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\(A_6\)-extensions of \(\mathbb Q\) and the mod \(p\) cohomology of \(\text{GL}_3(\mathbb Z)\). (English) Zbl 1104.11025

A special case of a generalization of a conjecture of Serre is as follows. Let \(\rho:G_{\mathbb{Q}}\rightarrow \text{GL}_3(\bar{\mathbb{F}}_p)\) be a continuous irreducible representation. If \(p>2\), we assume that \(\rho(\text{Frob}_\infty)\) has eigenvalues \(1,1,-1\) or \(1,-1,-1\). Let \(N\) be the level and \(\varepsilon\) the nebentype character associated to \(\rho\). Then for any weight \((a,b,c)\) associated to \(\rho\), there exists a Hecke eigenclass \(x\) in \(H^3(\Gamma_0(N),F(a,b,c)_\varepsilon)\) with \(\rho\) attachted.
As usual, \(G_{\mathbb{Q}}\) denotes the absolute Galois group of \({\mathbb{Q}}\). The level \(N=\prod_{q\neq p}q^{n_q}\) is defined via higher ramification groups \(G_i\) with \(n_q=\sum_{i=0}^\infty (|G_i|/|G_0|)\cdot \dim M/M^{G_i}\). A weight is a triple \((a,b,c)\) satisfying \(0\leq a-b,b-c\leq p-1\) and \(0\leq c\leq p-2\). \(\Gamma_0(N)\) is the subgroup of \(\text{Sl}_3({\mathbb{Z}})\) consisting of matrices whose first row is congruent to \((\star,0,0)\) modulo \(N\).
The authors present six examples of \(3\)-dimensional mod\(p\) Galois representations of projective type \(A_6\) for which they are able to obtain computational evidence for the conjecture. They looked at twelve \(A_6\)-extensions of \({\mathbb{Q}}\), which are ramified at most at two primes \(\leq 19\). They could feasible check the conjecture at six of these extensions. The checking of the conjecture requires studying the local behavior of these extensions. These computations were done entirely with PARI/GP.

MSC:

11F80 Galois representations
11R32 Galois theory

Software:

PARI/GP
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References:

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