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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 algebraic group $G$ over a global field $K$, there is a nonzero $n$-dimensional rational differential $K$-form which is invariant under left-translation. The differential form induces a Haar measure on $G(K_v)$ for each completion $K_v$ of $K$, hence it determines a Haar measure on the restricted direct product $G(\mathbb{A})$ as well. The differential form is unique up to scaling by $K^\times$, hence the obtained Haar measure on $G(\mathbb{A})$ is unique. When $G$ is simply connected and semisimple, Weil conjectured that $\mathrm{vol}(G(\mathbb{A})/G(K))=1$ for this particular measure.

In the case of $G=\mathrm{SL}_2$ and $K=\mathbb{Q}$, the invariant differential form is given by $\frac{dx\wedge dy\wedge dz}{x}$, where $\begin{pmatrix} x & y \\ z & t \end{pmatrix}$ are the usual coordinates on the group. For this form one obtains the numbers mentioned by Chandan Singh Dalawat (and also by Colmez's Un autre monde est possible), hence a proof of Weil's conjecture for $G=\mathrm{SL}_2$ and $K=\mathbb{Q}$. For details I recommend Platonov-Rapinchuk: Algebraic groups and number theory (Academic Press, 1994). See especially Example 3 on pp. 166-167, the Example on pp. 222-223, and the Example on p. 262.

Added. The OP is looking for a direct (global) proof of $\mathrm{vol}(\mathrm{SL}_2(\mathbb{A})/\mathrm{SL}_2(\mathbb{R}))=1$. For the analogous (local) statement that $\mathrm{vol}(\mathrm{SL}_2(\mathbb{R})/\mathrm{SL}_2(\mathbb{Z}))=\zeta(2)$ there are nice accounts via Eisenstein series, see for example here. No doubt this proof can be modified to yield $\mathrm{vol}(\mathrm{SL}_2(\mathbb{A})/\mathrm{SL}_2(\mathbb{R}))=1$ directly as well, but I don't know of any simple written account. Hopefully someone knows a good reference. In general, Langlands proved Weil's conjecture for simply connected Chevalley groups with the help of his general theory of Eisenstein series.

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 algebraic group $G$ over a global field $K$, there is a nonzero $n$-dimensional rational differential $K$-form which is invariant under left-translation. The differential form induces a Haar measure on $G(K_v)$ for each completion $K_v$ of $K$, hence it determines a Haar measure on the restricted direct product $G(\mathbb{A})$ as well. The differential form is unique up to scaling by $K^\times$, hence the obtained Haar measure on $G(\mathbb{A})$ is unique. When $G$ is simply connected and semisimple, Weil conjectured that $\mathrm{vol}(G(\mathbb{A})/G(K))=1$ for this particular measure.

In the case of $G=\mathrm{SL}_2$ and $K=\mathbb{Q}$, the invariant differential form is given by $\frac{dx\wedge dy\wedge dz}{x}$, where $\begin{pmatrix} x & y \\ z & t \end{pmatrix}$ are the usual coordinates on the group. For this form one obtains the numbers mentioned by Chandan Singh Dalawat (and also by Colmez's Un autre monde est possible), hence a proof of Weil's conjecture for $G=\mathrm{SL}_2$ and $K=\mathbb{Q}$. For details I recommend Platonov-Rapinchuk: Algebraic groups and number theory (Academic Press, 1994). See especially Example 3 on pp. 166-167, the Example on pp. 222-223, and the Example on p. 262.

Added. The OP is looking for a direct (global) proof of $\mathrm{vol}(\mathrm{SL}_2(\mathbb{A})/\mathrm{SL}_2(\mathbb{R}))=1$. For the analogous (local) statement that $\mathrm{vol}(\mathrm{SL}_2(\mathbb{R})/\mathrm{SL}_2(\mathbb{Z}))=\zeta(2)$ there are nice accounts via Eisenstein series, see for example here. No doubt this proof can be modified to yield $\mathrm{vol}(\mathrm{SL}_2(\mathbb{A})/\mathrm{SL}_2(\mathbb{Q}))=1$ directly as well, but I don't know of any simple written account. Hopefully someone knows a good reference. In general, Langlands proved Weil's conjecture for simply connected Chevalley groups with the help of his general theory of Eisenstein series.

