International Congress of Mathematicians

Exponential Sums and Diophantine Problems 215x e [—B,B] S . When s is sufficiently large in terms of d, the circle method shows undermodest geometric conditions that N(B) is asymptotic to the expected productof local densities. For fairly general polynomials, the condition on s is as severe ass > (d— l)2 rf , though for diagonal equations the methods of §2 relax this conditionto s > (1 + o(l))dlogd. However, there is a class of varieties with small dimensionrelative to degree, for which the circle method supplies non-trivial informationconcerning the density of rational points. The idea is to apply a descent processin order to interpret points on the original variety in terms of corresponding pointson a new variety, with higher dimension relative to degree, more amenable to thecircle method.To illustrate this principle, consider a field extension K of Q of degree n withassociated norm form N(x) £ Q[#i,... ,x„]. Also, let I and k be natural numberswith (k,l) = 1, and let a be a non-zero rational number. Then Heath-Brown andSkorobogatov [7] descend from the variety t l (l — t) k = aN(x) to the associatedvariety aN(u) + bN(v) = z n , for suitable integers a and 6. The circle methodestablishes weak approximation for the latter variety, and thereby it is shown thatthe Brauer-Manin obstruction is the only possible obstruction to the Hasse principleand weak approximation on any smooth projective model of the former variety.One can artificially construct further examples amenable to the circle method. Forexample, if we take linearly independent linear forms Fj(x) £ Q[xi,... ,x„] (1