Abstract
We consider two K3 surfaces defined over an arbitrary field, together with a smooth proper moduli space of stable sheaves on each. When the moduli spaces have the same dimension, we prove that if the étale cohomology groups (with \({\mathbb {Q}}_\ell \) coefficients) of the two surfaces are isomorphic as Galois representations, then the same is true of the two moduli spaces. In particular, if the field of definition is finite and the K3 surfaces have equal zeta functions, then so do the moduli spaces, even when the moduli spaces are not birational.
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Acknowledgements
I thank my advisor, Nicolas Addington, for supporting me and teaching me the material necessary to complete this project. Thanks also to Adrian Langer, François Charles, Max Lieblich, Eduard Looijenga, Valery Lunts, Mark de Cataldo, Luc Illusie, Andy Putman, and Katrina Honigs for the helpful discussions and correspondences, and the referee who suggested many improvements.
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Frei, S. Moduli spaces of sheaves on K3 surfaces and Galois representations. Sel. Math. New Ser. 26, 6 (2020). https://doi.org/10.1007/s00029-019-0530-7
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DOI: https://doi.org/10.1007/s00029-019-0530-7