On Deligne's conjecture for symmetric fourth $L$-functions of Hilbert modular forms

TL;DR

This paper proves an automorphic version of Deligne's conjecture for symmetric fourth L-functions associated with Hilbert modular forms, extending previous results to more general automorphic representations over CM-fields.

Contribution

It generalizes and refines existing results to establish Deligne's conjecture for a broader class of automorphic representations over CM-fields.

Findings

Automorphic analogue of Deligne's conjecture proven for symmetric fourth L-functions.

Extension of Morimoto's results to cohomological irreducible essentially conjugate self-dual automorphic representations.

Results apply to representations of GL_2 and GL_3 over CM-fields.

Abstract

We prove an automorphic analogue of Deligne's conjecture for symmetric fourth $L$-functions of Hilbert modular forms. We extend the result of Morimoto based on generalization and refinement of the results of Grobner and Lin to cohomological irreducible essentially conjugate self-dual cuspidal automorphic representations of ${\rm GL}_2$ and ${\rm GL}_3$ over CM-fields.