p-adic L-functions, geometry of eigenvarieties, Selmer groups

本课题是关于p-进L-函数、本征簇和Selmer群的。本征簇是给定约化群的自同构型的普适p-进数族。个体自同构形被猜想有一个p-进L函数，这是它们通常复的L函数的p进对应函数，并且很自然地期望给定还原群的个体自同构形的p进L-函数将适合于由本征簇所承载的解析族。然而，除了模形式的情况外，关于p-进L函数的存在性的研究很少，几乎没有关于它们的族的研究。即使对于模形式，仍然存在许多重要的问题，如许多临界p元L函数的定义和计算。该项目提出了一项解决其中一些问题的战略。它集中在最有算术意义的情况：“关键的”自同构形的情况。策略是考虑族中的自同构形式，其中关键的自同构形式可能具有较温和的兄弟姐妹。该项目的最终目的是将本征簇在某个点上的几何与相应自同构形式的p-进L函数的消失阶联系起来。这应该以这样的方式进行，结合Pi和Chenevier的早期工作，可以导致证明Bloch和Kato猜想的塞尔默群的秩之间的等式中的一个不等式，以及L函数的消失次序。现代数学的先驱们发现了一些非常显著的等式，例如所有正整数的平方的倒数之和等于单位圆盘面积的六分之一(欧拉)，这开启了一种数学研究的趋势，至今仍非常活跃。这些等式将解析边(有限级数中的和，微积分的对象)与若干几何性质乘以有理数的乘积的边联系起来(因此是数学家的研究对象)。这些等式，以及从那时起获得的大量著名结果，以及更多有待证明的结果，都包含在由Deligne、Beilinson、Bloch、Kato和Perrin-Riou建立的一个巨大的猜想框架中。在它们的现代和一般形式中，这些猜想仍然与一个被称为L函数的分析对象和一个被称为塞尔默群的数论对象有关。PI的项目旨在阐明这些猜想中的一个重要方面，即关于L函数的零点序的猜想。PI建议通过将双方与第三个对象相关联来实现这一点，该对象的出现时间要晚得多，即本征簇，它是自同构形式的普遍家族。

This project is about p-adic L-functions, eigenvarieties, and Selmer groups. Eigenvarieties are the universal p-adic families of automorphic forms for a given reductive group. Individual automorphic forms are conjectured to have a p-adic L-function, a p-adic counterpart of their usual complex L-function, and it is natural to expect that the p-adic L-functions of individual automorphic forms for a given reductive group will fit in an analytic family carried by the eigenvariety. However, besides the case of modular forms, very little is known on the existence of p-adic L-functions, and virtually nothing on their families. Even for modular forms, many important questions remain, such as the de finition and computation of many critical p-adic L-functions. This project proposes a strategy to address some of those questions. It focuses on the most arithmetically significant situation: the case of "critical" automorphic forms. The strategy is to consider automorphic forms in families, in which a critical automorphic forms may have milder siblings. The ultimate aim of the project is to relate the geometry of the Eigenvariety at some point to the order of vanishing of the p-adic L-function of the corresponding automorphic form. This should be done in such a way that, combined with earlier work of the PI and Chenevier, could lead to a proof of an inequality in the equality conjectured by Bloch and Kato between rank of Selmer groups, and order of vanishing of L-functions.The discovery, by the pioneers of mathematics of modern times, of some very remarkable equalities, like that the sum of the reciprocals of the square of all positive integers is equal to one sixth of the square of the area of a unit disc (Euler) have opened a trend of mathematical research which is still very active today. Those equalities relate an analytic side (the sum of an in finite series , an object of calculus) to a side which is a product of a number of geometric nature times a rational number (hence an object of study for number theorists). Those equalties, and a very great number of famous results obtained since then, as well as many more still to be proved, are all contained in a vast framework of conjectures built by Deligne, Beilinson, Bloch, Kato and Perrin-Riou. In their modern and general forms, those conjectures still relate an analytic object, called a L-function, and a number- theoretical one, called a Selmer group. The project of the PI intends to shed some light of one important aspect of those conjectures, the one concerning the order of the zeros of the L-functions. The PI proposes to do so by relating the two sides to a third object, whose appearance is much more recent, the Eigenvarieties, which are the universal families of automorphic forms.