Density of integer solutions to diagonal quadratic forms

Abstract

Let Q be a non-singular diagonal quadratic form in at least four variables. We provide upper bounds for the number of integer solutions to the equation Q = 0, which lie in a box with sides of length 2B, as B → ∞. The estimates obtained are completely uniform in the coefficients of the form, and become sharper as they grow larger in modulus.

Density of rational points on a quadric bundle in $\mathbb{P}^3\times \mathbb{P}^3$

“…The variety X defines a smooth hypersurface of bidegree (1,2) and has Picard group Pic(X) ∼ = Z 2 . If a point (x, y) ∈ X(Q) is represented by a vector (x, y) ∈ Z 4 prim × Z 4 prim , then we shall take H(x, y) = |x| 3 Manin Conjecture [9], one might expect that there is a Zariski open subset U ⊂ X such that N(U(Q), B) ∼ cB log B, as B → ∞, where c is the constant predicted by Peyre [19]. We certainly require U to exclude all subvarieties of the form x i = x j = x k = y l = 0 for {i, j, k, l} = {1, 2, 3, 4}, since the rational points on X which satisfy these constraints are easily seen to contribute ≫ B 3/2 to N(X(Q), B).…”

“…Our result adds to the small store of examples in which thin sets have been shown to exert a demonstrable influence on the distribution of rational points on Fano varieties. One of the first examples in this vein was discovered by Batyrev and Tschinkel [1], who showed that the split cubic surfaces in the biprojective hypersurface {x 1 y 3 1 + • • • + x 4 y 3 4 = 0} ⊂ P 3 × P 3 contribute significantly more than the Manin conjecture would predict for the number of rational points of bounded anticanonical height. More recently, Le Rudulier [18] has investigated Manin's conjecture for the Hilbert schemes Hilb 2 (P 1 ×P 1 ) and Hilb 2 (P 2 ), with the outcome that a thin set of rational points needs to be removed in order for the associated counting functions to behave as they should.…”

On the representation of integers by quadratic forms

“…The kernel of our work is [12, Theorems 1 and 2]. For any q ∈ N, and any c ∈ Z n , we define the sum 1) and the integral…”

“…When k = 0 it is of independent interest to try and obtain versions of this result for the counting function N w (Q; B) associated to a weight w : R n → R 0 that approximates the characteristic function of [−1, 1] n , since this amounts to counting rational points of bounded height on the quadric hypersurface Q = 0 in P n−1 . This line of enquiry has been pursued by the first author [1] for diagonal quadratic forms. A novel feature of this work is that quaternary forms are handled, these not being touched upon in the present work when k = 0.…”

Sums of four squareful numbers

“…We can write the integrand as e(±θΦ(x i )), where Φ( 2], and so we can apply the second derivative test as found in [26,Ch. 8,Proposition 2.3] to bound this integral by |θ| −1/2 .…”