Collaborative Research: Local Newforms for GSp(4)

本文研究了代数群PGSp（4，F）的不可约容许表示的新旧形式理论，其中F是特征为零的非阿基米德局部域。本研究的第一个目标是证明主要猜想，支持广泛的证据，这断言，一个通用的表示承认一个非零向量固定的一个paramodular组的一些水平;在最低水平，这样的一个向量是唯一的非零标量;并在最低水平，一个适当的规范化这样的一个向量，称为一个新的形式，计算L-因子的表示。第二个目标是证明一个关于一般表示中的旧型空间的结构的精确猜想，即，向量的空间由超过最小水平的水平的仿模群固定。第三，也是最后一个目标是调查是否厌恶的主要猜想持有的非通用表示在L-或A-packets的水平。简要地说，数论是数学的一部分，与研究积分解决代数方程。虽然数论的问题往往可以简单地说，解决方案可能需要广泛的理论发展和数学的其他部分的投入。广泛的特点，这项研究旨在tietogether，在一个有用的和新的方式，数论的两个部分。 第一部分，理论的自守表示，允许制定全面的说明和证明，然而，自守表示是抽象的。第二部分，Siegel模形式的理论，是重要的例子和应用;另一方面，Siegel模形式可以隐藏重要的一般现象。通过发展PGSp（4）的局部新旧形式理论，本研究将使从自同构表示系统地和概念性地构造Siegel模形式变得更容易，从而可能导致利用两种理论的优点的工作。 这项研究跨越了数学的几个分支学科，为了促进这项工作和其他工作，该项目将在万维网上创建一个关于GSp（4）现有文献的免费访问数据库。

This research is about a theory of new- and oldforms for irreducibleadmissible representations of the algebraic group PGSp(4,F), for F anonarchimedean local field of characteristic zero. The first objectiveof this research is to prove the main conjecture, supported byextensive evidence, which asserts that a generic representation admitsa nonzero vector fixed by a paramodular group of some level; that atthe minimal level such a vector is unique up to nonzero scalars; andthat at the minimal level a suitable normalization of such a vector,called a newform, computes the L-factor of the representation. Thesecond objective is to prove a precise conjecture about the structureof the spaces of oldforms in a generic representation, i.e., the spacesof vectors fixed by paramodular groups of level exceeding the minimallevel. The third, and final, objective is to investigate whether aversion of the main conjecture holds for nongeneric representations atthe level of L- or A-packets.Briefly summarized, number theory is the part of mathematics concernedwith the study of integral solutions to algebraic equations. Though theproblems of number theory often can be simply stated, solutions mayrequire extensive theoretical development and input from other parts ofmathematics. Broadly characterized, this research seeks to tietogether, in a useful and new way, two parts of number theory. Thefirst part, the theory of automorphic representations, allows theformulation of comprehensive conjectures and proofs; however,automorphic representations are abstract. The second part, the theoryof Siegel modular forms, is important for examples and applications; onthe other hand, Siegel modular forms can hide important generalphenomena. By developing the theory of local new- and oldforms forPGSp(4) this research will make it easier to systematically andconceptually construct Siegel modular forms from automorphicrepresentations, and thus may lead to work that draws on the strengthsof both theories. This research crosses several subdisciplines ofmathematics, and to facilitate this and other work this project willcreate a freely accessible database on the World Wide Web about theexisting literature on GSp(4).