As is well known, Euler proved the Basel identity $\displaystyle\sum\limits_{i=0}^{\infty} \frac{1}{n^2} = \frac{{\pi}^2}{6}$. By far the most illuminating explanation of this fact that I've seen is as follows:

By another theorem of Euler we can rewrite the identity as $\displaystyle \frac{{\pi}^2}{6}\cdot \prod_{p} \left(1 - \frac{1}{p^2}\right) = 1$, where $p$ ranges over all primes. The first term is the normalized volume of

$\displaystyle SL(2,\mathbb{R})/SL(2, \mathbb{Z})$

and the term corresponding to $p$ in the product is the normalized volume of

$SL(2, \mathbb{Z}_p)$.

With these replacements, the right hand side can be written as the normalized volume of

$\displaystyle SL(2,\mathbb{\mathbb{A}_{\mathbb{Q}}})/SL(2, \mathbb{Q})$.

where $\mathbb{A}_{\mathbb{Q}}$ denotes the adeles of $\mathbb{Q}$ But this last volume is equal to 1: this is a special case of the Weil Conjecture on Tamagawa Numbers.

I've been fascinated by this result for years, but have never been able to understand a proof of it (*from the adelic perspective*) even in cases as simple the one above. In this way, my question contrasts with that of Ben Weiland who asked about the theorem in more general settings.

I tried reading André Weil "Adeles and algebraic groups" with a view toward learning a proof but found the book unintelliglbe. I gathered that the idea of the proof is to show that the nonzero volume of some object is equal to the Tamagawa number multiplied by the original volume but beyond that understood nothing. My impression is that Marie-France Vigneras' book titled Arithmetique des algebres de quaternions has this material, but I don't read French.

What are some lucid sources that you would recommend for learning proofs of the some of the first few cases (including the case above) of the Weil Tamagawa Number conjecture from an adelic perspective?

1 I learned this material from Yuri Manin's "Reflections on Arithmetical Physics" and Maclachlan and Reid's "The Arithmetic of Hyperbolic 3-Manifolds"