Almost prime points on cubic surfaces

Sarnak and his collaborators initiated a program to investigate the distribution of points whose coordinates have few prime factors on varieties equipped with a group structure. During the talk we shall concentrate on the case of cubic surfaces defined by $F=0$, where $F$ is an integral cubic form in 4 variables. We prove that there exists an integer $r$ such that rational points for which the product of the coordinates has at most $r$ prime factors form a Zariski dense subset, provided that the cubic surface has one rational point. Moreover, we are able to obtain a rather small $r$ under the assumption that the cubic surface has certain geometric structure. Our approach relies on the circle method, weighted sieve arguments and results on linear equations in primes due to Green and Tao.

个人简介：王玉超，山东大学基础数学博士，上海大学理学院数学系讲师、硕士生导师，研究兴趣包括解析数论中的堆垒问题、丢番图几何等。在Int. Math. Res. Not.，J. Number Theory，Ramanujan J.等学术期刊上发表SCI论文8篇，主持国家自然科学基金青年科学基金项目1项。

邀请人：刘志新