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.

兰兰兹计划是当今数学研究中最引人注目的计划之一。它提出了代数，几何，数字理论和分析之间的深厚联系。迄今为止，该计划中最大的结果之一是安德鲁·威尔斯爵士（Andrew Wiles）爵士的最后定理证明，这引发了国家报纸的头条新闻。结果如下：让n成为大于2的整数，然后没有非零整数x，y和z，以使x^n + y^n = z^n。虽然这是一个相对可访问的陈述，但该结果的证明是在首次猜想的三个世纪以来出现的，并为其提供的努力导致了数字理论及以后的巨大创新。证明Fermat的最后一个定理的关键是数学对象之间的深厚连接称为模块化形式和椭圆曲线：模块化定理。我的工作与该理论密切相关的结果：Serre的模块化猜想。 Jean-Pierre Serre于1973年陈述，并于1987年进行了完善，实际上，Fermat的最后一个定理是其后果之一。 Fermat的最后定理中的方程式是用古希腊数学家毒理学命名的双志方程的一个例子。这些是具有整数系数的多项式方程，其解决方案也仅限于整数。求解双苯胺方程是数字理论的主要目标之一。尝试研究此类方程式的一种方法是研究Galois表示，即捕获双磷灰石方程对称的数学对象。 Serre发现可以通过查看模块化形式来研究此类表示，这些模块形式可以满足一些不错的对称属性。这称为Serre的模块化猜想的“弱”版本。 “强”版本描述了与任何给定的Galois表示相对应的模块化形式的属性。 Chandrashekhar Khare和Jean-Pierre Wintenberger最终证明了Serre的发现是正确的，这是许多其他数学家的作品。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。受凯文·巴扎德（Kevin Buzzard），弗雷德·戴蒙德（Fred Diamond）和弗雷泽·贾维斯（Frazer Jarvis）的作品的启发，我的目标是取得类似于塞尔（Serre）模块化概念的结果，但在更一般的情况下。这意味着我使用更复杂的Galois表示，这会导致许多复杂性和并发症，这就是我大多数研究所在。我进一步研究了威尔斯的最后一个定理证明，特别是几何对象，尤其是称为“ galois变形环”中的其他媒介。这些环是携带有关GALOIS表示的信息的数学对象。具体来说，他们告诉您当您进行Galois表示并尝试改变它时会发生什么。可以用所谓的表示理论来描述这种环的几何形状：这是一个名为Breuil-Mézard猜想的著名结果。 Matthew Emerton和Toby Gee发现，有可能对这些戒指的几何形状进行更精确的描述。到目前为止，我的工作有可能进一步增强艾默顿和吉的工作。迄今为止的工作，我的建议具有三个主要目标：（a）在模块化代表理论方面取得新的进步，（b）证明“弱” serre的重量版本暗示着“强烈”的“强”塞雷，这是受Fred Diamond and Shu Sasaki的启发而启发的，（c）将我的启发性地启发了我的启发。

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