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Archiv der Mathematik

, Volume 112, Issue 6, pp 599–613 | Cite as

Local to global principle for the moduli space of K3 surfaces

  • Gregorio BaldiEmail author
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Abstract

Recently S. Patrikis, J.F. Voloch, and Y. Zarhin have proven, assuming several well-known conjectures, that the finite descent obstruction holds on the moduli space of principally polarised abelian varieties. We show an analogous result for K3 surfaces, under some technical restrictions on the Picard rank. This is possible since abelian varieties and K3s are quite well described by ‘Hodge-theoretical’ results. In particular the theorem we present can be interpreted as follows: a family of \(\ell \)-adic representations that looks like the one induced by the transcendental part of the \(\ell \)-adic cohomology of a K3 surface (defined over a number field) determines a Hodge structure which in turn determines a K3 surface (which may be defined over a number field).

Keywords

K3 surfaces Galois representations Fontaine–Mazur conjecture 

Mathematics Subject Classification

Primary 11G35 14G35 11F80 

Notes

Acknowledgements

It is a pleasure to thank D. Valloni for reading a draft of this paper and A. Skorobogatov for useful discussions regarding the theory of K3 surfaces. We are grateful to an anonymous referee whose precious comments improved the exposition and Proposition 3.2.

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Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.University College LondonLondonUK

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