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.

When a holomorphic modular form is a newform, its L-function has nice analytic properties and associates a cuspidal automorphic representation, which is a restricted product of local representations. To recover the newform from the representation, Casselman considered the fixed line of the congruence subgroups of GL(2) at the conductor level on the local representations. A vector on this line shall encode the conductor, the L-function and the \epsilon-factor of the representation. This is called the theory of newforms for GL(2). Similar theory has been established for some groups of small ranks as well as GL(n). In this talk I will introduce one for SO(2n+1).