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).
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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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This work was supported by the Engineering and Physical Sciences Research Council [EP/L015234/1], the EPSRC Centre for Doctoral Training in Geometry and Number Theory (The London School of Geometry and Number Theory), University College London.
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Baldi, G. Local to global principle for the moduli space of K3 surfaces. Arch. Math. 112, 599–613 (2019). https://doi.org/10.1007/s00013-018-01295-1
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DOI: https://doi.org/10.1007/s00013-018-01295-1