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.

In joint work with Raf Cluckers, we propose a conjecture for exponential sums which generalizes both a conjecture by Igusa and a local variant by Denef and Sperber, in particular, it is without the homogeneity condition on the polynomial in the phase, and with new predicted uniform behavior. The exponential sums have summation sets consisting of integers modulo p^m lying p-adically close to y, and the proposed bounds are uniform in p, y, and m. We give evidence for the conjecture, by showing uniform bounds in p, y, and in some values for m. On the way, we prove new bounds for log-canonical thresholds which are closely related to the bounds predicted by the conjecture.