2007
DOI: 10.2140/ant.2007.1.1
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K3 surfaces with Picard number one and infinitely many rational points

Abstract: In general, not much is known about the arithmetic of K3 surfaces. Once the geometric Picard number, which is the rank of the Néron-Severi group over an algebraic closure of the base field, is high enough, more structure is known and more can be said. However, until recently not a single K3 surface was known to have geometric Picard number one. We give explicit examples of such surfaces over the rational numbers. This solves an old problem that has been attributed to Mumford. The examples we give also contain … Show more

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Cited by 91 publications
(90 citation statements)
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“…We conclude that ρ(S 3 ) 2. By [vLui07], we have ρ(S) ρ(S 3 ), so ρ(S) 2. It follows that S (and S) has Picard rank 2.…”
mentioning
confidence: 98%
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“…We conclude that ρ(S 3 ) 2. By [vLui07], we have ρ(S) ρ(S 3 ), so ρ(S) 2. It follows that S (and S) has Picard rank 2.…”
mentioning
confidence: 98%
“…Let = 3 be a prime and write φ(t) for the characteristic polynomial of the action of the absolute Frobenius on H 2 et (S 3 , Q ). Then ρ(S 3 ) is bounded from above by the number of roots of φ(t) that are of the form 3ζ, where ζ is a root of unity [vLui07,Prop. 2.3].…”
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confidence: 99%
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“…On the other hand, by counting points modulo 7 and 13 and using the method of van Luijk [31] (comparing the square classes of the discriminants of the Picard groups of these reductions) we obtain that the Picard number cannot be 30. Therefore ρ = 29 exactly.…”
Section: Elliptic Curvesmentioning
confidence: 99%
“…Real and complex multiplication for K3 surfaces. Our motivation comes from projective varieties X that are defined over a number field K. As was first noticed by R. van Luijk, from point counts on the reductions X p1 , X p2 modulo two primes [31,32,16,19] of good reduction or sometimes only one [18], one may determine the geometric Néron-Severi rank of X. This applies well even to varieties of general type [15], but a nontrivial case is provided already by K3 surfaces.…”
Section: Introductionmentioning
confidence: 99%