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Independence of $\boldsymbol{\ell}$ of monodromy groups

Author: CheeWhye Chin
Journal: J. Amer. Math. Soc. 17 (2004), 723-747
MSC (2000): Primary 14G10; Secondary 11G40, 14F20
Published electronically: March 30, 2004
MathSciNet review: 2053954
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Abstract: Let $X$ be a smooth curve over a finite field of characteristic $p$, let $E$ be a number field, and let $\mathbf{L} = \{\mathcal{L}_\lambda\}$ be an $E$-compatible system of lisse sheaves on the curve $X$. For each place $\lambda$ of $E$not lying over $p$, the $\lambda$-component of the system $\mathbf{L}$is a lisse $E_\lambda$-sheaf $\mathcal{L}_\lambda$ on $X$, whose associated arithmetic monodromy group is an algebraic group over the local field $E_\lambda$. We use Serre's theory of Frobenius tori and Lafforgue's proof of Deligne's conjecture to show that when the $E$-compatible system $\mathbf{L}$ is semisimple and pure of some integer weight, the isomorphism type of the identity component of these monodromy groups is ``independent of $\lambda$''. More precisely, after replacing $E$ by a finite extension, there exists a connected split reductive algebraic group $G_0$ over the number field $E$such that for every place $\lambda$ of $E$not lying over $p$, the identity component of the arithmetic monodromy group of $\mathcal{L}_\lambda$is isomorphic to the group $G_0$ with coefficients extended to the local field $E_\lambda$.

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  • [Ber75] P. Berthelot, Slopes of Frobenius in crystalline cohomology, Algebraic geometry (Proc. Sympos. Pure Math., Vol. 29, Humboldt State Univ., Arcata, Calif., 1974), Amer. Math. Soc., Providence, RI, 1975, pp. 315-328. MR 52:3167
  • [Bou58] N. Bourbaki, Éléments de mathématique. 23. Première partie: Les structures fondamentales de l'analyse. Livre II: Algèbre. Chapitre 8: Modules et anneaux semi-simples, Actualités Sci. Ind. no. 1261, Hermann, Paris, 1958. MR 20:4576
  • [Chin03a] C. Chin, Determining a connected split reductive group from its irreducible representations, math.RT/0211182, 2003.
  • [Chin03b] -, Independence of $\ell$ in Lafforgue's theorem, Adv. Math. 180 (2003), no. 1, 64-86, math.AG/0206001.
  • [Del74a] P. Deligne, La conjecture de Weil. I, Inst. Hautes Études Sci. Publ. Math. (1974), no. 43, 273-307. MR 49:5013
  • [Del74b] -, Théorie de Hodge. III, Inst. Hautes Études Sci. Publ. Math. (1974), no. 44, 5-77. MR 58:16653b
  • [Del80] -, La conjecture de Weil. II, Inst. Hautes Études Sci. Publ. Math. (1980), no. 52, 137-252. MR 83c:14017
  • [DG70] M. Demazure and P. Gabriel, Groupes algébriques. Tome I: Géométrie algébrique, généralités, groupes commutatifs, Masson & Cie, Éditeur, Paris, 1970, avec un appendice Corps de classes local par M. Hazewinkel. MR 46:1800
  • [dJ96] A. J. de Jong, Smoothness, semi-stability and alterations, Inst. Hautes Études Sci. Publ. Math. (1996), no. 83, 51-93. MR 98e:14011
  • [Katz99] N. M. Katz, Space filling curves over finite fields, Math. Res. Lett. 6 (1999), no. 5-6, 613-624, corrections in: 8 (2001), no. 5-6, 689-691. MR 2001e:11067
  • [KM74] -and W. Messing, Some consequences of the Riemann hypothesis for varieties over finite fields, Invent. Math. 23 (1974), 73-77. MR 48:11117
  • [Laf02] L. Lafforgue, Chtoucas de Drinfeld et correspondance de Langlands, Invent. Math. 147 (2002), no. 1, 1-241. MR 2002m:11039
  • [LP90] M. Larsen and R. Pink, Determining representations from invariant dimensions, Invent. Math. 102 (1990), no. 2, 377-398. MR 92c:22026
  • [LP92] -, On $\ell$-independence of algebraic monodromy groups in compatible systems of representations, Invent. Math. 107 (1992), no. 3, 603-636. MR 93h:22031
  • [LP95] -, Abelian varieties, $\ell$-adic representations, and $\ell$-independence, Math. Ann. 302 (1995), no. 3, 561-579. MR 97e:14057
  • [Ser65] J.-P. Serre, Zeta and $L$ functions, Arithmetical Algebraic Geometry (Proc. Conf. Purdue Univ., 1963), Harper & Row, New York, 1965, pp. 82-92. MR 33:2606
  • [Ser00] -, Lettres à Ken Ribet du 1/1/1981 et du 29/1/1981, \OEuvres. Collected papers. IV, Springer-Verlag, Berlin, 2000, 1985-1998, pp. viii+657. MR 2001e:01037

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Additional Information

CheeWhye Chin
Affiliation: Department of Mathematics, University of California, Berkeley, California 94720
Address at time of publication: The Broad Institute – MIT, 320 Charles Street, Cambridge, Massachusetts 02141

Keywords: Independence of $\ell$, monodromy groups, compatible systems
Received by editor(s): May 18, 2003
Published electronically: March 30, 2004
Dedicated: Dedicated to Nicholas M. Katz on his 60th birthday
Article copyright: © Copyright 2004 CheeWhye Chin

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