Collaborative Research: Local Newforms for GSp(4)

本文研究了特征为零的非阿基米德局部域的代数群PGSp（4，F）的不可约可导表示的新形式和旧形式理论。本研究的第一个目的是证明由大量证据支持的主要猜想，该猜想断言一般表示承认由某水平的旁模群固定的非零向量；在最小水平上，这样的向量在非零标量下是唯一的；在最小的水平上，这样一个向量的合适的归一化，称为新形式，计算表示的l因子。第二个目标是证明关于一般表示中旧形式空间结构的一个精确猜想，即由超过最小水平的顺模群所固定的向量空间。第三个，也是最后一个，目的是研究主猜想的厌恶是否在L-或a -包的水平上对非泛型表示成立。简而言之，数论是研究代数方程的积分解的数学部分。虽然数论的问题通常可以简单地陈述，但解决方案可能需要广泛的理论发展和数学其他部分的输入。概括地说，这项研究试图以一种有用的新方式将数论的两个部分联系在一起。第一部分，自同构表示理论，允许制定全面的猜想和证明；然而，自同构表示是抽象的。第二部分是西格尔模形式理论，具有重要的举例和应用价值；另一方面，西格尔模形式可以隐藏重要的一般现象。通过发展pgsp的局部新旧形式理论(4)，本研究将使从自同构表示系统地和概念性地构建西格尔模形式变得更容易，从而可能导致借鉴两种理论优势的工作。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现有文献的可免费访问的数据库(4)。

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).