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Let $F$ be a number field, and $G$ a connected semi-simple linear algebraic $F$-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.


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Do you want to assume $G$ is split here? Strong approximation fails heavily for compact forms of $SL_n$, for instance. – David Loeffler Dec 8 '11 at 11:22
@David Loeffler: thanks for reminding. Now the condition on isotropic factors is joined. – genshin Dec 8 '11 at 12:17
up vote 2 down vote accepted

The size of the double quotient can be bounded by a function exponential in the number of places where KG is not maximal compact. The base of the exponential will of course depend on the kernel of the central isogeny, but all the dependence on the field F can be moved into the implied constant.

The part about this that I don't know how to prove is to show that the double quotient is finite in the first place. I can only do this for PGLn, which I think is a worthwhile group to think about for this problem since you can see everything explicitly.

Let Z be the kernel of the central isogeny G'-->G. The natural map $G(\hat F)\rightarrow \prod'_v H^1(F_v,Z)$ has kernel $G'(\hat F)$ in which the rational points are dense by strong approximation. This map is surjective, so we can rewrite the adelic double quotient as a double quotient of this restricted direct product of H1's.

Now consider what happens when we shrink KG at one place v from a maximal compact to something smaller. Then the cardinality of the double quotient can increase by at most a factor of |H1(Fv,Z)|. To show the desired exponential growth, we simply need a bound on |H1(Fv,Z)| that is independent of v.

For Z=μn, we know explicitly $H^1(k,\mu_n)=k^\times/(k^\times)^n$ for any field k and the bound is easy. If G is split, then Z is a product of μn's and there is no more work to be done. For general G, pick a finite Galois extension K/F over which G splits. Then the inflation-restriction exact sequence together with the result for split groups implies the necessary uniform bound on |H1(Fv,Z)|.

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