Duke Mathematical Journal

A uniform open image theorem for -adic representations, I

Anna Cadoret and Akio Tamagawa

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Abstract

Let k be a field finitely generated over Q, and let X be a smooth, separated, and geometrically connected curve over k. Fix a prime . A representation ρ:π1(X)GLm(Z) is said to be geometrically Lie perfect if the Lie algebra of ρ(π1(X k¯)) is perfect. Typical examples of such representations are those arising from the action of π1(X) on the generic -adic Tate module T(Aη) of an abelian scheme A over X or, more generally, from the action of π1(X) on the -adic étale cohomology groups Héti(Yη¯,Q), i0, of the geometric generic fiber of a smooth proper scheme Y over X. Let G denote the image of ρ. Any k-rational point x on X induces a splitting x:Γk:=π1(Spec(k))π1(X) of the canonical restriction epimorphism π1(X)Γk so one can define the closed subgroup Gx:=ρx(Γk)G. The main result of this paper is the following uniform open image theorem. Under the above assumptions, for every geometrically Lie perfect representation ρ:π1(X)GLm(Z), the set Xρ of all xX(k) such that Gx is not open in G is finite and there exists an integer Bρ1 such that [G:Gx]Bρ for every xX(k)Xρ.

Article information

Source
Duke Math. J., Volume 161, Number 13 (2012), 2605-2634.

Dates
First available in Project Euclid: 11 October 2012

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1349960278

Digital Object Identifier
doi:10.1215/00127094-1812954

Mathematical Reviews number (MathSciNet)
MR2988904

Zentralblatt MATH identifier
1305.14016

Subjects
Primary: 14K15: Arithmetic ground fields [See also 11Dxx, 11Fxx, 11G10, 14Gxx]
Secondary: 14H30: Coverings, fundamental group [See also 14E20, 14F35] 22E20: General properties and structure of other Lie groups

Citation

Cadoret, Anna; Tamagawa, Akio. A uniform open image theorem for $\ell$ -adic representations, I. Duke Math. J. 161 (2012), no. 13, 2605--2634. doi:10.1215/00127094-1812954. https://projecteuclid.org/euclid.dmj/1349960278


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References

  • [1] K. Arai, On uniform lower bound of the Galois images associated to elliptic curves, J. Théor. Nombres Bordeaux 20 (2008), 23–43.
  • [2] M. Artin, A. Grothendieck, and J. L. Verdier, Théorie des topos et cohomologie étale des schéma, I, Séminaire de Géométrie Algébrique du Bois Marie 1963–1964 (SGA 4), Lecture Notes in Math. 269, Springer, Berlin, 1972; II, 270; III, 305, 1973.
  • [3] A. Borel, Linear Algebraic Groups, 2nd ed., Grad. Texts in Math. 126, Springer, New York, 1991.
  • [4] A. Cadoret and A. Tamagawa, “Torsion of abelian schemes and rational points on moduli spaces” in Algebraic Number Theory and Related Topics 2007, RIMS Kôkyûroku Bessatsu B12, Res. Inst. Math. Sci. (RIMS), Kyoto, 2009, 7–30.
  • [5] A. Cadoret and A. Tamagawa, Uniform boundedness of p-primary torsion of abelian schemes, Invent. Math. 188 (2012), 83–125.
  • [6] P. Deligne, Théorie de Hodge, II, Inst. Hautes Études Sci. Publ. Math. 40 (1971), 5–57.
  • [7] P. Deligne, La conjecture de Weil, II, Inst. Hautes Études Sci. Publ. Math. 52 (1980), 137–252.
  • [8] J. D. Dixon, M. P. F. du Sautoy, A. Mann, and D. Segal, Analytic Pro-$p$-groups, London Math. Soc. Lecture Note Ser. 157, Cambridge University Press, Cambridge, 1991.
  • [9] J. Ellenberg, C. Elsholtz, C. Hall, and E. Kowalski, Non-simple abelian varieties in a family: Geometric and analytic approaches, J. London Math. Soc. (2) 80 (2009), 135–154.
  • [10] G. Faltings and G. Wüstholz, eds., Rational Points, Aspects Math. E6, Vieweg, Braunschweig, Germany, 1984.
  • [11] A. Grothendieck and M. Reynaud, Revêtements étales et groupe fondamental, Séminaire de Géométrie Algébrique du Bois Marie 1960–1961 (SGA 1), Lecture Notes in Math. 224, Springer, Berlin, 1971.
  • [12] C.-Y. Hui, Specialization of monodromy group and $\ell$-independence, C. R. Math. Acad. Sci. Paris 350 (2012), 5–7.
  • [13] J.-P. Jouanolou, Théorèmes de Bertini et applications, Progr. Math. 42, Birkhäuser, Boston, 1983.
  • [14] A. Lubotsky, A group theoretic characterization of linear groups, J. Algebra 113 (1988), 207–214.
  • [15] D. Mumford and J. Fogarty, Geometric Invariant Theory, 2nd ed., Ergeb. Math. Grenzgeb. (2) 34, Springer, Berlin, 1982.
  • [16] J. Oesterlé, Réduction modulo $p^{n}$ des sous-ensembles analytiques fermés de $\mathbf{Z}_{p}^{N}$, Invent. Math. 66 (1982), 325–341.
  • [17] L. Ribes and P. Zalesskii, Profinite Groups, Ergeb. Math. Grenzgeb. (3) 40, Springer, Berlin, 2000.
  • [18] J.-P. Serre, Lie Algebras and Lie Groups, W. A. Benjamin, New York, 1965.
  • [19] J.-P. Serre, Abelian $l$-adic Representations and Elliptic Curves, W. A. Benjamin, New York, 1968.
  • [20] J.-P. Serre, Quelques applications du théorème de densité de Chebotarev, Inst. Hautes Études Sci. Publ. Math. 54 (1981), 323–401.
  • [21] J.-P. Serre, Lectures on the Mordell-Weil Theorem, Aspects Math. E15, Vieweg, Braunschweig, Germany, 1989.
  • [22] J.-P. Serre, “Lettres à Ken Ribet du 1/1/1981 et du 29/1/1981” in Œuvres, Collected papers, IV, 1985–1998, Springer, Berlin, 2000, 1–20.
  • [23] J. G. Zarhin, Torsion of abelian varieties in finite characteristic, Mat. Zametki 22 (1977), 3–11.
  • [24] Y. G. Zarhin, Abelian varieties having a reduction of $K3$ type, Duke Math. J. 65 (1992), 511–527.