Recognizing Galois representations of K3 surfaces
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Under the assumption of the Hodge, Tate and Fontaine–Mazur conjectures we give a criterion for a compatible system of \(\ell \)-adic representations of the absolute Galois group of a number field to be isomorphic to the second cohomology of a K3 surface. This is achieved by producing a motive M realizing the compatible system, using a local to global argument for quadratic forms to produce a K3 lattice in the Betti realization of M and then applying surjectivity of the period map for K3 surfaces to obtain a complex K3 surface. Finally we use a very general descent argument to show that the complex K3 surface admits a model over a number field.
It is a pleasure to thank Stefan Patrikis, for suggested this problem to me, for his patient guidance and for the many helpful discussions we had. I would also like to thank Domingo Toledo for some helpful discussions. The author was partially supported by NSF DMS 1246989.
- 1.André, Y.: Une introduction aux motifs: motifs purs, motifs mixtes, périodes. Société mathématique de France (2004)Google Scholar
- 2.Baldi, G.: Local to global principle for the moduli space of K3 surfaces. arXiv:1802.02042 (2018)
- 3.Bourbaki, N.: Algebra II: Chapters 4–7. Springer, New York (2013)Google Scholar
- 6.Huybrechts, D.: A global Torelli theorem for hyperkähler manifolds (after Verbitsky). Séminaire Bourbaki 1040 (2010–2011)Google Scholar
- 8.Huybrechts, D.: Motives of isogenous K3 surfaces. arXiv:1705.04063 (2017)
- 12.Moonen, B.: A remark on the Tate conjecture. arXiv:1709.04489 (2017)
- 15.Schneps, L., Lochak, P.: Geometric Galois Actions. In: London Mathematical Society Lecture Note Series, vol. 242 (1997)Google Scholar
- 18.Voisin, C.: Hodge theory and complex algebraic geometry. In: I. volume 76 of Cambridge Studies in Advanced Mathematics (2002)Google Scholar