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Let $H$ be an anisotropic $\mathbb{Q}$-form of $SL_2$, with $H(\mathbb{R}) = SL_2(\mathbb{R})$.

Are there any results or conjecture regarding the existence, or non-existence of intermediate subgroups $\Gamma$ between $H(\mathbb{Z})$ and $H(\mathbb{Z}[\sqrt{2}])$ such that $\Gamma$ is of infinite index in $H(\mathbb{Z}[\sqrt{2}])$ and $H(\mathbb{Z})$ is of infinite index in $\Gamma$ ?

When $H$ is isotropic, there are no such groups; this was proven by Venkataramana in "On some rigid subgroups of semisimple Lie groups." Israel J. Math. 89 (1995), no. 1-3, 227–236, doi: 10.1007/BF02808202. See Proposition 2.1.

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  • $\begingroup$ Just a remark, For algebraic number field $K$, I think the modular form theory of $H(K)$ should to be consider in $\mathbb{C}^{[K:\mathbb{Q}]}$. $\endgroup$
    – Hu xiyu
    Commented Jan 9, 2018 at 17:35
  • $\begingroup$ Either I am confused or it seems that Theorem 1 and 2 of Venkataramana's paper do not apply to the situation here: the hypotheses of either theorm ask for the group $H(\mathbb R)$ to have real rank $\ge 2$. $\endgroup$
    – Jean Raimbault
    Commented Jan 10, 2018 at 14:03
  • $\begingroup$ You're right. Proposition 2.1 is the one. I should have specified it. $\endgroup$
    – JadeSnail
    Commented Jan 10, 2018 at 15:16

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