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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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