Local root numbers, Bessel models, and a conjecture of Guo and Jacquet
Local root numbers, Bessel models, and a conjecture of Guo and Jacquet
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局部根数、贝塞尔模型以及郭和雅凯的猜想
DOI：
10.1016/j.jnt.2013.07.001
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发表时间：
2015
影响因子：
0.7
通讯作者：
Kimball
中科院分区：
文献类型：
--
作者：
Furusawa;Masaaki; Martin;Kimball
设E/F是数域的二次扩张，D是F上包含E的四元数代数。设π D是GL（n，D）的尖点自守表示，π是GL（2 n）的Jacquet-Langlands转移. Guo和Jacquet证明了如果π D由GL（n，E）区分，则π是辛的，L（1/2，π E）<$0，其中π E是π到E的基变.当n为奇数时，郭和Jacquet也给出了一个匡威。当n为偶数时，匡威并不总是成立，但我们猜想当且仅当满足某些局部根数条件时，逆命题成立，即当且仅当分裂特殊正交群SO（2 n+ 1）的相应一般表示具有特殊的E-Bessel模型。我们利用θ对应把SO（5）上的E-Bessel周期与GL（2，D）上的GL（2，E）-周期联系起来，并在n= 2时推出了部分猜想.
Let E/F be a quadratic extension of number fields and D a quaternion algebra over F containing E. Let π D be a cuspidal automorphic representation of GL (n, D) and π its Jacquet–Langlands transfer to GL (2 n). Guo and Jacquet conjectured that if π D is distinguished by GL (n, E), then π is symplectic and L (1/2, π E)≠ 0, where π E is the base change of π to E. When n is odd, Guo and Jacquet also conjectured a converse. The converse does not always hold when n is even, but we conjecture it holds if and only if certain local root number conditions are satisfied, which is if and only if the corresponding generic representation of the split special orthogonal group SO (2 n+ 1) has a special E-Bessel model. We use the theta correspondence to relate E-Bessel periods on SO (5) with GL (2, E)-periods on GL (2, D), and deduce part of our conjecture when n= 2.