Counting Rational Points on Algebraic Varieties

In these lectures we will be interested in solutions to Diophantine equations F (x1, . . . , xn) = 0, where F is an absolutely irreducible polynomial with integer coefficients, and the solutions are to satisfy (x1, . . . , xn) ∈ Z. Such an equation represents a hypersurface in A, and we may prefer to talk of integer points on this hypersurface, rather than solutions to the corresponding Diophantine equation. In many cases of interest the polynomial F is homogeneous, in which case the equation defines a hypersurface in Pn−1, and the non-zero integer solutions correspond to rational points on this hypersurface. In this situation the solutions of F (x1, . . . , xn) = 0 form families of scalar multiples, and each family produces a single rational point on the corresponding projective hypersurface. Occasionally we shall encounter systems of two or more equations, and these may correspond to varieties of codimension 2 or more, rather than hypersurfaces. Much work in the theory of Diophantine equations has been directed at showing that certain classes of equations have finitely many solutions. However we shall be interested in those cases where either we expect the number of solutions to be infinite, or we expect the number to be finite but cannot prove it. In these cases it is sensible to ask for bounds on the number of solutions which might lie in a large region max |xi| ≤ B, say.