Browning–Heath-Brown: Counting rational points on quadric surfaces

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Full paper in Markdown. The file's first page, the Discrete Analysis 2018:15 cover, prints "Licensed under a Creative Commons Attribution License (CC-BY)" and names no version, so the term is the Creative Commons Attribution license without a version; the file is arXiv's v3 (stamp "arXiv:1801.00979v3 [math.NT] 7 Sep 2018"), whose arXiv record names only arXiv's non-exclusive distribution license (https://arxiv.org/abs/1801.00979v3, read 2026-10-02), which neither upgrades nor contradicts the printed notice.

T. D. Browning, D. R. Heath-Brown, "Counting rational points on quadric surfaces," Discrete Analysis 2018:15 (2018), 29 pp., DOI.

Overview

Browning and Heath-Brown study the uniform counting function

$$N_Q(B)=\#\{\mathbf x\in\mathbb Z^4_{\mathrm{prim}}:Q(\mathbf x)=0, \ |\mathbf x|\le B\}$$

for a nonsingular integral quaternary quadratic form. Writing $\Delta_Q$ for its discriminant, $\|Q\|$ for its coefficient height, and $\Delta_{\mathrm{bad}}=\prod_{p^e\parallel\Delta_Q,\,e\ge2}p^e$, Theorem 1.1 proves, under $\Delta_{\mathrm{bad}}\le B^{1/20}$,

$$N_Q(B)\ll_\varepsilon \varpi(\Delta_Q)\Delta_{\mathrm{bad}}^{1/4+\varepsilon} \left(\frac{\|Q\|^4}{|\Delta_Q|}\right)^{5/8} \Pi_B\left(B^{4/3}+\frac{B^2}{|\Delta_Q|^{1/4}}\right),$$

where $\varpi(m)=\prod_{p\mid m}(1+p^{-1})$ and $\Pi_B=\prod_{p\le B}(1+\chi(p)/p)$, with $\chi$ induced by $(\Delta_Q/\cdot)$; see (1.1), (1.2), and Theorem 1.1. The implied constant depends only on $\varepsilon$. The result is uniform in the coefficients and removes the $B^\varepsilon$-loss, diagonality hypothesis, and square-free-discriminant hypothesis of the earlier result cited in the introduction. For a fixed form with a nontrivial zero, the asymptotics $c_QB^2$ for nonsquare discriminant and $c_QB^2\log B$ for square discriminant are cited from Heath-Brown [9, Theorems 6 and 7], not proved here. Conjecture 1.2, also not proved, proposes coefficient-independent bounds of these respective orders. The example $k(x_1^2+x_2^2+x_3^2-x_4^2)$ following (1.3) shows that any estimate of the displayed shape (1.3) must have exponent $\alpha\ge1/4$ on $\Delta_{\mathrm{bad}}$.

The proof begins with a Siegel-lemma slicing. Lemma 2.1 assigns every point to a primitive hyperplane $\mathbf c\cdot\mathbf x=0$ with $|\mathbf c|\ll B^{1/3}$, giving the sum (2.1) over $O(B^{4/3})$ ternary conics. For $Q^*(\mathbf c)\ne0$, Lemma 2.2 covers each real slice by logarithmically many ellipsoids whose volumes are controlled explicitly by $Q^*(\mathbf c)$, $\|Q\|$, $\Delta_Q$, $|\mathbf c|$, and $B$; the determinant identities driving this estimate are (2.3)–(2.5). Lemmas 2.3 and 2.4 convert these ellipsoids into coordinate boxes adapted to suitable lattice bases.

The arithmetic treatment of each conic combines the elementary box estimate of Lemma 2.5 with the local lattice decomposition of Lemma 2.6. The latter covers primitive zeros of a nonsingular ternary form $q$ by lattices satisfying the determinant lower bound (2.11), with the number of lattices governed by explicit local factors and possibly equal to zero when there is a local obstruction. For the hyperplane restriction $Q_{\mathbf c}$, Lemma 2.7 gives the key identity $\det M_{\mathbf c}=Q^*(\mathbf c)$, and Lemma 2.8 transfers the local determinant bound back to the original three-dimensional lattice. Lemma 2.9 then bounds the number of points on one slice in terms of the multiplicative function $R$ defined in (2.16). Summing the slice bounds yields the reduction (2.17), involving averages of $R(Q^*(\mathbf c))$.

Section 3 supplies the principal analytic input. Lemma 3.1 computes $\varrho(p)=p^3+(\Delta_Q/p)(p^2-p)$ away from $2\Delta_{\mathrm{bad}}$ and bounds all $\varrho(p^k)$. Theorem 3.2 is a Shiu-type short-box estimate: subject to the polynomial-size condition (3.2), for $h\mid\Delta_{\mathrm{bad}}^3$ and $h\le X^{1-\varepsilon}$, it bounds the average of $R(|Q^*(\mathbf x)|)$ over a box and the congruence $h\mid Q^*(\mathbf x)$ by

$$\ll_{A,\varepsilon}\Delta_{\mathrm{bad}}^\varepsilon h^{-1} (\Delta_{\mathrm{bad}}^3,h^4)^{1/4}\, \mathfrak S\frac{X^4}{\log X}.$$

Its proof uses the Selberg-sieve estimate in Lemma 3.3 and the Euler-factor analysis of Lemma 3.4.

In Section 4, Lemma 4.1 covers dyadic regions in $(\mathbf c,Q^*(\mathbf c))$ by boxes of side $X=B^{1/6}$; Lemma 4.2 disposes of forms with exceptionally large coefficients and reduces to primitive forms of controlled height. Applying Theorem 3.2 and estimating $\mathfrak S$ by Mertens’ theorem produces Theorem 1.1 for nonsquare discriminant. Section 5 handles square discriminant: rational lines can occur, but their Plücker points form a conic, and the resulting additional contribution is $O(B^2\log B)$; the paper shows that this is absorbed by the stated bound. Thus the scope is homogeneous nonsingular quadrics in four variables, with explicit but potentially large dependence on coefficient shape and square-full discriminant, and with the quantitative restriction $\Delta_{\mathrm{bad}}\le B^{1/20}$.

Relation to E940

This source bears on Problem 940.

The paper neither formulates nor proves a density statement for any of the sets in E940.