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{{expert needed|mathematics|talk=|reason=article requires a gentler encyclopedic introduction for the general reader|date=March 2019}}

In [[representation theory]] of mathematics, the '''Waldspurger formula''' relates the [[Special values of L-functions|special values]] of two [[L-function|''L''-functions]] of two related [[Admissible representation|admissible]] [[irreducible representation]]s. <!-- The following seems misplaced in the lede:-->  Let {{mvar|k}} be the base field, {{mvar|f}} be an [[automorphic form]] over {{mvar|k}}, {{pi}} be the representation associated via the [[Jacquet–Langlands correspondence]] with {{Var|f}}. [[Goro Shimura]] (1976) proved this formula, when <math>k = \mathbb{Q}</math> and {{mvar|f}} is a [[cusp form]]; [[Günter Harder]] made the same discovery at the same time in an unpublished paper.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 <math>k = \mathbb{Q}</math> and {{mvar|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 <math>k</math> be a [[number field]], <math>\mathbb{A}</math> be its [[adele ring]], <math>k^\times</math> be the [[subgroup]] of invertible elements of <math>k</math>, <math>\mathbb{A}^\times</math> be the subgroup of the invertible elements of <math>\mathbb{A}</math>, <math>\chi, \chi_1, \chi_2</math> be three quadratic characters over <math>\mathbb{A}^\times/k^\times</math>, <math>G = SL_2(k)</math>, <math>\mathcal{A}(G)</math> be the space of all [[cusp form]]s over <math>G(k)\backslash G(\mathbb{A})</math>, <math>\mathcal{H}</math> be the [[Hecke algebra of a locally compact group|Hecke algebra]] of <math>G(\mathbb{A})</math>. Assume that, <math>\pi</math> is an admissible irreducible representation from <math>G(\mathbb{A})</math> to <math>\mathcal{A}(G)</math>, the [[Center (algebra and category theory)|central character]] of π is trivial, <math>\pi_\nu \sim \pi[h_\nu] </math> when <math>\nu</math> is an archimedean place, <math>{A}</math> is a subspace of <math>{\mathcal{A}(G)}</math> such that <math> \pi|_\mathcal{H} : \mathcal{H} \to A</math>. We suppose further that, <math>\varepsilon(\pi\otimes\chi, 1/2)</math> is the Langlands <math>\varepsilon</math>-constant [ {{harv | Langlands | 1970 }}; {{harv | Deligne | 1972 }} ] associated to <math>\pi</math> and <math>\chi</math> at <math> s = 1/2 </math>. There is a <math>{\gamma \in k^\times}</math> such that <math>k(\chi) = k( \sqrt{\gamma} ) </math>.

Definition 1. The [[Legendre symbol]] <math>\left(\frac{\chi}{\pi}\right) = \varepsilon(\pi\otimes\chi, 1/2) \cdot \varepsilon(\pi, 1/2) \cdot \chi(-1).</math>
* 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 2. Let <math>{D_\chi}</math> be the [[Discriminant of an algebraic number field|discriminant]] of <math>\chi</math>. <math display="block"> p(\chi) = D_\chi^{1/2} \sum_{\nu\text{ archimedean}} \left\vert \gamma_\nu \right\vert_\nu^{h_\nu/2}. </math>

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

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

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

Definition 5. Let <math>D_1, D_2</math> be the discriminants of <math>\chi_1, \chi_2</math>.
: <math>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).</math>
* Comment. When the <math>\chi_1 = \chi_2</math>, the right hand side of Definition 5 becomes trivial.

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

{{math theorem | name = Theorem <ref>{{harv | Waldspurger | 1985 }}, Thm 4, p.&nbsp;235</ref> | math_statement = Let <math> k = \mathbb{Q} </math>. We assume that, (i) <math>L(\pi\otimes\chi_2, 1/2) \neq 0</math>; (ii) for <math>\nu \in \Sigma_s</math>, <math>\left(\frac{\chi_{1, \nu}} {\pi_\nu}\right) = \left(\frac{\chi_{2, \nu}} {\pi_\nu}\right)</math> . Then, there is a constant <math>{q \in \mathbb{Q}(\pi)}</math> such that <math display="block">L(\pi\otimes\chi_1, 1/2)  L(\pi\otimes\chi_2, 1/2)^{-1} = q p(\chi_1) p(\chi_2)^{-1} \prod_{\nu \in \Sigma_f} p(\pi_\nu, \chi_{1, \nu}, \chi_{2, \nu})</math>}}

Comments: {{ordered list | list-style-type = lower-roman
| 1 = 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.
| 2 = It is worthwhile to notice that, when the two characters are equal, the formula can be greatly simplified.
| 3 = [ {{harv | Waldspurger | 1985 }}, Thm 6, p.&nbsp;241 ] When one of the two characters is <math>{1}</math>, Waldspurger's formula becomes much more simple. Without loss of generality, we can assume that, <math>\chi_1 = \chi</math> and <math>\chi_2 = 1</math>. Then, there is an element <math>{ q \in \mathbb{Q}(\pi) }</math> such that <math>L(\pi\otimes\chi, 1/2) L(\pi, 1/2)^{-1} = q D_{\chi}^{1/2}.</math>}}

