On algebraicity of special values of symmetric 4-th and 6-th power L-functions for $$\mathrm {GL}(2)$$

Abstract

We prove an algebraicity of special values of symmetric 4-th and 6-th power L-functions of Hilbert cusp forms using its symmetric power lifting and several algebraicity results on special values of automorphic L-functions.

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

“…By the functoriality, we have Lps, Ψ , stdq " Lps, Sym 4 Π 1 b ω ´2 Π 1 q. It follows from Theorem 3.1 and our previous result [Che21b] on Deligne's conjecture for critical values of Lps, Sym 4 Π 1 b ω ´2 Π 1 q (see also [Mor21]), we obtain the period relation…”

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

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

“…We assume κ v ě 6 for all v P S 8 . By Deligne's conjecture for symmetric fourth L-function proved in [Che21d, Theorem 1.4], for all critical points m ě 1 of Lps, Π , Sym 4 b ω ´2 Π q (see also [Mor21]), we have…”

“…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]. More precisely, under the assumptions that κ ě 6, ω " 1, and AutpCq replaced by AutpC{Eq for some bi-quadratic extension E{Q, Morimoto proves that (1.1) holds up to square.…”

“…The conjecture is true when Π is of CM-type. In [Mor21], Conjecture 1.1 was proved (up to square when F " Q) by Morimoto under the following assumptions:…”