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 be the base field, be an automorphic form over , be the representation associated via the Jacquet–Langlands correspondence with . Goro Shimura (1976) proved this formula, when formula_1 and is a cusp form; Günter Harder made the same discovery at the same time in an unpublished paper.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 {formula_1 and 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 formula_3 be a number field, formula_4 be its adele ring, formula_5 be the subgroup of invertible elements of formula_3, formula_7 be the subgroup of the invertible elements of formula_4, formula_9 be three quadratic characters over formula_10, formula_11, formula_12 be the space of all cusp forms over formula_13, formula_14 be the Hecke algebra of formula_15. Assume that, formula_16 is an admissible irreducible representation from formula_15 to formula_12, the central character of π is trivial, formula_19 when formula_20 is an archimedean place, formula_21 is a subspace of formula_22 such that formula_23. We suppose further that, formula_24 is the Langlands formula_25-constant [ ; ] associated to formula_16 and formula_27 at formula_28. There is a formula_29 such that formula_30.
Definition 1. The Legendre symbol formula_31
Definition 2. Let formula_32 be the discriminant of formula_27. formula_34
Definition 3. Let formula_35. formula_36
Definition 4. Let formula_37 be a maximal torus of formula_38, formula_39 be the center of formula_38, formula_41. formula_42
Let formula_44 be a field such that formula_45. One can choose a K-subspaceformula_46 of formula_47 such that (i) formula_48; (ii) formula_49. De facto, there is only one such formula_50 modulo homothety. Let formula_51 be two maximal tori of formula_52 such that formula_53 and formula_54. We can choose two elements formula_55 of formula_50 such that formula_57 and formula_58.
Definition 5. Let formula_59 be the discriminants of formula_60.
We take formula_63 to be the set {all the finite formula_3-places formula_65 doesn't map non-zero vectors invariant under the action of formula_66 to zero}, formula_67 to be the set of { all formula_3-places formula_69 is real, or finite and special }.
Theorem [ , Thm 4, p. 235 ]. Let formula_70. We assume that, (i) formula_71; (ii) for formula_72, formula_73 . Then, there is a constant formula_74 such that
Comments:
The case when formula_81 and formula_82 is a metaplectic cusp form.
Let p be prime number, formula_83 be the field with "p" elements, formula_84 be the integer ring of formula_85. Assume that, formula_86, D is squarefree of even degree and coprime to "N", the prime factorization of formula_87 is formula_88. We take formula_89 to the set formula_90 formula_91 to be the set of all cusp forms of level "N" and depth 0. Suppose that, formula_92.
Definition 1. Let formula_93 be the Legendre symbol of "c" modulo "d", formula_94. Metaplectic morphism formula_95
Definition 2. Let formula_96. Petersson inner product formula_97
Definition 3. Let formula_98. Gauss sum formula_99
Let formula_100 be the Laplace eigenvalue of formula_82. There is a constant formula_102 such that formula_103
Definition 4. Assume that formula_104. Whittaker function
Definition 5. Fourier–Whittaker expansion formula_106. One calls formula_107 the Fourier–Whittaker coefficients of formula_82.
Definition 6. Atkin–Lehner operator formula_109 with formula_110
Definition 7. Assume that, formula_82 is a Hecke eigenform. Atkin–Lehner eigenvalue formula_112 with formula_113
Definition 8. formula_114
Let formula_115 be the metaplectic version of formula_91, formula_117 be a nice Hecke eigenbasis for formula_115 with respect to the Petersson inner product. We note the Shimura correspondence by formula_119
Theorem [ , Thm 5.1, p. 60 ]. Suppose that formula_120, formula_121 is a quadratic character with formula_122. Then