== The case when {{math|1='''F'''<sub>''p''</sub>(''T'')}} and {{math|''φ''}} is a metaplectic cusp form ==
Let p be prime number, <math>\mathbb{F}_p</math> be the field with ''p'' elements, <math>R = \mathbb{F}_p[T], k = \mathbb{F}_p(T), k_\infty = \mathbb{F}_p((T^{-1})), o_\infty</math> be the [[integer ring]] of <math>k_\infty, \mathcal{H} = PGL_2(k_\infty)/PGL_2(o_\infty), \Gamma = PGL_2(R)</math>. Assume that, <math>N, D\in R</math>, D is [[Square-free integer|squarefree]] of even degree and coprime to ''N'', the [[prime factorization]] of <math>N</math> is <math display="inline">\prod_\ell \ell^{\alpha_\ell}</math>. We take <math>\Gamma_0(N)</math> to the set <math display="inline">\left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \Gamma \mid c \equiv 0 \bmod  N \right\},</math> <math>S_0(\Gamma_0(N))</math> to be the set of all cusp forms of level ''N'' and depth 0. Suppose that, <math>\varphi, \varphi_1, \varphi_2 \in S_0(\Gamma_0(N))</math>.

Definition 1. Let <math>\left (\frac{c} {d} \right )</math> be the [[Legendre symbol]] of ''c'' modulo ''d'', <math> \widetilde{SL}_2(k_\infty) = Mp_2(k_\infty)</math>. Metaplectic morphism <math display="block">\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).</math>

Definition 2. Let <math> z = x + iy \in \mathcal{H}, d\mu = \frac{dx\,dy} {\left \vert y \right \vert^2}</math>. [[Petersson inner product]] <math display="block">\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.</math>

Definition 3. Let <math> n, P \in R</math>. [[Gauss sum]] <math display="block"> G_n(P) = \sum_{r \in R/PR} \left (\frac{r} {P} \right ) e(rnT^2). </math>

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

Definition 4. Assume that <math> v_\infty(a/b) = \deg(a) - \deg(b), \nu = v_\infty(y) </math>. [[Whittaker function]] <math display="block">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} </math>

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

Definition 6. [[Atkin–Lehner theory|Atkin–Lehner operator]] <math display="block"> W_{\alpha_\ell} = \begin{pmatrix} \ell^{\alpha_\ell} & b \\ N & \ell^{\alpha_\ell}d \end{pmatrix} </math> with <math> \ell^{2\alpha_\ell}d - bN = \ell^{\alpha_\ell}. </math>

Definition 7. Assume that, <math>\varphi</math> is a [[Eigenform|Hecke eigenform]]. Atkin–Lehner eigenvalue <math display="block"> w_{\alpha_\ell, \varphi} = \frac{ \varphi(W_{\alpha_\ell}z) } { \varphi(z) } </math> with <math> w_{\alpha_\ell, \varphi} = \pm 1. </math>

Definition 8. <math display="block"> L(\varphi, s) = \sum_{r \in R \backslash \{0\} } \frac{ \omega_\varphi(r) } { \left \vert r \right \vert_p^s }.</math>

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

Theorem [ {{harv | Altug | Tsimerman | 2010 }}, Thm 5.1, p.&nbsp;60 ]. Suppose that <math display="inline"> K_\varphi = \frac 1  { \sqrt{p} \left( \sqrt{p} - e^{-i\theta} \right) \left( \sqrt{p} - e^{i\theta} \right) } </math>, <math> \chi_D </math> is a quadratic character with <math> \Delta(\chi_D) = D </math>. Then
<math display="block"> \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). </math>

==References==
{{Reflist}}
*{{citation | last = Waldspurger | first = Jean-Loup | title = Sur les valeurs de certaines L-fonctions automorphes en leur centre de symétrie | journal = Compositio Mathematica | volume = 54 |  issue = 2 | year = 1985 | pages = 173–242 }}
*{{citation | last = Vignéras| first = Marie-France | chapter = Valeur au centre de symétrie des fonctions L associées aux formes modulaire | title = Séminarie de Théorie des Nombres, Paris 1979–1980 | series = Progress in Math.| publisher = Birkhäuser | year = 1981 | pages = 331–356 }}
*{{citation | last = Shimura | first = Gorô | title = On special values of zeta functions associated with cusp forms | journal = Communications on Pure and Applied Mathematics | volume = 29 | year = 1976 | pages = 783–804 | doi = 10.1002/cpa.3160290618 }}
*{{cite journal |last1= Altug |first1= Salim Ali |last2= Tsimerman |first2= Jacob | title = Metaplectic Ramanujan conjecture over function fields with applications to quadratic forms |journal= International Mathematics Research Notices |arxiv= 1008.0430 |date= 2010 |doi= 10.1093/imrn/rnt047 |s2cid= 119121964 }}
*{{cite book |last= Langlands |first= Robert |title= On the Functional Equation of the Artin L-Functions |url=https://publications.ias.edu/sites/default/files/a-ps.pdf|year= 1970|pages=1–287 }}
*{{cite conference |last= Deligne |first= Pierre | title = Les constantes des équations fonctionelle des fonctions L |book-title= Modular Functions of One Variable II |pages = 501–597 |conference= International Summer School on Modular functions |location= Antwerp |year= 1972 }}

[[Category:Representation theory]]
[[Category:Algebraic number theory]]
[[Category:Harmonic analysis]]
[[Category:Langlands program]]