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.

本课题是关于p进l函数、特征变数和Selmer群的。本征变是给定约化群的自同构形式的泛p进族。单个自同构形式被推测有一个p进l函数，一个它们通常的复l函数的p进对应物，并且很自然地期望给定约化群的单个自同构形式的p进l函数将适合于由本征变携带的解析族。然而，除了模形式的情况外，我们对p进l函数的存在性知之甚少，对它们的族也几乎一无所知。即使对于模形式，仍然存在许多重要的问题，例如许多关键p进l函数的定义和计算。本项目提出了解决其中一些问题的策略。它着重于最重要的算术情况：“临界”自同构形式的情况。策略是考虑家庭中的自同构形式，其中一个关键的自同构形式可能有较温和的兄弟姐妹。该项目的最终目的是将某点的特征变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。这应该以这样一种方式来完成，结合PI和Chenevier的早期工作，可以导致Bloch和Kato猜想的Selmer群的秩和l函数的消失阶之间的不等式的证明。近代数学的先驱们发现了一些非常了不起的等式，比如欧拉定理：所有正整数的平方的倒数之和等于单位圆盘面积平方的六分之一。这一发现开辟了数学研究的一个趋势，直到今天仍然非常活跃。这些等式将解析边（有限级数的和，微积分的对象）与几何性质数乘以有理数的乘积边（因此是数论学家的研究对象）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，都包含在由德列涅、贝林森、布洛赫、加藤和佩兰-里欧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，这些猜想仍然涉及一个解析对象，称为l -函数，和一个数论对象，称为塞尔默群。PI计划的目的是阐明这些猜想的一个重要方面，即关于l函数的零点的顺序。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，这个对象的出现要晚得多，即特征变体，它是自同构形式的普遍族。