### 11 december 2006,
Stephan Baier (IU Bremen, Germany),
The Sato-Tate and Lang-Trotter conjectures about elliptic curves on
average

**Abstract.**
Let $E$ be an elliptic curve over $\ratq$. For any prime number $p$ of
good reduction, let $\lambda_E(p)$ be the trace of the Frobenius
morphism of $E/\fie_p$. Then the number of points on the reduced curve
modulo $p$ equals $p+1- \lambda_E(p)$. By Hasse's theorem, there exists
a unique angle $0\le \theta\le \pi$ such that
$$\lambda_E(p)=\sqrt{p}\left(e^{i\theta_E(p)}+e^{-i\theta_E(p)}\right)=
2\sqrt{p}\cos \theta_E(p).$$
For the case when $E$ does not have complex multiplication, Sato and
Tate formulated a conjecture on the distribution of $\theta_E(p)$ as
$p$ varies. In a recent preprint, R. E. Taylor succeeded in proving the
Sato-Tate conjecture for elliptic curves $E$ that satisfy a certain
mild condition.\newline
In this talk, we will discuss the Sato-Tate distribution in {\it small}
sectors on average over a family of elliptic curves. Moreover, we will
talk about the
Lang-Trotter conjecture on average. The last-mentioned conjecture
predicts an asymptotics for the number of primes $p$ with
$\lambda_E(p)=r$, where $r$ is fixed. \newline
Some of these results are joined work with Liangyi Zhao.