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Zbl 0633.10029
Jacquet, Hervé
Sur un résultat de Waldspurger. II. (On a result of Waldspurger. II). (French)
[J] Compos. Math. 63, 315-389 (1987). ISSN 0010-437X; ISSN 1570-5846/e

A new kind of relative trace formula is proved for forms of GL(2) in this paper. Let E/F be a quadratic extension of number fields. Let $G\sb 0=GL(2)$ and let $G\sb 1$ be any form of $G\sb 0$ over F which splits over E. In $G\sb 0$ we have a split maximal torus $T\sb 0$ and in $G\sb 1$ a maximal torus $T\sb 1$ such that $T\sb 1(F)$ is isomorphic to $E\sp*$. Now a cuspidal kernel for $G\sb 0$ over E is integrated over the product of $G\sb 1(F)\setminus G\sb 1(F\sb A)$ and the space of idèle classes of $T\sb 0$ over E $(G\sb 1(F)$ is imbedded in $G\sb 0(E))$. This is done for all possible $G\sb 1$. The sum of those integrals is then equal to a sum of integrals of cuspidal kernels for the groups $G\sb 1$ over the square of the space of idèle classes of $T\sb 1.$ \par The formula reflects a correspondence between the disjoint union of the sets of double cosets $T\sb 0(E)\setminus G\sb 0(E)/G\sb 1(F)$ and the disjoint union of the sets $T\sb 1(F)\setminus G\sb 1(F)/T\sb 1(F)$. The formula is used to give a new proof of a result of Waldspurger concerning the nonvanishing of L-functions in the critical point.
[J.G.M.Mars]

MSC 2000:: *11F70 Representation-theoretic methods in automorphic theory
22E55 Repres. of Lie and linear algebraic groups over global fields

Keywords: automorphic representation; cuspidal representation; relative trace formula; forms of GL(2); cuspidal kernel; idèle classes; nonvanishing of L-functions

Cited in: Zbl 1026.11050

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