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Rigid local systems and motives of type G2. With an appendix by Michale Dettweiler and Nicholas M. Katz

Part of: Cycles and subschemes (Co)homology theory

Published online by Cambridge University Press:  24 March 2010

Michael Dettweiler  and
Stefan Reiter
Michael Dettweiler
Affiliation:
Interdisciplinary Center for Scientific Computing (IWR), University of Heidelberg, 69120 Heidelberg, Germany (email: [email protected])
Stefan Reiter
Affiliation:
Institut für Mathematik, Johannes Gutenberg Universität Mainz, D-55099, Mainz, Germany (email: [email protected])
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Abstract

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Using the middle convolution functor MCχ introduced by N. Katz, we prove the existence of rigid local systems whose monodromy is dense in the simple algebraic group G2. We derive the existence of motives for motivated cycles which have a motivic Galois group of type G2. Granting Grothendieck’s standard conjectures, the existence of motives with motivic Galois group of type G2 can be deduced, giving a partial answer to a question of Serre.

Keywords

middle convolutionlocal systemsmotives

MSC classification

Secondary: 14F05: Sheaves, derived categories of sheaves and related constructions 14C25: Algebraic cycles
Type
Research Article
Information
Compositio Mathematica , Volume 146 , Issue 4 , July 2010 , pp. 929 - 963
Copyright
Copyright © Foundation Compositio Mathematica 2010

