The density of S-integral points in projective space with respect to a quadric

The density of S-integral points in projective space with respect to a quadric by Nic Niedermowwe (Oxford)1. Introduction.Let S = {p 1 , . . ., p m , ∞} be a finite set of places of Q, including the archimedean one.The ring of S-integers is given by Its multiplicative subgroup O *S consists of the S-units, i.e. those elements of O S whose p-adic absolute value equals 1 for all p ∈ S. Given a smooth projective algebraic variety X over Q, and a smooth hypersurface D ⊂ X defined by a form F ∈ Z[x 1 , . . ., x n ], we say that x = (x 1 , . . ., x n ) ∈ X is a (D, S)-integral point, or an S-integral point with respect to D, if F (x) ≡ 0 (mod p) for all p ∈ S. We are interested in the asymptotic behaviour of the counting function-integral points of bounded height in the case where X = P n-1 and F is a quadratic form.Here P is a real parameter that tends to infinity, points in projective space are represented by primitive integral tuples, and B is some n-dimensional hyperrectangle in R n centred at the origin.In this form the problem corresponds to the degree two case of a question raised by Tschinkel [5, Problem 5.6].Before we can give a precise statement of our main result, it is necessary to introduce some notation.We write F for the matrix of F given by F (x) = 1 2 x T Fx, and let M be a real orthogonal matrix that diagonalises F. Accordingly, we choose B such that the edges of M T B are parallel to the coordinate axes.The set Λ shall consist of all primes p such that p | 2 det F but p ∈ S. We let ∆ be the set of all m + 1-tuples δ with entries in {0, 1}, and write p δ = (-1) δ 0 p δ 1 1 • • • p δm m for short.For a given prime p ∈ Λ ∪ S and