Abstract: A famous 1927 conjecture of Emil Artin states that every non-square integer different from -1 is a primitive root modulo infinitely many primes, and that the set of these primes has some explicit positive density. The conjecture was 'proved' after 40 years by Christopher Hooley, but only under assumption of the Generalised Riemann Hypothesis (GRH).

The analogous conjecture for points on elliptic curves over number fields, which goes by the name of Lang-Trotter, remains open.

I will discuss the phenomena in the associated Galois representation that prove the vanishing of the associated density and the finiteness of the set of primes in question. This is joint work with Nathan Jones (Chicago) and Francesco Pappalardi (Rome).

Introduction: Peter Stevenhagen是荷兰莱顿大学数学系教授，研究领域是代数数论。1988年，他于加州大学伯克利分校获得博士学位，博士论文为《Ray class groups and governing fields》 ，导师是Hendrik W. Lenstra, Jr.。

Stevenhagen教授在椭圆曲线理论及其算法方面取得了一系列成果，提出了在几乎多项式时间内构造椭圆曲线的算法，并在密码学中得到深刻应用 。他还在Artin的原根猜想中的校正因子等方面有深入研究。作为欧洲Algant项目主任，自2005年起，他与法、德、意等国伙伴合作，提供为期两年的代数、几何和数论硕士学位课程，特别是很多中国学生受次项目资助得以在欧洲学习代数与数论。他已培养众多优秀学生，桃李满天下。