Let $\chi$ be a totally odd character of a totally real number field. In 1981, B. Gross formulated a p-adic analogue of a conjecture of Stark which expresses the leading term at s=0 of the p-adic L-function attached to $\chi\omega$ as a product of a regulator and an algebraic number. Recently, Dasgupta-Darmon-Pollack proved Gross' conjecture in the rank one case under two assumptions: that Leopoldt's conjecture holds for F and p, and a certain technical condition when there is a unique prime above p in F. After giving some background and outlining their proof, I will explain how to remove both conditions, thus giving an unconditional proof of the conjecture. If there is extra time I will explain an application to the Iwasawa Main Conjecture which comes out of the proof, and make a few remarks on the higher rank case.

After defining exterior powers of \pi-divisible modules, we prove that the exterior powers of \pi-divisible modules of dimension at most one over any base scheme exist and their construction commute with arbitrary base change

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.