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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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