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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!

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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?

At least it seems that one could not expect the double quotient to be uniformly bounded when $K$ K_G\$ shrinks to the neutral element.

Thanks!

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# finiteness of class number: a bound for semi-simple groups?

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 $K$ is not a maximal compact open subgroup?

At least it seems that one could not expect the double quotient to be uniformly bounded when $K$ shrinks to the neutral element.

Thanks!
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