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Let $F$ be a number field, and $G$ a connected semi-simple linear algebraic $F$-group. F$-group, which does not contain anisotropic (simple)$F$-factors. Write$\hat{F}$for the ring of finite adeles$F\otimes\hat{\mathbb{Z}}$. Then the strong approximation theorem implies that the double coset$G(F)\backslash G(\hat{F}) / K_G$is finite for any compact open subgroup$K_G\subset G(\hat{F})$. In fact it is even equal to one (i.e. trivial double quotient) if$G$is simply connected as a semi-simple group. And in general, for$G$semi-simple but not simply-connected, how should one bound the growth of the size of the double quotient? At least we know that there is an isogeny$G'\rightarrow G$with$G'$semi-simple and simply connected. Can we expect the double quotient to be bounded by some function in terms of the degree of$G'\rightarrow G$and the set of finite places where$K_G$is not a maximal compact open subgroup? At least it seems that one could not expect the double quotient to be uniformly bounded when$K_G$shrinks to the neutral element. Thanks! 2 added 4 characters in body Let$F$be a number field, and$G$a connected semi-simple linear algebraic$F$-group. Write$\hat{F}$for the ring of finite adeles$F\otimes\hat{\mathbb{Z}}$. Then the strong approximation theorem implies that the double coset$G(F)\backslash G(\hat{F}) / K_G$is finite for any compact open subgroup$K_G\subset G(\hat{F})$. In fact it is even equal to one (i.e. trivial double quotient) if$G$is simply connected as a semi-simple group. And in general, for$G$semi-simple but not simply-connected, how should one bound the growth of the size of the double quotient? At least we know that there is an isogeny$G'\rightarrow G$with$G'$semi-simple and simply connected. Can we expect the double quotient to be bounded by some function in terms of the degree of$G'\rightarrow G$and the set of finite places where$K$K_G$ is not a maximal compact open subgroup?

Thanks!