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Modularity of Genus Two Curves

亏格两条曲线的模性

基本信息

批准号：
2001097
负责人：
Francesco Calegari
金额：
$ 65万
依托单位：
University of Chicago
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-08-01 至 2025-07-31
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2001097&HistoricalAwards=false
关键词：
Modularity Genus Two Curves

项目摘要

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.

数论的一个激励问题是理解多项式方程的整数解。一个著名的例子是费马，他证明了不存在正整数X、Y和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）利用了一类称为自守形式的函数。当人们试图理解一个简单的弦的振动时，傅立叶（大约200年前）有洞察力将任何这样的振动分解为纯波。自守形式是这些纯波的类似物，除了现在不是在一维（弦）中工作，而是在更高维度中考虑一种特殊的令人难以置信的对称配置。多项式方程组和自守“波”之间最普遍的联系仍然是高度抽象的，只有罗伯特·朗兰兹在20世纪60年代才提出。朗兰兹定理已经发现有深远的影响超出了原来的算术问题计数积分解决方案。在二元多项式的情况下，有一个数值不变量（亏格）来衡量它们的复杂性。朗兰兹对应的最简单例子（亏格为0时）在19世纪50年代由黎曼证明，而下一个例子（亏格为1时）在怀尔斯等人的著作中又过了150年才得到解决。当前项目的目标是解决亏格为2的情况。该项目的主要目标是建立有理数上亏格两曲线的模性。主要的方法是加强Boxer-Calegari-Gee-Pilloni最近的工作，证明了有理数上的亏格两条曲线是潜在的模。我们希望，技术成果所需的升级这一定理也应该有许多其他的应用程序中朗兰兹计划，我们打算pursued.This奖项反映了NSF的法定使命，并已被认为是值得的支持，通过评估使用基金会的知识价值和更广泛的影响审查标准。

项目成果

期刊论文数量（6）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

Rationality of twists of the Siegel modular variety of genus 2 and level 3

属 2 和级 3 的 Siegel 模变的扭曲合理性

DOI：
10.1090/proc/15854
发表时间：
2022
期刊：
Proceedings of the American Mathematical Society
影响因子：
1
作者：
Calegari, Frank;Chidambaram, Shiva
通讯作者：
Chidambaram, Shiva

Bloch groups, algebraic $K$-theory, units, and Nahm's Conjecture

布洛赫群、代数 $K$ 理论、单位和纳姆猜想

DOI：
10.24033/asens.2537
发表时间：
2023
期刊：
Annales scientifiques de l'École Normale Supérieure
影响因子：
0
作者：
CALEGARI, Frank;GAROUFALIDIS, Stavros;ZAGIER, Don
通讯作者：
ZAGIER, Don

Potential automorphy over CM fields

CM 场上的潜在自同构

DOI：
10.4007/annals.2023.197.3.2
发表时间：
2023
期刊：
Annals of Mathematics
影响因子：
4.9
作者：
Allen, Patrick;Calegari, Frank;Caraiani, Ana;Gee, Toby;Helm, David;Le Hung, Bao;Newton, James;Scholze, Peter;Taylor, Richard;Thorne, Jack
通讯作者：
Thorne, Jack

Abelian surfaces over totally real fields are potentially modular

完全真实域上的阿贝尔曲面可能是模块化的