Some results on binary forms and counting rational points on algebraic varieties

Abstract

In this thesis we study several problems related to the representation of integers by binary forms and counting rational points on algebraic varieties. In particular, we establish an asymptotic formula for $R_F(Z)$, the number of integers of absolute value up to $Z$ which can be represented by a binary form $F$ with integer coefficients, degree $d \geq 3$, and non-zero discriminant. We give superior results when $d = 3$ or $4$, which completely resolves the cases considered by Hooley. We establish an asymptotic formula for the number of pairs $(x,y) \in \bZ^2$ such that $F(x,y)$ is $k$-free, whenever $F$ satisfies certain necessary conditions and $k > 7d/18$. Finally, we give various results on the arithmetic of certain cubic and quartic surfaces as well as general methods to estimate the number of rational points of bounded height on algebraic varieties. In particular, we give a bound for the density of rational points on del Pezzo surfaces of degree $2$. These results depend on generalizations of Salberger's global determinant method in various settings.