Time: 2018年7月9日 15:00-16:00
Title: Almost prime points on unirational varieties
Abstract: We generalise the definition of the saturation number introduced by Bourgain, Gamburd and Sarnak to investigate the distribu-tion of rational points on unirational varieties whose coordinates have few prime factors. Let X ⊂Pn be a unirational variety defined over Q. Define the saturation number r(X) to be the least number r such that the set of x = (x0,...,xn) ∈Zn+1 for which [x] ∈X and the prim product x0 . . . xn has at most r prime factors, is Zariski dense in X . In this talk, we establish the finiteness of the saturation number for all unirational varieties. Our approach is based on unirationality ar- guments in combination with the weighted sieve. Moreover, we give explicit bounds for several special cases. This is joint work with E. Sofos (Max Planck Institute for Mathematics).