As far as I know, there are two versions of Waldspurger's formula (classical and adelic), which can be vaguely stated as follows
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(Classical version) Let $f$ be a half-integral weight modular form of weight $k/2$ (where $k$ is odd), level $N$ and character $\chi$ and $F = \operatorname{Sh}(f)$ be its Shimura correspondent in $S_{k-1}(N/2, \chi^2)$. Then $|a_d(f)|^2$, (square of) Fourier coefficients of $f$ for square-free $d$, can be represented as $L(1/2, F \otimes \chi_d)$, the special $L$-value of quadratic twist of $F$.
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(Adelic version, brought from Horawa's notes on Zydor's lectures on periods of automorphic forms) Let $\pi$ be an automorphic cuspidal representation of $\mathrm{GL}_2(\mathbb{A}_F)$ and $\phi = \bigotimes_v \phi_v \in\pi$ be an automorphic form. Let $T \leq \mathrm{GL}_2$ be a non-split torus. Then the square of the toric period is equal to product of special $L$-values and local periods:
$$
|\mathcal{P}_T(\phi)|^2 = \bigg|\int_{T(\mathbb{A}_F) / T(F)} \phi(t) dt\bigg|^2 = c\cdot L(1/2, \pi) \cdot L(1/2, \pi \otimes \eta) \cdot \prod_v \mathcal{L}_v(\phi_v).
$$
I wonder how the adelic version can be thought as a generalization of the classical version. To be precise, I can't find a connection with toric periods and Fourier coefficients of half-integral weight modular forms (there's even nothing about Shimura correspondence in the adelic statement. Note that Waldspurger mention about the classical result in the introduction of his original paper). Is it possible to somewhat adelize the classical statement as follows: for an automorphic cuspidal representation $\pi'$ of $\operatorname{Mp}_2(\mathbb{A}_F)$ (or maybe 2-cover of $\operatorname{GL}_2(\mathbb{A}_F)$), and a corresponding automorphic representation $\pi$ of $\mathrm{GL}_2(\mathbb{A}_F)$ via Shimura correspondence, can we related a certain kind of period of $\pi'$ with special $L$-values of quadratic twists of $\pi$?
In the above note by Zydor, it does not mention about half-integral weight modular forms, instead it consider sum over Heegner points. Maybe this is the right classical statement that corresponds to the above adelic statement, but I still don't get why this corresponds to the toric periods.
Edit: Based on Kimball's answer and Peter Humphrey's comment, they are separate theorems and the second is NOT an adelic version of the first one. For completion, here are the list of relevant papers from Waldspurger (found on mathscinet)
- Correspondance de Shimura ((J. Math. Pures. Appl. 1980): Adelic version of Shimura correspondence.
- Sur les coefficients de Fourier des formes modulaires de poids demi-entier (J. Math. Pures. Appl. 1981): Classical Waldspurger's formula for half-integral weight modular forms and $L$-values.
- Sur les valeurs de certaines fonctions $L$ automorphes en leur centre de symetrie (Compositio, 1985): Above adelic results on toric periods and $L$-values.
- Correspondences de Shimura et quaternions (Forum math, 1991): Shimura correspondence between $\mathrm{Mp}_2$ and $D^\times$, viewed as a composition of the original Shimura correspondence and Jacquet-Langlands correspondence?
Best Answer
These are two separate theorems, proved in different papers of Waldspurger (I think in 1980/1981 and 1985, respectively), so you shouldn't conflate them. The first theorem can be viewed as an "$L$-value correspondence between automorphic representations of GL(2) and Mp(2) (Shimura correspondence), and the second between automorphic representations of GL(2) and $B^\times$ where $B$ is a quaternion algebra (Jacquet-Langlands correspondence).
So really Waldspurger proved a triangle of relations between three things: $L$-values of quadratic twists, Fourier coefficients of half-integral weight forms, and toric periods.