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The Manin conjecture for \(x_0y_0+\dots+x_sy_s=0\). (English) Zbl 1171.11054

This paper gives a proof of the Manin conjecture for the variety \[ x_0 y_0+\cdots+ x_s y_s= 0 \] in \(\mathbb P^s(\mathbb Q)\times \mathbb P^s(\mathbb Q)\). This is a flag variety, and for such varieties the conjecture has been proved in general by J. Franke, Yu. I. Manin and Y. Tschinkel [Invent. Math. 95, No. 2, 421–435 (1989; Zbl 0674.14012)]. However the goal of the present paper is to demonstrate that the circle method can be used in this particular example. Let \[ N(B)=\#\{({\mathbf x},{\mathbf y})\in \mathbb P^s(\mathbb Q)\times \mathbb P^s(\mathbb Q): {\mathbf x}.{\mathbf y}= 0,\;x_0\cdots x_s y_0\cdots y_s\neq 0,\;H({\mathbf x},{\mathbf y})\leq B\}, \] where \[ H({\mathbf x},{\mathbf y})= \max_{0\leq j\leq s}|x_i y_j|^s \] if we choose representatives for \([{\mathbf x}]\), \([{\mathbf y}]\) for which \[ \text{h.c.f}(x_0,\dots, x_s)= \text{h.c.f}(y_0,\dots, y_s)= 1. \] It is then shown using the classical circle method, that if \(s\geq 2\) then \[ N(B)= c_s B\log B+ O_s(B) \] for a certain positive constant \(c_s\). The constant is given explicitly, but it is not explored whether or not it agrees with Peyre’s prediction.
The principal novelty in the proof lies in the minor arc treatment, which has to cope with the rather awkward height function.

MSC:

11P55 Applications of the Hardy-Littlewood method
11G35 Varieties over global fields
11D72 Diophantine equations in many variables

Citations:

Zbl 0674.14012
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References:

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