## 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.

### MSC:

 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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