11-13【Sinnou David】五教5402 数论讨论班系列报告


题目:On Lehmer problem on semi-abelian varieties

报告人:Sinnou David, 巴黎六大(Université Pierre et Marie Curie)



摘要:The classical Lehmer problem states that the Weil height of a (non torsion) algebraic number of degree $d$ over the rational numbers is at least $c/d$ where $c$ is universal. While still open, good partial results are known. This conjecture is also known to generalize to general multiplicative groups (for these questions it is enough to consider a power of $G_m$ as well as to abelian varieties, provided one replaces the Weil height by the Néron-Tate height. However, while a semi-abelian variety over a number field can also be endowed with a normalized height, natural generalisations of Lehmer's question fail to hold due to a natural obstruction reminiscent of unlikely intersections appearing in Pink-Zilber conjectures. We propose a generalisation of Lehmer's conjecture taking it into account and prove partial results for semi-abelian varieties having a CM (small dimension) base.