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, 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this formula, when $k=\mathbb {Q}$ and f is a cusp form; Günter Harder made the same discovery at the same time in an unpublished paper.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 $\beta$ 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).

Theorem  — Let $k=\mathbb {Q}$. We assume that, (i) $L(\pi \otimes \chi _{2},1/2)\neq 0$; (ii) for $\nu \in \Sigma _{s}$, $\left({\frac {\chi _{1,\nu }}{\pi _{\nu }}}\right)=\left({\frac {\chi _{2,\nu }}{\pi _{\nu }}}\right)$ . Then, there is a constant ${q\in \mathbb {Q} (\pi )}$ such that $$L(\pi \otimes \chi _{1},1/2)L(\pi \otimes \chi _{2},1/2)^{-1}=qp(\chi _{1})p(\chi _{2})^{-1}\prod _{\nu \in \Sigma _{f}}p(\pi _{\nu },\chi _{1,\nu },\chi _{2,\nu })$$

Comments:

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. [ (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, $\chi _{1}=\chi$ and $\chi _{2}=1$. Then, there is an element ${q\in \mathbb {Q} (\pi )}$ such that $L(\pi \otimes \chi ,1/2)L(\pi ,1/2)^{-1}=qD_{\chi }^{1/2}.$

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 ${\textstyle \prod _{\ell }\ell ^{\alpha _{\ell }}}$. We take $\Gamma _{0}(N)$ to the set ${\textstyle \left\{{\begin{pmatrix}a&b\\c&d\end{pmatrix}}\in \Gamma \mid c\equiv 0{\bmod {N}}\right\},}$ $S_{0}(\Gamma _{0}(N))$ 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 ${\textstyle K_{\varphi }={\frac {1}{{\sqrt {p}}\left({\sqrt {p}}-e^{-i\theta }\right)\left({\sqrt {p}}-e^{i\theta }\right)}}}$, $\chi _{D}$ 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 (PDF). pp. 1–287.
• Deligne, Pierre (1972). "Les constantes des équations fonctionelle des fonctions L". Modular Functions of One Variable II. International Summer School on Modular functions. Antwerp. pp. 501–597.