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Recognizing Galois representations of K3 surfaces

Abstract

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.

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References

  1. 1.

    André, Y.: Une introduction aux motifs: motifs purs, motifs mixtes, périodes. Société mathématique de France (2004)

  2. 2.

    Baldi, G.: Local to global principle for the moduli space of K3 surfaces. arXiv:1802.02042 (2018)

  3. 3.

    Bourbaki, N.: Algebra II: Chapters 4–7. Springer, New York (2013)

    Google Scholar 

  4. 4.

    Cassels, J.W.S.: Rational Quadratic Forms. Courier Dover Publications, Mineola (2008)

    MATH  Google Scholar 

  5. 5.

    Derome, G.: Descente algébriquement close. J. Algebra 266(2), 418–426 (2003)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Huybrechts, D.: A global Torelli theorem for hyperkähler manifolds (after Verbitsky). Séminaire Bourbaki 1040 (2010–2011)

  7. 7.

    Huybrechts, D.: Lectures on K3 Surfaces, vol. 158. Cambridge University Press, Cambridge (2016)

    Book  Google Scholar 

  8. 8.

    Huybrechts, D.: Motives of isogenous K3 surfaces. arXiv:1705.04063 (2017)

  9. 9.

    Ihara, Y., Nakamura, H.: Some illustrative examples for anabelian geometry in high dimensions. Lond. Math. Soc. Lect. Note Ser. 1(242), 127–138 (1997)

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Jannsen, U.: Motives, numerical equivalence, and semi-simplicity. Invent. Math. 107(1), 447–452 (1992)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Ma, S.: Twisted Fourier–Mukai number of a 3 surface. Trans. Am. Math. Soc. 362(1), 537–552 (2010)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Moonen, B.: A remark on the Tate conjecture. arXiv:1709.04489 (2017)

  13. 13.

    Patrikis, S., Voloch, J., Zarhin, Y.: Anabelian geometry and descent obstructions on moduli spaces. Algebra Number Theory 10(6), 1191–1219 (2016)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Rizov, J.: Moduli stacks of polarized K3 surfaces in mixed characteristic. Serdica Math. J. 32, 131–178 (2006)

    MathSciNet  MATH  Google Scholar 

  15. 15.

    Schneps, L., Lochak, P.: Geometric Galois Actions. In: London Mathematical Society Lecture Note Series, vol. 242 (1997)

  16. 16.

    Tankeev, S.G.: K3 surfaces over number fields and the Mumford–Tate conjecture. Math. USSR Izvestiya 37(1), 191 (1991)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Toledo, D.: Projective varieties with non-residually finite fundamental group. Publ. Math. l’Inst. Hautes Études Sci. 77(1), 103–119 (1993)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Voisin, C.: Hodge theory and complex algebraic geometry. In: I. volume 76 of Cambridge Studies in Advanced Mathematics (2002)

  19. 19.

    Zarhin, Y.G.: Hodge groups of K3 surfaces. J. Reine Angew. Math. 341(193–220), 54 (1983)

    MathSciNet  Google Scholar 

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Author's contributions

Acknowledgements

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.

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Correspondence to Christian Klevdal.

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Klevdal, C. Recognizing Galois representations of K3 surfaces. Res. number theory 5, 16 (2019). https://doi.org/10.1007/s40993-019-0154-1

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