Rationality results for the exterior and the symmetric square L-function (with an appendix by Nadir Matringe)

Abstract

Let G = GL 2n over a totally real number field F and n ≥ 2. Let be a cuspidal automorphic representation of G(A), which is cohomological and a functorial lift from SO(2n + 1). The latter condition can be equivalently reformulated that the exterior square L-function of has a pole at s = 1. In this paper, we prove a rationality result for the residue of the exterior square L-function at s = 1 and also for the holomorphic value of the symmetric square L-function at s = 1 attached to . As an application of the latter, we also obtain a period-free relation between certain quotients of twisted symmetric square L-functions and a product of Gauß sums of Hecke characters.

Special values of $L$-functions and the refined Gan-Gross-Prasad conjecture

“…Thm. 6.4 can be rewritten as an Aut(C)-equivariant relation of the residue Res s=1 (L S (s, Π × Π ∨ )) with an archimedean period, the bottom-degree Whittaker period p(Π) and a top-degree version of the latter, as explained in [Gro17b], Thm. 8.5.…”

Algebraicity of the near central non-critical value of symmetric fourth $L$-functions for Hilbert modular forms

“…A crude form of Theorem 1.1 is also given in [BR17,(3.4.3)]. See also the results of Grobner-Harris-Lapid [GHL16, § 6] and Grobner [Gro18] on the algebraicity of non-critical L-values in terms of top degree Whittaker and Shalika periods.…”