Deformations of Saito-Kurokawa type Galois representations

该方案旨在证明虚二次域上的阿贝尔曲面和椭圆曲线的模性，这是朗兰兹程序中连接代数几何和自同构形式的下一个主要挑战。数学家罗伯特·朗兰兹（Robert Langlands）在上世纪六七十年代提出的这一系列猜想预测了三种看似无关的物体之间的精确联系。它们来自表示理论（以模形式的形式），数论（伽罗瓦表示）和代数几何（例如椭圆曲线或阿贝尔曲面）。这个项目将使我们更好地理解阿贝尔曲面和西格尔模形式的算法，这不仅是数论学家和几何学家感兴趣的，也是物理学家和密码学家感兴趣的。在朗兰兹程序中建立联系，使数论学家能够更深入地了解所涉及对象的性质，并使他们能够证明定理，例如1994年怀尔斯和泰勒证明费马大定理的著名例子。怀尔斯证明的关键是建立了一种模形式，其相关的伽罗瓦表示与椭圆曲线的伽罗瓦表示一致。本提案将研究阿贝尔曲面的模性，从椭圆曲线的情况向上一维。对于这种情况，布鲁默和克莱默最近提出了一个精确的猜想，预测阿贝尔曲面应该对应于权值2的旁模西格尔模形式。我们提出在不假设残模性的情况下证明这个“副模猜想”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的有理扭转点的情况。这意味着相应的p进伽罗瓦表示模为p可约。当这个剩余表示有三个不可约成分时，Serre猜想（kare - wintenberger的一个定理）告诉我们，它的半简化与通过Saito-Kurokawa提升椭圆模形式得到的Siegel模形式相关的伽罗瓦表示同构。我们称这种残差表示为“sk型”。Wiles率先证明伽罗瓦表示的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该表示模p，并表明它们都来自模形式。然而，残余可约情况对变形研究提出了重大挑战。在与Krzysztof Klosin的联合工作中，PI开发了一种具有两个残差块的剩余可约伽罗瓦表示的模块化的新方法，表明模块化通常来自于模形式和Bloch-Kato猜想的实例之间的同余。通过推广我们的方法，我们将证明残差为SK型的p进伽罗瓦表示的R=T定理，并建立其所有变形的模性。除了在剩余可约伽罗瓦表示的变形理论中开发新的工具外，这还需要研究Saito-Kurokawa举的p进性质。特别地，我们将构造Saito-Kurokawa升降机与其他Siegel模形式之间的同余。为了获得非上同权2的情况，经典技术不适用，我们将证明p进族的这种同余。这将允许我们证明具有有理p-扭转的阿贝尔曲面的旁模猜想。此外，我们将研究虚二次场上椭圆曲线的Bianchi模性，这是另一个著名的迄今为止难以努力的例子，通过证明由它们的基变化Q给出的阿贝尔曲面的顺模性。

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.