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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 [ (Langlands 1970); (Deligne 1972) ] associated to and at . There is a such that .
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).
Theorem [1] — Let . We assume that, (i) ; (ii) 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.
[ (Waldspurger 1985), Thm 6, p. 241 ] 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 Fp(T) and φ 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
Waldspurger, Jean-Loup (1985), "Sur les valeurs de certaines L-fonctions automorphes en leur centre de symétrie", Compositio Mathematica, 54 (2): 173–242
Vignéras, Marie-France (1981), "Valeur au centre de symétrie des fonctions L associées aux formes modulaire", Séminarie de Théorie des Nombres, Paris 1979–1980, Progress in Math., Birkhäuser, pp. 331–356
Shimura, Gorô (1976), "On special values of zeta functions associated with cusp forms", Communications on Pure and Applied Mathematics, 29: 783–804, doi:10.1002/cpa.3160290618
Altug, Salim Ali; Tsimerman, Jacob (2010). "Metaplectic Ramanujan conjecture over function fields with applications to quadratic forms". International Mathematics Research Notices. arXiv:1008.0430. doi:10.1093/imrn/rnt047. S2CID119121964.
Deligne, Pierre (1972). "Les constantes des équations fonctionelle des fonctions L". Modular Functions of One Variable II. International Summer School on Modular functions. Antwerp. pp. 501–597.