## Experimental Mathematics

### Experimental indications of three-dimensional Galois representations from the cohomology of {${\rm SL}(3,{\bf Z})$}

#### Abstract

Conjecturally, any "algebraic'' automorphic representation on $\GL(n)$ should have an $n$-dimensional Galois representation attached. Many examples of algebraic automorphic representations come from the cohomology over $\bold C$ of congruence subgroups of $\GL(n,\bold Z)$. On the other hand, the first author has conjectured that for any Hecke eigenclass in the mod $p$ cohomology of a congruence subgroup of $\GL(n,\Z)$ there should be an attached $n$-dimensional Galois representation.

By computer, we found Hecke eigenclasses in the mod $p$ cohomology of certain congruence subgroups of $\SL(3,\bold Z)$. In a range of examples, we then found a Galois representation (uniquely determined up to isomorphism by our data) that seemed to be attached to the Hecke eigenclass.

#### Article information

Source
Experiment. Math., Volume 1, Issue 3 (1992), 209-223.

Dates
First available in Project Euclid: 25 March 2003

https://projecteuclid.org/euclid.em/1048622024

Mathematical Reviews number (MathSciNet)
MR1203875

Zentralblatt MATH identifier
0780.11029

Subjects
Primary: 11F75: Cohomology of arithmetic groups
Secondary: 11F80: Galois representations

#### Citation

Ash, Avner; McConnell, Mark. Experimental indications of three-dimensional Galois representations from the cohomology of {${\rm SL}(3,{\bf Z})$}. Experiment. Math. 1 (1992), no. 3, 209--223. https://projecteuclid.org/euclid.em/1048622024