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
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
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
The nonvanishing hypothesis at infinity for Rankin-Selberg convolutions
We prove the nonvanishing hypothesis at infinity for Rankin-Selberg convolutions for $\GL(n)\times \GL(n-1)$.
A Simple Proof of Rationality of Siegel-Weil Eisenstein Series
We study rationality properties of Eisenstein series on quasi-split unitary similitude groups.
On the critical values of Hecke L-series
L’accès aux archives de la revue « Mémoires de la S. M. F. » ( http://smf. emath.fr/Publications/Memoires/Presentation.html), implique l’accord avec les conditions générales d’utilisation (
...
...