I will report on joint work with M. Rapoport and B. Smithling, on an "arithmetic transfer conjecture" (ATC). Similar to the arithmetic fundamental lemma conjecture (for unramified unitary groups over a p-adic field), the transfer conjecture connects the derivative of a relative orbital integral to an intersection number on a formal moduli space of $p$-divisible groups associated to a special parahoric of a ramified unitary group. Miraculously, this formal moduli space has good reduction.

I'll speak about ongoing joint work with Bruinier, Kudla, Rapoport, and Yang on the modularity of certain generating series of cycles on unitary Shimura varieties, and applications to new cases of Colmez's conjecture on the Faltings heights of CM abelian varieties.

The Hodge-Tate period map is an important, new tool for studying the geometry of Shimura varieties, p-adic automorphic forms and torsion classes in the cohomology of Shimura varieties. It is a G(A_f)-equivariant map from a perfectoid Shimura variety into a flag variety with only an action of G(Q_p) and can be thought of as a p-adic analogue of the Borel embedding. In this talk, I will describe a canonical construction of the Hodge-Tate period map and of automorphic vector bundles for Shimura varieties of Hodge type. This is part of ongoing joint work with Peter Scholze.

I will report on a generalization of the classical Koecher's principle for the coherent cohomology of Shimura varieties.

We prove some new vanishing results for the Bloch-Kato Selmer groups of Adjoint motives. This is joint work with David Geraghty.

I show how Beilinson's Eisenstein symbol give rise to motivic cohomology classes attached to pairs of modular forms of weight >= 2. These motivic cohomology classes can be used to construct an Euler system -- a compatible family of global cohomology classes -- attached to pairs of modular forms, related to the critical values of the corresponding Rankin-Selberg L-function. This is joint work with Kings and Loeffler, extending my previous work with Lei and Loeffler for weight 2 forms. This Euler system has several arithmetic applications, including one divisibility in the Iwasawa main conjecture for modular forms over imaginary quadratic fields, and cases of the finiteness of Tate--Shafarevich groups for elliptic curves twisted by dihedral Artin representations. 

This talk will describe some recent applications of p-adic L-functions and Iwasawa theory to the arithmetic of elliptic curves, especially to establishing the p-parts of the the Birch-Swinnerton-Dyer formula in cases of rank at most one.