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

Question

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.

Thanks!

Answer 1

The size of the double quotient can be bounded by a function exponential in the number of places where K_G 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 PGL_n, 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 K_G 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(F_v,Z)|. To show the desired exponential growth, we simply need a bound on |H¹(F_v,Z)| that is independent of v.

For Z=μ_n, 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 μ_n'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(F_v,Z)|.