Heath-Brown: The density of rational points on cubic surfaces

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corollary_p3: For every epsilon > 0, at most O(x^{4/9+epsilon}) positive integers up to x have two or more distinct representations as a sum of two cubes of nonnegative integers.

theorem_1: For a nonsingular integral cubic form F in four variables whose surface F = 0 contains three rational coplanar lines, the number of integer vectors x with F(x) = 0 and Euclidean length at most P that lie on no rational line of the surface is O(P^{4/3+epsilon}), the implied constant depending only on F and epsilon.

theorem_2: For an integral ternary quadratic form q with nonzero determinant Delta and with Delta_0 the highest common factor of the 2 by 2 minors of its matrix, the number of primitive integer zeros in the box |x_i| at most R_i is O({1 + (R_1 R_2 R_3 Delta_0^2 / Delta)^{1/2}} d_3(Delta)).

theorem_3: If q is a nonsingular integral ternary quadratic form with coefficients bounded by ||q|| and the binary form q(0, x_2, x_3) is nonsingular, then for every integer k the equation q(x) = 0 has O((||q|| R)^epsilon) primitive integer solutions in the cube |x_i| at most R with x_1 = k.

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D. Heath-Brown, "The density of rational points on cubic surfaces," Acta Arithmetica, 79(1), 17-30, 1997. https://doi.org/10.4064/aa-79-1-17-30

Read status. Claims checked: Theorems 1, 2 and 3 and the Corollary were read clause by clause on the printed pages (pp. 2-5), as were the statements of Lemmas 1-6 summarized below. The proofs were read for their structure, not checked step by step.

Bears on. #940: the Corollary bounds by $O(x^{4/9+\varepsilon})$ the integers up to $x$ with two or more distinct representations as a sum of two cubes of nonnegative integers, one restricted collision set inside the case $r=3$, since cubes are $3$-powerful; it bounds neither the integers with one representation nor sums of three cubes or of general $3$-powerful numbers, and settles no part of the problem.

Results. Theorem 1 (p. 2); the Corollary (p. 3, unnumbered); Theorem 2 (p. 4); Theorem 3 (p. 5). Lemmas 1-6 (pp. 5-11) are proof steps, summarized in the overview.

Overview

The paper studies the number

$$N_F(P)=\#\{\mathbf x\in\mathbb Z^4:F(\mathbf x)=0,\ |\mathbf x|\le P\}$$

for a cubic form $F$, where $|\mathbf x|$ is the Euclidean length, after removing points lying on rational lines of the cubic surface. The resulting count is denoted $N^{(0)}(P)$. The motivating, explicitly conjectural bound is $N^{(0)}(P)\ll_{F,\varepsilon}P^{1+\varepsilon}$; the paper does not prove this conjecture (§1, p. 1). For the Fermat cubic

$$W^3+X^3+Y^3+Z^3=0 \tag{1}$$

the removed lines account for the trivial equalities between two sums of two cubes.

The principal theorem states that if $F\in\mathbb Z[W,X,Y,Z]$ is nonsingular and $F=0$ contains three rational coplanar lines, then

$$N^{(0)}(P)\ll_{F,\varepsilon}P^{4/3+\varepsilon}$$

(Theorem 1, §1, p. 2). This improves the previously known exponent $5/3+\varepsilon$ for (1). Applied to (1), it gives the stated corollary that at most $O_\varepsilon(x^{4/9+\varepsilon})$ positive integers $n\le x$ have two or more distinct representations as a sum of two nonnegative cubes (Corollary, p. 3). The paper separately cites Hooley for the lower order-of-magnitude bound $x^{1/3}\log x$; that lower bound is background, not proved here (p. 3).

The geometric hypothesis supplies the central normal form. After a rational linear change of variables, the three lines lie in $Z=0$, and $F$ becomes either

$$F=WXY-ZQ(W,X,Y,Z) \tag{2}$$

or

$$F=WX(W+X)-ZQ(W,X,Y,Z), \tag{3}$$

according as the defining linear forms of the lines are independent or dependent (§1, p. 3). For (1), an explicit transformation to form (2) is displayed on pp. 3–4. The argument first counts primitive vectors. Writing $W=aU$ and $Z=bU$, with $(a,b)=1$, converts $F=0$ into the ternary quadratic equation

$$q(U,X,Y)=2aXY-2bQ(aU,X,Y,bU)=0 \tag{4}$$

or

$$q(U,X,Y)=2aX(aU+X)-2bQ(aU,X,Y,bU)=0. \tag{5}$$

The main uniform counting input is Theorem 2 (§1 and §2, pp. 4–8). If $q$ is an integral ternary quadratic form with matrix $M$, $\Delta=|\det M|\ne0$, and $\Delta_0$ is the gcd of the $2\times2$ minors of $M$, then the number of primitive zeros in $|x_i|\le R_i$ is

$$\ll \left\{1+\left(\frac{R_1R_2R_3\Delta_0^2}{\Delta}\right)^{1/2}\right\}d_3(\Delta).$$

