K3 surfaces with real or complex multiplication

Eva Bayer-Fluckiger; Bert van Geemen; Matthias Schütt

K3 surfaces with real or complex multiplication

Algebraic Geometry 2025-10-21 v3 Number Theory

Authors: Eva Bayer-Fluckiger , Bert van Geemen , Matthias Schütt

Abstract

Let $E$ be a totally real number field of degree $d$ and let $m \geqslant 3$ be an integer. We show that if $md \leqslant 21$ then there exists an $(m-2)$ -dimensional family of complex projective $K3$ surfaces with real multiplication by $E$ . Analogous results are proved for CM number fields and also for all known higher-dimensional hyperk\"ahler manifolds.

Keywords

algebraic surfaces manifolds class number of quadratic fields

Cite

@article{arxiv.2401.04072,
  title  = {K3 surfaces with real or complex multiplication},
  author = {Eva Bayer-Fluckiger and Bert van Geemen and Matthias Schütt},
  journal= {arXiv preprint arXiv:2401.04072},
  year   = {2025}
}

Comments

32 pages; v3: revision adding section on Brauer groups and details for motivation and on Witt rings; minor corrections and clarifications throughout

K3 surfaces with real or complex multiplication

Abstract

Keywords

Cite

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