FRG: Collaborative Research: Automorphic forms, Galois representations, periods and p-adic L-functions

“该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。PI将研究代数数论中的一些基本问题，特别是与伽罗瓦表示和L函数的特殊值之间的深层联系有关的问题（如Birch-Swinnerton-Dyer猜想和Bloch-Kato猜想中所述）。更具体地说，PI将研究以下主题：（i）Mod-p Galois表示和mod-p模形式;（ii）构建与自守形式相关的Galois表示和动机;（iii）Iwasawa主猜想;（iv）自守形式的p-adic族及其应用;（iv）代数循环，p-adic L-函数和Euler系统。因此，主要的重点将放在自守形式和伽罗瓦表示理论中的p-adic方法上。PI将安排短期访问，以便在他们自己、附属研究人员和该领域的新来者之间进行合作研究，积极参与研究生培训和博士后咨询，并组织两次研讨会和一次最后会议，为传播结果准备会议记录。在非技术性术语中，PI将研究的问题涉及显示两个数论对象的惊人相等，一个是解析定义，另一个是代数定义。通过这种方式，要研究的问题被两个共同的哲学线索联系在一起：互反律的概念，它在数论中有着悠久而深厚的传统，可以追溯到高斯的二次互反律，以及类数公式的概念，可以追溯到狄利克雷的基本思想。此外，这种以先验不同方式定义的数学对象的等式不仅具有理论意义，而且往往具有非常具体的应用，最近最引人注目的是费马最后定理和Sato-Tate猜想的解决方案。研讨会，最后的会议，以及研究生和博士后咨询将对该领域新研究人员的形成和促进新的合作产生重要影响。

"This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5)."The PIs will study some fundamental problems in algebraic number theory, particularly problems related to the deep links between Galois representations and special values of L-functions (as conjectured in the Birch-Swinnerton-Dyer Conjecture and the Bloch-Kato Conjectures). More specifically, the PIs will study the following themes: (i) Mod-p Galois representations and mod-p modular forms; (ii) Constructing Galois representations and motives associated to automorphic forms; (iii) The Iwasawa Main Conjecture; (iv) p-adic families of automorphic forms and applications; (iv) Algebraic cycles, p-adic L-functions and Euler systems. Thus the main focus will be on p-adic methods in the theory of automorphic forms and Galois representations. The PIs will arrange short-term visits for collaborative research purposes between themselves, affiliated researchers and new comers in the area, be actively involved in graduate training and postdoctoral advising, and organize two workshops and a final conference,with preparation of a proceedings for dissemination of the results.In nontechnical terms, the problems that the PIs will study involve showing the surprising equality of two number-theoretic objects, one defined analytically and the other algebraically. In this way, the problems to be studied are linked by two common philosophical threads: the notion of a reciprocity law, which has a long and deep tradition in number theory, going back to the quadratic reciprocity law of Gauss, and the notion of a class number formula, which goes back to the fundamental ideas of Dirichlet. Further, such equalities of mathematical objects defined in a priori different ways are not just of theoretical interest but tend to have extremely concrete applications, the most striking recent ones being the resolution of Fermat's last theorem and the Sato-Tate conjecture. The workshops, the final conference, and the graduate and post-doctoral advising will have an important impact on the formation of new researchers in the field and on the promotion of new collaborations.