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Source. Theorem 1.6, pp. 4--5, of Victor Y. Wang, Sums of cubes and the Ratios Conjectures, arXiv:2108.03398v2 (19 April 2023), the edition named on the source card. The hypotheses are Conjectures 1.2 (p. 3), 1.4 and 1.5 (p. 4) and 1.8 (p. 5).

Read depth. Claims checked: the statement, the definitions (1.2)--(1.5) and (1.11), and the four conjectures it assumes were read clause by clause on pp. 2--5; the proof on p. 57 was read for its structure. Nothing here is independently reviewed.

Statement

Fix a cubic form $F\in\mathbb Z[x_1,\ldots,x_6]$ with nonzero discriminant, let $V$ be the hypersurface $F=0$ in $\mathbb P^5_{\mathbb Q}$, and let $\Upsilon$ be the set of $3$-dimensional subspaces $L\subseteq\mathbb Q^6$ on which $F$ vanishes (p. 2). For $w\in C_c^\infty(\mathbb R^6)$ and real $X\ge1$ put

$$N_{F,w}(X)=\sum_{\mathbf x\in\mathbb Z^6:\,F(\mathbf x)=0}w(\mathbf x/X),\qquad E_{F,w}(X)=N_{F,w}(X)-\mathfrak S_F\,\sigma_{\infty,F,w}\,X^3-\sum_{L\in\Upsilon}\ \sum_{\mathbf x\in L\cap\mathbb Z^6}w(\mathbf x/X),$$

with $\mathfrak S_F$ the singular series and $\sigma_{\infty,F,w}$ the real density of (1.4) (displays (1.2)--(1.4), pp. 2--3). The asymptotic (1.5) is $\lim_{X\to\infty}X^{-3}E_{F,w}(X)=0$.

Theorem 1.6 (pp. 4--5). Let $m=6$ and let $F$ be diagonal. Assume Conjectures 1.2, 1.4, 1.5 and 1.8. Then (1.5) holds for every $w\in C_c^\infty(\mathbb R^m)$ whose support satisfies

$$\overline{\{\mathbf x\in\mathbb R^m: w(\mathbf x)\ne0\}}\subseteq\{\mathbf x\in\mathbb R^m: x_1\cdots x_m\ne0\}$$

(display (1.11)). Hence the Hasse principle holds for $V$. Moreover, if $F=x_1^3+\cdots+x_6^3$, then 100% of the integers $a\not\equiv\pm4\pmod 9$ lie in $F_0(\mathbb Z^3)$, where $F_0(x,y,z)=x^3+y^3+z^3$.

Conjectures 1.2, 1.4 and 1.5 are stated on the Theorem 1.3 page. Conjecture 1.8 (RA1$o$, p. 5), for even $m$ and assuming Conjecture 1.2, is a first-moment prediction: for $M\ge1$, a modulus $n_0\in[1,M]$ and $\mathbf a,\mathbf b\in\mathbb Z^m\cap[-M,M]^m$, the sum of $\Phi^{\mathbf c,1}(s)$ over $\mathbf c\in\mathcal S_1$ in the dilated box $Z\cdot\mathcal B_M(\mathbf b)$ of (1.12) with $\mathbf c\equiv\mathbf a\bmod n_0$ equals the sum over the same $\mathbf c$ of $(1+o_{F,M;Z\to\infty}(1))A^{\mathbf a,n_0}_{F,1}(s)$, at $s=\sigma(Z)+it$ with $\sigma(Z)=1/2+1/\log Z$ and $t\in[-\log Z,\log Z]$, for $Z\ge2$ (display (1.14)); here $A^{\mathbf a,n_0}_{F,1}(s)$ is the Euler product of §6.3.1, absolutely convergent in $\operatorname{Re}(s)>1/3$. All four conjectures are unproved, so every conclusion is conditional.

The third conclusion concerns integer cubes of either sign. It says nothing about sums of three nonnegative cubes, and it gives no representation of any particular integer.

Proof pointer

§10.3, p. 57. Unconditionally, (1.3), (2.10) and Theorem 2.5 give $E_{F,w}(X)/X^3=O(X^{2.75+\epsilon})/X^3+\Sigma^\natural(X,\mathcal S_1)$ (display (10.30)). Under (1.11), Theorem 10.7 (p. 56) gives $\Sigma^\natural(X,\mathcal S_1)=o_{X\to\infty}(X^{(6-m)/4})$, its moment hypotheses supplied by Propositions 9.7 and 9.9; with $m=6$ this is (1.5). The Hasse principle follows by choosing $w$ with $\sigma_{\infty,F,w}>0$. The density statement for $F_0(\mathbb Z^3)$ is deduced from (1.5) through Wang's earlier work (Theorem 1.1 of the paper cited as [Wan23c], or Theorem 2.1.8 of [Wan22]), not proved here.

Dependencies

Conjectures 1.2, 1.4, 1.5 and 1.8 as hypotheses; within the paper, Theorem 2.5, Propositions 9.7 and 9.9, and Theorem 10.7; outside it, the deduction cited above from [Wan23c] or [Wan22].

Bears on

• Problem 940: no direct bearing. The 100% statement counts representations $a=x^3+y^3+z^3$ with $x,y,z\in\mathbb Z$ of either sign, while the problem's sums use positive $r$-powerful numbers; the conditional relation at $r=3$ comes from Theorem 1.3.