Finite period vectors and Gauss sums

IF 1.2 3区 数学 Q1 MATHEMATICS
Yeongseong Jo
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引用次数: 0

Abstract

We study four sums including the Jacquet–Piatetski-Shapiro–Shalika, Flicker, Bump–Friedberg, and Jacquet–Shalika sums associated to irreducible cuspidal representations of general linear groups over finite fields. By computing explicitly, we relate Asai and Bump–Friedberg gamma factors over finite fields to those over nonarchimedean local fields through level zero supercuspidal representation. Via Deligne–Kazhdan close field theory, we prove that exterior square and Bump–Friedberg gamma factors agree with corresponding Artin gamma factors of their associated tamely ramified representations through local Langlands correspondence. We also deduce product formulæ for Asai, Bump–Friedberg, and exterior square gamma factors in terms of Gauss sums. By combining these results, we examine Jacquet–Piatetski-Shapiro–Shalika, Flicker–Rallis, Jacquet–Shalika, and Friedberg–Jacquet periods and vectors and their connections to Rankin–Selberg, Asai, exterior square, and Bump–Friedberg gamma factors, respectively.

有限周期向量和高斯和
我们研究了与有限域上一般线性群的不可还原尖顶表示相关的四个和,包括 Jacquet-Piatetski-Shapiro-Shalika、Flicker、Bump-Friedberg 和 Jacquet-Shalika 和。通过显式计算,我们将有限域上的浅井伽马因子和布姆普-弗里德伯格伽马因子与非拱顶局部域上的伽马因子通过零级超pidal 表示联系起来。通过德利涅-卡兹丹近场理论,我们证明了外部平方和布姆普-弗里德伯格伽马因数通过局部朗兰兹对应关系与它们相关的驯化斜面表示的相应阿廷伽马因数一致。我们还用高斯和推导出了浅井、布姆普-弗里德伯格和外部平方伽马因数的乘积公式。结合这些结果,我们研究了 Jacquet-Piatetski-Shapiro-Shalika、Flicker-Rallis、Jacquet-Shalika 和 Friedberg-Jacquet 周期和向量,以及它们分别与 Rankin-Selberg、Asai、外部平方和及 Bump-Friedberg 伽马因数的联系。
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来源期刊
CiteScore
2.00
自引率
20.00%
发文量
133
审稿时长
6-12 weeks
期刊介绍: Finite Fields and Their Applications is a peer-reviewed technical journal publishing papers in finite field theory as well as in applications of finite fields. As a result of applications in a wide variety of areas, finite fields are increasingly important in several areas of mathematics, including linear and abstract algebra, number theory and algebraic geometry, as well as in computer science, statistics, information theory, and engineering. For cohesion, and because so many applications rely on various theoretical properties of finite fields, it is essential that there be a core of high-quality papers on theoretical aspects. In addition, since much of the vitality of the area comes from computational problems, the journal publishes papers on computational aspects of finite fields as well as on algorithms and complexity of finite field-related methods. The journal also publishes papers in various applications including, but not limited to, algebraic coding theory, cryptology, combinatorial design theory, pseudorandom number generation, and linear recurring sequences. There are other areas of application to be included, but the important point is that finite fields play a nontrivial role in the theory, application, or algorithm.
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