Related papers: Period relations for automorphic forms on unitary …

This is a survey of recent work on values of Rankin-Selberg $L$-functions of pairs of cohomological automorphic representations that are {\it critical} in Deligne's sense. The base field is assumed to be a CM field. Deligne's conjecture is…

The present paper is devoted to the relations between Deligne's conjecture on critical values of motivic $L$-functions and the multiplicative relations between periods of arithmetically normalized automorphic forms on unitary groups. As an…

The present paper is devoted to the relations between Deligne's conjecture on critical values of motivic $L$-functions and the multiplicative relations between periods of arithmetically normalized automorphic forms on unitary groups. In the…

In this paper, we prove Deligne's conjecture for symmetric sixth $L$-functions of Hilbert modular forms. We extend the result of Morimoto based on a different approach. We define automorphic periods associated to globally generic…

The goal of this paper is to provide a refinement of a formula proved by the first author which expresses some critical values of automorphic $L$-functions on unitary groups as Petersson norms of automorphic forms. Here we provide a Galois…

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 values of automorphic L-functions for…

We prove Deligne's conjecture for central critical values of certain automorphic $L$-functions for ${\rm GL}(3)\times {\rm GL}(2)$. The proof is base on rationality results for central critical values of triple product $L$-functions, which…

In this paper we prove a version of Deligne's conjecture for potentially automorphic motives, twisted by certain algebraic Hecke characters. The Hecke characters are chosen in such a way that we can use automorphic methods in the context of…

Let $K$ be a quadratic imaginary field. Let $\Pi$ (resp. $\Pi'$) be a regular algebraic cuspidal representation of $GL_{n}(K)$ (resp. $GL_{n-1}(K)$) which is moreover cohomological and conjugate self-dual. In \cite{harris97}, M. Harris has…

This paper is the first in a series of two dedicated to the study of period relations of the type $$ L(\frac{1}{2}+k,\Pi)\;\in\;(2\pi i)^{d\cdot k}\Omega_{(-1)^k}{\mathbb Q}(\Pi),\quad \frac{1}{2}+k\;\text{critical}, $$ for certain…

Let $\mathcal K$ be an imaginary quadratic field. Let $\Pi$ and $\Pi'$ be irreducible generic cohomological automorphic representation of $GL(n)/{\mathcal K}$ and $GL(n-1)/{\mathcal K}$, respectively. Each of them can be given two natural…

The purpose of this semi-expository article is to give another proof of a classical theorem of Shimura on the critical values of the standard L-function attached to a Hilbert modular form. Our proof is along the lines of previous work of…

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 irreducible regular algebraic cuspidal…

Let $F$ be a number field and $D$ a quaternion algebra over $F$. Take a cuspidal automorphic representation $\pi$ of $D_{\mathbb{A}}^\times$ with trivial central character and a cusp form $\phi$ in $\pi$. Using the prehomogeneous zeta…

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 the Deligne conjecture in terms of these…

Given a curve $C$ over a number field $K$ equipped with the action of a finite group $G$ by $K$-automorphisms, one obtains a factorisation of $L(C,s)$ into a product of $L$-functions of `motivic pieces of curves' associated to irreducible…

We study the automorphic period associated to a $G$-Hamiltonian variety $M$ whose dual is $\check{M} = T^*(\check{G}/\check{L})$, where $\check{G}$ is a general linear group and $\check{L}$ is a Levi subgroup. For certain cuspidal…

We relate analytically defined deformations of modular curves and modular forms from the literature to motivic periods via cohomological descriptions of deformation theory. Leveraging cohomological vanishing results, we prove the existence…

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 originally defined by Harder when $n=2$, and later…

Waldspurger's formula gives an identity between the norm of a torus period and an L-function of the twist of an automorphic representation on GL(2). For any two Hecke characters of a fixed quadratic extension, one can consider the two torus…