Its proof imposes local lattice conditions at every prime dividing $\Delta$, combines them by the Chinese remainder theorem, rescales the resulting lattices, and applies successive minima together with the elementary zero estimate of Lemma 2 (pp. 6–8). Lemma 1 bounds primitive zeros of a nonzero binary form; Lemma 2 gives $O_d(1+(X_1X_2X_3)^{1/2})$ primitive zeros of a ternary form without a rational linear factor; and Lemma 3 records the nonuniform $O_q(P)$ bound for a fixed nonsingular ternary quadratic form (§2, pp. 5–6).

The complementary input, Theorem 3 (§1–2, pp. 5 and 8), treats a fixed first coordinate: if both $q$ and $q(0,x_2,x_3)$ are nonsingular, then for every integer $k$ there are only $O((\lVert q\rVert R)^\varepsilon)$ primitive zeros in $|x_i|\le R$ with $x_1=k$. The proof diagonalizes rationally and reduces to divisor-type bounds for representations by a binary quadratic form.

The exceptional cases are separated geometrically. Lemma 4 (§3, p. 9) shows that if the specialized ternary form $q(U,X,Y;a,b)$ is singular, then it has only two primitive zeros or its zeros lift to points on rational lines of $F=0$, hence do not contribute to $N^{(0)}(P)$. The failure of nonsingularity for $q(0,X,Y;a,b)$ occurs for at most four coprime pairs $(a,b)$, and these pairs, as well as $a=0$, contribute $O(P)$ (§3, p. 9).

For the generic case, a factorization of $Z$ produces an index satisfying

$$Z/a_i,\quad L_i/a_i\ll P^{2/3}, \tag{7}$$

and hence parameters

$$a,b\ll P^{2/3}. \tag{8}$$

(§4, p. 10). Lemma 5 proves the crucial uniformity $\Delta_0\ll_F1$ for coprime $(a,b)$, using elimination theory and the nonsingularity of $F$ (§4, pp. 10–11). The determinant has the form $\Delta=|G(a,b)|$, where $G$ is an integral binary form of degree five. Lemma 6 controls the number of dyadic pairs for which $|G(a,b)|$ is small (§4, pp. 11–12). Theorem 2 then contributes, for each pair,

$$\ll P^{3/2+\varepsilon}\{\Delta\max(|a|,b)\}^{-1/2}, \tag{9}$$

while Theorem 3 gives the alternative $O(P^{1+\varepsilon}/\max(|a|,b))$. Interpolating these estimates and applying Lemma 6 over dyadic ranges yields $O(P^{4/3+\varepsilon})$, completing Theorem 1 (§4, pp. 12–13). The paper does not treat arbitrary cubic surfaces; extending the theorem to singular surfaces is mentioned only as a possibility (pp. 2–3).

Relation to E940

This source bears on Problem 940.

Write

$$\mathcal P_r=\{m\ge1: p^e\Vert m\Rightarrow e\ge r\}$$

and

$$\mathcal S_r=\{n:n=m_1+\cdots+m_k,\ 0\le k\le r,\ m_i\in\mathcal P_r\}.$$

E940 asks, for $r\ge3$, whether infinitely many positive integers lie outside $\mathcal S_r$ and whether $\#(\mathcal S_r\cap[1,x])=o(x)$.

The paper bears directly only on a restricted component of the case $r=3$. Every positive cube is 3-powerful, and two distinct representations

$$a^3+b^3=c^3+d^3$$

produce an integral point $(a,b,-c,-d)$ on (1). After the rational-line solutions encoding trivial permutations are removed, Theorem 1 gives

$$\#\{n\le x:n\text{ has two or more distinct representations }n=a^3+b^3,\ a,b\ge0\} \ll_\varepsilon x^{4/9+\varepsilon}$$

(Corollary, p. 3). Thus the corollary is a bound on the exceptional multiplicity set for sums of two cubes, not on all integers represented by three cubes or by three arbitrary 3-powerful numbers.

The paper does not resolve any case of E940.

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