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(1976) proved this formula, when and is a cusp form; Günter Harder made the same discovery at the same time in an unpublished paper.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 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 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 [ ; ] associated to and at . There is a such that .

Definition 1. The Legendre symbol

• 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 .

Definition 3. Let .

Definition 4. Let be a maximal torus of , be the center of , .

• 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 .

• 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).

Comments:

The case when and 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

Definition 2. Let . Petersson inner product

Definition 3. Let . Gauss sum

Let be the Laplace eigenvalue of . There is a constant such that

Definition 4. Assume that . Whittaker function

Definition 5. Fourier–Whittaker expansion One calls the Fourier–Whittaker coefficients of .

Definition 6. Atkin–Lehner operator with

Definition 7. Assume that, is a Hecke eigenform. Atkin–Lehner eigenvalue with

Definition 8.

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

Theorem [ , Thm 5.1, p. 60 ]. Suppose that , is a quadratic character with . Then

References