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

PIS将研究代数数论中的一些基本问题，特别是与L函数的伽罗瓦表示和特定值之间的深层联系的问题(如在Birch-Swinnerton-Dyer猜想和Bloch-Kato猜想中所猜想的)。更具体地说，PI将研究以下主题：(I)mod-p伽罗瓦表示和mod-p模形式；(Ii)构造与自同构形式有关的伽罗瓦表示和动机；(Iii)岩泽主要猜想；(Iv)p-进自同构形式及其应用；(Iv)代数环、p-进L函数和欧拉系。因此，主要的焦点将集中在自同构形式和伽罗瓦表示理论中的p-进方法上。私人投资机构将安排他们自己、附属研究人员和该领域的新来者之间为合作研究目的而进行的短期访问，积极参与研究生培训和博士后建议，并组织两次研讨会和一次最终会议，并准备发布结果的会议记录。在非技术术语中，个人投资机构将研究的问题涉及证明两个数论对象令人惊讶地相等，一个用解析定义，另一个用代数定义。通过这种方式，要研究的问题通过两条共同的哲学线索联系在一起：倒易律的概念，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易定律，以及类数公式的概念，它可以追溯到狄利克莱特的基本思想。此外，以先验的不同方式定义的这些数学对象的等式不仅具有理论上的兴趣，而且往往具有非常具体的应用，最近最引人注目的是费马大定理和佐藤泰特猜想的解。讲习班、最后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.