Algebraicity of the central critical values of twisted triple product $L$-functions

TL;DR

This paper proves the algebraicity of central critical values of twisted triple product L-functions for motivic Hilbert cusp forms over a totally real cubic algebra, linking it to cohomological periods and generalizing previous results on Deligne's conjecture.

Contribution

It establishes the algebraicity of these L-values in terms of cohomological periods and extends prior work on automorphic L-functions for GL(3)×GL(2).

Findings

Algebraicity of central critical values expressed via cohomological periods.

Generalization of Deligne's conjecture for specific automorphic L-functions.

Relation of cohomological periods under twisting by algebraic Hecke characters.

Abstract

We study the algebraicity of the central critical values of twisted triple product $L$-functions associated to motivic Hilbert cusp forms over a totally real \'etale cubic algebra in the totally unbalanced case. The algebraicity is expressed in terms of the cohomological period constructed via the theory of coherent cohomology on quaternionic Shimura varieties developed by Harris. As an application, we generalize our previous result on Deligne's conjecture for certain automorphic $L$-functions for ${\rm GL}_3 \times {\rm GL}_2$. We also establish a relation for the cohomological periods under twisting by algebraic Hecke characters.