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's not obvious though, in fact, 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..
Comment. When the two characters coincide with each other, the right hand side of Definition 5 becomes trivial.
We take to be the set, to be the set of. Theorem . Let. We assume that, ; for, . Then, there is a constant such that
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.
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.
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,
