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One does not need that the class number is $1$, the construction in Brian Conrad's notes works in general. See Propositions 6-7 in Ch.V-4 of Weil: Basic Number Theory. The class group enters for $A_K^ …

In order to talk about the volume you need to fix the measure first. Weil observed that there is a canonical choice for the Haar measure on $G(\mathbb{A})$. For a connected $n$-dimensional semisimple …

It is clear from the definition and right-invariance of the measure $dh$ that $Pf$ is left-invariant under $SL_2(\mathbb{A}_F)$. As Paul Garrett kindly explained, $Pf$ is also left-invariant under $GL …

This is Theorem 1 in Chapter IV-1 of Weil: Basic Number Theory. See also Corollaries 1-2 after the proof of this theorem.

Yes, $G(\mathbb A_S) G(k) = G(\mathbb A)$ holds when $S$ is sufficiently large and contains the set of archimedean places $\infty$. This is because the double coset space $G(A_\infty)\backslash G(\mat …

To every local admissible representation $\pi_v$ of $\mathrm{GL}_n(k_v)$, there is a local $L$-function $L(s,\pi_v)$. For a global admissible representation $\pi=\otimes_v \pi_v$ of $\mathrm{GL}_n(\ma …

More is true: the generating set $\{f_p:\text{$p$ is a prime}\}$ is dense in $\smash{\widehat{\mathbb Z}}^\times$. To see this, it suffices to show that, for any finite set of primes $S$, the projecti …

The ray class group is a more general object than the one considered in the original post. It is defined as $$K^\times \backslash \mathbb{A}_{K}^\times /(U_\infty U_{K,I}),$$ where $U_\infty$ is an op …