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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}\tag{1} \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}\tag{2} \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 \begin{align}\tag{3} \mathcal{P}(f_1) \mathcal{P}(f_2) \sim \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} relies on the Shimizu lifting and the Siegel-Weil formula.

Is there a direct proof of the factorization of the global period integral (1) into local integrals (2) that avoids resorting to the toric integral of automorphic forms on the left hand side of (3)? 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?

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    $\begingroup$ What do you mean by "direct proof"? There is a proof by Herve Jacquet and Chen Nan that uses the relative trace formula, but I wouldn't say that this is any more direct than using the theta correspondence. $\endgroup$
    – Peter Humphries
    Commented Sep 5 at 20:21
  • $\begingroup$ Thank you for the response! Both the relative trace formula approach by Hervé Jacquet and Chen Nan, as well as the theta correspondence (or Shimizu lifting), provide proofs of Waldspurger’s formula. What I was trying to clarify is whether the equality of the factorized global period integral with the special value of the L-function (as seen in Waldspurger’s formula) can be viewed as equivalent to the formula itself, or is it better understood as a corollary of Waldspurger’s formula? $\endgroup$
    – Alvin
    Commented Sep 5 at 20:49
  • $\begingroup$ Both $\int_{\mathbb{A}_F^{\times} E^{\times} \backslash \mathbb{A}_E^{\times}} \langle \pi(h) \cdot f_1,f_2\rangle \, dh$ and $\prod_v \int_{F_v^{\times} \backslash E_v^{\times}} \langle \pi_v(h_v) \cdot f_{1,v},f_{2,v}\rangle \, dh_v$ define elements of $\mathrm{Hom}_{\mathbb{A}_E^{\times}}(\pi \otimes \widetilde{\pi},1)$, which is one-dimensional, so they must be equal up to a scalar. $\endgroup$
    – Peter Humphries
    Commented Sep 6 at 0:09
  • $\begingroup$ I wouldn't call (2) a "factorisation" per se, because $L(\pi_E, 1/2)$ is outside the region of convergence of the Euler product formula. So you shouldn't think of (2) purely as a product of local terms: you need the analytic continuation of the L-function to make sense of it. $\endgroup$
    – David Loeffler
    Commented Sep 6 at 5:46

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