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

p 进 L 函数、特征变量几何、Selmer 群

• 批准号: 0801205
• 负责人: Joel Bellaiche
• 金额: $ 13.8万
• 依托单位: Brandeis University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-07-01 至 2012-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0801205&HistoricalAwards=false
• 关键词: adic; functions; geometry; eigenvarieties; Selmer

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.

这个项目是关于p-adic L-函数，特征簇和塞尔默群。特征簇是给定约化群的自同构形式的泛p进族。个别的自守形式被证明有一个p-adic L-函数，一个p-adic对应于它们通常的复L-函数，并且很自然地期望，对于一个给定的约化群，个别自守形式的p-adic L-函数将适合于一个由本征簇携带的解析族。然而，除了模形式的情况之外，人们对p-adic L-函数的存在知之甚少，对其族几乎一无所知。即使对于模形式，也有许多重要的问题，如许多临界p-adic 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。它集中在算术上最重要的情况：“临界”自守形式的情况。该策略是考虑家庭中的自守形式，其中一个关键的自守形式可能有温和的兄弟姐妹。该项目的最终目的是将特征变量在某点的几何关系与相应自守形式的p进L-函数的消失阶联系起来。这应该这样做的方式，结合早期的工作PI和Chenevier，可以导致一个不等式的证明，在平等之间的排名布洛赫和加藤塞尔默集团，并下令消失的L-功能。发现，由先驱的数学的现代，一些非常显着的等式，如所有正整数平方的倒数之和等于单位圆盘面积平方的六分之一（欧拉），开创了数学研究的一个潮流，至今仍十分活跃。这些等式将一个解析面（一个有限级数的和，微积分的一个对象）与一个几何性质的数乘以一个有理数的乘积（因此是数论家的研究对象）联系起来。 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，都包含在由德利涅、贝林森、布洛赫、加藤和佩林-里乌建立的一个庞大的结构框架中。在它们的现代和一般形式中，这些理论仍然涉及一个被称为L-函数的分析对象和一个被称为塞尔默群的数论对象。PI的项目旨在阐明这些理论的一个重要方面，即关于L-函数零点的阶数。PI建议通过将这两个方面与第三个对象联系起来来做到这一点，第三个对象的出现要晚得多，即本征变种，它是自守形式的普遍家族。