On a conjecture of Jacquet

الأصل · EN

In this note, we prove in full generality a conjecture of Jacquet concerning the nonvanishing of the triple product L-function at the central point. Let be a number field and let πᵢ, i=1, 2, 3 be cuspidal automorphic representations of GL₂() such that the product of their central characters is trivial. Then the central value L(12,π₁⊗π₂⊗π₃) of the triple product L--function is nonzero if and only if there exists a quaternion algebra B over and automorphic forms fᵢᵇ∈ πᵢᵇ, such that the integral of the product f₁ᵇ f₂ᵇ f₃ᵇ over the diagonal Z(A) B×() B×(A) is nonzero, where πᵢᵇ is the representation of B×() corresponding to πᵢ. In a previous paper, we proved this conjecture in the special case where = and the πᵢ's correspond to a triple of holomorphic newforms. Recent improvement on the Ramanujan bound due to Kim and Shahidi, results about the local L-factors due to Ikeda and Ramakrishnan, results of Chen-bo Zhu and Sahi about invariant distributions and degenerate principal series in the complex case, and an extension of the Siegel--Weil formula to similitude groups allow us to carry over our method to the general case.