Waldspurger Formula as a Torus Integral

Question

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
• 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.

Answer 1

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.