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.

模形式算术理论自一个多世纪前由印度著名数学家拉马努金提出以来，已成为数学研究的一个主要领域，在我们对数论和代数几何（动机）的现代理解中发挥着核心和统一的作用，并与数学的各个领域都有联系。模形式对其他领域有用的最简单和最古老的原因是，数论、组合学、代数和表示论、代数几何和理论物理学中感兴趣的许多整数序列具有令人惊讶的特性，即它们的生成函数满足函数方程，从而使其成为模形式。因此，对这些序列算术性质的研究就成为模形式算术性质研究的一部分。特别是，为了研究以素数 p 为模的序列的整除性或同余性，必须研究以 p 为模的模形式理论。该项目提出了一种新的基础研究方法，基于对作用于以 p 为模的模形式空间的大赫克代数的结构的确定，以及附加到每个模形式的伽罗瓦伪表示的存在，不一定是一个 Hecke 算子的特征向量。这种方法应该允许弱全纯模形式的系数之间的同余的系统理论，不仅产生（通常情况下孤立的同余示例），而且导致对此类同余的完整分类。它还将允许完全描述全纯模形式模 p 的系数的渐近行为，目前为止，这仅在某些非常特殊的情况下才知道，并且在弱全纯模形式的更神秘的情况下，可以更好地理解推测潜伏在这些同余背后的“混沌”，特别是导致关于配分函数的约简模素数 p 的几个突出猜想的证明。该项目的第二个方面与第一个方面不同，但使用类似的工具，旨在证明与动机的 L 函数和 Selmer 群的特殊值相关的 Bloch-Kato 猜想的重要案例，并发展 p 进 L 函数理论。从长远来看，这两个方面应该重新统一成一个非常大的混合特征自守形式普适族理论，同时包括伽罗瓦、L-函数和系数方面。