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Inventiones mathematicae

, Volume 214, Issue 2, pp 605–639 | Cite as

Arithmetic representations of fundamental groups I

  • Daniel Litt
Article

Abstract

Let X be a normal algebraic variety over a finitely generated field k of characteristic zero, and let \(\ell \) be a prime. Say that a continuous \(\ell \)-adic representation \(\rho \) of \(\pi _1^{\acute{\mathrm{e}}\text {t}}(X_{\bar{k}})\) is arithmetic if there exists a finite extension \(k'\) of k, and a representation \(\tilde{\rho }\) of \(\pi _1^{\acute{\mathrm{e}}\text {t}}(X_{k'})\), with \(\rho \) a subquotient of \(\tilde{\rho }|_{\pi _1(X_{\bar{k}})}\). We show that there exists an integer \(N=N(X, \ell )\) such that every nontrivial, semisimple arithmetic representation of \(\pi _1^{\acute{\mathrm{e}}\text {t}}(X_{\bar{k}})\) is nontrivial mod \(\ell ^N\). As a corollary, we prove that any nontrivial \(\ell \)-adic representation of \(\pi _1^{\acute{\mathrm{e}}\text {t}}(X_{\bar{k}})\) which arises from geometry is nontrivial mod \(\ell ^N\).

Notes

Acknowledgements

This work owes a great deal to discussions with Johan de Jong, Ravi Vakil, David Hansen, and Daniel Halpern-Leistner. Its genesis was a question by Lisa S. on Mathoverflow [28] and a talk by Benjamin Bakker on his work with Jacob Tsimerman [3, 4]. George Boxer provided the author with Example 4.5. Will Sawin suggested the use of Lemma 2.10 after reading an earlier version of this paper, and pointed the author to [5], which dramatically improved the main results of the paper—I am deeply grateful to him. I would also like to thank the anonymous referee. The author completed this work while supported by an NSF Postdoctoral Research Fellowship, Award No. 1502386.

