Automorphy and adjoint Selmer groups over CM fields

现代代数数论的最大进展之一是证明了Shimura-Taniyama猜想，该猜想的陈述是每条有理椭圆曲线都是模的。有理椭圆曲线可以被认为是满足简单多项式方程的点的集合，而模形式是复杂上半平面上的某种类型的全纯函数。Shimura-Taniyama猜想粗略地说，对于除有限多个素数p之外的所有素数p，描述p的方程的模p的解的数目可以从某种模形式的傅里叶系数中读出。这个惊人的想法是，一个算术对象，即椭圆曲线，应该与一个(乍一看！)分析对象，即模数形式，现在被视为朗兰兹哲学的一部分，在过去的40年里，该哲学强烈地影响了数论和相关领域。 值得注意的是，在Wiles的突破性工作25年后，我们仍然不能证明虚二次域上的自然推广。然而，在与许多合著者的合作中，我们刚刚开始在这个问题上取得重大进展，事情正在以令人兴奋的速度发展。例如，在与Caraiani、Calegari、Gee、Helm、Le Hung、牛顿、Scholze、Taylor和Thorne的联合工作中，我们证明了CM域上椭圆曲线的潜在自同构，即椭圆曲线在有限扩张上成为模。进一步证明了Cm域上椭圆曲线的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的正比例椭圆曲线的实际自同构，特别地，暗示了它们的L函数允许解析连续。 这些作品蕴含着强大的成果和丰硕的新思想。本提案涉及这些新技术和成果的一些应用。特别是，我们的目标是在朗兰兹程序中提高我们对局部-全局相容的了解，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，我们提出了一个多方面的方法来研究朗兰兹程序和Venkatesh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.