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308    15. Elliptic Curves over Global Fields and  -Adic Representations

        (8.3) Proposition. Let G be a closed subgroup of     GL 2 (Z   ) and denote by G

                                                                ¯
        its projection to GL 2 (Z   ) and the reduction of G   into GL 2 (F   ) by G   .Then G is
        open in the product     GL 2 (Z   ) if the following three conditions hold:

         (a) For all  , the subgroup G   is open in GL 2 (Z   ).

         (b) The image of G under det :  GL 2 (Z   ) →  Z is open.
                                                    ∗

         (c) For all but a finite number of  , the subgroup G   contains the subgroup SL 2 (Z   )
            of GL 2 (Z   ).
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