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.