## Experimental Mathematics

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

### Galois representations, Hecke operators, and the mod-$p$ cohomology of ${\rm GL}(3,\mathbb Z)$ 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 $\mathbb F_p$, 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 $GL(3,\,\bar{\mathbb F}_p$). 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 ${\rm GL}(3,\mathbb Z)$ with twisted coefficients. Experiment. Math. 7 (1998), no. 4, 361--390. https://projecteuclid.org/euclid.em/1047674153