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

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 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 GSp 4 ˆGL 2 can be expressed in terms of these periods. In the case of Kim-Ramakrishnan-Shahidi lifts of 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 GSp 4 .

Algebraicity of ratios of special values of Rankin-Selberg $L$-functions and applications to Deligne's conjecture

“…For r " 2, we prove the conjecture assuming κ ě 6 in [Che22b]. For symmetric even power L-functions, we have the results [Stu80] and [Stu89] of Sturm for Sym 2 , and [Mor21], [Che21b], [Che21c] for Sym 4 and Sym 6 due to Morimoto and the author. For higher r, our result is as follows:…”

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

“…‚ n " 4, 6: Morimoto [Mor21], C.- [Che21b], [Che21c]. In this paper, we consider the case when n " 5.…”