References

  1. 1.
    Anderson, G., Ihara, Y.: Pro-\(l\) branched coverings of \({ P}^1\) and higher circular \(l\)-units. Ann. Math. (2) 128(2), 271–293 (1988)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Anderson, G.W., Ihara, Y.: Pro-\(l\) branched coverings of \({ P}^1\) and higher circular \(l\)-units. II. Int. J. Math. 1(2), 119–148 (1990)CrossRefGoogle Scholar
  3. 3.
    Bakker, B., Tsimerman, J.: The geometric torsion conjecture for abelian varieties with real multiplication. ArXiv e-prints (2015)Google Scholar
  4. 4.
    Bakker, B., Tsimerman, J.: \(p\)-torsion monodromy representations of elliptic curves over geometric function fields. Ann. Math. (2) 184(3), 709–744 (2016)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bogomolov, F.A.: Sur l’algébricité des représentations \(l\)-adiques. C. R. Acad. Sci. Paris Sér. A-B 290(15), A701–A703 (1980)zbMATHGoogle Scholar
  6. 6.
    Brunebarbe, Y.: A strong hyperbolicity property of locally symmetric varieties. ArXiv e-prints (2016)Google Scholar
  7. 7.
    Cadoret, A., Moonen, B.: Integral and adelic aspects of the Mumford–Tate conjecture. ArXiv e-prints (2015)Google Scholar
  8. 8.
    Cadoret, A.: Note on the gonality of abstract modular curves. In: The Arithmetic of Fundamental Groups—PIA 2010, volume 2 of Contributions in Mathematical and Computational Science, pp. 89–106. Springer, Heidelberg (2012)zbMATHGoogle Scholar
  9. 9.
    Cadoret, A., Tamagawa, A.: On a weak variant of the geometric torsion conjecture. J. Algebra 346, 227–247 (2011)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Cadoret, A., Tamagawa, A.: A uniform open image theorem for \(\ell \)-adic representations. I. Duke Math. J. 161(13), 2605–2634 (2012)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Cadoret, A., Tamagawa, A.: A uniform open image theorem for \(\ell \)-adic representations. II. Duke Math. J. 162(12), 2301–2344 (2013)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Cadoret, A., Tamagawa, A.: Gonality of abstract modular curves in positive characteristic. Compos. Math. 152(11), 2405–2442 (2016)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Deligne, P.: Le groupe fondamental de la droite projective moins trois points. In: Galois groups over \({Q}\) (Berkeley, CA, 1987), volume 16 of Mathematical Science Research Institute Publications, pp. 79–297. Springer, New York (1989)Google Scholar
  14. 14.
    Deligne, P.: Le groupe fondamental du complément d’une courbe plane n’ayant que des points doubles ordinaires est abélien (d’après W. Fulton). In: Bourbaki Seminar, vol. 1979/80, volume 842 of Lecture Notes in Mathematics, pp. 1–10. Springer, Berlin (1981)Google Scholar
  15. 15.
    Deligne, P.: Théorie de Hodge. I, pp. 425–430 (1971)Google Scholar
  16. 16.
    Deligne, P.: La conjecture de Weil. II. Inst. Hautes Études Sci. Publ. Math. 52, 137–252 (1980)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Hain, R., Matsumoto, M.: Weighted completion of Galois groups and Galois actions on the fundamental group of \(\mathbb{P}^1-\{0,1,\infty \}\). Compos. Math. 139(2), 119–167 (2003)CrossRefGoogle Scholar
  18. 18.
    Hain, R., Matsumoto, M.: Galois actions on fundamental groups of curves and the cycle \(C-C^-\). J. Inst. Math. Jussieu 4(3), 363–403 (2005)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Hironaka, H.: Resolution of singularities of an algebraic variety over a field of characteristic zero. I. Ann. Math. (2) 79, 109–203 (1964)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Hironaka, H.: Resolution of singularities of an algebraic variety over a field of characteristic zero. II. Ann. Math. (2) 79, 205–326 (1964)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Hwang, J.-M., To, W.-K.: Uniform boundedness of level structures on abelian varieties over complex function fields. Math. Ann. 335(2), 363–377 (2006)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Ihara, Y.: Braids, Galois groups, and some arithmetic functions. In: Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990), pp. 99–120. Mathematical Society of Japan, Tokyo (1991)Google Scholar
  23. 23.
    Ihara, Y.: Some arithmetic aspects of Galois actions in the pro-\(p\) fundamental group of \(\mathbb{P}^{1}-\{0,1,\infty \}\). In: Arithmetic fundamental Groups and Noncommutative Algebra (Berkeley, CA, 1999), volume 70 of Proceedings of Symposium Pure Mathematics, pp. 247–273. American Mathematical Society, Providence, RI (2002)Google Scholar
  24. 24.
