Toric periods for a p-adic quaternion algebra

IF 1 3区 数学 Q1 MATHEMATICS
U. K. Anandavardhanan, Basudev Pattanayak
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引用次数: 0

Abstract

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

$$\begin{aligned} c(\pi \text {;}\,H,K) = |\langle v_H,v_K \rangle _\pi |^2. \end{aligned}$$

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 \(\varepsilon (\pi ) = 1\).

p-adic 四元数代数的 Toric 周期
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。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
CiteScore
1.60
自引率
0.00%
发文量
236
审稿时长
3-6 weeks
期刊介绍: "Mathematische Zeitschrift" is devoted to pure and applied mathematics. Reviews, problems etc. will not be published.
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