On Manin’s conjecture for singular del Pezzo surfaces of degree 4. I. (English) Zbl 1132.14019

Browning, T. D.; de la Bretèche, R.

Mich. Math. J. 55, No. 1, 51-80 (2007).

This paper concerns Manin’s conjecture for a certain singular del Pezzo surface \(X\) of degree 4, given by the equations \[ x_0x_1- x^2_2= x_0x_4- x_1x_2+ x^2_3= 0. \] This has just one singular point, which is of type \(D_5\). Let \(N(B)\) be the standard counting function for rational points of height at most \(B\), for the open subset of \(X\) for which one excludes points on the line \(x_0= x_2= x_3= 0\).
It is shown that Manin’s conjecture holds for this surface, in the strong form \[ N(B)= BP(\log B)+ O(B^\theta), \] for any \(\theta> 11/12\). Here \(P\) is a certain polynomial of degree 5, whose leading term is explicitly calculated, and shown to agree with the prediction by Peyre. Moreover, it is shown that the height zeta-function has a meromorphic continuation to \(\sigma> 5/6\), and is holomorphic for \(\sigma> 9/10\) apart from a sixth-order pole at \(s= 1\).
For the proof, the authors establish a bijection between the points under consideration and the integral points on the universal torsor associated to \(X\). These latter points are then handled by analytic techniques, including an estimate for the average of the fractional part of \((a- bx^2)/q\), as the integer \(x\) varies.

Reviewer: Roger Heath-Brown (Oxford)

MSC:

14G05	Rational points
14J26	Rational and ruled surfaces
11D45	Counting solutions of Diophantine equations

Keywords:

Manin’s conjecture; degree 4; singular; asymptotic formula; height zeta-function

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