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Chebotarev–Sato–Tate distribution for abelian surfaces potentially of \(\textrm{GL}_2\)-type

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Abstract

We state a hybrid Chebotarev–Sato–Tate conjecture for abelian varieties and we prove it in several particular cases using current potential automorphy theorems.

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References

  1. Allen, P.B., Calegari, F., Caraiani, A., Gee, T., Helm, D., Le Hung, B.V., Newton, J., Scholze, P., Taylor, R., Thorne, J.A.: Potential automorphy over CM fields. Ann. Math. 197(3), 897–1113 (2023)

    Article  MathSciNet  Google Scholar 

  2. Amri, M.A.: On the Lang–Trotter conjecture for a class of non-generic abelian surfaces (2022). ar**v:2211.10523

  3. Arthur, J., Clozel, L.: Simple Algebras, Base Change, and the Advanced Theory of the Trace Formula. Annals of Mathematics Studies, vol. 120. Princeton University Press, Princeton (1990)

    Google Scholar 

  4. Barnet-Lamb, T., Gee, T., Geraghty, D.: The Sato–Tate conjecture for Hilbert modular forms. J. Amer. Math. Soc. 24(2), 411–469 (2011)

    Article  MathSciNet  Google Scholar 

  5. Barnet-Lamb, T., Gee, T., Geraghty, D., Taylor, R.: Potential automorphy and change of weight. Ann. Math. 179(2), 501–609 (2014)

    Article  MathSciNet  Google Scholar 

  6. Brauer, R.: On Artin’s \(L\)-series with general group characters. Ann. Math. 48(2), 502–514 (1947)

    Article  MathSciNet  Google Scholar 

  7. Clozel, L.: The Sato–Tate conjecture. In: Jenison, D., et al. (eds.) Current Developments in Mathematics, 2006, pp. 1–34. International Press, Somerville (2006)

    Google Scholar 

  8. Clozel, L., Harris, M., Taylor, R.: Automorphy for some \(l\)-adic lifts of automorphic \( {{\rm mod }} \,l\) Galois representations. Publ. Math. Inst. Hautes Études Sci. 108, 1–181 (2008)

    Article  Google Scholar 

  9. Cogdell, J.W., Kim, H.H., Murty, M.R.: Lectures on Automorphic \(L\)-Functions. Fields Institute Monographs, vol. 20. American Mathematical Society, Providence (2004)

    Google Scholar 

  10. Fité, F., Kedlaya, K.S., Rotger, V., Sutherland, A.V.: Sato–Tate distributions and Galois endomorphism modules in genus 2. Compositio Math. 148(5), 1390–1442 (2012)

    Article  MathSciNet  Google Scholar 

  11. Grbac, N., Shahidi, F.: Endoscopic transfer for unitary groups and holomorphy of Asai \(L\)-functions. Pacific J. Math. 276(1), 185–211 (2015)

    Article  MathSciNet  Google Scholar 

  12. Harris, M.: Potential automorphy of odd-dimensional symmetric powers of elliptic curves and applications. In: Tschinkel, Yu., Zarhin, Yu. (eds.) Algebra, Arithmetic, and Geometry: in Honor of Yu.I. Manin. Vol. II. Progress in Mathematics, vol. 270, pp. 1–21. Birkhäuser, Boston (2009)

  13. Harris, M., Shepherd-Barron, N., Taylor, R.: A family of Calabi–Yau varieties and potential automorphy. Ann. Math. 171(2), 779–813 (2010)

    Article  MathSciNet  Google Scholar 

  14. Hida, H.: Modular Forms and Galois Cohomology. Cambridge Studies in Advanced Mathematics, vol. 69. Cambridge University Press, Cambridge (2000)

    Book  Google Scholar 

  15. Johansson, C.: On the Sato–Tate conjecture for non-generic abelian surfaces. Trans. Amer. Math. Soc. 369(9), 6303–6325 (2017)

    Article  MathSciNet  Google Scholar 

  16. Langlands, R.P.: Problems in the theory of automorphic forms. In: Taam, C.T. (ed.) Lectures in Modern Analysis and Applications, Vol. III. Lecture Notes in Mathematics, vol. 170, pp. 18–61. Springer, Berlin (1970)

