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On the nonvanishing of some L-functions. (English) Zbl 0659.10031
The main subject of this paper is a relative trace formula. It relates the kernels K and \(K'\) of two convolution operators. K describes an operator in \(H=L^ 2(Z(F_ A)G(F)\setminus G(F_ A))\) with F a number field, \(F_ A\) its adeles, \(G=GL_ 2\) and Z the center of \(GL_ 2\). The operator corresponding to \(K'\) acts in the genuine part \(H'\) of \(L^ 2(SL_ 2(F)\setminus \tilde G_ 1(F_ 1))\) with \(\tilde G_ 1(F_ A)\) the metaplectic cover of \(SL_ 2(F_ A)\). Suppose K, resp. \(K'\) is the kernel of convolution by f, resp. \(f'\). The main theorem states that under certain matching conditions on f and \(f'\) \[ I(f;\eta,\psi_ 1)=J(f';\epsilon,\psi) \] with \[ I(f;\eta,\psi_ 1)=\int_{F_ A/F}\int_{F^*_ A/F^*}K\left( \begin{pmatrix} a & 0\\ 0 & 1\end{pmatrix},\begin{pmatrix} 1 & u\\ 0 & 1\end{pmatrix} \right) \eta(a) d^*a \psi_ 1(-u) du \] where \(\psi\) is a non-trivial character of \(F_ A/F\), \(\psi_ 1(u)=\psi (2\epsilon u)\), \(\epsilon\) a suitable element of \(F^*\) and \(\eta\) a quadratic idele character; \[ J(f';\epsilon,\psi)=\int_{(F_ A/F)^ 2}K'\left( \begin{pmatrix} 1 & u\\ 0 & 1 \end{pmatrix},\begin{pmatrix} 1 & v \\ 0 & 1 \end{pmatrix} \right)\quad \psi (- \epsilon u) \psi (-\epsilon v) du dv, \] with a suitable interpretation of \(\begin{pmatrix} 1 & * \\ 0 & 1 \end{pmatrix}\). Although I and J are not defined in exactly the same way, careful computations in \(GL_ 2(F_ v)\) and \(\tilde G_ 1(F_ v)\) for the places v of F lead to matching conditions for the factors at v of f and \(f'\) sufficient for equality of I and J. In this proof the sum \(\sum f(x^{-1}\xi y)\) over \(\xi\in Z(F)\setminus G(F)\) defining K and the corresponding sum for \(K'\) are taken apart. So the proof may be called geometrical.
The kernels K and \(K'\) may also be split up according to the decomposition of the spaces H and \(H'\) into invariant components for \(G(F_ A)\), resp. \(\tilde G_ 1(F_ A)\). The term \(I_{\pi}\) of I corresponding to a cuspidal component \(\pi\) of H is a multiple of \(L(\frac{1}{2},\pi \otimes \eta)\). It is sketched how to derive a characterization of the quadratic characters \(\eta\) such that \(L(\frac{2}{2},\pi \otimes \eta)\neq 0\) similar to the result of Waldspurger.
Reviewer: R.W.Bruggeman

MSC:
11F70 Representation-theoretic methods; automorphic representations over local and global fields
22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings
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