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 algebraic group $G$ over a global field $K$, there is a nonzero $n$-dimensional rational differential $K$-form which is invariant under left-translation. The differential form induces a Haar measure on $G(K_v)$ for each completion $K_v$ of $K$, hence it determines a Haar measure on the restricted direct product $G(\mathbb{A})$ as well. The differential form is unique up to scaling by $K^\times$, hence the obtained Haar measure on $G(\mathbb{A})$ is unique. When $G$ is simply connected and semisimple, Weil conjectured that $\mathrm{vol}(G(\mathbb{A})/G(K))=1$ for this particular measure.

In the case of $G=\mathrm{SL}_2$ and $K=\mathbb{Q}$, the invariant differential form is given by $\frac{dx\wedge dy\wedge dz}{x}$, where $\begin{pmatrix} x & y \\ z & t \end{pmatrix}$ are the usual coordinates on the group. For this form one obtains the numbers mentioned by Chandan Singh Dalawat (and also by Colmez's Un autre monde est possible), hence a proof of Weil's conjecture for $G=\mathrm{SL}_2$ and $K=\mathbb{Q}$. For details I recommend Platonov-Rapinchuk: Algebraic groups and number theory (Academic Press, 1994). See especially Example 3 on pp. 166-167, the Example on pp. 222-223, and the Example on p. 262.

Added. The OP is looking for a direct (global) proof of $\mathrm{vol}(\mathrm{SL}_2(\mathbb{A})/\mathrm{SL}_2(\mathbb{R}))=1$. For the analogous (local) statement that $\mathrm{vol}(\mathrm{SL}_2(\mathbb{R})/\mathrm{SL}_2(\mathbb{Z}))=\zeta(2)$ there are nice accounts via Eisenstein series, see for example here. No doubt this proof can be modified to yield $\mathrm{vol}(\mathrm{SL}_2(\mathbb{A})/\mathrm{SL}_2(\mathbb{R}))=1$ as well, but I don't know of any simple written account. Hopefully someone knows a good reference. In general, Langlands proved Weil's conjecture for simply connected Chevalley groups with the help of his general theory of Eisenstein series.

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 algebraic group $G$ over a global field $K$, there is a nonzero $n$-dimensional rational differential $K$-form which is invariant under left-translation. The differential form induces a Haar measure on $G(K_v)$ for each completion $K_v$ of $K$, hence it determines a Haar measure on the restricted direct product $G(\mathbb{A})$ as well. The differential form is unique up to scaling by $K^\times$, hence the obtained Haar measure on $G(\mathbb{A})$ is unique. When $G$ is simply connected and semisimple, Weil conjectured that $\mathrm{vol}(G(\mathbb{A})/G(K))=1$ for this particular measure.

In the case of $G=\mathrm{SL}_2$ and $K=\mathbb{Q}$, the invariant differential form is given by $\frac{dx\wedge dy\wedge dz}{x}$, where $\begin{pmatrix} x & y \\ z & t \end{pmatrix}$ are the usual coordinates on the group. For this form one obtains the numbers mentioned by Chandan Singh Dalawat (and also by Colmez's Un autre monde est possible), hence a proof of Weil's conjecture for $G=\mathrm{SL}_2$ and $K=\mathbb{Q}$. For details I recommend Platonov-Rapinchuk: Algebraic groups and number theory (Academic Press, 1994). See especially Example 3 on pp. 166-167, the Example on pp. 222-223, and the Example on p. 262.

