Maximality of Galois actions for compatible systems. (English) Zbl 0912.11026

From the introduction: The maximality of Galois groups associated with cohomology of varieties was first discussed by Serre. He proved that if \(K\) is a number field and \(E/K\) an elliptic curve not potentially of CM-type, then for all \(\ell \gg 0\), the homomorphism \(\text{Gal} (\overline K/K)\to \text{GL} (T_\ell (E))\), giving the Galois action on the Tate module of \(E\), is surjective. Here the author would like to generalize this to the case of a compatible system of \(n\)-dimensional representations \[ \rho_\ell: \text{Gal} (\overline K/K) \to\text{GL}_n (\mathbb{Q}_\ell), \] in the sense of Serre. Of course, the image of \(\rho_\ell\) is not generally Zariski-dense in \(\text{GL}_n (\mathbb{Q}_\ell)\), so the maximality condition must be formulated relative to the Zariski closure, \(G_\ell\), of \(\rho_\ell (\text{Gal} (\overline K/K))\). One might hope that the image of \(\rho_\ell\) is a maximal compact subgroup of \(G_\ell (\mathbb{Q}_\ell)\), but this is too optimistic: the center causes problems, and we do not even know a priori that \(G_\ell\) is reductive.
To formulate a maximality conjecture that avoids such problems, we introduce maps \[ G^0_\ell @>\sigma>> G_\ell^{\text{ad}} @>>\tau> G_\ell^{\text{sc}}, \] where \(G^0_\ell\) denotes the identity component of \(G_\ell\), \(G_\ell^{\text{ad} }\) the quotient of \(G^0_\ell\) by its radical, and \(G_\ell^{\text{sc}}\) the simply connected cover of \(G_\ell^{\text{ad}}\). We expect that for \(\ell \gg 0\), \[ \tau^{-1} \Bigl(\sigma \biggl( \rho_\ell\bigl( \text{Gal} (\overline K/K) \bigr) \cap G^0_\ell (\mathbb{Q}_\ell) \biggr) \Bigr) \tag{1} \] should be a hyperspecial maximal compact subgroup of \(G_\ell^{\text{sc}}\). This technical condition implies, in particular, that with respect to Haar measure, the group (1) is of maximal volume.
The main result of this paper is a weaker claim, namely that (1) is a hyperspecial maximal compact for a set of primes \(\ell\) of Dirichlet density 1. The argument is valid not only for systems of cohomology representations, but for all compatible systems of \(\ell\)-adic Galois representations.
The proof makes essential use of a group-theoretical result that may be of some independent interest. Given a connected semisimple algebraic group \(G/ \mathbb{Q}\), we can extend \(G\) to a smooth group scheme over \(\mathbb{Z} [1/N]\) for \(N\) sufficiently divisible. We consider the family of finite groups \(G(\mathbb{F}_\ell)\) for \(\ell \gg 0\). The theorem says that for \(\ell\gg 0\), every maximal proper subgroup of the abstract group \(G(\mathbb{F}_\ell)\) is actually algebraic in a suitable sense. The proof depends on the classification of finite simple groups.
The two other main ingredients in the proof of the main theorem are the Bruhat-Tits theory of \(p\)-adic groups, and the theory of algebraic monodromy of compatible systems of Galois representations developed earlier.


11G09 Drinfel’d modules; higher-dimensional motives, etc.
11G10 Abelian varieties of dimension \(> 1\)
11S37 Langlands-Weil conjectures, nonabelian class field theory
20G25 Linear algebraic groups over local fields and their integers
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