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Goro Shimura proved this formula, when and is a cusp form; Günter Harder made the same discovery at the same time in an unpublished paper.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 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.
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 ; . 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.
We take to be the set, to be the set of.
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 . Suppose that, is a quadratic character with. Then