# Whittaker rational structures and special values of the Asai $L$-function

@article{Grobner2014WhittakerRS,
title={Whittaker rational structures and special values of the Asai \$L\$-function},
author={Harald Grobner and Michael Harris and Erez Lapid},
journal={arXiv: Number Theory},
year={2014}
}
• Published 8 August 2014
• Mathematics
• arXiv: Number Theory
Let $F$ be a totally real number field and $E/F$ a totally imaginary quadratic extension of $F$. Let $\Pi$ be a cohomological, conjugate self-dual cuspidal automorphic representation of $GL_n(\mathbb A_E)$. Under a certain non-vanishing condition we relate the residue and the value of the Asai $L$-functions at $s=1$ with rational structures obtained from the cohomologies in top and bottom degrees via the Whittaker coefficient map. This generalizes a result in Eric Urban's thesis when $n = 2… Special values of adjoint L-functions and congruences for automorphic forms on GL(n) over a number field • Mathematics • 2014 We prove an integrality result for the value at$s=1$of the adjoint$L$-function associated to a cohomological cuspidal automorphic representation on${\rm GL}(n)$over any number field. We then RATIONALITY RESULTS FOR THE EXTERIOR AND THE SYMMETRIC SQUARE L-FUNCTION Let G = GL2n over a totally real number field F and n ≥ 2. Let Π be a cuspidal automorphic representation of G(A), which is cohomological and a functorial lift from SO(2n + 1). The latter condition Rationality results for the exterior and the symmetric square L-function (with an appendix by Nadir Matringe) Let $$G=\mathrm{GL}_{2n}$$G=GL2n over a totally real number field F and $$n\ge 2$$n≥2. Let $$\Pi$$Π be a cuspidal automorphic representation of $$G(\mathbb {A})$$G(A), which is cohomological and a On Deligne's conjecture for symmetric fourth$L$-functions of Hilbert modular forms We prove Deligne's conjecture for symmetric fourth$L$-functions of Hilbert modular forms. We extend the result of Morimoto based on generalization and refinement of the results of Grobner and Lin to Special values of L-functions and the refined Gan-Gross-Prasad conjecture • Mathematics • 2017 We prove explicit rationality-results for Asai-$L$-functions,$L^S(s,\Pi',{\rm As}^\pm)$, and Rankin-Selberg$L$-functions,$L^S(s,\Pi\times\Pi')$, over arbitrary CM-fields$F$, relating critical Period Relations and Special Values of Rankin-Selberg L-Functions • Mathematics • 2017 This is a survey of recent work on values of Rankin-Selberg L-functions of pairs of cohomological automorphic representations that are critical in Deligne’s sense. The base field is assumed to be a PERIOD RELATIONS AND SPECIAL VALUES OF RANKIN-SELBERG L-FUNCTIONS This is a survey of recent work on values of Rankin-Selberg L-functions of pairs of cohomological automorphic representations that are critical in Deligne’s sense. The base field is assumed to be a On the Beilinson–Bloch–Kato conjecture for Rankin–Selberg motives • Mathematics Inventiones mathematicae • 2022 In this article, we study the Beilinson-Bloch-Kato conjecture for motives corresponding to the Rankin-Selberg product of conjugate self-dual automorphic representations, within the framework of the WHITTAKER PERIODS, MOTIVIC PERIODS, AND SPECIAL VALUES OF TENSOR PRODUCT$L$-FUNCTIONS • Mathematics Journal of the Institute of Mathematics of Jussieu • 2015 Let${\mathcal{K}}$be an imaginary quadratic field. Let${\rm\Pi}$and${\rm\Pi}^{\prime }$be irreducible generic cohomological automorphic representation of$\text{GL}(n)/{\mathcal{K}}$and ## References SHOWING 1-10 OF 34 REFERENCES Special values of adjoint L-functions and congruences for automorphic forms on GL(n) over a number field • Mathematics • 2014 We prove an integrality result for the value at$s=1$of the adjoint$L$-function associated to a cohomological cuspidal automorphic representation on${\rm GL}(n)$over any number field. We then On the arithmetic of Shalika models and the critical values of L-functions for GL2n • Mathematics • 2011 Let$\Pi$be a cohomological cuspidal automorphic representation of${\rm GL}_{2n}(\Bbb{A})$over a totally real number field$F$. Suppose that$\Pi$has a Shalika model. We define a rational cohomology of arithmetic groups, parabolic subgroups and the special values of$l$-functions on gl$_{n}$• J. Mahnkopf • Mathematics Journal of the Institute of Mathematics of Jussieu • 2005 let$\pi$be a cuspidal automorphic representation of$\mathrm{gl}_n(\mathbb{a}_{\mathbb{q}})$with non-vanishing cohomology. under a certain local non-vanishing assumption we prove the rationality On Certain Period Relations for Cusp Forms on GLn • Mathematics • 2007 Let$\pi$be a regular algebraic cuspidal automorphic representation of${\rm GL}_n({\mathbb A}_F)$for a number field$F$. We consider certain periods attached to$\pi$. These periods were Critical values of the twisted tensor$L$-function in the imaginary quadratic case The twisted tensor L-function of f , which we denote by G(s, f), is a certain Dirichlet series associated to a quadratic extension of number fields K/F , and a cuspidal automorphic function f over K. On the Special Values of Certain Rankin-Selberg L-Functions and Applications to Odd Symmetric Power L-Functions of Modular Forms We prove an algebraicity result for the central critical value of certain Rankin-Selberg L-functions for GL n × GL n−1 . This is a generalization and refinement of the results of Harder [14], Critical values of automorphic L-functions for GL(r)×GL(r) Abstract. We compute, up to an element of a fixed number field, the critical values of the L-function of a pair of automorphic, cuspidal, cohomological representations of any GL(r). The result is WHITTAKER PERIODS, MOTIVIC PERIODS, AND SPECIAL VALUES OF TENSOR PRODUCT$L$-FUNCTIONS • Mathematics Journal of the Institute of Mathematics of Jussieu • 2015 Let${\mathcal{K}}$be an imaginary quadratic field. Let${\rm\Pi}$and${\rm\Pi}^{\prime }$be irreducible generic cohomological automorphic representation of$\text{GL}(n)/{\mathcal{K}}\$ and
On some arithmetic properties of automorphic forms of GL(m) over a division algebra
• Mathematics
• 2011
In this paper we investigate arithmetic properties of automorphic forms on the group G' = GL_m/D, for a central division-algebra D over an arbitrary number field F. The results of this article are
CONDITIONAL BASE CHANGE FOR UNITARY GROUPS
Introduction. It has been known for many years that the stabilization of the Arthur-Selberg trace formula would, or perhaps we should write “will,” have important consequences for the Langlands