Automorphy and adjoint Selmer groups over CM fields

现代代数数论的一大进步是Shimura-Taniyama猜想的证明，该猜想的陈述是每个有理椭圆曲线都是模。有理椭圆曲线可以理解为满足简单多项式方程的点的集合，而模形式则是复上半平面上的一类全纯函数。Shimura-Taniyama猜想粗略地说，对于除了有限个素数p之外的所有素数p，对描述p的方程取模的解的个数可以从某种模形式的傅里叶系数中得到。这个令人惊奇的想法是，一个算术对象，椭圆曲线，应该与一个（乍一看！）解析对象，模形式相关联，现在被视为朗兰兹哲学的一部分，在过去的40年里，它强烈地影响了数论和相关领域。值得注意的是，在怀尔斯的突破性工作25年后，我们仍然不能证明虚数二次域的自然泛化。然而，在与许多合著者的共同努力下，我们现在才刚刚开始在这个问题上取得重大突破，事情正在以令人兴奋的速度发展。例如，在与Caraiani， Calegari, Gee, Helm, Le Hung, Newton, Scholze， Taylor和Thorne的联合工作中，我们证明了CM场上椭圆曲线的潜在自同构，即椭圆曲线在有限扩展上成为模。进一步证明了CM场上椭圆曲线的Sato-Tate猜想和Ramanujan猜想的新情形。在最近与Khare和Thorne的联合工作中，我们证明了虚二次域上椭圆曲线的正比例的实际自同构，特别是暗示了它们的l函数允许解析延拓。这些作品包含了强有力的成果和富有成效的新思想。本提案涉及这些新技术和成果的一些应用。特别是，我们的目标是提高我们对Langlands方案的局部-全局兼容性的认识，并处理Bloch-Kato和Perrin-Riou的猜想（在伴随情况下）。建立了虚二次域上半稳定椭圆曲线的自同构。最后，我们提出了一种多方面的方法来研究朗兰兹程序和文卡特什程序中的派生结构。

One of the great advances in modern algebraic number theory was the proof of the Shimura-Taniyama conjecture, the statement of which is that every rational elliptic curve is modular. A rational elliptic curve can be thought of as the set of points satisfying a simple polynomial equation, while a modular form is a certain type of holomorphic function on the complex upper half plane. The Shimura-Taniyama conjecture roughly says that for all but finitely many primes p, the number of solutions modulo p of the equation describing the can be read off from the Fourier coefficients of some modular form. This amazing idea that an arithmetic object, the elliptic curve, should be related to an (at first glance!) analytic object, the modular form, is now viewed as part of a philosophy of Langlands that has strongly influenced number theory and related areas in the past 40 years. It's remarkable that 25 years after Wiles's breakthrough work, we still can't prove the natural generalization over imaginary quadratic fields. However, in joint work with many coauthors, we are just now starting to break serious ground on this problem, and things are moving at an exciting pace. For example, in joint work with Caraiani, Calegari, Gee, Helm, Le Hung, Newton, Scholze, Taylor, and Thorne, we prove potential automorphy of elliptic curves over CM fields, i.e. that an elliptic curve becomes modular over finite extension. We furthermore prove the Sato-Tate conjecture for elliptic curves over CM fields, and new cases of the Ramanujan conjecture. In more recent joint work with Khare and Thorne, we prove actual automorphy for a positive proportion of elliptic curves over imaginary quadratic fields, in particular implying that their L-functions admit analytic continuation. These works contain powerful results and fruitful new ideas. This proposal deals with a number of applications of these new techniques and results. In particular, we aim to improve our knowledge of local-global compatibility in the Langlands program and treat conjectures of Bloch-Kato and Perrin-Riou (in the adjoint case). We also aim to establish the automorphy of semistable elliptic curves over imaginary quadratic fields. Finally, we propose a multifaceted approach to studying derived structure in Langlands program and Venkatesh's program.