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and f is a cusp form; Günter Harder made the same discovery at the same time in an unpublished paper.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
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.
be a number field,
be the subgroup of invertible elements of
be the subgroup of the invertible elements of
be the space of all cusp forms over
is an admissible irreducible representation from
, the central character of π is trivial,
is an archimedean place,
ε ( π ⊗ χ , 1
-constant [ (Langlands 1970); (Deligne 1972) ] associated to
The Legendre symbol
= ε ( π ⊗ χ , 1
De facto, there is only one such
doesn't map non-zero vectors invariant under the action of
is real, or finite and special).
Comments: Let p be prime number,
be the field with p elements,
be the integer ring of
, D is squarefree of even degree and coprime to N, the prime factorization of
to be the set of all cusp forms of level N and depth 0.
be the Legendre symbol of c modulo d,
be the Laplace eigenvalue of
be a nice Hecke eigenbasis for
with respect to the Petersson inner product.
We note the Shimura correspondence by
Theorem [ (Altug & Tsimerman 2010), Thm 5.1, p. 60 ].