    Google Scholar 

  17. Milne, J.S.: Abelian varieties. In: Cornell, G., Silverman, J.H. (eds.) Arithmetic Geometry, pp. 103–150. Springer, New York (1986)

    Chapter  Google Scholar 

  18. Murty, M.R., Murty, V.K.: The Sato–Tate conjecture and generalizations. In: Mukunda, N. (ed.) Current Trends in Science, pp. 639–646. Indian Academy of Sciences, Bengaluru (2009)

    Google Scholar 

  19. Ribet, K.A.: Abelian Varieties over \({\mathbb{Q} }\) and modular forms. In: Hahn, S.G., Suh, D.Y. (eds.) Algebra and Topology 1992, pp. 53–79. Korea Advanced Institute of Science and Technology, Taejŏn (1992)

    Google Scholar 

  20. Serre, J.-P.: Abelian \(l\)-Adic Representations and Elliptic Curves. McGill University Lecture Notes written with the collaboration of Willem Kuyk and John Labute, W.A. Benjamin, New York (1968)

    Google Scholar 

  21. Serre, J..-P..: Propriétés conjecturales des groupes de Galois motiviques et des reprśentations \(l\)-adiques. In: Jannsen, U., et al. (eds.) Motives. Proceedings of Symposia in Pure Mathematics, vol. 55.1, pp. 377–400. American Mathematical Society, Providence (1994)

    Google Scholar 

  22. Serre, J.-P.: Complex Semisimple Lie Algebras. Springer Monographs in Mathematics, Springer, Berlin (2012)

    Google Scholar 

  23. Serre, J.-P.: Lectures on \(N_X(p)\). Chapman & Hall/CRC Research Notes in Mathematics, vol. 11. CRC Press, Boca Raton (2012)

    Google Scholar 

  24. Shahidi, F.: On certain \(L\)-functions. Amer. J. Math. 103(2), 297–355 (1981)

    Article  MathSciNet  Google Scholar 

  25. Shimura, G.: On analytic families of polarized abelian varieties and automorphic functions. Ann. Math. 78(1), 149–192 (1963)

    Article  MathSciNet  Google Scholar 

  26. Sutherland, A.V.: Sato–Tate distributions. In: Analytic Methods in Arithmetic Geometry. Contemporary Mathematics, vol. 740, pp. 197–248. American Mathematical Society, Providence (2019)

  27. Tate, J.: Algebraic cycles and poles of zeta functions. In: Schilling, O.F.G. (ed.) Arithmetical Algebraic Geometry, pp. 93–110. Harper & Row, New York (1965)

    Google Scholar 

  28. Tate, J.: Endomorphisms of abelian varieties over finite fields. Invent. Math. 2(2), 134–144 (1966)

    Article  MathSciNet  Google Scholar 

  29. Taylor, N.: Sato–Tate distributions on Abelian surfaces. Trans. Amer. Math. Soc. 373(5), 3541–3559 (2020)

    Article  MathSciNet  Google Scholar 

  30. Taylor, R.: Remarks on a conjecture of Fontaine and Mazur. J. Inst. Math. Jussieu 1(1), 125–143 (2002)

    Article  MathSciNet  Google Scholar 

  31. Taylor, R.: Automorphy for some \(l\)-adic lifts of automorphic \( {{\rm mod }}\, l\) Galois representations. II. Publ. Math. Inst. Hautes Études Sci. 108, 183–239 (2008)

    Article  Google Scholar 

  32. Wong, P.-J.: On the Chebotarev–Sato–Tate phenomenon. J. Number Theory 196, 272–290 (2019)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The author wishes to thank Gabor Wiese for drawing his attention to this problem, as well as for the invaluable comments on an earlier version. The author would also like to thank Christian Johansson and Noah Taylor for their helpful conversations.

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  1. Department of Mathematics, Ibn Tofail University, Kenitra, Morocco

    Mohammed Amin Amri

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Amri, M.A. Chebotarev–Sato–Tate distribution for abelian surfaces potentially of \(\textrm{GL}_2\)-type. European Journal of Mathematics 9, 86 (2023). https://doi.org/10.1007/s40879-023-00682-5

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