Experimental Mathematics

Galois representations, Hecke operators, and the mod-{$p$} cohomology of {${\rm GL}(3,\bold Z)$} with twisted coefficients

Gerald Allison, Avner Ash, and Eric Conrad

Abstract

We compute the degree 3 homology of GL(3,\,\funnyZ) with coefficients in the module of homogeneous polynomials in three variables of degree $g$ over \funnyF$_p$, for $g\leq 200$ and $p\leq 541$. The homology has a "boundary part'' and a "quasicuspidal'' part which we determine.

By conjecture a Hecke eigenclass in the homology has an attached Galois representation into GL(3,\,{\mathversion{normal}$\bar{\funnyF}$}$_p$). The conjecture is proved for the boundary part and explored experimentally for the quasicuspidal part.

Article information

Source
Experiment. Math., Volume 7, Issue 4 (1998), 361-390.

Dates
First available in Project Euclid: 14 March 2003

Permanent link to this document
https://projecteuclid.org/euclid.em/1047674153

Mathematical Reviews number (MathSciNet)
MR1678079

Zentralblatt MATH identifier
0923.11083

Subjects
Primary: 11F80: Galois representations
Secondary: 11F60: Hecke-Petersson operators, differential operators (several variables) 11F75: Cohomology of arithmetic groups 11R39: Langlands-Weil conjectures, nonabelian class field theory [See also 11Fxx, 22E55]

Keywords
Galois representations Hecke operators mod-$p$ cohomology modular symbols

Citation

Allison, Gerald; Ash, Avner; Conrad, Eric. Galois representations, Hecke operators, and the mod-{$p$} cohomology of {${\rm GL}(3,\bold Z)$} with twisted coefficients. Experiment. Math. 7 (1998), no. 4, 361--390. https://projecteuclid.org/euclid.em/1047674153


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