I have a research-level but not necessarily new question about certain equidistribution problems. If $\phi \in L^2(S^2)$ then we could define the Weyl sums:
$$ \int \phi \, \mu\_d = \frac{1}{|\mathcal{G}\_d|} \int\_{S^2} \phi \left( \frac{a}{\sqrt{d}}, \frac{b}{\sqrt{d}} , \frac{c}{\sqrt{d}}\right) $$


where $a^2 + b^2 + c^2 = d$ and $a,b,c \in \mathbb{Z}$. These are integer points on the sphere. The Waldspurger formula says these Weyl sums could be written as:


$$ W(\phi , d ) = \int\_{\mathbf{T}(\mathbb{Q}) \backslash z\_d . \mathbf{T}(\mathbb{A}) / K\_{T\_d} } \phi(z\_d.t) \, dt $$


These integer points on the sphere could be a torus orbit. So I kept trying to understand these torus:


$$ \mathbf{T}(\mathbb{Q}) \backslash z\_d . \mathbf{T}(\mathbb{A}) \,/ \,\mathbf{K}\_{T\_d} 
 \subset\mathbf{G}(\mathbb{Q}) \backslash \mathbf{G}(\mathbb{A}) \,/\, \mathbf{K} $$


They seem to refer to a specific torus construction:


$$ \mathbf{T}\_d = \mathrm{res}\_{K/Q} \big( \mathbb{G}\_m / \mathbb{G}\_m \big) \subset \mathbf{G} = \mathrm{PG}(B^{(2,\infty)}) $$
I believe "res" is short for "restriction" which could mean anything really. Possibly Weil's "restriction of coefficients" And I'm not sure what it means to restrict from $K = \mathbb{Q}(\sqrt{d})$ to a quadratic form $Q = a^2 + b^2 + c^2$. 


On the $\mathbf{G}$ side there some kind of quaternion algbra where $\sqrt{d}$ gets represented in as the quaternion $z\_d = a \mathbf{i} + b \mathbf{j} + c \mathbf{k}$ for a given solution $d = a^2 + b^2 + c^2$.


Then they say things like: $K\_\infty = \mathbf{T}\_d(\mathbb{R}) \simeq SO(2, \mathbb{R}) $, or $K\_{\mathbf{T}\_d} = \mathbf{T}\_d(\mathbf{A}\_\mathbb{Q}) \cap \mathbf{K}$


We an keep reading and learing... I'm not an expert on algebraic groups and I've run out of time trying to decipher the notation. So **I can never fully understand the group they are talking about** 


In the beginning, it sounds like a very nice idea. I'm sure there is something, but is there an easier way to understand this torus orbit they are describing?




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A few possible sources:


* [Equidistribution, L-functions and ergodic theory](http://www.mathunion.org/ICM/ICM2006.2/Main/icm2006.2.0421.0458.ocr.pdf)
* [Algebraic Groups](http://www.jmilne.org/math/CourseNotes/iAG200.pdf)
* [Introduction to Arithmetic Groups](https://arxiv.org/abs/math/0106063)
* [Sur les valeurs de certaines fonctions L automorphes en leur centre de symétrie](https://eudml.org/doc/89702)


These notations, unfortunately, exclude qualified readers from other fields who potentially like to understand. 



Waldspurger Formula as a Torus Integral
The $K/Q$ is clearly a typo: $(\mathrm{res}\_{K/Q})$ refers to restriction of scalars from the quadratic field $K$ to the rational field $\mathbf{Q}$. (And it is $\mathrm{Res}\_{K/\mathbf{Q}}(\mathbf{G}\_m)/\mathbf{G}\_m$ that appears). And the the torus orbit is described precisely (a certain quotient). It seems hard to say anything else about this definition... (except translate it back to a concrete problem in special cases, which is what Michel and Venkatesh do at the beginning of the paper you are liberally quoting from).


The formula you write is *not* the Waldspurger formula. What the Waldspurger formula does (as explained on line 9 of 2.2.1) is give an expression for the Weyl sum in terms of L-functions.


