Counting rational points on quadric surfaces

Abstract

We give an upper bound for the number of rational points of height at most B, lying on a surface defined by a quadratic form Q. The bound shows an explicit dependence on Q. It is optimal with respect to B, and is also optimal for typical forms Q.

A uniform estimate for the density of rational points on quadrics

“…11D45 (11D09, 11E12). Theorem 1.1 confirms a conjecture made by Browning and Heath-Brown in [2]. In fact, the main result in [2] gives an estimate for N(Q, B) which actually gets sharper than Theorem 1.1 for suitably generic quaternary quadratic forms (namely those for which the discriminant is close to being square-free and of order Q ).…”

“…The argument involved for the case of four variables is quite similar to the last section of [2], with a slight modification due to the fact that we do not consider the same collection of planes, so we do not get the same number of lines. Consider the planes generated by the various sublattices of dimension 3 that we constructed above.…”

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

“…This is explained further in §4. 5. At this point it is crucial that the quadratic form F (x) is diagonal.…”

Sums of four squareful numbers

“…2 ) implies that there exists integers d, r with dr = k, such that d|x 2 and r|(y 3 1 − y 3 2 ). Observe that since k is square-free, d|x 2 if and only if d|x.…”