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

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扬提出以来，已成为数学研究的一个重要领域，在我们对数论和代数几何(动机)的现代理解中起着核心和统一的作用，并与数学的各个领域相联系。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、代数和表示理论、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使其成为模形式的函数方程。因此，对这些序列的算术性质的研究成为模形式算术性质研究的一部分。特别地，要研究模为素数p的序列的可除性或同余性质，必须研究模p的模形式理论。该项目提出了一种新的基本研究方法，其基础是确定了作用于模p的模形式空间上的大Hecke代数的结构，并且每个模形式上都存在Galois伪表示，而不一定是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的同余的孤立例子，而且导致这种同余的完整分类。它还将允许完整地描述模p的全纯模形式的系数的渐近行为，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，更好地理解猜想地潜藏在这些同余背后的“混沌”，特别是导致关于配分函数的约化模素数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，但使用了类似的工具，目的是证明与L的特殊值有关的Bloch-Kato猜想的重要案例-动机函数和Selmer群的函数，并发展p-进L函数理论。从长远来看，这两个方面应该结合到一个非常大的混合特征的自同构型普适族理论中，同时包括伽罗瓦函数、L函数和系数方面。

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.