Galois representations and Hecke operators associated with the mod $p$ cohomology of $GL(1,\mathbb {Z})$ and $GL(2,\mathbb {Z})$
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- by Avner Ash PDF
- Proc. Amer. Math. Soc. 125 (1997), 3209-3212 Request permission
Abstract:
We prove that any Hecke eigenclass in the mod $p$ cohomology of a congruence subgroup of $GL(1,\mathbb {Z})$ or $GL(2,\mathbb {Z})$ has attached to it a mod $p$ Galois representation such that the characteristic polynomial of a Frobenius element at a prime $l$ equals the Hecke polynomial at $l$.References
- Avner Ash, Galois representations attached to mod $p$ cohomology of $\textrm {GL}(n,\textbf {Z})$, Duke Math. J. 65 (1992), no. 2, 235–255. MR 1150586, DOI 10.1215/S0012-7094-92-06510-0
- A. Ash, Galois representations and cohomology of $GL(n,\mathbb {Z} )$, Seminaire de Theorie des Nombres, Paris, 1989-90, (S. David, ed.), Birkhauser, Boston (1992), 9-22.
- A. Ash and R. Manjrekar, Galois Representations and Hecke Operators associated with the mod-$p$ cohomology of $GL(m(p-1),\mathbb {Z} )$, to appear in Math. Zeit.
- Avner Ash and Mark McConnell, Experimental indications of three-dimensional Galois representations from the cohomology of $\textrm {SL}(3,\textbf {Z})$, Experiment. Math. 1 (1992), no. 3, 209–223. MR 1203875
- Avner Ash and Glenn Stevens, Cohomology of arithmetic groups and congruences between systems of Hecke eigenvalues, J. Reine Angew. Math. 365 (1986), 192–220. MR 826158
- Pierre Deligne and Jean-Pierre Serre, Formes modulaires de poids $1$, Ann. Sci. École Norm. Sup. (4) 7 (1974), 507–530 (1975) (French). MR 379379
- Jean-Pierre Serre, Sur les représentations modulaires de degré $2$ de $\textrm {Gal}(\overline \textbf {Q}/\textbf {Q})$, Duke Math. J. 54 (1987), no. 1, 179–230 (French). MR 885783, DOI 10.1215/S0012-7094-87-05413-5
- Goro Shimura, Introduction to the arithmetic theory of automorphic functions, Kanô Memorial Lectures, No. 1, Iwanami Shoten Publishers, Tokyo; Princeton University Press, Princeton, N.J., 1971. Publications of the Mathematical Society of Japan, No. 11. MR 0314766
Additional Information
- Avner Ash
- Affiliation: The Ohio State University, Department of Mathematics, 231 W. 18th Ave, Columbus, Ohio 43210
- MR Author ID: 205374
- Email: ash@math.ohio-state.edu
- Received by editor(s): June 23, 1996
- Additional Notes: Research partially supported by NSA grant MDA-904-94-2030. This manuscript is submitted for publication with the understanding that the United States government is authorized to reproduce and distribute reprints.
- Communicated by: William W. Adams
- © Copyright 1997 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 125 (1997), 3209-3212
- MSC (1991): Primary 11F75
- DOI: https://doi.org/10.1090/S0002-9939-97-04085-9
- MathSciNet review: 1422842