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, π 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this formula, when k = \mathbb{Q} and f is a cusp form; Günter Harder made the same discovery at the same time in an unpublished paper.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 [ (Langlands 1970); (Deligne 1972) ] 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).
- Comment. Because all the terms in the right either have value +1, or have value −1, the term in the left can only take value in the set {+1, −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}.
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
\betais 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 formula in the theorem is the well-known Waldspurger formula. It is of global-local nature, in the left with a global part, in the right with a local part. By 2017, mathematicians often call it the classic Waldspurger's formula.
- It is worthwhile to notice that, when the two characters are equal, the formula can be greatly simplified.
- [ (Waldspurger 1985), Thm 6, p. 241 ] When one of the two characters is
{1}, Waldspurger's formula becomes much more simple. Without loss of generality, we can assume that,\chi_1 = \chiand\chi_2 = 1. Then, there is an element{ q \in \mathbb{Q}(\pi) }such thatL(\pi\otimes\chi, 1/2) L(\pi, 1/2)^{-1} = q D_{\chi}^{1/2}.
The case when Fp(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 [ (Altug & Tsimerman 2010), 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).
References
- Waldspurger, Jean-Loup (1985), "Sur les valeurs de certaines L-fonctions automorphes en leur centre de symétrie", Compositio Mathematica. 54 (2): 173–242
- Vignéras, Marie-France (1981), "Valeur au centre de symétrie des fonctions L associées aux formes modulaire", "Séminarie de Théorie des Nombres, Paris 1979–1980", Progress in Math., Birkhäuser, pp. 331–356
- Shimura, Gorô (1976), "On special values of zeta functions associated with cusp forms", Communications on Pure and Applied Mathematics. 29: 783–804, doi:10.1002/cpa.3160290618
- Altug, Salim Ali & Tsimerman, Jacob (2010). "Metaplectic Ramanujan conjecture over function fields with applications to quadratic forms". International Mathematics Research Notices. arXiv:1008.0430. doi:10.1093/imrn/rnt047. S2CID 119121964
- Langlands, Robert (1970). On the Functional Equation of the Artin L-Functions. pp. 1–287.
- Deligne, Pierre (1972). Les constantes des équations fonctionelles des fonctions L. International Summer School on Modular functions. pp. 501–597.