Automorphy and adjoint Selmer groups over CM fields

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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.