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

Abstract

We prove Deligne’s conjecture for symmetric fifth L-functions of elliptic newforms of weight greater than 5. As a consequence, we establish period relations between motivic periods associated to an elliptic newform and the Betti–Whittaker periods of its symmetric cube functorial lift to GL 4 {\operatorname{GL}_{4}} .

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

“…We extend the result of Garrett and Harris to κ " 3, 4 in [Che21a]. 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.…”

“…We extend the result of Garrett and Harris to κ " 3, 4 in [Che21a]. For r " 2, we prove the conjecture assuming κ ě 6 in [Che22b]. We prove in this paper the following theorem regarding Deligne's conjecture for arbitrary r: Theorem B (Theorem 5.11).…”

“…In practice, compared with known methods in the literature, Conjecture 1.1 provides another approach to prove algebraicity results for automorphic/motivic L-functions. For instance, we have apply Conjecture 1.1 for GL 4 ˆGL 3 in [Che22b] to prove Deligne's conjecture for symmetric fifth L-functions of modular forms. Applications to Deligne's conjecture for motivic L-functions of higher degree are given in § 5 below.…”