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

这项拨款涉及几个由l函数的p进方面所激发的项目，l函数是数论研究的一个中心对象。项目的第一部分涉及某些p进微分算子和p进测度（爱森斯坦测度）的发展，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先前关于这些主题的结果之上。这些爱森斯坦测度也被推测为给出了某些流形的同伦论不变量，推广了Witten格。项目的第二部分涉及提案第一部分中开发的工具的应用。其主要应用是构造p进l函数，该函数可以p进插值自同构形式族上的l函数的特殊值。这个应用程序的一部分是由Michael Harris、Jian-Shu Li和Christopher Skinner共同完成的。该项目的这一部分也应用于Iwasawa理论，这是一种用于研究某些算术数据的p进理论，反过来又有望与Birch和Swinnerton-Dyer猜想有关。l函数是一类在数论中占有重要地位的复值函数。它们在某些点上的值满足显著同余条件，并与许多开放问题有关。研究l函数的一种方法是使用p进数（取决于素数p），它是有理数的一种扩展，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进方面的技术。所提出的研究在代数数论、解析数论和代数拓扑学方面具有预期的结果。

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.