L-functions and Eisenstein series: p-adic aspects and applications

这笔拨款涉及几个项目，这些项目是由数论中的核心研究对象L函数的p-进方面所激发的。该项目的第一部分涉及发展某些p-进微分算子和p-进测度(艾森斯坦测度)，PI将使用这些技术工具来构造p-进L-函数。这些p-进微分算子和p-进测度建立在PI关于这些主题的先前结果的基础上。这些Eisenstein测度也被猜想给出了推广Witten亏格的某些流形的同伦理论不变量。该项目的第二部分涉及提案第一部分中开发的工具的应用。它的主要应用是构造p元L函数，它对隶属于自同构族的L函数的特殊值进行p元插值。本申请的一部分是与迈克尔·哈里斯、李建树和克里斯托弗·斯金纳共同申请的。该项目的这一部分也应用于岩泽理论，这是一种研究某些算术数据的p-adar理论，而该理论又有望与Birch和Swinnerton-Dyer猜想有关。L函数是在数论中起重要作用的某些复值函数。它们在某些点上的值满足惊人的同余条件，并与许多公开问题有关。研究L函数的一种方法是使用p-进数(取决于素数p)，它是有理数的推广，类似于实数，但不同于实数。这一建议涉及进一步发展研究L函数的p元方面的技术。所提出的研究在代数数论、解析数论和代数拓扑学方面都有预期的结果。

This grant concerns several projects motivated by p-adic aspects of L-functions, a central object of study in number theory. The first part of the project involves the development of certain p-adic differential operators and p-adic measures (Eisensteinmeasures), technical tools that the PI will use to construct p-adic L-functions. These p-adic differential operators and p-adic measures build on the PI's prior results on these topics. These Eisenstein measures are also conjectured to give a homotopy-theoretic invariant of certain manifolds that generalizes the Witten genus. The second part of the project concerns applications of the tools developed in the first part of the proposal. The main application is the construction of p-adic L-functions that p-adically interpolate the special values of L-functions attached to families of automorphic forms. One portion of this application is joint with Michael Harris, Jian-Shu Li, and Christopher Skinner. This part of the project also has applications to Iwasawa theory, a p-adic theory for studying certain arithmetic data, which in turn is expected to relate to the Birch and Swinnerton-Dyer Conjecture. L-functions are certain complex-valued functions that play an important role in number theory. Their values at certain points satisfy striking congruence conditions and relate to many open problems. One approach to studying L-functions uses p-adic numbers (depending on a prime number p), an extension of the rational numbers analogous to but different from the real numbers. This proposal involves further developing techniques for studying p-adic aspects of L-functions. The proposed research has expected consequences in algebraic number theory, analytic number theory, and algebraic topology.