Related papers: Periods detecting Eisenstein series and sums of $L…

In this article, we study the co-period integral attached to an automorphic form on $\GL(2)$ and two exceptional theta series on the cubic Kazhdan-Patterson cover of $\GL(2)$. In the local aspect, we show the $\Hom$-space is always of one…

We establish results on the rationality of ratios of successive critical values of Langlands-Shahidi $L$-functions, as they appear in the constant term of the Eisenstein series associated with the exceptional group of type $G_2$ over a…

Following a strategy suggested by Michel--Venkatesh, we study the cubic moment of automorphic $L$-functions on $\operatorname{PGL}_2$ using regularized diagonal periods of products of Eisenstein series. Our main innovation is to produce…

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 expressing the critical values in the range of…

Using the relations between rational functions and Eisenstein series, as well as the inferences for cotangent sums and period polynomials, we work out a precise description for Eisenstein series whose $L$-series vanish at certain critical…

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…

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…

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…

In this article, we compute the Gan-Gross-Prasad period integral of Klingen Eisenstein series over the unitary group $\mathrm{U}(m+1, n+1)$ with a cuspidal automorphic form over $\mathrm{U}(m+1, n)$, and show that it is related to certain…

We study the Eisenstein series and constant term functors in the framework of geometric theory of automorphic functions. Our main result says that for a parabolic P in G with Levi quotient M, the !-constant term functor CT_!:D-mod(Bun_G)->…

We derive a precise relation of poles of Eisenstein series associated to the cuspidal datum $\chi\otimes\sigma$ and the lowest occurrence of theta lifts of a cuspidal automorphic representation $\sigma$ of a unitary group, where $\chi$ is a…

We study conjectures of Ben-Zvi--Sakellaridis--Venkatesh that categorify the relationship between automorphic periods and $L$-functions in the context of the Geometric Langlands equivalence. We provide evidence for these conjectures in some…

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…

In this paper, we introduce a new family of period integrals attached to irreducible cuspidal automorphic representations $\sigma$ of symplectic groups $\mathrm{Sp}_{2n}(\mathbb{A})$, which detects the right-most pole of the $L$-function…

We prove a refinement of the global Gan-Gross-Prasad conjecture proposed by Ichino-Ikeda and N. Harris for unitary groups under some local conditions. We need to assume some expected properties of L-packets and some part of the local…

By the unfolding method, Rankin-Selberg L-functions for ${\rm GL}(n)\times{\rm GL}(m)$ can be expressed in terms of period integrals. These period integrals actually define invariant forms on tensor products of the relevant automorphic…

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…

We calculate a $\mathrm{G}_2$-period of a Fourier coefficient of a cuspidal Eisenstein series on the split simply-connected group $\mathrm{E}_6$, and relate this period to the Ginzburg-Rallis period of cusp forms on $\mathrm{GL}_6$. This…

In recent years, Rogers and Zudilin developed a method to write $L$-values attached to elliptic curves as periods. In order to apply this method to a broader collection of $L$-values, we study Eisenstein series and determine their Fourier…

Fix a number field k with its adele ring A. Let G=O(n+3) be an orthogonal group of k-rank 1 and H=O(n+2) a k-anisotropic subgroup. We have previously [arXiv:0908.3521] described how to factor the global period of a spherical Eisenstein…