Abstract
In this note we prove a simultaneous extension of the author’s joint result with M. Harris for critical values of Rankin–Selberg L-functions L(s,Pi times Pi ') (Grobner and Harris in J Inst Math Jussieu 15:711–769, 2016, Thm. 3.9) to (i) general CM-fields F and (ii) cohomological automorphic representations Pi '=Pi _1boxplus cdots boxplus Pi _k which are the isobaric sum of unitary cuspidal automorphic representations Pi _i of general linear groups of arbitrary rank over F. In this sense, the main result of these notes, cf. Theorem 1.9, is a generalization, as well as a complement, of the main results in Raghuram (Forum Math 28:457–489, 2016; Int Math Res Not 2:334–372, 2010. https://doi.org/10.1093/imrn/rnp127), and Mahnkopf (J Inst Math Jussieu 4:553–637, 2005).
Highlights
As both, the Whittaker model and the above cohomological model, are irreducible representations, their rational structures are unique up to multiplication by non-zero complex numbers
GLn(AF ), which is cohomological with respect to Eμ and let = 1 · · · k by an isobaric automorphic representation of GLn−1(AF ), fully induced from distinct unitary cuspidal automorphic representations i, 1 ≤ i ≤ k, which is cohomological with respect to Eλ and of central character ω
The reader may want to compare this corollary to the main result of [15], where a similar result on quotients of consecutive critical values of Rankin–Selberg L-functions attached to cuspidal representations and over totally real fields has been established
Summary
The purpose of this note is to prove a broad generalization of our own rationality-result,. [12, Thm. 3.9], established ibidem for critical values of the Rankin–Selberg L-function L(s, × ) of certain automorphic representations ⊗ of GLn × GLn−1 over an imaginary quadratic field K. Our generalization of this result will be in terms of the nature of the base field K, and even more importantly, of the nature of the automorphic representation. Our rationality-theorem [12, Thm. f ), asserting that all critical values of L(s, × ) are a product of transcendental periods and a Gauß-sum, up to a factor coming out of a concrete number field, namely.
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