Mod p and p-Adic Aspects of Modular and Automorphic Forms

模和自同构形式的 Mod p 和 p-Adic 方面

• 批准号: 1405993
• 负责人: Joel Bellaiche
• 金额: $ 20.66万
• 依托单位: Brandeis University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-07-15 至 2018-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1405993&HistoricalAwards=false
• 关键词: Mod; Adic; Aspects; Modular; Automorphic

Since it was initiated by the famous Indian mathematician Ramanujan more than a century ago, the arithmetic theory of modular forms has become a major field of mathematical research that plays a central and unifying role in our modern understanding of number theory and algebraic geometry (motives) and has connections with various fields of mathematics. The simplest and oldest reason for which modular forms are useful to those other fields is that many sequences of integers of interest in number theory, combinatorics, algebra and representation theory, algebraic geometry, and theoretical physics have the surprising property that their generating function satisfies a functional equation that makes it a modular form. Hence the study of the arithmetic properties of these sequences becomes a part of the study of arithmetic property of modular forms. In particular, to study divisibility or congruence properties of those sequences modulo a prime number p, one must study the theory of modular forms modulo p.The project proposes a new approach to this fundamental study, based on the determination of the structure of the big Hecke algebras acting on space of modular forms modulo p, and the existence of a Galois pseudo-representation attached to every modular form, not necessarily an eigenvector for the Hecke operators. This approach should allow for a systematic theory of congruences between the coefficients of weakly holomorphic modular forms, not only producing, as was often the case isolated examples of congruences, but leading to a complete classification of such congruences. It will also allow to completely describe the asymptotic behavior of coefficients of holomorphic modular forms modulo p, which is only known in certain very particular cases as of now, and, in the more mysterious case of weakly holomorphic modular forms, to better understand the "chaos" that conjecturally lurks behind these congruences, in particular leading to the proof of several outstanding conjectures concerning the reduction modulo primes p of the partition function. A second aspect of the project, which is distinct from the first but uses similar tools, aims at proving important cases of the Bloch-Kato conjecture relating special values of L-functions of motives and Selmer groups, and at developing the theory of p-adic L-functions. In the long run, those two aspects should be reunited into one very large theory of universal families of automorphic forms in mixed characteristic, including at the same time the Galois, L-functions, and coefficients aspects.

自从印度著名数学家拉马努金在一个多世纪前提出模形式的算术理论以来，它已经成为数学研究的一个主要领域，在我们对数论和代数几何的现代理解中起着中心和统一的作用，并与数学的各个领域有联系。模形式对其他领域有用的最简单和最古老的原因是，在数论、组合学、代数和表示论、代数几何和理论物理中，许多感兴趣的整数序列具有令人惊讶的性质，即它们的生成函数满足一个函数方程，使其成为模形式。因此，研究这些序列的算术性质成为研究模形式的算术性质的一部分。特别是，要研究这些序列模素数p的整除或同余性质，必须研究模p的模形式理论。该项目提出了一种新的方法来进行这一基础研究，基于确定作用于模p的模形式空间上的大Hecke代数的结构，以及每个模形式的Galois伪表示的存在性，不一定是Hecke算子的本征向量。这种方法应该允许一个系统的理论之间的系数弱全纯模形式的同余，不仅生产，因为往往是孤立的例子，同余，但导致一个完整的分类等同余。它也将允许完全描述全纯模形式模p的系数的渐近行为，到目前为止，这只在某些非常特殊的情况下是已知的，并且，在更神秘的弱全纯模形式的情况下，为了更好地理解隐藏在这些同余后面的“混沌”，特别是导致证明了几个突出的命题有关减少模素数p的分区函数。该项目的第二个方面，这是不同于第一，但使用类似的工具，旨在证明重要案件的布洛赫-加藤猜想有关特殊价值的L-功能的动机和塞尔默集团，并在发展理论的p进L-功能。从长远来看，这两个方面应该重新结合成一个非常大的混合特征自守形式的普适族理论，同时包括伽罗瓦，L-函数和系数方面。