Linear growth for Châtelet surfaces. (English) Zbl 1253.11069
In this paper the author proves an upper bound on the expected number of points of bounded height on Châtelet surfaces defined over \(\mathbb{Q}\). Namely, let \(X\) be the rational surface defined by a smooth proper model of the affine surface \(y^2-az=f(x)\), where \(a\in\mathbb{Z}\) is not a square and \(f(x)\in\mathbb{Z}[x]\) is separable of degree 3 or 4.
The main result is Theorem 1: Let \(\rho_X\) be the rank of the Picard group of \(X\) and for a real number \(B\) let \(N(B)\) be the number of points \(x\in X(\mathbb{Q})\) such that \(H(\psi(x))\leq B\), where \(\psi:X\to\mathbb{P}^4\) is the map corresponding to the anticanonical bundle on \(X\) and \(H\) is the exponential height on \(\mathbb{P}^4\) metrized by some choice of norm. Then \[ N(B)=\mathcal{O}\left(B(\log B)^{\rho_X-1}\right) \] if \(a<0\). This result is predicted by the Manin conjecture.
The proof of Theorem 1 uses a method originally due to Selberger and crucially relies on the conic bundle structure of \(X/\mathbb{P}^1\). This allows the author to reduce the counting problem on \(X\) to a counting problem on the corresponding conics over \(p\in\mathbb{P}^1\). The latter is solved by computing the average order of a certain arithmetic function \(\bar{\omega}\) as it ranges over binary quartic forms using a result due to the author and R. de la Bretèche [Acta Arith. 125, No. 3, 291–304 (2007; Zbl 1159.11035)] from analytic number theory.
It should be noted that in joint work with R. de la Bretèche and E. Peyre [Ann. Math. (2) 175, No. 1, 297–343 (2012; Zbl 1237.11018)], the author has verified the full Manin conjecture, including the constant predicted by Peyre, in the special case \(a=-1\) and \(f\) totally reducible of degree 3.
The main result is Theorem 1: Let \(\rho_X\) be the rank of the Picard group of \(X\) and for a real number \(B\) let \(N(B)\) be the number of points \(x\in X(\mathbb{Q})\) such that \(H(\psi(x))\leq B\), where \(\psi:X\to\mathbb{P}^4\) is the map corresponding to the anticanonical bundle on \(X\) and \(H\) is the exponential height on \(\mathbb{P}^4\) metrized by some choice of norm. Then \[ N(B)=\mathcal{O}\left(B(\log B)^{\rho_X-1}\right) \] if \(a<0\). This result is predicted by the Manin conjecture.
The proof of Theorem 1 uses a method originally due to Selberger and crucially relies on the conic bundle structure of \(X/\mathbb{P}^1\). This allows the author to reduce the counting problem on \(X\) to a counting problem on the corresponding conics over \(p\in\mathbb{P}^1\). The latter is solved by computing the average order of a certain arithmetic function \(\bar{\omega}\) as it ranges over binary quartic forms using a result due to the author and R. de la Bretèche [Acta Arith. 125, No. 3, 291–304 (2007; Zbl 1159.11035)] from analytic number theory.
It should be noted that in joint work with R. de la Bretèche and E. Peyre [Ann. Math. (2) 175, No. 1, 297–343 (2012; Zbl 1237.11018)], the author has verified the full Manin conjecture, including the constant predicted by Peyre, in the special case \(a=-1\) and \(f\) totally reducible of degree 3.
Reviewer: Jan Steffen Müller (Hamburg)
MSC:
| 11G50 | Heights |
| 11D45 | Counting solutions of Diophantine equations |
| 14G05 | Rational points |
| 11G35 | Varieties over global fields |
References:
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