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

TL;DR

This paper proves Deligne's conjecture for symmetric sixth L-functions of Hilbert modular forms by relating algebraic critical values to automorphic periods and extending previous results with a new approach.

Contribution

It introduces a novel approach to prove Deligne's conjecture for symmetric sixth L-functions, defining automorphic periods for GSp4 and establishing period relations for Hilbert modular forms.

Findings

Proved Deligne's conjecture for symmetric sixth L-functions of Hilbert modular forms.

Defined automorphic periods for GSp4 over totally real fields.

Established period relations linking automorphic periods and Petersson norms.

Abstract

In this paper, we prove Deligne's conjecture for symmetric sixth $L$-functions of Hilbert modular forms. We extend the result of Morimoto based on a different approach. We define automorphic periods associated to globally generic $C$-algebraic cuspidal automorphic representations of ${\rm GSp}_4$ over totally real number fields whose archimedean components are (limits of) discrete series representations. We show that the algebraicity of critical $L$-values for ${\rm GSp}_4 \times {\rm GL}_2$ can be expressed in terms of these periods. In the case of Kim-Ramakrishnan-Shahidi lifts of ${\rm GL}_2$, we establish period relations between the automorphic periods and powers of Petersson norm of Hilbert modular forms. The conjecture for symmetric sixth $L$-functions then follows from these period relations and our previous work on the algebraicity of critical values for the adjoint $L$-functions for ${\rm GSp}_4$.