Base change and triple product 𝐿-series

Let π i \pi _i be an irreducible cuspidal automorphic representation of G L 2 \mathrm {GL}_2 with central character ω i \omega _i . When ω 1 ω 2 ω 3 \omega _1\omega _2\omega _3 is trivial, Atsushi Ichino proved a formula for the central value L ( 1 2 , π 1 × π 2 × π 3 ) L(\frac {1}{2}, \pi _1\times \pi _2\times \pi _3) of the triple product L L -series in terms of global trilinear forms. We will extend this formula to the case when ω 1 ω 2 ω 3 \omega _1\omega _2\omega _3 is a quadratic character, giving a non-vanishing criterion of a local trilinear form in terms of the central value of the gamma factor.