Let $\pi$ be the irreducible cuspidal automorphic representation of $\mathrm{GL}_2$. Let $E/F$ be a quadratic extension with given embedding $E^{\times} \to \mathrm{GL}_2(F)$. For $f_1 \in \pi$, $f_2 \in \tilde{\pi}$, the global version of Waldspurger’s model is defined as \begin{align*} \int_{\mathbb{A}^\times_FE^\times\backslash \mathbb{A}_E^\times} \langle\pi(h)f_1, f_2\rangle\, dh, \end{align*} where $\langle, \rangle$ denotes the Peterson inner product on $\mathrm{GL}_2(F) \backslash \mathrm{GL}_2(\mathbb{A})$.

Does this global period integral factor into a product of local integrals over each place $v$ of $F$? From the celebrated Waldspurger's formula relating the toric period and the special values of automorphic L-functions, we know this global period integral can be factored into \begin{align*} \frac{L(\pi_E, 1/2)}{L(\pi, \mathrm{Ad}, 1)} \, \prod_v \int_{F_v^\times\backslash E_v^\times} \langle\pi_v(h_v)f_{1,v}, f_{2,v}\rangle\, dh_v. \end{align*} The original proof of Waldspurger’s formula relies on the Shimizu lifting and the Siegel-Weil formula.

Is there a direct proof of the factorization of the global period integral into local integrals that avoids using the toric integral of automorphic forms? In other words, does the factorization naturally imply the full Waldspurger formula, or is there an additional layer in Waldspurger’s argument that goes beyond the factorization of the global period integral?