Vol. 8, No. 1, 2014

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On the Picard number of K3 surfaces over number fields

François Charles

Vol. 8 (2014), No. 1, 1–17

We discuss some aspects of the behavior of specialization at a finite place of Néron–Severi groups of K3 surfaces over number fields. We give optimal lower bounds for the Picard number of such specializations, thus answering a question of Elsenhans and Jahnel. As a consequence of these results, we show that it is possible to compute explicitly the Picard number of any given K3 surface over a number field.

K3 surfaces, Picard number, Néron–Severi group
Mathematical Subject Classification 2010
Primary: 14J28
Secondary: 14C22, 14G25, 11G35
Received: 1 February 2012
Revised: 20 September 2012
Accepted: 4 November 2012
Published: 20 April 2014
François Charles
Institut de Recherche Mathématiques de Rennes
Université de Rennes 1
Campus de Beaulieu
35042 Rennes