K3 surfaces with Picard number one and infinitely many rational points

By Ronald van Luijk. 2005

129 citations recorded by the source corpus.

In general, not much is known about the arithmetic of K3 surfaces. Once the geometric Picard number, which is the rank of the Neron-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 infinitely many rational points, thereby answering a question of Swinnerton-Dyer and Poonen.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/28c0786cd1a6b01ad0b78d09882b6a1d9cfcea76

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Citations in the corpus

Bounding Picard numbers of surfaces using p-adic cohomology (2006, background): “We can thus control the size of NS( X K ) by controlling NS( X k ); in some cases, one can gain further control by reducing modulo more than one prime of good reduction [50].”
On the Picard number of K3 surfaces over number fields (2011, methodology): “Using the Weil conjectures [5], we can compute the characteristic polynomial of Frobenius at any finite place of k, see [27], [10].”
K3 surfaces and equations for Hilbert modular surfaces (2012, methodology): “We found the method of [van Luijk 2007] quite useful in determining the ranks of the Néron–Severi lattices of these surfaces.”
Elliptic Surfaces (2009, methodology): “Kloosterman achieved this as follows: he first determined a surface (over Q) where by means of good reduction mod p, the Lefschetz fixed point formula implied ρ ≤ 18 after point counting over Fq for q = p, p(2), p(3); then he compared the reduction modulo another good prime q to prove ρ = 17 by an e”
Lectures on K3 Surfaces (2016, background): “In [606] van Luijk also shows that the set of quartic K3 surfaces over Q of geometric Picard number one and with infinitely many rational points is in fact dense in the moduli space of quartics.”
Lines on Fermat surfaces (2008, unknown): “Key words: Fermat surface, Néron-Severi group, supersingular reduction.”
Transcendental obstructions to weak approximation on general K3 surfaces (2010, methodology): “We use a method of van Luijk, together with the work of Elsenhans and Jahnel on K3 surfaces of degree 2, to assemble a K3 surface for which we can prove that this is the case [vL07, EJ08a, EJ08b].”