The algebraicity of critical values of triple product
-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
-functions
of Hilbert modular forms.
Keywords
special values of $L$-functions, triple product
$L$-functions
