Related papers: Period relations for Rankin-Selberg convolutions f…

We formulate and prove the archimedean period relations for Rankin-Selberg convolutions for $\mathrm{GL}(n)\times \mathrm{GL}(n-1)$. As a consequence, we prove the period relations for critical values of the Rankin-Selberg L-functions for…

We formulate and prove the archimedean period relations for Rankin-Selberg convolutions of $\text{GL}(n)\times \text{GL}(n)$ and $\text{GL}(n)\times \text{GL}(n-1)$, for all generic cohomological representations. As a consequence, we prove…

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…

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…

We refine and extend previous constructions of $p$-adic $L$-functions for Rankin-Selberg convolutions on $\GL(n)\times\GL(n-1)$ for regular algebraic representations over totally real fields. We also prove an intrinsic functional equation…

In this article, we give a cohomological interpretation of (a special case of) the integrals constructed by the second named author and Q. Zhang \cite{YanZhang2023} which represent the product of Rankin-Selberg $L$-functions of…

We unconditionally construct cyclotomic p-adic L-functions for Rankin-Selberg convolutions for GL(n+1) x GL(n) over arbitrary number fields, and show that they satisfy an expected functional equation.

We compute the second moment of a certain family of Rankin-Selberg $L$-functions L(f x g, 1/2) where f and g are Hecke-Maass cusp forms on GL(n). Our bound is as strong as the Lindel\"of hypothesis on average, and recovers individually the…

In this article we prove some period relations for the ratio of Deligne's periods for certain tensor product motives. These period relations give a motivic interpretation for certain algebraicity results for ratios of successive critical…

Let $\mathsf k$ be a local field. Let $I_\nu$ and $I_{\nu'}$ be smooth principal series representations of $\mathrm{GL}_n(\mathsf k)$ and $\mathrm{GL}_{n-1}(\mathsf k)$ respectively. The Rankin-Selberg integrals yield a continuous bilinear…

We give a construction of a wide class of modular symbols attached to reductive groups. As an application we construct a p-adic distribution interpolating the special values of the twisted Rankin-Selberg L-function attached to cuspidal…

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 successive critical values for…

We prove an algebraicity result for the central critical value of certain Rankin-Selberg L-functions for GL(n) x GL(n-1). This is a generalization and refinement of some results of Harder, Kazhdan-Mazur-Schmidt, Mahnkopf, and…

The aim of this paper is to determine all algebraic relations among various special gamma values over function fields, and prove a Chowla-Selberg-type formula for quasi-periods of CM abelian $t$-modules. Our results are based on the…

We prove an asymptotic expansion of the second moment of the central values of the $\mathrm{GL}(n)\times\mathrm{GL}(n)$ Rankin--Selberg $L$-functions $L(1/2,\pi\otimes\pi_0)$, for a fixed cuspidal automorphic representation $\pi_0$, over…

This is an announcement of certain rationality results for the critical values of the degree-2n L-functions attached to GL(1) $\times$ SO(n, n) over $\mathbb Q$ for an even positive integer n. The proof follows from studying the rank-one…

In this paper we prove rationality results of critical values for $L$-functions attached to representations in the residual spectrum of $\mathrm{GL}_4(\mathbb{A})$. We use the Jacquet-Langlands correspondence to describe their partial…

After introducing the notion of uniform integrality of critical values of the Rankin-Selberg $L$-functions for $\mathrm{GL}_{n}\times \mathrm{GL}_{n-1}$, we study it when the base field is totally imaginary. For this purpose, we adopt…

For an even positive integer $n$, we study rank-one Eisenstein cohomology of the split orthogonal group ${\rm O}(2n+2)$ over a totally real number field $F.$ This is used to prove a rationality result for the ratios of successive critical…

In a previous article we had proved an algebraicity result for the central critical value for L-functions for GL(n) x GL(n-1) over Q assuming the validity of a nonvanishing hypothesis involving archimedean integrals. The purpose of this…