Sieving on Projective Varieties

Abstract

We consider a system of quadratic forms F_1,...,F_R with integer coefficients. Generalizing the work of Myerson we find an asymptotic formula for the number of integral zeros within a growing box which also lie in suitable residue classes, i.e. fix natural numbers m,n and let Omega_p be a chosen subset of (Z/p^mZ)^n then we count all x=(x_1,...,x_n) in Z^n with |x_i|<P such that F_1(x),..,F_R(x)=0 and x \in Omega_p. This is done by using Selberg's sieve. As an application we study the number of rational points that lie in a thin set in the intersection of a system of quadratic forms F_1,...,F_R. Lastly we find a lower bound for the number of x in Z^n with |x_i|<P such that F_1(x),...,F_R(x)=0 and x_1*...* x_n is a positive integer with at most r prime divisors outside some given finite set.