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?

A few possible sources:

- Equidistribution, L-functions and ergodic theory
- Algebraic Groups
- Introduction to Arithmetic Groups
- Sur les valeurs de certaines fonctions L automorphes en leur centre de symétrie

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