Waldspurger formula

Let ${\displaystyle k}$ be a number field, ${\displaystyle \mathbb {A} }$ be its adele ring, ${\displaystyle k^{\times }}$ be the subgroup of invertible elements of ${\displaystyle k}$, ${\displaystyle \mathbb {A} ^{\times }}$ be the subgroup of the invertible elements of ${\displaystyle \mathbb {A} }$, ${\displaystyle \chi ,\chi _{1},\chi _{2}}$ be three quadratic characters over ${\displaystyle \mathbb {A} ^{\times }/k^{\times }}$, ${\displaystyle G=SL_{2}(k)}$, ${\displaystyle {\mathcal {A}}(G)}$ be the space of all cusp forms over ${\displaystyle G(k)\backslash G(\mathbb {A} )}$, ${\displaystyle {\mathcal {H}}}$ be the Hecke algebra of ${\displaystyle G(\mathbb {A} )}$. Assume that, ${\displaystyle \pi }$ is an admissible irreducible representation from ${\displaystyle G(\mathbb {A} )}$ to ${\displaystyle {\mathcal {A}}(G)}$, the central character of π is trivial, ${\displaystyle \pi _{\nu }\sim \pi [h_{\nu }]}$ when ${\displaystyle \nu }$ is an archimedean place, ${\displaystyle {A}}$ is a subspace of ${\displaystyle {{\mathcal {A}}(G)}}$ such that ${\displaystyle \pi |_{\mathcal {H}}:{\mathcal {H}}\to A}$. We suppose further that, ${\displaystyle \varepsilon (\pi \otimes \chi ,1/2)}$ is the Langlands ${\displaystyle \varepsilon }$-constant [ (Langlands 1970); (Deligne 1972) ] associated to ${\displaystyle \pi }$ and ${\displaystyle \chi }$ at ${\displaystyle s=1/2}$. There is a ${\displaystyle {\gamma \in k^{\times }}}$ such that ${\displaystyle k(\chi )=k({\sqrt {\gamma }})}$.

Definition 1. The Legendre symbol ${\displaystyle \left({\frac {\chi }{\pi }}\right)=\varepsilon (\pi \otimes \chi ,1/2)\cdot \varepsilon (\pi ,1/2)\cdot \chi (-1).}$

• 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 ${\displaystyle {D_{\chi }}}$ be the discriminant of ${\displaystyle \chi }$. ${\displaystyle p(\chi )=D_{\chi }^{1/2}\sum _{\nu {\text{ archimedean}}}\left\vert \gamma _{\nu }\right\vert _{\nu }^{h_{\nu }/2}.}$

Definition 3. Let ${\displaystyle f_{0},f_{1}\in A}$. ${\displaystyle b(f_{0},f_{1})=\int _{x\in k^{\times }}f_{0}(x)\cdot {\overline {f_{1}(x)}}\,dx.}$

Definition 4. Let ${\displaystyle {T}}$ be a maximal torus of ${\displaystyle {G}}$, ${\displaystyle {Z}}$ be the center of ${\displaystyle {G}}$, ${\displaystyle \varphi \in A}$. ${\displaystyle \beta (\varphi ,T)=\int _{t\in Z\backslash T}b(\pi (t)\varphi ,\varphi )\,dt.}$

• Comment. It is not obvious though, that the function ${\displaystyle \beta }$ is a generalization of the Gauss sum.

Let ${\displaystyle K}$ be a field such that ${\displaystyle k(\pi )\subset K\subset \mathbb {C} }$. One can choose a K-subspace${\displaystyle {A^{0}}}$ of ${\displaystyle A}$ such that (i) ${\displaystyle A=A^{0}\otimes _{K}\mathbb {C} }$; (ii) ${\displaystyle (A^{0})^{\pi (G)}=A^{0}}$. De facto, there is only one such ${\displaystyle A^{0}}$ modulo homothety. Let ${\displaystyle T_{1},T_{2}}$ be two maximal tori of ${\displaystyle G}$ such that ${\displaystyle \chi _{T_{1}}=\chi _{1}}$ and ${\displaystyle \chi _{T_{2}}=\chi _{2}}$. We can choose two elements ${\displaystyle \varphi _{1},\varphi _{2}}$ of ${\displaystyle A^{0}}$ such that ${\displaystyle \beta (\varphi _{1},T_{1})\neq 0}$ and ${\displaystyle \beta (\varphi _{2},T_{2})\neq 0}$.

