Algebraicity of critical values of triple product $L$-functions in the balanced case

Abstract

The algebraicity of critical values of triple product L-functions in the balanced case was proved by Garrett and Harris, under the assumption that the critical points are on the right and away from center of the critical strip. The missing right-half critical points correspond to certain holomorphic Eisenstein series outside the range of absolute convergence. The remaining difficulties are construction of these holomorphic Eisenstein series and verification of the non-vanishing of the corresponding non-archimedean local zeta integrals. In this paper, we address these problems and complement the result of Garrett and Harris to all critical points. As a consequence, we obtain new cases of Deligne's conjecture for symmetric cube L-functions of Hilbert modular forms.

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

“…For r " 1, the conjecture was proved by Garrett and Harris [GH93] assuming κ ě 5. 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].…”

“…In the balanced case, the conjecture was proved by Garrett and Harris in [GH93] under some conditions which were lifted by the author in [Che21a]. We refer to the introduction of [Che21a] for a survey of known results in this case. In the unbalanced case, the conjecture is partially proved.…”

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

“…Note that the results are special cases of Theorem 4.5. On the other hand, when we take Σ be the Kim-Ramakrishnan-Shahidi lift of Π , the algebraicity of these critical values can be expressed in terms of powers of }f Π } using the results [GH93], [GL16], [Mor18], [Che21c], [Che21d], [Har21], [JST21]. We then obtain the period relation (1.3).…”

“…We then use results in the literature [GL16], [Har21], and [JST21], together with Deligne's conjecture for symmetric cube L-functions of Π [GH93] and [Che21c] to study the algebraicity of the L-function on the right-hand side of (5.1). As a consequence, we obtain period relation between p S8 pΣ q and }f Π } 4 .…”

“…As a consequence, they proved the conjecture for n " 3 under the assumptions that κ ě 6 and m is on the right and away from the center of the critical strip. We complement the result of Garrett and Harris in [Che21c] and relaxed the assumption to κ ě 3. Based on symmetric power functoriality and various algebraicity results on critical L-values in the literature, especially [GL21], Morimoto proved the algebraicity of the critical L-values for n " 4, 6 in [Mor21].…”