Eigencurve was introduced by Coleman and Mazur to parametrize
modular forms varying p-adically. It is a rigid analytic curve such that
each point corresponds to an overconvegent eigenform. In this talk, we
discuss a result on the geometry of the eigencurve: over the boundary
annuli of the weight space, the eigencurve breaks up into infinite disjoint
union of connected components and the weight map is finite and flat on each
component. This was first observed by Buzzard and Kilford by an explicit
computation in the case of p=2 and tame level 1. We will explain a
generalization to the definite quaternion case with no restriction on p and
the tame level. This is a joint work with Ruochuan Liu and Daqing Wan,
based on an idea of Robert Coleman.

The Birch--Swinnerton-Dyer conjecture is now a theorem, under some mild hypotheses, for elliptic curves over Q with analytic rank ≤ 1. One of the main ingredients in the proof is Kolyvagin's theory of Euler systems: compatible families of cohomology classes which can be seen as an "arithmetic avatar'' of an L-function. The existence of Euler systems in other settings would have similarly strong arithmetical applications, but only a small number of examples are known.

In this talk, I'll introduce Euler systems and their uses, and I'll describe the construction of a new Euler system, which is attached to the Rankin--Selberg convolution of two modular forms; this is joint work with Antonio Lei and David Loeffler. I'll also explain recent work with Loeffler and Guido Kings where we prove an explicit reciprocity law for this Euler system, and use this to prove cases of the BSD conjecture and the finiteness of Tate--Shafarevich groups.

For a field of definition $k$ of an abelian variety $\Av$ and prime ideal $\ip$ of $k$ which is of a good reduction for $\Av$, the structure of $\Av(\F_{\ip})$ as abelian group is:

    \Av(\F_{\ip})\simeq \Z/d_1(\ip)\Z\oplus\cdots\oplus\Z/d_g(\ip)\Z\oplus\Z/e_1(\ip)\Z\oplus\cdots\oplus\Z/e_g(\ip)\Z,

    where $d_i(\ip)|d_{i+1}(\ip)$, $d_g(\ip)|e_1(\ip)$, and $e_i(\ip)|e_{i+1}(\ip)$ for $1\leq i<g$.

    We are interested in finding an asymptotic formula for the number of prime ideals $\ip$ with $N\ip<x$, $\Av$ has a good reduction at $\ip$, $d_1(\ip)=1$. We succeed in this under the assumption of the Generalized Riemann Hypothesis (GRH). Unconditionally, we achieve a short range asymptotic for abelian varieties of CM type, and the full cyclicity theorem for elliptic curves over a number field containing CM field.

We will describe an explicit reciprocity law for generalised Heegner cycles, relating the images of certain twists of these classes under the Bloch-Kato dual exponential map to certain Rankin-Selberg L-values, and explain the applications of this formula to the proof of certain rank 0 cases of the Bloch-Kato conjecture. This is a joint work with M.-L. Hsieh.