Definition 5. Let ${\displaystyle D_{1},D_{2}}$ be the discriminants of ${\displaystyle \chi _{1},\chi _{2}}$.

${\displaystyle p(\pi ,\chi _{1},\chi _{2})=D_{1}^{-1/2}D_{2}^{1/2}L(\chi _{1},1)^{-1}L(\chi _{2},1)L(\pi \otimes \chi _{1},1/2)L(\pi \otimes \chi _{2},1/2)^{-1}\beta (\varphi _{1},T_{1})^{-1}\beta (\varphi _{2},T_{2}).}$

• Comment. When the ${\displaystyle \chi _{1}=\chi _{2}}$, the right hand side of Definition 5 becomes trivial.

We take ${\displaystyle \Sigma _{f}}$ to be the set {all the finite ${\displaystyle k}$-places ${\displaystyle \nu \mid \ \pi _{\nu }}$ doesn't map non-zero vectors invariant under the action of ${\displaystyle {GL_{2}(k_{\nu })}}$ to zero}, ${\displaystyle {\Sigma _{s}}}$ to be the set of { all ${\displaystyle k}$-places ${\displaystyle \nu \mid \nu }$ is real, or finite and special }.

Theorem [ (Waldspurger 1985), Thm 4, p. 235 ]. Let ${\displaystyle k=\mathbb {Q} }$. We assume that, (i) ${\displaystyle L(\pi \otimes \chi _{2},1/2)\neq 0}$; (ii) for ${\displaystyle \nu \in \Sigma _{s}}$, ${\displaystyle \left({\frac {\chi _{1,\nu }}{\pi _{\nu }}}\right)=\left({\frac {\chi _{2,\nu }}{\pi _{\nu }}}\right)}$ . Then, there is a constant ${\displaystyle {q\in \mathbb {Q} (\pi )}}$ such that

${\displaystyle L(\pi \otimes \chi _{1},1/2)L(\pi \otimes \chi _{2},1/2)^{-1}=qp(\chi _{1})p(\chi _{2})^{-1}\prod _{\nu \in \Sigma _{f}}p(\pi _{\nu },\chi _{1,\nu },\chi _{2,\nu })}$

Comments:

• (i) 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.
• (ii) It is worthwhile to notice that, when the two characters are equal, the formula can be greatly simplified.
• (iii) [ (Waldspurger 1985), Thm 6, p. 241 ] When one of the two characters is ${\displaystyle {1}}$, Waldspurger's formula becomes much more simple. Without loss of generality, we can assume that, ${\displaystyle \chi _{1}=\chi }$ and ${\displaystyle \chi _{2}=1}$. Then, there is an element ${\displaystyle {q\in \mathbb {Q} (\pi )}}$ such that ${\displaystyle L(\pi \otimes \chi ,1/2)L(\pi ,1/2)^{-1}=qD_{\chi }^{1/2}.}$

Let p be prime number, ${\displaystyle \mathbb {F} _{p}}$ be the field with p elements, ${\displaystyle R=\mathbb {F} _{p}[T],k=\mathbb {F} _{p}(T),k_{\infty }=\mathbb {F} _{p}((T^{-1})),o_{\infty }}$ be the integer ring of ${\displaystyle k_{\infty },{\mathcal {H}}=PGL_{2}(k_{\infty })/PGL_{2}(o_{\infty }),\Gamma =PGL_{2}(R)}$. Assume that, ${\displaystyle N,D\in R}$, D is squarefree of even degree and coprime to N, the prime factorization of ${\displaystyle N}$ is ${\displaystyle \prod _{\ell }\ell ^{\alpha _{\ell }}}$. We take ${\displaystyle \Gamma _{0}(N)}$ to the set ${\displaystyle \left\{{\begin{pmatrix}a&b\\c&d\end{pmatrix}}\in \Gamma \mid c\equiv 0{\bmod {N}}\right\},}$ ${\displaystyle S_{0}(\Gamma _{0}(N))}$ to be the set of all cusp forms of level N and depth 0. Suppose that, ${\displaystyle \varphi ,\varphi _{1},\varphi _{2}\in S_{0}(\Gamma _{0}(N))}$.

