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ALGEBRAIC CYCLES ON COMPACT QUATERNIONIC SHIMURA FOURFOLDS AND POLES OF L-FUNCTIONS

Published online by Cambridge University Press:  09 December 2010

CRISTIAN VIRDOL
CRISTIAN VIRDOL*
Affiliation:
Department of Mathematics, Columbia University, Room 509, MC 4406, 2990 Broadway, New York, NY 10027, USA e-mail: virdol@math.columbia.edu
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Abstract

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In this article we prove Tate conjecture for a large class of compact quaternionic Shimura fourfolds.

Keywords

11F4111F8011R4211R8014C25
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 53 , Issue 2 , May 2011 , pp. 359 - 367
Copyright
Copyright © Glasgow Mathematical Journal Trust 2010

References

REFERENCES

1.Blasius, D., Hilbert modular forms and the Ramanujan conjecture, in Noncommutative geometry and number theory (Aspects Math. vol. E37) (Vieweg, Wiesbaden, 2006) 35–56.CrossRefGoogle Scholar
2.Carayol, H., Sur la mauvaise réduction des courbes de Shimura, Compositio Mathematica 59 (2) (1986), 151–230.Google Scholar
3.Flicker, Y. Z. and Hakim, J. L., Quaternionic distinguished representations, Am. J. Math. 116 (3) (June 1994), 683–736.CrossRefGoogle Scholar
4.Harder, G., Langlands, R. P. and Rapoport, M., Algebraische Zycklen auf Hilbert–Blumenthal–Flächen, J. Reine Angew. Math. 366 (1986), 53–120.Google Scholar
5.Jacquet, H. and Gelbart, S., A relation between automorphic representations of GL(2) and GL(3), Ann. Sci. École Norm. Sup. 11 (1979), 471–542.Google Scholar
6.Jacquet, H. and Lai, K., A relative trace formula, Compositio Math. 54 (1985), 243–310.Google Scholar
7.Jacquet, H., Piatetski-Shapiro, I. I. and Shalika, J. A., Rankin–Selberg convolutions, Am. J. Math. 105 (2) (1983), 367–464.CrossRefGoogle Scholar
8.Klingenberg, C., Die Tate-Vermutungen für Hilbert–Blumenthal–Flächen, Invent. Math. 89 (1987), 291–317.CrossRefGoogle Scholar
9.Lai, K. F., Algebraic cycles on compact Shimura surface, Math. Z. 189 (1985), 593–602.CrossRefGoogle Scholar
10.Langlands, R. P., Base change for GL(2) (Ann. Math. Studies vol. 96) (Princeton University Press, Princeton, NJ, 1980).Google Scholar
11.Murty, V. K. and Ramakrishnan, D., Period relations and the Tate conjecture for Hilbert modular surfaces, Invent. Math. 89 (1987), 319–345.CrossRefGoogle Scholar
12.Ramakrishnan, D., Algebraic cycles on Hilbert modular fourfolds and poles of L-functions, in Proceedings of the International Conference on Algebraic Groups and Arithmetic (2005), 271–274.Google Scholar
13.Ramakrishnan, D., Modular curves, modular surfaces and modular fourfolds, in Algebraic cycles and motives, Volume 1, 278–292, dedicated to Jacob Murre, Cambridge University Press, London Math. Soc. Lecture Notes 343 (2007).CrossRefGoogle Scholar
14.Ramakrishnan, D., Modularity of solvable Artin representations of GO(4)-type, Int. Math. Res. Not. (1) (2002), 1–54.Google Scholar
15.Rogawski, J. D. and Tunnell, J. B., On Artin L-functions associated to Hilbert modular forms of weight one, Invent. Math. 74 (1983), 1–43.CrossRefGoogle Scholar
16.Tate, J., Algebraic cycles and poles of zeta functions, in Arithmetical algebraic geometry (Schilling, O. D. G., Editor) (Harper and Row, New York, 1966).Google Scholar
17.Taylor, R., On Galois representations associated to Hilbert modular forms, Invent. Math. 98 (1989), 265–280.CrossRefGoogle Scholar