let <i>G</i> be a compact group with two given subgroups <i>H</i> and <i>K</i>. Let <span>\\\\(\\\\pi \\\\)</span> be an irreducible representation of <i>G</i> such that its space of <i>H</i>-invariant vectors as well as the space of <i>K</i>-invariant vectors are both one dimensional. Let <span>\\\\(v_H\\\\)</span> (resp. <span>\\\\(v_K\\\\)</span>) denote an <i>H</i>-invariant (resp. <i>K</i>-invariant) vector of unit norm in a given <i>G</i>-invariant inner product <span>\\\\(\\\\langle ~,~ \\\\rangle _\\\\pi \\\\)</span> on <span>\\\\(\\\\pi \\\\)</span>. We are interested in calculating the correlation coefficient </p><span>$$\\\\begin{aligned} c(\\\\pi \\\\text {;}\\\\,H,K) = |\\\\langle v_H,v_K \\\\rangle _\\\\pi |^2. \\\\end{aligned}$$</span><p>In this paper, we compute the correlation coefficient of an irreducible representation of the multiplicative group of the <i>p</i>-adic quaternion algebra with respect to any two tori. In particular, if <span>\\\\(\\\\pi \\\\)</span> is such an irreducible representation of odd minimal conductor with non-trivial invariant vectors for two tori <i>H</i> and <i>K</i>, then its root number <span>\\\\(\\\\varepsilon (\\\\pi )\\\\)</span> is <span>\\\\(\\\\pm 1\\\\)</span> and <span>\\\\(c(\\\\pi \\\\text {;}\\\\, H, K)\\\\)</span> is non-vanishing precisely when <span>\\\\(\\\\varepsilon (\\\\pi ) = 1\\\\)</span>.</p>\",\"PeriodicalId\":18278,\"journal\":{\"name\":\"Mathematische Zeitschrift\",\"volume\":\"41 1\",\"pages\":\"\"},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2024-07-14\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Mathematische Zeitschrift\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s00209-024-03551-3\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematische Zeitschrift","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00209-024-03551-3","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
让 G 是一个紧凑群,有两个给定的子群 H 和 K。让 \(\pi \) 是 G 的不可还原表示,使得它的 H 不变向量空间和 K 不变向量空间都是一维的。让 \(v_H\) (resp. \(v_K\)) 表示给定 G 不变内积 \(\langle ~,~ \rangle _\pi \) 在 \(\pi \) 上的单位法的 H 不变(或 K 不变)向量。我们感兴趣的是计算相关系数 $$\begin{aligned} c(\pi \text {;}\,H,K) = |\langle v_H,v_K \rangle _\pi |^2。\end{aligned}$$ 在本文中,我们计算 p-adic 四元数代数的乘法群的不可还原表示与任意两个环的相关系数。特别地,如果\(\pi \)是这样一个奇数最小导体的不可还原表示,它对于两个环 H 和 K 具有非难变向量,那么它的根((\varepsilon (\pi )\)是(\pm 1\ ),并且(c(\pi \text {;}\, H, K)\)恰好在(\(\varepsilon (\pi)= 1\ )时是非递减的。
Let G be a compact group with two given subgroups H and K. Let \(\pi \) be an irreducible representation of G such that its space of H-invariant vectors as well as the space of K-invariant vectors are both one dimensional. Let \(v_H\) (resp. \(v_K\)) denote an H-invariant (resp. K-invariant) vector of unit norm in a given G-invariant inner product \(\langle ~,~ \rangle _\pi \) on \(\pi \). We are interested in calculating the correlation coefficient
In this paper, we compute the correlation coefficient of an irreducible representation of the multiplicative group of the p-adic quaternion algebra with respect to any two tori. In particular, if \(\pi \) is such an irreducible representation of odd minimal conductor with non-trivial invariant vectors for two tori H and K, then its root number \(\varepsilon (\pi )\) is \(\pm 1\) and \(c(\pi \text {;}\, H, K)\) is non-vanishing precisely when \(\varepsilon (\pi ) = 1\).