Definition 1. Let ${\displaystyle \left({\frac {c}{d}}\right)}$ be the Legendre symbol of c modulo d, ${\displaystyle {\widetilde {SL}}_{2}(k_{\infty })=Mp_{2}(k_{\infty })}$. Metaplectic morphism ${\displaystyle \eta :SL_{2}(R)\to {\widetilde {SL}}_{2}(k_{\infty }),{\begin{pmatrix}a&b\\c&d\end{pmatrix}}\mapsto \left({\begin{pmatrix}a&b\\c&d\end{pmatrix}},\left({\frac {c}{d}}\right)\right).}$

Definition 2. Let ${\displaystyle z=x+iy\in {\mathcal {H}},d\mu ={\frac {dx\,dy}{\left\vert y\right\vert ^{2}}}}$. Petersson inner product ${\displaystyle \langle \varphi _{1},\varphi _{2}\rangle =[\Gamma$ :\Gamma _{0}(N)]^{-1}\int _{\Gamma _{0}(N)\backslash {\mathcal {H}}}\varphi _{1}(z){\overline {\varphi _{2}(z)}}\,d\mu .}

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

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

Definition 4. Assume that ${\displaystyle v_{\infty }(a/b)=\deg(a)-\deg(b),\nu =v_{\infty }(y)}$. Whittaker function

${\displaystyle W_{0,i\theta }(y)={\begin{cases}{\frac {\sqrt {p}}{e^{i\theta }-e^{-i\theta }}}\left[\left({\frac {e^{i\theta }}{\sqrt {p}}}\right)^{\nu -1}-\left({\frac {e^{-i\theta }}{\sqrt {p}}}\right)^{\nu -1}\right],&{\text{when }}\nu \geq 2;\\0,&{\text{otherwise}}\end{cases}}}$.

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

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

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

Definition 8. ${\displaystyle L(\varphi ,s)=\sum _{r\in R\backslash \{0\}}{\frac {\omega _{\varphi }(r)}{\left\vert r\right\vert _{p}^{s}}}.}$

Let ${\displaystyle {\widetilde {S}}_{0}({\widetilde {\Gamma }}_{0}(N))}$ be the metaplectic version of ${\displaystyle S_{0}(\Gamma _{0}(N))}$, ${\displaystyle \{E_{1},\ldots ,E_{d}\}}$ be a nice Hecke eigenbasis for ${\displaystyle {\widetilde {S}}_{0}({\widetilde {\Gamma }}_{0}(N))}$ with respect to the Petersson inner product. We note the Shimura correspondence by ${\displaystyle \operatorname {Sh} .}$

Theorem [ (Altug & Tsimerman 2010), Thm 5.1, p. 60 ]. Suppose that ${\displaystyle K_{\varphi }={\frac {1}{{\sqrt {p}}({\sqrt {p}}-e^{-i\theta })({\sqrt {p}}-e^{i\theta })}}}$, ${\displaystyle \chi _{D}}$ is a quadratic character with ${\displaystyle \Delta (\chi _{D})=D}$. Then

${\displaystyle \sum _{\operatorname {Sh} (E_{i})=\varphi }\left\vert \omega _{E_{i}}(D)\right\vert _{p}^{2}={\frac {K_{\varphi }G_{1}(D)\left\vert D\right\vert _{p}^{-3/2}}{\langle \varphi ,\varphi \rangle }}L(\varphi \otimes \chi _{D},1/2)\prod _{\ell }\left(1+\left({\frac {\ell ^{\alpha _{\ell }}}{D}}\right)w_{\alpha _{\ell },\varphi }\right).}$

• 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 Math., 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:1008.0430v3. doi:10.1093/imrn/rnt047. S2CID 119121964.CS1 maint: ref=harv (link)
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