Waldspurger formula

In representation theory of mathematics, the Waldspurger formula relates the special values of two L-functions of two related admissible irreducible representations. Let $k$ be the base field, $f$ be an automorphic form over $k$, $\pi$ be the representation associated via the Jacquet–Langlands correspondence with f. Goro Shimura (1976) proved this formula, when $$k = \mathbb{Q}$$ and $f$ is a cusp form; Günter Harder made the same discovery at the same time in an unpublished paper. Marie-France Vignéras (1980) proved this formula, when $$k = \mathbb{Q}$$ and $f$ is a newform. Jean-Loup Waldspurger, for whom the formula is named, reproved and generalized the result of Vignéras in 1985 via a totally different method which was widely used thereafter by mathematicians to prove similar formulas.

Statement
Let $$k$$ be a number field, $$\mathbb{A}$$ be its adele ring, $$k^\times$$ be the subgroup of invertible elements of $$k$$, $$\mathbb{A}^\times$$ be the subgroup of the invertible elements of $$\mathbb{A}$$, $$\chi, \chi_1, \chi_2$$ be three quadratic characters over $$\mathbb{A}^\times/k^\times$$, $$G = SL_2(k)$$, $$\mathcal{A}(G)$$ be the space of all cusp forms over $$G(k)\backslash G(\mathbb{A})$$, $$\mathcal{H}$$ be the Hecke algebra of $$G(\mathbb{A})$$. Assume that, $$\pi$$ is an admissible irreducible representation from $$G(\mathbb{A})$$ to $$\mathcal{A}(G)$$, the central character of π is trivial, $$\pi_\nu \sim \pi[h_\nu] $$ when $$\nu$$ is an archimedean place, $${A}$$ is a subspace of $${\mathcal{A}(G)}$$ such that $$ \pi|_\mathcal{H} : \mathcal{H} \to A$$. We suppose further that, $$\varepsilon(\pi\otimes\chi, 1/2)$$ is the Langlands $$\varepsilon$$-constant [ ; ] associated to $$\pi$$ and $$\chi$$ at $$ s = 1/2 $$. There is a $${\gamma \in k^\times}$$ such that $$k(\chi) = k( \sqrt{\gamma} ) $$.

Definition 1. The Legendre symbol $$\left(\frac{\chi}{\pi}\right) = \varepsilon(\pi\otimes\chi, 1/2) \cdot \varepsilon(\pi, 1/2) \cdot \chi(-1).$$ Definition 2. Let $${D_\chi}$$ be the discriminant of $$\chi$$. $$ p(\chi) = D_\chi^{1/2} \sum_{\nu\text{ archimedean}} \left\vert \gamma_\nu \right\vert_\nu^{h_\nu/2}. $$
 * Comment. Because all the terms in the right either have value +1, or have value &minus;1, the term in the left can only take value in the set {+1, &minus;1}.

Definition 3. Let $$f_0, f_1 \in A$$. $$ b(f_0, f_1) = \int_{x\in k^\times} f_0(x) \cdot \overline{f_1(x)} \, dx.$$

Definition 4. Let $${T}$$ be a maximal torus of $${G}$$, $${Z}$$ be the center of $${G}$$, $$\varphi \in A$$. $$\beta (\varphi, T) = \int_{t \in Z\backslash T} b(\pi (t)\varphi, \varphi) \, dt .$$
 * Comment. It is not obvious though, that the function $$\beta$$ is a generalization of the Gauss sum.

Let $$K$$ be a field such that $$k(\pi)\subset K\subset\mathbb{C}$$. One can choose a K-subspace$${A^0}$$ of $$A$$ such that (i) $$A = A^0 \otimes_K\mathbb{C}$$; (ii) $$(A^0)^{\pi(G)} = A^0$$. De facto, there is only one such $$A^0$$ modulo homothety. Let $$T_1, T_2$$ be two maximal tori of $$G$$ such that $$\chi_{T_1} = \chi_1$$ and $$\chi_{T_2} = \chi_2$$. We can choose two elements $$\varphi_1, \varphi_2 $$ of $$A^0$$ such that $$\beta(\varphi_1, T_1) \neq 0$$ and $$\beta(\varphi_2, T_2) \neq 0$$.

Definition 5. Let $$D_1, D_2$$ be the discriminants of $$\chi_1, \chi_2$$.
 * $$p(\pi, \chi_1, \chi_2) = D_1^{-1/2} D_2^{1/2} L(\chi_1, 1)^{-1} L(\chi_2, 1) L(\pi\otimes\chi_1, 1/2) L(\pi\otimes\chi_2, 1/2)^{-1} \beta(\varphi_1, T_1)^{-1} \beta(\varphi_2, T_2).$$


 * Comment. When the $$\chi_1 = \chi_2$$, the right hand side of Definition 5 becomes trivial.