Added. The OP is looking for a direct (global) proof of $\mathrm{vol}(\mathrm{SL}_2(\mathbb{A})/\mathrm{SL}_2(\mathbb{R}))=1$. For the analogous (local) statement that $\mathrm{vol}(\mathrm{SL}_2(\mathbb{R})/\mathrm{SL}_2(\mathbb{Z}))=\zeta(2)$ there are nice accounts via Eisenstein series, see for example here. No doubt this proof can be modified to yield $\mathrm{vol}(\mathrm{SL}_2(\mathbb{A})/\mathrm{SL}_2(\mathbb{R}))=1$ directly as well, but I don't know of any simple written account. Hopefully someone knows a good reference. In general, Langlands proved Weil's conjecture for simply connected Chevalley groups with the help of his general theory of Eisenstein series.

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 algebraic group $G$ over a global field $K$, there is a nonzero $n$-dimensional rational differential $K$-form which is invariant under left-translation. The differential form induces a Haar measure on $G(K_v)$ for each completion $K_v$ of $K$, hence it determines a Haar measure on the restricted direct product $G(\mathbb{A})$ as well. The differential form is unique up to scaling by $K^\times$, hence the obtained Haar measure on $G(\mathbb{A})$ is unique. When $G$ is simply connected and semisimple, Weil conjectured that $\mathrm{vol}(G(\mathbb{A})/G(K))=1$ for this particular measure.

In the case of $G=\mathrm{SL}_2$ and $K=\mathbb{Q}$, the invariant differential form is given by $\frac{dx\wedge dy\wedge dz}{x}$, where $\begin{pmatrix} x & y \\ z & t \end{pmatrix}$ are the usual coordinates on the group. For this form one obtains the numbers mentioned by Chandan Singh Dalawat (and also by Colmez's Un autre monde est possible), hence a proof of Weil's conjecture for $G=\mathrm{SL}_2$ and $K=\mathbb{Q}$. For details I recommend Platonov-Rapinchuk: Algebraic groups and number theory (Academic Press, 1994). See especially Example 3 on pp. 166-167, the Example on pp. 222-223, and the Example on p. 262.

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 algebraic group $G$ over a global field $K$, there is a nonzero $n$-dimensional rational differential $K$-form which is invariant under left-translation. The differential form induces a Haar measure on $G(K_v)$ for each completion $K_v$ of $K$, hence it determines a Haar measure on the restricted direct product $G(\mathbb{A})$ as well. The differential form is unique up to scaling by $K^\times$, hence the obtained Haar measure on $G(\mathbb{A})$ is unique. When $G$ is simply connected and semisimple, Weil conjectured that $\mathrm{vol}(G(\mathbb{A})/G(K))=1$ for this particular measure.

In the case of $G=\mathrm{SL}_2$ and $K=\mathbb{Q}$, the invariant differential form is given by $\frac{dx\wedge dy\wedge dz}{x}$, where $\begin{pmatrix} x & y \\ z & t \end{pmatrix}$ are the usual coordinates on the group. For this form one obtains the numbers mentioned by Chandan Singh Dalawat (and also by Colmez's Un autre monde est possible), hence a proof of Weil's conjecture for $G=\mathrm{SL}_2$ and $K=\mathbb{Q}$. For details I recommend Platonov-Rapinchuk: Algebraic groups and number theory (Academic Press, 1994). See especially Example 3 on pp. 166-167, the Example on pp. 222-223, and the Example on p. 262.

Added. The OP is looking for a direct (global) proof of $\mathrm{vol}(\mathrm{SL}_2(\mathbb{A})/\mathrm{SL}_2(\mathbb{R}))=1$. For the analogous (local) statement that $\mathrm{vol}(\mathrm{SL}_2(\mathbb{R})/\mathrm{SL}_2(\mathbb{Z}))=\zeta(2)$ there are nice accounts via Eisenstein series, see for example here. No doubt this proof can be modified to yield $\mathrm{vol}(\mathrm{SL}_2(\mathbb{A})/\mathrm{SL}_2(\mathbb{R}))=1$ as well, but I don't know of any simple written account. Hopefully someone knows a good reference. In general, Langlands proved Weil's conjecture for simply connected Chevalley groups with the help of his general theory of Eisenstein series.