On Jacquet's conjecture: The split period case (Q2708370)

Let \(G=\text{GL}(2,{\mathbb{A}})\), where \({\mathbb{A}}\) denotes the adele ring of a totally real number field \(F\). Let \(\pi_i\) denote irreducible automorphic cuspidal representations of \(G\), \(i=1,2,3\), for which the central character of the tensor product \(\pi_1\otimes \pi_2\otimes \pi_3\) is trivial. It was conjectured by Jacquet that \(L(1/2,\pi_1\otimes \pi_2\otimes \pi_3)\not= 0\) if and only if NEWLINE\[NEWLINE \int_{{\mathbb{A}}^\times D(F)\setminus D({\mathbb{A}})} \varphi_1(g)\varphi_2(g)\varphi_3(g) dg\not= 0NEWLINE\]NEWLINE for some \(\varphi_i\) in the space of \(\pi^D_i\) (\(i=1,2,3\)), where \(D\) is a quaternion algebra and \(\pi\) corresponds to \(\pi^D\) via the Jacquet-Langlands correspondence. NEWLINENEWLINEThe author, completing his earlier work [Nonvanishing of the central critical value of the triple product \(L\)-functions, Int. Math. Res. Not. 1998, No. 2, 73--84 (1998; Zbl 0909.11022)], proves this conjecture in the case where \(D=\text{GL}_2\).