Related papers: A multivariate integral representation on $\mathrm…

We construct new irreducible components in the discrete automorphic spectrum of symplectic groups. The construction lifts a cuspidal automorphic representation of $\mathrm{GL}_{2n}$ with a linear period to an irreducible component of the…

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…

A fundamental question, first raised by Langlands, is to know whether the Rankin-Selberg product of two (not necessarily holomorphic) cusp forms f and g is modular, i.e., if there exists an automorphic form f box g on GL(4)/Q whose standard…

In this work we develop an integral representation for the partial $L$-function of a pair $\pi\times\tau$ of genuine irreducible cuspidal automorphic representations, $\pi$ of the $m$-fold covering of Matsumoto of the symplectic group…

Based on Furusawa's theory, we present an integral representation for the L-function L(s,\pi \times \tau), where \pi is a cuspidal automorphic representation on GSp(4) related to a holomorphic Siegel modular form, and where \tau is an…

In this note, we use a basic identity, derived from the generalized doubling integrals of \cite{C-F-G-K1}, in order to explain the existence of various global Rankin-Selberg integrals for certain $L$-functions. To derive these global…

We introduce a new invariant, the conductor exponent, of a generic irreducible Casselman-Wallach representation of $\mathrm{GL}_n$ that quantifies the extent to which this representation may be ramified. We also determine a distinguished…

We show that certain paramodular cuspidal automorphic irreducible representations of $\mathrm{GSp}(4,\mathbb{A}_\mathbb{Q})$, which are not CAP, are globally generic. This implies a multiplicity one theorem for paramodular cuspidal…

Let $f$ and $g$ be holomorphic or Maass cusp forms for $\rm SL_2(\mathbb{Z})$ with normalized Fourier coefficients $\lambda_f(n)$ and $\lambda_g(n)$, respectively. In this paper, we prove nontrivial estimates for the sum $$…

Let $k$ be a local field of characteristic zero. Rankin-Selberg's local zeta integrals produce linear functionals on generic irreducible admissible smooth representations of $GL_n(k)\times GL_r(k)$, with certain invariance properties. We…

In this paper, we study top Fourier coefficients of certain automorphic representations of $\mathrm{GL}_n(\mathbb{A})$. In particular, we prove a conjecture of Jiang on top Fourier coefficients of isobaric automorphic representations of…

Let $\mathbb{E}$ be a quadratic extension of a number field $\mathbb{F}$. Let $E(g, s)$ be an Eisenstein series on $GL_2(\mathbb{E})$, and let $F$ be a cuspidal automorphic form on $GL_2(\mathbb{F})$. We will consider in this paper the…

We prove that the local Rankin--Selberg integrals for principal series representations of the general linear groups agree with certain simple integrals over the Rankin--Selberg subgroups, up to certain constants given by the local gamma…

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…

We prove a prime number theorem first for the classical Rankin-Selberg L-function $L(s,\pi\times\pi')$ over any Galois extension with $\pi$ and $\pi'$ unitary automorphic cuspidal representations of $GL_n$ and $GL_m$ respectively with at…

We prove a conjectural formula relating the Bessel period of certain automorphic forms on $\mathrm{GSp}_4$ to a central $L$-value. This formula is proposed by Liu \cite{liu} as the refined Gan-Gross-Prasad conjecture for the groups…

Let $f$ be a normalized holomorphic cusp form for $SL_2(\mathbb{Z})$ of weight $k$ with $k\equiv0\bmod 4$. By the Kuznetsov trace formula for $GL_3(\mathbb R)$, we obtain the first moment of central values of $L(s,f\otimes \phi)$, where…

Let $\tau_1^{(r)}$, $\tau_2^{(r)}$ be two genuine cuspidal automorphic representations on $r$-fold covers of the adelic points of the general linear groups $GL_{n_1}$, $GL_{n_2}$, resp., and let $E(g,s)$ be the associated Eisenstein series…

In this paper, we prove the fundamental properties of gamma factors defined by Rankin-Selberg integrals of Shimura type for pairs of generic representations $(\pi, \tau)$ of $\mathrm{U}_{2\ell}(F)$ and $\mathrm{GL}_n(E)$ for a local field…

Let L^S(\pi,s,st) be a partial L-function of degree 7 of a cuspidal automorphic representation \pi of the exceptional group G_2. Here we construct a Rankin-Selberg integral for representations having certain Fourier coefficient.