Serre-type conjectures and mod p Langlands correspondences

Serre 型猜想和 mod p Langlands 对应

• 批准号: 0902044
• 负责人: Francesco Calegari
• 金额: $ 9.05万
• 依托单位: Northwestern University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-08-01 至 2013-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0902044&HistoricalAwards=false
• 关键词: Serre; type; conjectures; mod; Langlands

"This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5)."The aim of this project is to investigate some mod p aspects of the Langlands program. For GL_2 over the rational numbers, Serre's conjecture, now a theorem of Khare-Wintenberger and Kisin, predicts the modularity of certain mod p Galois representations. Generalizations of Serre's conjecture have been put forth for other reductive groups by various mathematicians, including the PI. One main goal is to prove strong new results towards the weight parts of such conjectures. Another main goal is to make progress towards the formulation of a local mod p (and ultimately also p-adic) Langlands correspondence for reductive p-adic groups, in particular for the group GL_3 over the p-adic numbers. These correspondences are expected to be closely related to the weights in Serre-types conjectures, but have only been understood so far for GL_2 over the p-adic numbers.The broader context of this project is the area of number theory, one of the oldest branches of mathematics. Number theory concerns the study of whole numbers and the solvability of equations in whole numbers. Its most important practical application has been to cryptography, which is crucial for secure data transmission on the internet. Through the amazing conjectural framework of the Langlands Program, number theory interacts with the seemingly unrelated areas of group theory (study of symmetries), analysis and geometry. This has proved to be very fruitful and led to some of the recent successes such as the proof of Fermat's Last Theorem.

“该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。“这个项目的目的是调查朗兰兹纲领的一些现代方面。对于有理数上的GL_2，Serre猜想，现在是Khare-Wintenberger和Kisin的定理，预言了某些mod p Galois表示的模性。塞尔猜想的推广已经被包括PI在内的各种数学家提出用于其他约化群。一个主要的目标是证明强有力的新结果对重量的部分，这样的结构。另一个主要目标是在约化p-adic群（特别是p-adic数上的群GL_3）的局部mod p（最终也是p-adic）朗兰兹对应的公式化方面取得进展。这些对应关系被认为与Serre型代数中的权密切相关，但到目前为止，人们只理解了p-adic数上GL_2的对应关系。这个项目的更广泛的背景是数论领域，数论是数学最古老的分支之一。数论涉及研究整数和整数方程的可解性。它最重要的实际应用是密码学，这对互联网上的安全数据传输至关重要。通过朗兰兹纲领惊人的结构框架，数论与群论（对称性研究）、分析和几何等看似无关的领域相互作用。这已被证明是非常富有成效的，并导致了一些最近的成功，如证明费马大定理。