New Directions in Modularity

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变量X的素数小于X的个数。解决方法出人意料：这个*离散*计数问题被证明与欧拉首先研究的*连续*函数的性质有关，但现在被称为Riemann Zeta函数。这一突破取决于黎曼发现的这个函数的一个重要性质，即它具有一个隐藏的对称性，该对称性将“S”点上的函数值与“1-S”点上的函数值联系起来。这种隐藏的对称性被证明只是冰山一角；有一大类相似的函数表现出相似的对称性(至少在猜想上是这样)，这些函数出现在数学和物理的许多领域。通过证明某类这样的函数具有这种对称性，Andrew Wiles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Riemann Zeta函数相同的对称性。更具体地说，存在由自然数g索引的这类函数的自然集合。当g为零时，存在欧拉和Riemann所考虑的单个函数。当g为1时，函数类正是Wiles所研究的函数类。PI计划研究当g为2时产生的函数。PI的研究(包括与Emerton、Geraghty和Venkatesh的合作)的一个首要主题是上同调中的扭类的研究，包括算术群的Betti上同调和Shimura变种的凝聚上同调，以及它们与朗兰兹程序的猜想关系。PI的工作表明，理解扭转是理解超越下村变化的互惠的核心。该项目的主要长期技术目标之一是证明隶属于亏格两条曲线的L函数满足期望的函数方程。对于亏格1的曲线，这是著名的谷山-下村猜想，现在是一个定理，在许多情况下首先由Wiles证明。对于亏格为零的曲线，这是一个著名的黎曼定理，即黎曼Zeta函数的函数方程。为了解决这个问题，PI(和David Geraghty)发展了Taylor-Wiles方法的一个推广，可以想象它可以在这种情况下应用。他的大部分努力将致力于试图克服进行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，以及某些范围外上同调群的消失。更广泛地说，PI打算在其他上下文中进一步发展这种一般方法来解决模性问题，包括猜想与GL(N)的有理数上的自同构形式相关联的伽罗瓦表示。他还计划研究这些结果在Bloch-Kato猜想中的应用，以及在K理论、完全上同调和朗兰兹计划之间建立联系。

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.