Modularity of Genus Two Curves

数论的一个激励问题是理解多项式方程的所有整数解。一个著名的例子来自于费马，他猜想不存在正整数X，Y， Z的三元组，使得当N大于2时，X^N + Y^N = Z^N。这个问题的最终解决方案（由Andrew Wiles在1994年提出）利用了一类被称为自同构形式的函数。当一个人试图理解一根简单弦的振动时，傅里叶（大约200年前）有了将任何这种振动分解成纯波的洞察力。自同构形式是这些纯波的类似物，除了现在不是在一维（弦）中工作，而是在高维中考虑一种特殊的难以置信的对称构型。多项式方程组和自同构“波”之间最普遍的联系仍然是高度推测性的，直到20世纪60年代才由罗伯特·朗兰兹（Robert Langlands）表述出来。朗兰兹猜想的意义已经远远超出了计算积分解的原始算术问题。在两个变量多项式的情况下，有一个数值不变量（属）来衡量它们的复杂性。朗兰兹对应最简单的例子（当属为零时）在19世纪50年代由黎曼证明，下一个例子（当属为1时）在150年后的怀尔斯和其他人的工作中才得到解决。当前项目的目标是解决属为2的情况。该奖项将支持研究生的研究训练。本课题的主要目标是建立有理数上的两条曲线的模性。主要的方法是加强Boxer-Calegari-Gee-Pilloni最近的工作，他们证明了有理数上的两条曲线是潜在模的。我们期望升级这个定理所需要的技术结果在我们打算追求的朗兰兹程序中也有许多其他的应用。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

One of the motivating problems of number theory is to understand all the solutions in integers to polynomial equations. One famous example is due to Fermat, who conjectured that there were no triples of positive integers X,Y, and Z such that X^N + Y^N = Z^N whenever N greater than 2. The eventual solution of this problem (by Andrew Wiles in 1994) exploited a class of functions known as automorphic forms. When one tries to understand the vibration of a simple string, Fourier (some 200 years ago) had the insight to decompose any such vibration into pure waves. Automorphic forms are the analogs of these pure waves except now instead of working in one dimension (a string) one considers a particular incredibly symmetric configuration in higher dimensions. The most general link between systems of polynomial equations and automorphic “waves" remains highly conjectural, and was only formulated by Robert Langlands in the 1960s. The Langlands conjectures have since found to have far reaching implications beyond the original arithmetic problems of counting integral solutions. In the case of polynomials in two variables, there is a numerical invariant (the genus) which measures their complexity. The simplest instance of the Langlands correspondence (when the genus is zero) was proved by Riemann in the 1850s, and the next case (when the genus is one) was not resolved for another 150 years in the work Wiles and others. The goal of the current project is to resolve the case when the genus is two. The award will support research training of graduate students.The main goal of this project is to established the modularity of genus two curves over the rational numbers. The main approach is to strengthen recent work of Boxer-Calegari-Gee-Pilloni who prove that genus two curves over the rational numbers are potentially modular. We expect that the technical results required to upgrade this theorem should also have many other applications in the Langlands program which we intend to pursue.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.