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New Directions in Modularity

模块化的新方向

基本信息

批准号：
1404620
负责人：
Francesco Calegari
金额：
$ 30.6万
依托单位：
Northwestern University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2014
资助国家：
美国
起止时间：
2014-10-01 至 2016-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1404620&HistoricalAwards=false
关键词：
New Directions Modularity

项目摘要

The problem of understanding prime numbers goes back to antiquity. One natural problem which arose in the 19th century was to count the number of primes less than X for some variable quantity X. The resolution came in an unexpected way : this *discrete* counting problem turned out to be related to the properties of a *continuous* function first studied by Euler but now known as the Riemann zeta function. The breakthrough hinged on a crucial property of this function discovered by Riemann, namely, that it had a hidden symmetry which related the value of the function evaluated at the point "s" to the value evaluated at the point "1-s". This hidden symmetry turns out to be merely the tip of the iceberg; there are a wide class of similar functions which exhibit similar symmetries (at least conjecturally), and these functions arise in many areas of mathematics and physics. By proving that a certain class of such functions had this symmetry, Andrew Wiles in 1994 was able to prove Fermat's Last Theorem. The PI's work aims to prove that a certain class of functions exhibit the same symmetry as the Riemann zeta function. More specifically, there are a natural collections of such functions indexed by a natural number g. When g is zero, there is a single function which was the one considered by Euler and Riemann. When g is one, the class of functions are exactly the ones studied by Wiles. The PI plans to study the functions arising when g is two. An overarching theme of the PI's research (including collaborations with Emerton, Geraghty, and Venkatesh) has been the study of torsion classes in cohomology, both in the Betti cohomology of arithmetic groups and the coherent cohomology of Shimura varieties, and their conjectural relationship with the Langlands program. The PI's work suggests that understanding torsion is at the heart of understanding reciprocity beyond Shimura varieties. One of the main long term technical goals of the proposed project is to prove that L-functions attached to genus two curves satisfy the expected functional equations. For curves of genus one, this is the famous Taniyama-Shimura conjecture, now a theorem, which was first proved in many cases by Wiles. For curves of genus zero, this is a famous theorem of Riemann, namely, the functional equation of the Riemann zeta function. In order to approach this problem, the PI has (with David Geraghty) developed a generalization of the Taylor-Wiles method which may conceivably apply in this case. Much of his effort will be devoted to trying to overcome the numerous technical obstacles which are required to carry out this argument, including local-global compatibility for Galois representations, and vanishing of cohomology groups outside certain ranges. More generally, the PI intends to further develop this general approach to modularity questions in other contexts, including Galois representations conjecturally associated to automorphic forms for GL(n) over the rationals. He also plans to study applications of these results to the Bloch-Kato conjecture and establishing links between K-theory, completed cohomology, and the Langlands program.

理解质数的问题可以追溯到古代。19世纪出现的一个自然问题是计算某个变量X的小于X的素数的数量。解决方法出人意料：这个离散计数问题原来与一个连续函数的性质有关，这个连续函数最初是由欧拉研究的，现在被称为黎曼ζ函数。这个突破取决于黎曼发现的这个函数的一个关键性质，即它有一个隐藏的对称性，它将函数在s点的值与1-s点的值联系起来。这种隐藏的对称性只是冰山一角；有一大类相似的函数表现出相似的对称性（至少在推测上），这些函数出现在数学和物理的许多领域。通过证明一类这样的函数具有这种对称性，安德鲁·怀尔斯在1994年证明了费马大定理。PI的工作旨在证明某类函数表现出与黎曼ζ函数相同的对称性。更具体地说，存在一个以自然数g为索引的函数的自然集合。当g为零时，存在一个由欧拉和黎曼考虑的函数。当g = 1时，这类函数就是Wiles研究过的。PI计划研究g = 2时的函数。PI的研究（包括与Emerton， Geraghty和Venkatesh的合作）的一个总体主题是对上同调中的扭转类的研究，包括在算法群的Betti上同调和Shimura变种的相干上同调中，以及它们与Langlands程序的推测关系。PI的工作表明，理解扭转是理解超越志村变异的互惠的核心。提出的项目的主要长期技术目标之一是证明附属于属二曲线的l函数满足预期的函数方程。对于属1的曲线，这是著名的谷山-志村猜想，现在是一个定理，它在很多情况下是由怀尔斯首先证明的。对于属零的曲线，这是著名的黎曼定理，即黎曼ζ函数的泛函方程。为了解决这个问题，PI（与David Geraghty）开发了一种泰勒-怀尔斯方法的推广方法，可以想见，这种方法可能适用于这种情况。他的大部分努力将致力于克服实现这一论证所需的众多技术障碍，包括伽罗瓦表示的局部-全局兼容性，以及在一定范围外上同群的消失。更一般地说，PI打算在其他情况下进一步发展这种通用方法来解决模块化问题，包括与理性上的GL(n)的自同构形式推测相关的伽罗瓦表示。他还计划研究这些结果在布洛赫-加藤猜想中的应用，并建立k理论、完全上同论和朗兰兹计划之间的联系。