The Gross-Kudla-Schoen modified diagonal cycle on the triple product of the modular curve with itself provides a wealth of arithmetic information about modular forms, including derivatives of complex L-functions, special values of p-adic L-functions, and points on elliptic curves, known as Chow-Heegner points. In this talk, I will discuss a formula expressing the p-adic logarithm of a Chow-Heegner point in terms of the coefficients of the ordinary projection of certain p-adic modular forms.

We prove a B-SD conjecture for elliptic curves (for the p^infinity Selmer groups with arbitrary rank) a la Mazur-Tate and Darmon in anti-cyclotomic setting, for certain primes p. This is done, among other things, by proving a conjecture of Kolyvagin in 1991 on p-indivisibility of (derived) Heegner points over ring class fields. Some applications follow, for example, the p-part of the refined B-SD conjecture in the rank one case.

We give a Chabauty-like method for finding p-adic approximations to
integral points on hyperelliptic curves when the Mordell-Weil rank of
the Jacobian equals the genus. The method uses an interpretation of
the component at p of the p-adic height pairing in terms of iterated
Coleman integrals.  This is joint work with Amnon Besser and Steffen
Mueller.

Given an ordinary differential equation whose coefficients are
meromorphic functions of a complex variable, the only obstruction to
convergence of local solutions in a disc is the presence of
singularities within the disc. It was observed decades ago that this
fails if one replaces "complex" by "p-adic", e.g., consider the
exponential function. In recent work of Baldassarri, Poineau, Pulita,
and the speaker, it has emerged that the convergence properties of such
solutions in the p-adic case can be described quite simply in terms of
Berkovich analytic geometry. We will give this description (without
assuming any prior familiarity with Berkovich's theory) and mention some
applications to studying wild ramification of covers of p-adic curves.

The local Langlands correspondence is a relationship between representations of the Galois group of a p-adic field F and the rerepresentations of GL_n(F).  Understanding the behavior of the local Langlands correspondence as one  varies Galois representations in families is an important ingredient in  Emerton's recent proof of many cases of the Fontaine-Mazur conjecture. I will explain this question, and its connection to questions involving the Bernstein center, an algebra that acts naturally on a category of representations of GL_n(F).