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$$K3$$ surfaces with Picard number one and infinitely many rational points. (English) Zbl 1123.14022
The author constructs infinitely many $$K3$$-surfaces $$X/\mathbb Q$$ such that $$X(\mathbb Q)$$ is infinite and the geometric Picard number of $$X$$ equals 1. A sketch of the proof is as follows. The author starts with a family of degree 4 surfaces $$X$$ in $$\mathbb P^2$$, such that all members of this family reduce to the same surface modulo 6.
Let $$X_p$$ denote the reduction of $$X$$ modulo $$p$$. Using the Lefschetz trace formula the author shows that the geometric Picard number of $$X_2$$ and of $$X_3$$ equal two. This implies that the geometric Picard number of $$X$$ is at most 2. He shows that $$X_2$$ contains a line and $$X_3$$ contains a conic. This implies that the discriminant of the Néron-Severi lattice of $$X_2$$ and of $$X_3$$ differ by a non-square. He then observes that if $$X$$ would have geometric Picard number 2 then both discriminants would differ by a square, hence $$X$$ has geometric Picard number 2.
He considers a subfamily of $$X$$ such that each member has a hyperplane section with two nodes. The normalization $$E$$ of such a curve is an elliptic curve. The author finds two rational points $$O,T$$ on $$E$$, taking $$O$$ as the identity he checks whether $$2520T=O$$, since this is not the case it follows from a Theorem of Mazur that $$T$$ has infinite order, hence $$X$$ contains infinitely many rational points.

MSC:
 14J28 $$K3$$ surfaces and Enriques surfaces 14C22 Picard groups 14G05 Rational points
Keywords:
Picard group of $$K3$$ surfaces
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