Coset representatives of $\mathbb{A}/\mathbb{Q}$ and cusp forms

Question

I'm reading Deitmar's "Automorphic form" and it said that a function $\varphi\in L^{2}(\mathrm{GL}_{2}(\mathbb{Q})\backslash \mathrm{GL}_{2}(\mathbb{A})^{1})$ is a cusp form if $$ \int_{N(\mathbb{Q})\backslash N(\mathbb{A})}\varphi(ng)dn = \int_{\mathbb{A}/\mathbb{Q}} \varphi\left(\begin{pmatrix}1&x\\&1\end{pmatrix}g\right)dx=0 $$ for almost all $g\in \mathrm{GL}_{2}(\mathbb{A})^{1}$, where $\mathrm{GL}_{2}(\mathbb{A})^{1}=\{g\in \mathrm{GL}_{2}(\mathbb{A})\,:\, |\det(g)|=1\}$.

I think this definition should be compatible with the classical definition of holomorphic cusp forms, so the following formula will holds for $\varphi=\phi_{f}$ where $f\in S_{k}(\mathrm{SL}_{2}(\mathbb{Z}))$ is a cusp form of weight $k$ and $$ \phi_{f}(g) = f(g_{\infty}i)j(g_{\infty}, i)^{-k} $$ where $g_{\infty}$ is a $\infty$-part of $g$ and $j(g_{\infty}, z) = c_{\infty}z+d_{\infty}$. I tried to show this, but I don't know how to integrate over $\mathbb{A}/\mathbb{Q}$. So my question can be summarized as follows:

1. What is a set of coset representatives of $\mathbb{A}/\mathbb{Q}$ and how to integrate over it?
2. Why we need a condition that the integral vanishes for almost all $g$, not all $g$?

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Thank you for your answers, and I just found that we have $$ \mathbb{A}/\mathbb{Q}\simeq \prod_{p<\infty}\mathbb{Z}_{p}\times [0,1) $$ The proof may be found at here. (I think I saw this few years ago but I completely forgot about this..)

Answer 1

• 1. It is easy to see (by definition of restricted direct product) that $$ \mathbb A = \mathbb Q(\hat {\mathbb Z} \times \mathbb R). $$ Since the intersection of $\mathbb Q$ (diagonally embedded) and $(\hat {\mathbb Z} \times \mathbb R)$ are the rational numbers which are $p$-adic integers everywhere, this intersection is $\mathbb Z$. So we get $$ \mathbb A/ \mathbb Q = (\hat {\mathbb Z} \times \mathbb R) / \mathbb Z \simeq \hat {\mathbb Z} \times (\mathbb R / \mathbb Z).$$
• 2. The "almost all" condition is because a "for all" condition doesn't make sense for $L^2$-functions.

Also, you seem to be missing a determinant factor in your definition of $j(g,z)$.

The passage from classical modular forms to adelic automorphic forms is explained many places, e.g., Bump's book, Gelbart's book, Cogdell's Fields Institute notes, Kudla's notes in "Introduction to the Langlands Program."