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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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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


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 $l$.
  • 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
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Additional Information
  • Avner Ash
  • Affiliation: The Ohio State University, Department of Mathematics, 231 W. 18th Ave, Columbus, Ohio 43210
  • MR Author ID: 205374
  • Email:
  • 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:
  • MathSciNet review: 1422842