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On Manin’s conjecture for certain Châtelet surfaces. (Sur la conjecture de Manin pour certaines surfaces de Châtelet.) (French. English summary) Zbl 1295.11029
This paper considers Châtelet surfaces over \(\mathbb{Q}\) of the form \[ y^2+z^2=P(x,1) \] where \(P(x,y)\in\mathbb{Z}[x,y]\) is either a quartic form irreducible over \(\mathbb{Q}(i)\), or splits over \(\mathbb{Q}(i)\) as a product of two non-proportional quadratic forms, each irreducible over \(\mathbb{Q}(i)\). For this surface Manin’s conjecture predicts that the counting function \(N(B)\) for rational points of height at most \(B\) should grow like \(C_PB\log B\), where \(C_P\) is the constant described by E. Peyre [Duke Math. J. 79, No. 1, 101–218 (1995; Zbl 0901.14025)]. For this one has to use an appropriate height function, which is described in the paper.
The main theorems prove the required asymptotic formulae in the two cases described above. This is a very significant achievement, and completes the treatment of Manin’s conjecture for the different possible factorizations of \(P\), the remaining cases, which are hard but easier, having been dealt with by R. de la Bretèche and T. D. Browning [J. Reine Angew. Math. 646, 1–44 (2010; Zbl 1204.11158)] and [Isr. J. Math. 191, 973–1012 (2012; Zbl 1293.11058)], and by R. de la Bretèche, T. D. Browning and E. Peyre [Ann. Math. (2) 175, 297–343 (2012; Zbl 1237.11018)]. It seems plausible that all these results could be extended by replacing \(y^2+z^2\) with an arbitrary irreducible quadratic form \(Q(y,z)\), provided that one interprets suitably the notion of “the number of representations by \(Q\)”.
The proof is long and difficult. The most important new tool is the authors’ average bound for the generalized Hooley \(\Delta\)-function [C. Hooley, J. Lond. Math. Soc., II. Ser. 85, No. 3, 669–693 (2012; Zbl 1258.11086)]. Very roughly this says that for an non-principal character \(\chi\), the function \[ \Delta(n,\chi):=\sup_{u\in\mathbb{R},\,0\leq u\leq 1}\Biggl|\sum_{\substack{ d\mid n \\ e^u<d\leq e^{u+v}}}\chi(d)\Biggr| \] has mean square size \((\log n)^{o(1)}\), even after weighting by suitable arithmetic factors.

MSC:
11D45 Counting solutions of Diophantine equations
11D25 Cubic and quartic Diophantine equations
14G05 Rational points
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