Algebraicity of critical values of triple product $L$-functions in the balanced case

TL;DR

This paper extends the algebraicity results of triple product L-values to all critical points by constructing necessary Eisenstein series and verifying local integrals, leading to new cases of Deligne's conjecture.

Contribution

It addresses the missing right-half critical points in the algebraicity of triple product L-values, completing the previous results by Garrett and Harris.

Findings

Extended algebraicity to all critical points of triple product L-functions.

Constructed holomorphic Eisenstein series outside the convergence range.

Verified non-vanishing of local zeta integrals for new cases.

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 center of the critical strip. The missing right-half critical points correspond to certain holomorphic Eisenstein series outside the range of absolute convergence. The remaining difficulties are construction of these holomorphic Eisenstein series and verification of the non-vanishing of the corresponding non-archimedean local zeta integrals. In this paper, we address these problems and complement the result of Garrett and Harris to all critical points. As a consequence, we obtain new cases of Deligne's conjecture for symmetric cube $L$-functions of Hilbert modular forms.