Explicit families of $K3$ surfaces having real multiplication

Andreas-Stephan Elsenhans; Jörg Jahnel

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arXiv · 2002.00233

Explicit families of $K3$ surfaces having real multiplication

Andreas-Stephan Elsenhans·Jörg Jahnel

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Abstract

For families of $K3$ surfaces, we establish a sufficient criterion for real or complex multiplication. Our criterion is arithmetic in nature. It may show, at first, that the generic fibre of the family has a nontrivial endomorphism field. Moreover, the endomorphism field does not shrink under specialisation. As an application, we present two explicit families of $K3$ surfaces having real multiplication by $\bbQ(\sqrt{2})$ and $\bbQ(\sqrt{5})$, respectively.

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Andreas-Stephan Elsenhans, Jörg Jahnel. 2020-02-01. Explicit families of $K3$ surfaces having real multiplication. https://arxiv.org/abs/2002.00233

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