Abstract

The algebraicity of critical values of triple product $L$-functions in the balanced case was proved by Garrett and Harris, under the assumption that the critical points are on the right and away from the center of the critical strip. The missing right-half critical points correspond to certain holomorphic Eisenstein series outside of the range of absolute convergence. The remaining difficulties are the construction of these holomorphic Eisenstein series and the verification of the nonvanishing of the corresponding nonarchimedean local zeta integrals. In this paper, we address these problems and extend the result of Garrett and Harris to all critical points. As a consequence, we obtain new cases of an automorphic variant of a generalization of Deligne’s conjecture for symmetric cube $L$-functions of Hilbert modular forms.

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