On the Picard number of K3 surfaces over number fields

TL;DR

This paper investigates the specialization behavior of Néron-Severi groups of K3 surfaces over number fields, providing optimal bounds for Picard numbers and enabling explicit computation of these invariants.

Contribution

It offers the first optimal lower bounds for Picard numbers of specializations and demonstrates how to explicitly compute the Picard number of any K3 surface over a number field.

Findings

Established optimal lower bounds for Picard numbers of specializations.

Proved the computability of the Picard number for any K3 surface over a number field.

Answered a question posed by Elsenhans and Jahnel.

Abstract

We discuss some aspects of the behavior of specialization at a finite place of N\'eron-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 explicitly compute the Picard number of any given K3 surface over a number field.