Modular representation theory, Hilbert modular forms and the geometric Breuil-Mézard conjecture.

模表示理论、希尔伯特模形式和几何布勒伊-梅扎德猜想。

• 批准号: EP/W001683/1
• 负责人: Hanneke Wiersema
• 金额: $ 38.97万
• 依托单位: University of Cambridge
• 依托单位国家: 英国
• 项目类别: Fellowship
• 财政年份: 2022
• 资助国家: 英国
• 起止时间: 2022 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FW001683%2F1
• 关键词: Modular; representation; theory; Hilbert; modular

The Langlands program is one of the most striking programs in mathematical research today. It suggests deep connections between algebra, geometry, number theory and analysis. One of the biggest results in this program to date is the proof of Fermat's Last Theorem by Sir Andrew Wiles, which sparked headlines in the national newspapers. The result is as follows: let n be an integer greater than 2, then there are no non-zero integers x, y and z such that x^n + y^n = z^n. While this is a relatively accessible statement, the proof of this result came more than three centuries after it was first conjectured and the efforts towards proving it led to huge innovations in number theory and beyond. The key to proving Fermat's Last Theorem was a deep connection between mathematical objects called modular forms and elliptic curves: the modularity theorem. My work is on a result closely related to this theorem: Serre's modularity conjecture. This was stated by Jean-Pierre Serre in 1973 and refined in 1987, and in fact, has Fermat's Last Theorem, as one of its consequences. The equation in Fermat's Last Theorem is an example of a Diophantine equation, named after the ancient Greek mathematician Diophantus. These are polynomial equations with integer coefficients whose solutions are also restricted to integers. Solving Diophantine equations is one of the main goals of number theory. One way to try to study such equations is to investigate Galois representations, which are mathematical objects capturing symmetries of Diophantine equations. Serre discovered that such representations can be studied by looking at modular forms, which are functions satisfying some nice symmetry properties. This is called the "weak" version of Serre's modularity conjecture. The "strong" version describes the properties of the modular form that corresponds to any given Galois representation. Serre's discoveries were eventually proved correct by Chandrashekhar Khare and Jean-Pierre Wintenberger, building on work of many other mathematicians. The proof relies heavily on earlier work showing that the "weak" version implies the "strong" version. Inspired by work of Kevin Buzzard, Fred Diamond and Frazer Jarvis, I aim to achieve results similar to Serre's modularity conjecture, but in a more general context. This means I work with more complex Galois representations and this causes many intricacies and complications, and this is wherein most of my research lies. I further study other ingredients that also featured in Wiles' proof of Fermat's Last Theorem, in particular geometric objects called "Galois deformation rings". These rings are mathematical objects that carry information about Galois representations. Specifically, they tell you what happens when you take a Galois representation and try to alter it a bit. The geometry of such rings can be described in terms of so-called representation theory: this is a famous result called the Breuil-Mézard conjecture. Matthew Emerton and Toby Gee discovered it was possible to give a more precise description of this geometry of these rings. My work so far has the potential to further enhance Emerton and Gee's work.Building on my work to date, my proposal has three primary goals: (A) to make new advances in modular representation theory,(B) to prove a weight version of "weak" Serre implies "strong" Serre inspired by work of Fred Diamond and Shu Sasaki,(C) apply my work to refine the geometric Breuil-Mézard conjecture.

朗兰兹纲领是当今数学研究中最引人注目的纲领之一。它暗示了代数、几何、数论和分析之间的深刻联系。迄今为止，该计划最大的成果之一是安德鲁·怀尔斯爵士对费马大定理的证明，这一证明在全国性报纸上引起了轰动。结果如下：设n是大于2的整数，则不存在非零整数x、y和z使得x^n + y^n = z^n。虽然这是一个相对容易理解的陈述，但这一结果的证明是在它首次被证明之后的三个多世纪，并且证明它的努力导致了数论和其他领域的巨大创新。证明费马大定理的关键是在称为模形式的数学对象和椭圆曲线之间的深刻联系：模性定理。我的工作是密切相关的结果这个定理：塞尔的模块性猜想。这是由Jean-Pierre Serre在1973年提出并在1987年完善的，事实上，费马大定理是其结果之一。费马大定理中的方程是丢番图方程的一个例子，以古希腊数学家丢番图的名字命名。这些是具有整数系数的多项式方程，其解也仅限于整数。求解丢番图方程是数论的主要目标之一。尝试研究此类方程的一种方法是研究伽罗瓦表示，它是捕获丢番图方程对称性的数学对象。塞尔发现，这种表示可以通过查看模形式来研究，模形式是满足一些很好的对称性质的函数。这被称为塞尔模块性猜想的“弱”版本。“强”版本描述了对应于任何给定伽罗瓦表示的模形式的性质。塞尔的发现最终被证明是正确的Karrashekhar Khare和让-皮埃尔·温滕贝格，建立在工作的许多其他数学家。这个证明在很大程度上依赖于早期的工作，表明“弱”版本意味着“强”版本。受到Kevin Buzzard，Fred Diamond和弗雷泽贾维斯工作的启发，我的目标是实现类似于Serre的模块性猜想的结果，但在更一般的背景下。这意味着我的工作涉及更复杂的伽罗瓦表示，这会导致许多复杂和并发症，这就是我的大部分研究所在。我进一步研究了怀尔斯证明费马大定理的其他要素，特别是被称为“伽罗瓦变形环”的几何对象。这些环是携带伽罗瓦表示信息的数学对象。具体地说，它们告诉你，当你取一个伽罗瓦表示，并试图改变它一点时，会发生什么。这样的环的几何可以用所谓的表示论来描述：这是一个著名的结果，称为Breuil-Mézard猜想。马修·埃默顿和托比·吉发现有可能对这些环的几何形状给出更精确的描述。我的工作到目前为止有可能进一步加强Emerton和Gee的工作。在我迄今为止的工作的基础上，我的建议有三个主要目标：（A）在模表示理论方面取得新的进展，（B）证明“弱”Serre的重量版本意味着“强”Serre受到Fred Diamond和Shu Sasaki工作的启发，（C）应用我的工作来改进几何Breuil-Mézard猜想。

项目成果

Integrality of twisted $L$-values of elliptic curves

椭圆曲线扭曲$L$值的积分

• DOI: 10.4171/dm/x25
• 发表时间: 2022
• 期刊: Documenta Mathematica
• 影响因子: 0.9
• 作者: Wiersema H