Counting rational points on smooth cubic surfaces

• Main Author: Sofos, Efthymios
• Published: University of Bristol 2015
• Subjects: 515
• Online Access: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.702750

We develop a method that is capable of proving lower bounds that are consistent with Manin's conjecture for the number of rational points of bounded height on Fano varieties for which establishing Manin's conjecture is far out of reach with current technology. More specifically we develop the fibration method from the perspective of analytic number theory and as an application we provide correct lower bounds for families of smooth cubic surfaces with a rational line. The key ingredient of our approach is an asymptotic formula with a precise error term for Manin's conjecture for general smooth conics. Using the geometry of conic bundles we transform the average of the Peyre constants of the fibers into divisor sums over values of quintic binary forms. The underlying divisor functions include Dirichlet convolutions of quadratic characters whose modulus is unbounded, an element which is new in the large body of work on divisors sums. In the concluding chapter we combine the fibration method with sieve techniques to attack the saturation number problem. We show that rational points with at most 22 prime factors counted with multiplicity form a Zariski dense subset of each smooth cubic surface with two rational skew lines. This chapter is part of joint work with Yuchao Wang.