Period Relations and Special Values of Rankin-Selberg L-Functions

@article{Harris2017PeriodRA,
  title={Period Relations and Special Values of Rankin-Selberg L-Functions},
  author={Michael Harris and Jiezhu Lin},
  journal={arXiv: Number Theory},
  year={2017},
  pages={235-264}
}
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 CM field. Deligne’s conjecture is stated in the language of motives over \(\mathbb{Q}\), and expresses the critical values, up to rational factors, as determinants of certain periods of algebraic differentials on a projective algebraic variety over homology classes. The results that can be proved by… 
Deligne's conjecture for automorphic motives over CM-fields, Part I: factorization
This is the first of two papers devoted to the relations between Deligne's conjecture on critical values of motivic $L$-functions and the multiplicative relations between periods of arithmetically
Period Relations for Standard $L$-functions of Symplectic Type
This article is to understand the critical values of $L$-functions $L(s,\Pi\otimes \chi)$ and to establish the relation of the relevant global periods at the critical places. Here $\Pi$ is an
Special Values of L-functions for GL(n) Over a CM Field
  • A. Raghuram
  • Mathematics
    International Mathematics Research Notices
  • 2021
We prove a Galois-equivariant algebraicity result for the ratios of successive critical values of $L$-functions for ${\textrm GL}(n)/F,$ where $F$ is a totally imaginary quadratic extension of a
L -values of Elliptic Curves twisted by Hecke Grössencharacters
(over equation (over number fields) Elliptic curves twisted by Grössencharacters Modular forms associated to Grössencharacters Abstract Let E/K be an elliptic curve over an imaginary quadratic field K
Special values of L-functions and the refined Gan-Gross-Prasad conjecture
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
Archimedean period relations and period relations for Rankin-Selberg convolutions
We prove the Archimedean period relations for Rankin-Selberg convolutions for GL(n) × GL(n − 1). This implies the period relations for critical values of the Rankin-Selberg L-functions for

References

SHOWING 1-10 OF 22 REFERENCES
L-functions and periods of polarized regular motives.
It is probably fair to say that all known results on special values of L-functions of motives over number fields are proved by identifying the L-function in question, more or less explicitly, with an
Period relations for automorphic forms on unitary groups and critical values of $L$-functions
In this paper we explore some properties of periods attached to automorphic representations of unitary groups over CM fields and the critical values of their $L$-functions. We prove a formula
Period relations for automorphic induction and applications, I
Eisenstein Cohomology for GL(N) and ratios of critical values of Rankin-Selberg L-functions - I
The aim of this article is to study rank-one Eisenstein cohomology for the group GL(N)/F, where F is a totally real field extension of Q. This is then used to prove rationality results for ratios of
Whittaker rational structures and special values of the Asai $L$-function
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
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
An automorphic version of the Deligne conjecture
In this paper we introduce an automorphic variant of the Deligne conjecture for tensor product of two motives over a quadratic imaginary field. On one hand, we define some motivic periods and rewrite
Special values of automorphic $L$-functions for $GL_{n}\times GL_{n'}$ over CM fields, factorization and functoriality of arithmetic automorphic periods
Michael HARRIS defined the arithmetic automorphic periods for certain cuspidal representations of $GL_{n}$ over quadratic imaginary fields in his Crelle paper 1997. He also showed that critical
WHITTAKER PERIODS, MOTIVIC PERIODS, AND SPECIAL VALUES OF TENSOR PRODUCT $L$ -FUNCTIONS
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
Motives over totally real fields and $p$-adic $L$-functions
© Annales de l’institut Fourier, 1994, tous droits réservés. L’accès aux archives de la revue « Annales de l’institut Fourier » (http://annalif.ujf-grenoble.fr/) implique l’accord avec les conditions
...
...