Abstract
Let G=mathrm{GL}_{2n} over a totally real number field F and nge 2. Let Pi be a cuspidal automorphic representation of G(mathbb {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 Pi 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 Pi . 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.
Highlights
1.1 General backgroundLet F be an algebraic number field and let be a cuspidal automorphic representation of GL2(AF )
For the scope of this paper, we would like to mention Manin and Shimura, who were the first to study special values of L(s, ) in the particular case, when F is totally real, i.e., when comes from a Hilbert modular form, cf. [27] and [33], and Kurchanov, who treated the case of a CM-field F in a series of papers, cf. [24,25]
In [15], Harder considered the case of an arbitrary number field F, while in [16], he extended the methods of the above authors to some automorphic representations, which do not necessarily come from cusp forms
Summary
Let F be an algebraic number field and let be a cuspidal automorphic representation of GL2(AF ). Guided by the above methods, there is a growing number of results that have been proved about the rationality of special values of certain automorphic Lfunctions attached to GLn. As a selection of examples, relevant to the present paper, we refer to Raghuram [28,29], Harder–Raghuram [17], Grobner–Harris [11]; Grobner– Raghuram [14], Grobner–Harris–Lapid [12] and Balasubramanyam–Raghuram [2]. While most of the aforementioned papers deal with special values of the Rankin–Selberg L-function (by some variation or the other), the principal L-function, or the Asai L-functions, here we would like to study the algebraicity of the exterior square L-function and the symmetric square L-function, attached to a cuspidal automorphic representation of the general linear group
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