Algebra & Number Theory

K3 surfaces with Picard rank 20

Matthias Schütt

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We determine all complex K3 surfaces with Picard rank 20 over . Here the Néron–Severi group has rank 20 and is generated by divisors which are defined over . Our proof uses modularity, the Artin–Tate conjecture and class group theory. With different techniques, the result has been established by Elkies to show that Mordell–Weil rank 18 over is impossible for an elliptic K3 surface. We apply our methods to general singular K3 surfaces, that is, those with Néron–Severi group of rank 20, but not necessarily generated by divisors over .

Article information

Algebra Number Theory, Volume 4, Number 3 (2010), 335-356.

Received: 21 July 2009
Revised: 14 November 2009
Accepted: 31 December 2009
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14J28: $K3$ surfaces and Enriques surfaces
Secondary: 11F11: Holomorphic modular forms of integral weight 11G15: Complex multiplication and moduli of abelian varieties [See also 14K22] 11G25: Varieties over finite and local fields [See also 14G15, 14G20] 11R29: Class numbers, class groups, discriminants

singular K3 surface Artin–Tate conjecture complex multiplication modular form class group


Schütt, Matthias. K3 surfaces with Picard rank 20. Algebra Number Theory 4 (2010), no. 3, 335--356. doi:10.2140/ant.2010.4.335.

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