We take $$\Sigma_f$$ to be the set {all the finite $$k$$-places $$\nu \mid \ \pi_\nu$$ doesn't map non-zero vectors invariant under the action of $${GL_2(k_\nu)}$$ to zero}, $${\Sigma_s}$$ to be the set of (all $$k$$-places $$\nu \mid \nu$$ is real, or finite and special).

$$

Comments:

The case when $F_{p}(T)$ and $φ$ is a metaplectic cusp form
Let p be prime number, $$\mathbb{F}_p$$ be the field with p elements, $$R = \mathbb{F}_p[T], k = \mathbb{F}_p(T), k_\infty = \mathbb{F}_p((T^{-1})), o_\infty$$ be the integer ring of $$k_\infty, \mathcal{H} = PGL_2(k_\infty)/PGL_2(o_\infty), \Gamma = PGL_2(R)$$. Assume that, $$N, D\in R$$, D is squarefree of even degree and coprime to N, the prime factorization of $$N$$ is $\prod_\ell \ell^{\alpha_\ell}$. We take $$\Gamma_0(N)$$ to the set $\left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \Gamma \mid c \equiv 0 \bmod N \right\},$  $$S_0(\Gamma_0(N))$$ to be the set of all cusp forms of level N and depth 0. Suppose that, $$\varphi, \varphi_1, \varphi_2 \in S_0(\Gamma_0(N))$$.

Definition 1. Let $$\left (\frac{c} {d} \right )$$ be the Legendre symbol of c modulo d, $$ \widetilde{SL}_2(k_\infty) = Mp_2(k_\infty)$$. Metaplectic morphism $$\eta : SL_2(R) \to \widetilde{SL}_2(k_\infty), \begin{pmatrix} a & b \\ c & d \end{pmatrix} \mapsto \left( \begin{pmatrix} a & b \\ c & d \end{pmatrix}, \left (\frac{c} {d} \right )\right).$$

Definition 2. Let $$ z = x + iy \in \mathcal{H}, d\mu = \frac{dx\,dy} {\left \vert y \right \vert^2}$$. Petersson inner product $$\langle \varphi_1, \varphi_2\rangle = [\Gamma : \Gamma_0(N)]^{-1} \int_{\Gamma_0(N) \backslash \mathcal{H}} \varphi_1(z) \overline{\varphi_2(z)} \, d\mu.$$

Definition 3. Let $$ n, P \in R$$. Gauss sum $$ G_n(P) = \sum_{r \in R/PR} \left (\frac{r} {P} \right ) e(rnT^2). $$

Let $$\lambda_{\infty, \varphi} $$ be the Laplace eigenvalue of $$\varphi$$. There is a constant $$\theta \in \mathbb{R}$$ such that $$ \lambda_{\infty, \varphi} = \frac { e^{-i\theta} + e^{i\theta} } { \sqrt{p} }. $$

Definition 4. Assume that $$ v_\infty(a/b) = \deg(a) - \deg(b), \nu = v_\infty(y) $$. Whittaker function $$W_{0, i\theta}(y) = \begin{cases} \frac{ \sqrt{p} } { e^{i\theta} - e^{-i\theta} } \left[ \left(\frac{ e^{i\theta} } { \sqrt{p} }\right)^{\nu - 1} - \left(\frac{ e^{-i\theta} } { \sqrt{p} }\right)^{\nu - 1} \right], & \text{when } \nu \geq 2; \\ 0, & \text{otherwise}. \end{cases} $$

Definition 5. Fourier–Whittaker expansion $$ \varphi(z) = \sum_{ r \in R } \omega_\varphi(r) e(rxT^2) W_{0, i\theta}(y). $$ One calls $$ \omega_\varphi(r) $$ the Fourier–Whittaker coefficients of $$\varphi$$.

Definition 6. Atkin–Lehner operator $$ W_{\alpha_\ell} = \begin{pmatrix} \ell^{\alpha_\ell} & b \\ N & \ell^{\alpha_\ell}d \end{pmatrix} $$ with $$ \ell^{2\alpha_\ell}d - bN = \ell^{\alpha_\ell}. $$

Definition 7. Assume that, $$\varphi$$ is a Hecke eigenform. Atkin–Lehner eigenvalue $$ w_{\alpha_\ell, \varphi} = \frac{ \varphi(W_{\alpha_\ell}z) } { \varphi(z) } $$ with $$ w_{\alpha_\ell, \varphi} = \pm 1. $$

Definition 8. $$ L(\varphi, s) = \sum_{r \in R \backslash \{0\} } \frac{ \omega_\varphi(r) } { \left \vert r \right \vert_p^s }.$$

Let $$ \widetilde{S}_0(\widetilde{\Gamma}_0(N)) $$ be the metaplectic version of $$S_0(\Gamma_0(N))$$, $$ \{ E_1, \ldots, E_d \}$$ be a nice Hecke eigenbasis for $$ \widetilde{S}_0(\widetilde{\Gamma}_0(N)) $$ with respect to the Petersson inner product. We note the Shimura correspondence by $$\operatorname{Sh}.$$

Theorem [, Thm 5.1, p. 60 ]. Suppose that $ K_\varphi = \frac 1 { \sqrt{p} \left( \sqrt{p} - e^{-i\theta} \right) \left( \sqrt{p} - e^{i\theta} \right) } $, $$ \chi_D $$ is a quadratic character with $$ \Delta(\chi_D) = D $$. Then $$ \sum_{\operatorname{Sh}(E_i) = \varphi} \left \vert \omega_{E_i}(D) \right \vert_p^2 = \frac{ K_\varphi G_1(D) \left \vert D \right \vert_p^{-3/2} } { \langle \varphi, \varphi\rangle } L(\varphi \otimes \chi_D, 1/2) \prod_\ell \left( 1 + \left (\frac{ \ell^{\alpha_\ell} } D \right ) w_{\alpha_\ell, \varphi} \right). $$