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

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.

### MSC:

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

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