# A question about the quotient measure on the ideles and the adeles

Let $F$ be a local field, then the additive Haar measure is easy to relate to the multiplicative Haar measure:

$$\int_F f(x) d^+ x = \int_{F^\times} f(x) |x| d^+ x.$$

I know that the ideles have zero measure in the adeles, so there is no way in comparing the additive Haar measure with the multiplicative measure of the ring of adeles.

Can we relate the measure on the compact quotients $F^\times \backslash \mathbb{A}^1$ and $F \backslash \mathbb{A}$?

Strong approximation comes to my mind, but I am not sure how to apply it in a reasonable manner. Any suggestions?

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I don't know why you need this, but you could try thinking of the ideles as a hyperbola in the square of the adeles. In a local sense it is not very hard to see the relationshop of the measures. –  Charles Matthews Mar 10 '12 at 12:21
I do not get, what you mean by hyperbolas in the squares? Could you please elaborate or indicate, where this idea originates from? $x \mapsto x^2$ is also not well behaved in resiude characteristic $2$. Of course, if I work in characteristic zero, then this is no problem, but for positive characteristic.... –  Marc Palm Mar 10 '12 at 17:04
pm, Charles might mean that the ideles are the solution to the "hyperbola" $xy=1$ in $\mathbb A\times\mathbb A$. This is where you get the topology of the ideles from; I don't know if it helps in relating the measures on the quotients (because I haven't thought about it). –  B R Mar 10 '12 at 18:10
Weil's book "Basic Number Theory" is a good reference for such questions. –  GH from MO Mar 13 '12 at 22:00

Rather than choose fundamental domains, the measure on a quotient $G/H$ of abelian topological groups is completely determined by the measure on $G$ and that on $H$, by $\int_G f = \int_{G/H} \int_H f(gh)\,dh\;dg$ for compactly supported continuous $f$.
The measure on the ideles is determined by the local measures everywhere, which at finite places give the local units measure 1. Then $J/k^\times$ has a uniquely-determined measure determined by counting measure on $k^\times$ and the measure on $J$. Then $(J/k^\times)/(J^1/k^\times)\approx (0,+\infty)$ is given the usual measure $dx/x$ via that natural isomorphism. By the relation of measures, as above, this uniquely specifies the measure on $J^1/k^\times$.
It is a further exercise to show that with this canonically defined measure the total measure of $J^1/k^\times$ is the usual $2^{r_1}(2\pi)^{r_2}hR/D^{1/2}w$.