Manin's conjecture for Châtelet surfaces

Let $\Delta\in\mathbb{Q}^\times$ satisfy $\sqrt{-\Delta}\notin\mathbb{Q}$, and let $f(z)\in\mathbb{Z}[z]$ be a separable polynomial of degree $3$ or $4$ that decomposes as a product of $r$ irreducible polynomials $f=f_1\cdots f_r$. Let $X_{\Delta,f}/\mathbb{Q}$ be a Châtelet surface defined by

$$x^2+\Delta y^2=f(z).$$

Assume that $X_{\Delta,f}(\mathbb{Q})\neq\emptyset$. Manin's conjecture. As $B\to\infty$,

$$\widetilde{N}(X_{\Delta,f},B)\sim C_{\Delta,f}B\log(B)^{\varrho_{\Delta,f}-1},$$

where $C_{\Delta,f}>0$ and

$$\varrho_{\Delta,f}=2+\#\{1\leq i\leq r:\sqrt{-\Delta}\in\mathbb{Q}[z]/(f_i(z))\}.$$

This is a precise form of the Batyrev–Manin prediction for the point-counting function associated with the natural height on a singular degree-four del Pezzo model of the Châtelet surface; Peyre's work interprets the leading constant through local densities and the Brauer group. The paper's abstract states that the conjecture is resolved for all Châtelet surfaces over $\mathbb{Q}$, including the case $\Delta<0$; the result therefore establishes the asserted asymptotic in this setting.