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 Template:Mvar be the base field, Template:Mvar be an automorphic form over Template:Mvar, Template:Pi be the representation associated via the Jacquet–Langlands correspondence with Template:Var. Goro Shimura (1976) proved this formula, when $k=\mathbb{Q}$ and Template:Mvar is a cusp form; Günter Harder made the same discovery at the same time in an unpublished paper.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 Template:Mvar 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, $𝔸$ be its adele ring, $k^{\times }$ be the subgroup of invertible elements of $k$, ${𝔸}^{\times }$ be the subgroup of the invertible elements of $𝔸$, $\chi ,{\chi }_1,{\chi }_2$ be three quadratic characters over ${𝔸}^{\times }/k^{\times }$, $G=S{L}_2(k)$, $𝒜(G)$ be the space of all cusp forms over $G(k)\mathrm{\setminus }G(𝔸)$, $\mathscr{H}$ be the Hecke algebra of $G(𝔸)$. Assume that, $\pi$ is an admissible irreducible representation from $G(𝔸)$ to $𝒜(G)$, the central character of π is trivial, ${\pi }_{\nu }\sim \pi [h_{\nu }]$ when $\nu$ is an archimedean place, $A$ is a subspace of $𝒜(G)$ such that $\pi {|}_{\mathscr{H}}:\mathscr{H}\to A$. We suppose further that, $\epsilon (\pi \otimes \chi ,1/2)$ is the Langlands $\epsilon$-constant [ Template:Harv; Template:Harv ] 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 $\left(\frac{\chi }{\pi }\right)=\epsilon (\pi \otimes \chi ,1/2)\cdot \epsilon (\pi ,1/2)\cdot \chi (-1).$

• 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 $\chi$. $$p(\chi )=D_{\chi }^{1/2}\sum _{\nu \text{ archimedean}}{\left|{\gamma }_{\nu }\right|}_{\nu }^{h_{\nu }/2}.$$

Definition 3. Let $f_0,f_1\in A$. $b(f_0,f_1)=\underset{x\in k^{\times }}{\int }{f}_0(x)\cdot \overline{f_1(x)}dx.$

Definition 4. Let $T$ be a maximal torus of $G$, $Z$ be the center of $G$, $\phi \in A$. $$\beta (\phi ,T)=\underset{t\in Z\mathrm{\setminus }T}{\int }b(\pi (t)\phi ,\phi )dt.$$

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

Let $K$ be a field such that $k(\pi )\subset K\subset ℂ$. One can choose a K-subspace$A^0$ of $A$ such that (i) $A=A^0{\otimes }_Kℂ$; (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 ${\phi }_1,{\phi }_2$ of $A^0$ such that $\beta ({\phi }_1,T_1)\ne 0$ and $\beta ({\phi }_2,T_2)\ne 0$.

Definition 5. Let $D_1,D_2$ be the discriminants of ${\chi }_1,{\chi }_2$.

$p(\pi ,{\chi }_1,{\chi }_2)=D_1^{-1/2}{D}_2^{1/2}L({\chi }_1,1{)}^{-1}L({\chi }_2,1)L(\pi \otimes {\chi }_1,1/2)L(\pi \otimes {\chi }_2,1/2{)}^{-1}\beta ({\phi }_1,T_1{)}^{-1}\beta ({\phi }_2,T_2).$

• Comment. When the ${\chi }_1={\chi }_2$, the right hand side of Definition 5 becomes trivial.

We take ${\mathrm{\Sigma }}_f$ to be the set {all the finite $k$-places $\nu \mid {\pi }_{\nu }$ doesn't map non-zero vectors invariant under the action of $G{L}_2(k_{\nu })$ to zero}, ${\mathrm{\Sigma }}_s$ to be the set of (all $k$-places $\nu \mid \nu$ is real, or finite and special).

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The case when Template:Math and Template:Math is a metaplectic cusp form

Let p be prime number, ${𝔽}_p$ be the field with p elements, $R={𝔽}_p[T],k={𝔽}_p(T),k_{\mathrm{\infty }}={𝔽}_p((T^{-1})),o_{\mathrm{\infty }}$ be the integer ring of $k_{\mathrm{\infty }},\mathscr{H}=PG{L}_2(k_{\mathrm{\infty }})/PG{L}_2(o_{\mathrm{\infty }}),\mathrm{\Gamma }=PG{L}_2(R)$. 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 ${\mathrm{\Gamma }}_0(N)$ to the set $\{\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\in \mathrm{\Gamma }\mid c\equiv 0modN\},$ $S_0({\mathrm{\Gamma }}_0(N))$ to be the set of all cusp forms of level N and depth 0. Suppose that, $\phi ,{\phi }_1,{\phi }_2\in S_0({\mathrm{\Gamma }}_0(N))$.

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

Definition 2. Let $z=x+iy\in \mathscr{H},d\mu =\frac{dxdy}{{\left|y\right|}^2}$. Petersson inner product $$⟨{\phi }_1,{\phi }_2⟩=[\mathrm{\Gamma }:{\mathrm{\Gamma }}_0(N){]}^{-1}\underset{{\mathrm{\Gamma }}_0(N)\mathrm{\setminus }\mathscr{H}}{\int }{\phi }_1(z)\overline{{\phi }_2(z)}d\mu .$$

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

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

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

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

Definition 6. Atkin–Lehner operator $$W_{{\alpha }_{\ell }}=\left(\begin{array}{cc}{\ell }^{{\alpha }_{\ell }}& b\\ N& {\ell }^{{\alpha }_{\ell }}d\end{array}\right)$$ with ${\ell }^{2{\alpha }_{\ell }}d-bN={\ell }^{{\alpha }_{\ell }}.$

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

Definition 8. $$L(\phi ,s)=\sum _{r\in R\mathrm{\setminus }\{0\}}\frac{{\omega }_{\phi }(r)}{{\left|r\right|}_p^s}.$$

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

Theorem [ Template:Harv, Thm 5.1, p. 60 ]. Suppose that $K_{\phi }=\frac{1}{\sqrt{p}(\sqrt{p}-e^{-i\theta })(\sqrt{p}-e^{i\theta })}$, ${\chi }_D$ is a quadratic character with $\mathrm{\Delta }({\chi }_D)=D$. Then $$\sum _{\mathrm{Sh}(E_i)=\phi }{|{\omega }_{E_i}(D)|}_p^2=\frac{K_{\phi }{G}_1(D){\left|D\right|}_p^{-3/2}}{⟨\phi ,\phi ⟩}L(\phi \otimes {\chi }_D,1/2)\prod _{\ell }(1+\left(\frac{{\ell }^{{\alpha }_{\ell }}}{D}\right)w_{{\alpha }_{\ell },\phi }).$$

References

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