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Deformations of Saito-Kurokawa type Galois representations

Saito-Kurokawa型伽罗瓦表示的变形

基本信息

批准号：
EP/R006563/1
负责人：
Tobias Berger
金额：
$ 42.29万
依托单位：
University of Sheffield
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2017
资助国家：
英国
起止时间：
2017 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FR006563%2F1
关键词：
Deformations Saito Kurokawa type Galois

项目摘要

This proposal sets out to prove the modularity of abelian surfaces and of elliptic curves over imaginary quadratic fields, the next major challenges in the Langlands programme linking algebraic geometry and automorphic forms. This series of conjectures made by the mathematician Robert Langlands in the 1960s and 70s predicts precise links between three seemingly unrelated classes of objects. These come from representation theory (in the form of modular forms), number theory (Galois representations) and algebraic geometry (e.g. elliptic curves or abelian surfaces). This project will lead to a much better understanding of the arithmetic of abelian surfaces and of Siegel modular forms, which are of interest not only to number theorists and geometers, but also physicists and cryptographers. Establishing links in the Langlands programme enables number theorists to understand more deeply the properties of the objects involved and allows them to prove theorems, such as the famous example of the proof of Fermat's last theorem by Wiles and Taylor in 1994. The key ingredient in Wiles' proof was to establish that there is a modular form whose associated Galois representation agrees with that of an elliptic curve. This proposal will study the modularity of abelian surfaces, one dimension up from the case of elliptic curves. A precise conjecture for this case was recently formulated by Brumer and Kramer predicting that abelian surfaces should correspond to paramodular Siegel modular forms of weight 2. We propose to prove the first general result for this "paramodular conjecture" without assuming residual modularity.For this we will study cases where the abelian surface has a rational torsion point of a prime order p. This means that the corresponding p-adic Galois representation becomes reducible modulo p. When this residual representation has three irreducible constituents, Serre's conjecture (a theorem of Khare-Wintenberger) tells us that its semi-simplification is isomorphic to the Galois representation associated to the Siegel modular form obtained by lifting an elliptic modular form via the Saito-Kurokawa lift. We call such residual representations "of SK-type". The approach pioneered by Wiles for proving the modularity of a Galois representation is to consider deformations of its residual representation, i.e. p-adic Galois representations reducing to this representation modulo p, and to show that they all arise from modular forms. The residually reducible situation, however, poses major challenges for the study of deformations. In joint work with Krzysztof Klosin the PI developed a new approach to the modularity of residually reducible Galois representations with two residual pieces, showing that modularity often follows from congruences between modular forms and instances of the Bloch-Kato conjectures. By generalizing our method we are going to prove so-called R=T theorems for p-adic Galois representations that residually are of SK type, establishing the modularity of all their deformations. In addition to developing new tools in the deformation theory of residually reducible Galois representations this requires studying the p-adic properties of Saito-Kurokawa lifts. In particular, we will construct congruences between Saito-Kurokawa lifts and other Siegel modular forms. To access the non-cohomological weight 2 case, for which classical techniques do not apply, we will prove such congruences for p-adic families. This will allow us to prove the paramodular conjecture for abelian surfaces with rational p-torsion. In addition, we will study the Bianchi modularity of elliptic curves over imaginary quadratic fields, another famous case that has resisted efforts so far, by proving the paramodularity of the abelian surface given by their base change to Q.

该提案旨在证明虚二次域上的阿贝尔曲面和椭圆曲线的模性，这是连接代数几何和自守形式的朗兰兹计划中的下一个主要挑战。数学家罗伯特·朗兰兹 (Robert Langlands) 在 20 世纪 60 年代和 70 年代提出的这一系列猜想预测了三类看似不相关的物体之间的精确联系。这些来自表示论（以模形式）、数论（伽罗瓦表示）和代数几何（例如椭圆曲线或阿贝尔曲面）。该项目将导致人们更好地理解阿贝尔曲面和西格尔模形式的算术，这不仅引起数论学家和几何学家的兴趣，而且引起物理学家和密码学家的兴趣。在朗兰兹纲领中建立联系使数论学家能够更深入地理解所涉及对象的属性，并允许他们证明定理，例如怀尔斯和泰勒在 1994 年证明费马大定理的著名例子。怀尔斯证明的关键要素是确定存在一个模形式，其相关的伽罗瓦表示与椭圆曲线的表示一致。该提案将研究阿贝尔曲面的模性，即椭圆曲线情况的一维。 Brumer 和 Kramer 最近提出了针对这种情况的精确猜想，预测阿贝尔曲面应该对应于权重 2 的拟模西格尔模形式。我们建议在不假设残差模性的情况下证明这种“拟模猜想”的第一个一般结果。为此，我们将研究阿贝尔曲面具有素数阶 p 有理扭转点的情况。这意味着相应的 p 进伽罗瓦表示变成可约模 p。当这个残差表示具有三个不可约成分时，塞尔猜想（Khare-Witenberger 的定理）告诉我们，它的半简化与与通过 Saito-Kurokawa 升力提升椭圆模形式获得的 Siegel 模形式相关的 Galois 表示是同构的。我们将这种残差表示称为“SK 型”。 Wiles 首创的证明伽罗瓦表示模块化的方法是考虑其残差表示的变形，即 p 进伽罗瓦表示简化为以 p 为模的表示，并证明它们都源自模块化形式。然而，残余可还原情况给变形研究带来了重大挑战。在与 Krzysztof Klosin 的合作中，PI 开发了一种新的方法来处理具有两个残差部分的残差可约伽罗瓦表示的模块化，表明模块化通常来自于模块化形式和布洛赫-加藤猜想的实例之间的同余。通过推广我们的方法，我们将证明 p 进伽罗瓦表示的所谓 R=T 定理，该表示仍属于 SK 类型，从而建立其所有变形的模块化。除了在剩余可约伽罗瓦表示的变形理论中开发新工具之外，还需要研究 Saito-Kurokawa 升力的 p-adic 性质。特别是，我们将在 Saito-Kurokawa 电梯和其他 Siegel 模块化形式之间构建一致性。为了访问经典技术不适用的非上同调权重 2 的情况，我们将证明 p-adic 族的此类同余性。这将使我们能够证明具有有理 p 扭转的阿贝尔曲面的拟模猜想。此外，我们将通过证明由 Q 的基数变化给出的阿贝尔曲面的拟模性来研究虚二次域上椭圆曲线的 Bianchi 模性，这是迄今为止尚未成功的另一个著名案例。