    Ihara, Y.: On Galois representations arising from towers of coverings of \({ P}^1{\setminus }\{0,1,\infty \}\). Invent. Math. 86(3), 427–459 (1986)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Kim, M.: The motivic fundamental group of \(\mathbb{P}^1{\setminus }\{0,1,\infty \}\) and the theorem of Siegel. Invent. Math. 161(3), 629–656 (2005)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Kisin, M.: Potential semi-stability of \(p\)-adic étale cohomology. Isr. J. Math. 129, 157–173 (2002)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Kunrui, Y.: \(p\)-Adic logarithmic forms and group varieties. III. Forum Math. 19(2), 187–280 (2007)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Lisa, S.: Are there nonisotrivial elliptic curves over \(\mathbb{G}_m\)? MathOverflow. http://mathoverflow.net/q/190530 (version: 2014-12-12)
  29. 29.
    Nadel, A.M.: The nonexistence of certain level structures on abelian varieties over complex function fields. Ann. Math. (2) 129(1), 161–178 (1989)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Neukirch, J., Schmidt, A., Wingberg, K.: Cohomology of number fields, volume 323 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 2nd edn. Springer, Berlin (2008)Google Scholar
  31. 31.
    Noguchi, J.: Moduli space of abelian varieties with level structure over function fields. Int. J. Math. 2(2), 183–194 (1991)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Quillen, D.: Rational homotopy theory. Ann. Math. (2) 90, 205–295 (1969)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Revêtements étales et groupe fondamental (SGA 1). Documents Mathématiques (Paris) [Mathematical Documents (Paris)], 3. Société Mathématique de France, Paris, 2003. Séminaire de géométrie algébrique du Bois Marie 1960–61. [Algebraic Geometry Seminar of Bois Marie 1960–61], Directed by A. Grothendieck, With two papers by M. Raynaud, Updated and annotated reprint of the 1971 original [Lecture Notes in Math., 224, Springer, Berlin; MR0354651 (50 #7129)]Google Scholar
  34. 34.
    Ribes, L., Zalesskii, P.: Profinite groups, volume 40 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], 2nd edn. Springer, Berlin (2010)Google Scholar
  35. 35.
    Sawin, W.: Personal communication (2016)Google Scholar
  36. 36.
    Serre, J.-P.: Oeuvres/Collected papers. IV. 1985–1998. Springer Collected Works in Mathematics. Springer, Heidelberg (2013). Reprint of the 2000 edition [MR1730973]Google Scholar
  37. 37.
    Serre, J.-P.: Propriétés conjecturales des groupes de Galois motiviques et des représentations \(l\)-adiques. In: Motives (Seattle, WA, 1991), volume 55 of Proceedings of Symposium on Pure Matheamtics, pp. 377–400. American Mathematical Society, Providence, RI (1994)Google Scholar
  38. 38.
    Tate, J.: Endomorphisms of abelian varieties over finite fields. Invent. Math. 2, 134–144 (1966)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Wickelgren, K.: \(n\)-nilpotent obstructions to \(\pi_1\) sections of \(\mathbb{P}^{1}-\{0,1,\infty \}\) and Massey products. In: Galois–Teichmüller Theory and Arithmetic Geometry, volume 63 of Advances Studies in Pure Mathematics, pp. 579–600. Mathematical Society of Japan, Tokyo (2012)Google Scholar
  40. 40.
    Wickelgren, K.: On 3-nilpotent obstructions to \(\pi_1\) sections for \(\mathbb{P}^{1}_\mathbb{Q}-\{0,1,\infty \}\). In: The Arithmetic of Fundamental Groups—PIA 2010, volume 2 of Contributions in Mathematical and Computational Science, pp. 281–328. Springer, Heidelberg (2012)Google Scholar
  41. 41.
    Wintenberger, J.-P.: Démonstration d’une conjecture de Lang dans des cas particuliers. J. Reine Angew. Math. 553, 1–16 (2002)MathSciNetzbMATHGoogle Scholar
  42. 42.
    Wojtkowiak, Z.: On \(\ell \)-adic iterated integrals V: linear independence, properties of \(\ell \)-adic polylogarithms, \(\ell \)-adic sheaves. In: The Arithmetic of Fundamental Groups—PIA 2010, volume 2 of Contributions in Mathematical Computer Science, pp. 339–374. Springer, Heidelberg (2012)zbMATHGoogle Scholar
  43. 43.
    Wojtkowiak, Z.: On \(l\)-adic iterated integrals. I. Analog of Zagier conjecture. Nagoya Math. J. 176, 113–158 (2004)MathSciNetCrossRefGoogle Scholar
  44. 44.
    Wojtkowiak, Z.: On \(l\)-adic iterated integrals. II. Functional equations and \(l\)-adic polylogarithms. Nagoya Math. J. 177, 117–153 (2005)MathSciNetCrossRefGoogle Scholar
  45. 45.
    Wojtkowiak, Z.: On \(l\)-adic iterated integrals. III. Galois actions on fundamental groups. Nagoya Math. J. 178, 1–36 (2005)MathSciNetCrossRefGoogle Scholar
  46. 46.
    Wojtkowiak, Z.: On \(l\)-adic iterated integrals. IV. Ramification and generators of Galois actions on fundamental groups and torsors of paths. Math. J. Okayama Univ. 51, 47–69 (2009)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsColumbia UniversityNew YorkUSA

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