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
and f is a cusp form; Günter Harder made the same discovery at the same time in an unpublished paper.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
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
be a number field,
be its adele ring,
be the subgroup of invertible elements of
,
be the subgroup of the invertible elements of
,
be three quadratic characters over
,
,
be the space of all cusp forms over
,
be the Hecke algebra of
. Assume that,
is an admissible irreducible representation from
to
, the central character of π is trivial,
when
is an archimedean place,
is a subspace of
such that
. We suppose further that,
is the Langlands
-constant [ (Langlands 1970); (Deligne 1972) ] associated to
and
at
. There is a
such that
.
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).}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
- 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
be the discriminant of
. ![{\displaystyle p(\chi )=D_{\chi }^{1/2}\sum _{\nu {\text{ archimedean}}}\left\vert \gamma _{\nu }\right\vert _{\nu }^{h_{\nu }/2}.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Definition 3. Let
. ![{\displaystyle b(f_{0},f_{1})=\int _{x\in k^{\times }}f_{0}(x)\cdot {\overline {f_{1}(x)}}\,dx.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Definition 4. Let
be a maximal torus of
,
be the center of
,
. ![{\displaystyle \beta (\varphi ,T)=\int _{t\in Z\backslash T}b(\pi (t)\varphi ,\varphi )\,dt.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
- Comment. It is not obvious though, that the function
is a generalization of the Gauss sum.
Let
be a field such that
. One can choose a K-subspace
of
such that (i)
; (ii)
. De facto, there is only one such
modulo homothety. Let
be two maximal tori of
such that
and
. We can choose two elements
of
such that
and
.
Definition 5. Let
be the discriminants of
.
![{\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}).}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
- Comment. When the
, the right hand side of Definition 5 becomes trivial.
We take
to be the set {all the finite
-places
doesn't map non-zero vectors invariant under the action of
to zero},
to be the set of (all
-places
is real, or finite and special).
Theorem [1] — Let
. We assume that, (i)
; (ii) for
,
. Then, there is a constant
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 })}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Comments:
- 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.
- It is worthwhile to notice that, when the two characters are equal, the formula can be greatly simplified.
- [ (Waldspurger 1985), Thm 6, p. 241 ] When one of the two characters is
, Waldspurger's formula becomes much more simple. Without loss of generality, we can assume that,
and
. Then, there is an element
such that ![{\displaystyle L(\pi \otimes \chi ,1/2)L(\pi ,1/2)^{-1}=qD_{\chi }^{1/2}.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
The case when Fp(T) and φ is a metaplectic cusp form
Let p be prime number,
be the field with p elements,
be the integer ring of
. Assume that,
, D is squarefree of even degree and coprime to N, the prime factorization of
is
. We take
to the set
to be the set of all cusp forms of level N and depth 0. Suppose that,
.
Definition 1. Let
be the Legendre symbol of c modulo d,
. 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).}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Definition 2. Let
. 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 .}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Definition 3. Let
. Gauss sum ![{\displaystyle G_{n}(P)=\sum _{r\in R/PR}\left({\frac {r}{P}}\right)e(rnT^{2}).}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Let
be the Laplace eigenvalue of
. There is a constant
such that ![{\displaystyle \lambda _{\infty ,\varphi }={\frac {e^{-i\theta }+e^{i\theta }}{\sqrt {p}}}.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Definition 4. Assume that
. 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}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Definition 5. Fourier–Whittaker expansion
One calls
the Fourier–Whittaker coefficients of
.
Definition 6. Atkin–Lehner operator
with ![{\displaystyle \ell ^{2\alpha _{\ell }}d-bN=\ell ^{\alpha _{\ell }}.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Definition 7. Assume that,
is a Hecke eigenform. Atkin–Lehner eigenvalue
with ![{\displaystyle w_{\alpha _{\ell },\varphi }=\pm 1.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Definition 8. ![{\displaystyle L(\varphi ,s)=\sum _{r\in R\backslash \{0\}}{\frac {\omega _{\varphi }(r)}{\left\vert r\right\vert _{p}^{s}}}.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Let
be the metaplectic version of
,
be a nice Hecke eigenbasis for
with respect to the Petersson inner product. We note the Shimura correspondence by ![{\displaystyle \operatorname {Sh} .}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Theorem [ (Altug & Tsimerman 2010), Thm 5.1, p. 60 ]. Suppose that
,
is a quadratic character with
. 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).}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
References
- ^ (Waldspurger 1985), Thm 4, p. 235
- 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.