Waldspurger formula

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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 ${\displaystyle k=\mathbb {Q} }$ and f is a cusp form; Günter Harder made the same discovery at the same time in an unpublished paper.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 ${\displaystyle k=\mathbb {Q} }$ and f 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.