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。

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.