The general Ramanujan Conjectures for congruence subgroups of arithmetic groups, and approximations that have been proven towards them, are central to many diophantine applications. Recently analogous results have been established for quite general subgroups of GL(n,Z) called  "thin groups ". We will describe some of these and review some of their applications (mainly diophantine) as well as the ubiquity of thin groups.

A curve is a one dimensional space cut out by polynomial equations. In particular, one can consider curves over finite fields, which means the polynomial equations should have coefficients in some finite field and that points on the curve are given by values of the variables in the finite field that satisfy the given polynomials. A basic question is how many points such a curve has, and for a family of curves one can study the distribution of this statistic. We will give concrete examples of families in which this distribution is known or predicted, and give a sense of the different kinds of mathematics that are used to study different families.

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.