References

[1]André, Y., Pour une théorie inconditionelle des motifs, Publ. Math. Inst. Hautes Études Sci. 83 (1996), 5–49.Google Scholar
[2]André, Y., Une introduction aux motifs, Panoramas et Synthèses, vol. 17 (Soc. Math. France, Paris, 2004).Google Scholar
[3]Aschbacher, M., Chevalley groups of type G 2 as the group of a trilinear form, J. Algebra 109 (1987), 193–259.CrossRefGoogle Scholar
[4]Bernstein, J., Beilinson, A. A. and Deligne, P., Faisceaux pervers, Astérisque 100 (1982).Google Scholar
[5]Beukers, F. and Heckman, G., Monodromy for the hypergeometric function nF n−1, Invent. Math. 95 (1989), 325–354.CrossRefGoogle Scholar
[6]Borel, A., Linear algebraic groups, Graduate Texts in Mathematics, vol. 126 (Springer, New York, 1991).CrossRefGoogle Scholar
[7]Carter, R. W., Finite groups of Lie type: conjugacy classes and complex characters (John Wiley and Sons, Chichester, 1993).Google Scholar
[8]Deligne, P., Equations differéntielles à points singuliers réguliers, Lecture Notes in Mathematics, vol. 163 (Springer, Berlin, 1970).CrossRefGoogle Scholar
[9]Deligne, P., Variétés de Shimura: interprétation modulaire, et techniques de construction de modèles canoniques, in Automorphic forms, representations and L-functions, Proceedings of Symposia in Pure Mathematics, vol. 33, eds Borel, A. and Casselman, W. (American Mathematical Society, Providence, RI, 1979), 247–290.CrossRefGoogle Scholar
[10]Deligne, P., La conjecture de Weil. II, Publ. Math. Inst. Hautes Études Sci. 52 (1980), 137–252.CrossRefGoogle Scholar
[11]Deligne, P., Catégories tannakiennes, in The Grothendieck Festschrift, Vol. II, Progress in Mathematics, vol. 87 (Birkhäuser, Boston, 1990), 111–195.Google Scholar
[12]Deligne, P., Milne, J., Ogus, A. and Shih, K., Hodge cycles, motives, and Shimura varieties, Lecture Notes in Mathematics, vol. 900 (Springer, Berlin, 1982).Google Scholar
[13]Esnault, H., Schechtman, V. and Viehweg, E., Cohomology of local systems on the complement of hyperplanes, Invent. Math. 109 (1992), 557–561.Google Scholar
[14]Faltings, G., p-adic Hodge theory, J. Amer. Math. Soc. 1 (1988), 255–299.Google Scholar
[15]Feit, W. and Fong, P., Rational rigidity of G 2(p) for any prime p>5, in Proceedings of the Rutgers group theory year, 1983–1984, eds Aschbacher, M.et al. (Cambridge University Press, Cambridge, 1985), 323–326.Google Scholar
[16]Gross, B. H. and Savin, G., Motives with Galois group of type G 2: an exceptional theta-correspondence, Compositio Math. 114 (1998), 153–217.Google Scholar
[17]Grothendieck, A., Théorie des topos et cohomologie étale des schémas (SGA 4), Lecture Notes in Mathematics, vol. 269, 270, 305 (Springer, Berlin, 1972–1973) (with M. Artin and J.-L. Verdier).CrossRefGoogle Scholar
[18]Grothendieck, A., Cohomologie ℓ-adique et fonctions L (SGA V), Lecture Notes in Mathematics, vol. 589 (Springer, Berlin, 1977).Google Scholar
[19]Hironaka, H., Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II, Ann. of Math. (2) 79 (1964), 109–203, 205–326.CrossRefGoogle Scholar
[20]Ince, E. L., Ordinary differential equations (Dover Publications, Mineola, NY, 1956).Google Scholar
[21]Jannsen, U., Kleiman, S. and Serre eds, J.-P., Motives I, II, Proceedings of Symposia in Pure Mathematics, vol. 55 (American Mathematical Society, Providence, RI, 1994).Google Scholar
[22]Katz, N. M., Sommes exponentielles, Astérisque 79 (1980).Google Scholar
[23]Katz, N. M., Gauss sums, Kloosterman sums, and monodromy groups, Annals of Mathematics Studies, vol. 116 (Princeton University Press, Princeton, 1988).Google Scholar
[24]Katz, N. M., Exponential sums and differential equations, Annals of Mathematics Studies, vol. 124 (Princeton University Press, Princeton, 1990).Google Scholar
[25]Katz, N. M., Rigid local systems, Annals of Mathematics Studies, vol. 139 (Princeton University Press, Princeton, NJ, 1996).CrossRefGoogle Scholar
[26]Kisin, M. and Wortmann, S., A note on Artin motives, Math. Res. Lett. 10 (2003), 375–389.Google Scholar
[27]Kneser, M., Galois-Kohomologie halbeinfacher algebraischer Gruppen über p-adischen Körpern I, II, Math. Z. 88 (1965), 259–276 and 89 (1965), 250–272.CrossRefGoogle Scholar
[28]Langlands, R. P., Automorphic representations, Shimura varieties, and motives. Ein Märchen, in Automorphic forms, representations and L-functions, Proceedings of Symposia in Pure Mathematics, vol. 33, eds Borel, A. and Casselman, W. (American Mathematical Society, Providence, RI, 1979), 205–246.Google Scholar
[29]Lawther, R., Jordan block sizes of unipotent elements in exceptional algebraic groups, Comm. Algebra 23 (1995), 4125–4156.CrossRefGoogle Scholar
[30]Onishchik, A. L. and Vinberg, E. B., Lie groups and algebraic groups (Springer, Berlin, 1990).Google Scholar
[31]Riemann, B., Beiträge zur Theorie der durch die Gauss’sche Reihe F(α,β,γ,x) darstellbaren Functionen, Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen 7 (1857), 3–32.Google Scholar
[32]Saavedra Rivano, N., Categories Tannakiennes, Lecture Notes in Mathematics, vol. 265 (Springer, Berlin, 1972).CrossRefGoogle Scholar
[33]Scholl, A., On some ℓ-adic representations of attached to noncongruence subgroups, Bull. London Math. Soc. 38 (2006), 561–567.CrossRefGoogle Scholar
[34]Serre, J.-P., Lectures on the Mordell–Weil theorem, Aspects of Mathematics, vol. 15 (Vieweg, Wiesbaden, 1989).CrossRefGoogle Scholar
[35]Serre, J.-P., Propriétés conjecturales des groupes de Galois motiviques et des représentations ℓ-adiques, in Motives I, Proceedings of Symposia in Pure Mathematics, vol. 55 eds Jannsen, U., Kleiman, S. and Serre, J.-P. (American Mathematical Society, Providence, RI, 1994), 377–400.CrossRefGoogle Scholar
[36]Serre, J.-P., Cohomologie galoisienne: progrès et problèmes, Astérisque 227 (1995), 229–257 (Séminaire Bourbaki 1993/94).Google Scholar
[37]Thompson, J. G., Rational rigidity of G 2(5), in Proceedings of the Rutgers group theory year, 1983–1984, eds Aschbacher, M.et al. (Cambridge University Press, Cambridge, 1985), 321–322.Google Scholar