If we knew that BM controls HP for Châtelet surfaces then...

Appears in collection : Skorobogatov 60

... a positive proportion of Châtelet surfaces would have a rational point. Here we fixed a norm form N, any integer d and we considered the surface N=f for a random degree d integer polynomial f. I will talk about our recent preprint arXiv:2005.02998 with Alexei Skorobogatov, where positive proportion is proved unconditionally.