Analytic Methods for the Distribution of Rational Points on Algebraic Varieties

The most important analytic method for handling equidistribution questions about rational points on algebraic varieties is undoubtedly the HardyLittlewood circle method. There are a number of good texts available on the circle method, but the reader may particularly wish to study the books by Davenport [4] and Vaughan [11]. In this lecture we shall consider an irreducible form F (X1, . . . , Xn) ∈ Z[X1, . . . , Xn] of degree d which defines a hypersurface F = 0 in Pn−1. The fundamental questions will be: are there any rational points on the hypersurface? If so, are they equidistributed in a suitable sense? We shall address these issues by looking at integer points on the affine cone. Thus to ask about equidistribution with respect to measures on Pn−1(R), for example, we may take a small box R := n ∏