New Approaches To Modularity

纵观数论的历史，一个共同的主题是算术和分析之间意想不到的联系。连接算术和分析的一个关键统一概念是L函数的概念。定义L-函数有两种根本不同的方法。一种方法是计算某些多项式方程模素数的解的数量，然后将所得的数字放入某个生成函数中。另一种方法是从对称流形上的某些微分方程的高度对称解（自守形式）开始，然后使用分析根据这些解定义L函数。朗兰兹纲领中的互易性概念是这样一种猜想，即所有用多项式方程定义的L-函数也可以用自守形式定义。这种理论上的联系在数论和调和分析中都有着深远的意义。 本研究计划旨在进一步探讨倒易猜想及其意义。算术中一些最基本的有趣的L-函数类源于二元整数系数多项式方程（有理数上的代数曲线）。与这样的多项式相关联的是一个不变量g（亏格），它是一个非负整数。如果亏格g为零，则所得到的L-函数可以用黎曼zeta函数表示，在这种情况下，互易性来自黎曼的工作。如果亏格g是1，则互易猜想等价于谷山-志村猜想，现已得到验证。这个项目的目标是在亏格为2的情况下取得重大进展。

Throughout the history of number theory, a common theme is the unexpected link between arithmetic and analysis. A key unifying concept that links arithmetic and analysis is the notion of an L-function. There are two fundamentally different ways to define L-functions. One way is to count the number of solutions to certain polynomial equations modulo primes, and then to put the resulting numbers into a certain generating function. The other way is to start with highly symmetric solutions to certain differential equations on symmetric manifolds (automorphic forms) and then to define L-functions in terms of these solutions using analysis. The idea of reciprocity in the Langlands program is the conjecture that all L-functions defined in terms of polynomial equations can also be defined in terms of automorphic forms. This conjectural link is expected to have deep implications in both number theory and harmonic analysis. This research project aims to further explore the reciprocity conjecture and its implications.Some of the most basic interesting classes of L-functions in arithmetic arise from polynomial equations with integer coefficients in two variables (algebraic curves over the rational numbers). Associated to such a polynomial is an invariant g (the genus) which is a non-negative integer. If the genus g is zero, then the resulting L-function can be expressed in terms of the Riemann zeta function, and reciprocity in this case follows from Riemann's work. If the genus g is one, then the reciprocity conjecture is equivalent to the Taniyama-Shimura conjecture, now verified. The goal of this project is to make significant progress on the case when the genus g is two.