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 ${\displaystyle {k=\mathbb {Q} }}$ and f is a cusp form; Günter Harder made the same discovery at the same time in an unpublished paper.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 ${\displaystyle {k=\mathbb {Q} }}$ 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 ${\displaystyle {k}}$ 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, ${\displaystyle {\epsilon (\pi \otimes \chi ,1/2)}}$ is the Langlands ${\displaystyle {\epsilon }}$-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 ${\displaystyle {({\frac {\chi }{\pi }})=\epsilon (\pi \otimes \chi ,1/2)\cdot \epsilon (\pi ,1/2)\cdot \chi (-1)}}$.

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

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 ${\displaystyle {G}}$, ${\displaystyle {\phi \in A}}$. ${\displaystyle {\beta (\phi ,T)=\int _{t\in Z\backslash T}b(\pi (t)\phi ,\phi )dt}}$.

• Comment. It's not obvious though, in fact, 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 ${\displaystyle {\phi _{1},\phi _{2}}}$ of ${\displaystyle {A^{0}}}$ such that ${\displaystyle {\beta (\phi _{1},T_{1})\neq 0}}$ and ${\displaystyle {\beta (\phi _{2},T_{2})\neq 0}}$.

Definition 5. Let ${\displaystyle {D_{1},D_{2}}}$ be the discriminants of ${\displaystyle {\chi _{1},\chi _{2}}}$. ${\displaystyle {p(\pi ,\chi _{1},\chi _{2})=D_{1}^{-1/2}D_{2}^{1/2}L(\chi _{1},1)^{-1}L(\chi _{2},1)L(\pi \otimes \chi _{1},1/2)L(\pi \otimes \chi _{2},1/2)^{-1}\beta (\phi _{1},T_{1})^{-1}\beta (\phi _{2},T_{2})}}$.

• Comment. When the two characters coincide with each other, 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 ${\displaystyle {\nu \in \Sigma _{s}}}$, ${\displaystyle {({\frac {\chi _{1,\nu }}{\pi _{\nu }}})=({\frac {\chi _{2,\nu }}{\pi _{\nu }}})}}$ . 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 ${\displaystyle {L(\pi \otimes \chi ,1/2)L(\pi ,1/2)^{-1}=qD_{\chi }^{1/2}}}$.

The case when ${\displaystyle {k=\mathbb {F} _{p}(T)}}$ and ${\displaystyle {\phi }}$ is a metaplectic cusp form

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 ${\displaystyle {\prod _{l}l^{\alpha _{l}}}}$. We take ${\displaystyle {\Gamma _{0}(N)}}$ to the set { ${\displaystyle {{\begin{pmatrix}a&b\\c&d\end{pmatrix}}\in \Gamma \mid c\equiv 0\ mod\ N}}$ }, ${\displaystyle {S_{0}(\Gamma _{0}(N))}}$ to be the set of all cusp forms of level N and depth 0. Suppose that, ${\displaystyle {\phi ,\phi _{1},\phi _{2}\in S_{0}(\Gamma _{0}(N))}}$.

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 ${\displaystyle {\eta :SL_{2}(R)\to {\widetilde {SL}}_{2}(k_{\infty }),{\begin{pmatrix}a&b\\c&d\end{pmatrix}}\mapsto ({\begin{pmatrix}a&b\\c&d\end{pmatrix}},\left({\frac {c}{d}}\right))}}$.

Definition 2. Let ${\displaystyle {z=x+iy\in {\mathcal {H}},d\mu ={\frac {dxdy}{\left\vert y\right\vert ^{2}}}}}$. Petersson inner product ${\displaystyle {<\phi _{1},\phi _{2}>=[\Gamma$ :\Gamma _{0}(N)]^{-1}\int _{\Gamma _{0}(N)\backslash {\mathcal {H}}}\phi _{1}(z){\overline {\phi _{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 ${\displaystyle {\lambda _{\infty ,\phi }}}$ be the Laplace eigenvalue of ${\displaystyle {\phi }}$. There is a constant ${\displaystyle {\theta \in \mathbb {R} }}$ such that ${\displaystyle {\lambda _{\infty ,\phi }={\frac {e^{-i\theta }+e^{i\theta }}{\sqrt {p}}}}}$.

Definition 4. Assume that, ${\displaystyle {v_{\infty }(a/b)=deg(a)-deg(b),\nu =v_{\infty }(y)}}$. Whittaker function ${\displaystyle {W_{0,i\theta }(y)={\begin{cases}{\frac {\sqrt {p}}{e^{i\theta }-e^{-i\theta }}}[({\frac {e^{i\theta }}{\sqrt {p}}})^{\nu -1}-({\frac {e^{-i\theta }}{\sqrt {p}}})^{\nu -1}],when\ \nu \geq 2;\\0,otherwise\end{cases}}}}$.

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

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

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

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

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 ${\displaystyle {Sh}}$.

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

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

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 Math., 29: 783–804
• Altug, Salim Ali; Tsimerman, Jacob (2010). "Metaplectic Ramanujan conjecture over function fields with applications to quadratic forms". arXiv:1008.0430v3.CS1 maint: ref=harv (link)
• Langlands, Robert (1970). On the Functional Equation of the Artin L-Functions.CS1 maint: ref=harv (link)
• 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.CS1 maint: ref=harv (link)