# Analytic Twists of GL3 × GL2 Automorphic Forms

@article{Lin2021AnalyticTO,
title={Analytic Twists of GL3 × GL2 Automorphic Forms},
author={Yongxiao Lin and Qingfeng Sun},
journal={International Mathematics Research Notices},
year={2021}
}
• Published 20 December 2019
• Mathematics
• International Mathematics Research Notices
Let $\pi$ be a Hecke–Maass cusp form for $\textrm{SL}_3(\mathbb{Z})$ with normalized Hecke eigenvalues $\lambda _{\pi }(n,r)$. Let $f$ be a holomorphic or Maass cusp form for $\textrm{SL}_2(\mathbb{Z})$ with normalized Hecke eigenvalues $\lambda _f(n)$. In this paper, we are concerned with obtaining nontrivial estimates for the sum \begin{align*}& \sum_{r,n\geq 1}\lambda_{\pi}(n,r)\lambda_f(n)e\left(t\,\varphi(r^2n/N)\right)V\left(r^2n/N\right), \end{align*}where $e(x)=e^{2\pi ix}$, $V(x… Algebraic twists of$\mathrm{GL}_3\times \mathrm{GL}_2L$-functions. • Mathematics • 2019 We prove that the coefficients of a$\mathrm{GL}_3\times \mathrm{GL}_2$Rankin--Selberg$L$-function do not correlate with a wide class of trace functions of small conductor modulo primes, Uniform bounds for$\rm GL(3) \times GL(2)L$-functions Abstract. In this paper, we prove uniform bounds for GL(3) × GL(2) L-functions in the GL(2) spectral aspect and the t aspect by a delta method. More precisely, let φ be a Hecke–Maass cusp form for Hybrid subconvexity bounds for twists of$\rm GL (3)\times GL(2)L$-functions The subconvexity problem of automorphic L-functions on the critical line is one of the central problems in number theory. In general, let C denote the analytic conductor of the relevant L-function, HYBRID SUBCONVEXITY BOUNDS FOR TWISTS OF GL(3)×GL(2) L-FUNCTIONS The subconvexity problem of automorphic L-functions on the critical line is one of the central problems in number theory. Subconvexity bounds have many very important applications such as the Uniform subconvex bounds for Rankin-Selberg$L$-functions Let f be a Maass cusp form for SL2(Z) with Laplace eigenvalue 1/4 + μ 2 f , μf > 0. Let g be an arbitrary but fixed holomorphic or Maass cusp form for SL2(Z). In this paper, we establish the UNIFORM SUBCONVEXITY BOUNDS FOR GL(3)×GL(2) L-FUNCTIONS The subconvexity problem of automorphic L-functions on the critical line is one of the central problems in number theory, which have very important applications to equidistribution problems. The Averages of coefficients of a class of degree 3 L-functions • Mathematics The Ramanujan Journal • 2021 In this note, we give a detailed proof of an asymptotic for averages of coefficients of a class of degree three L -functions which can be factorized as a product of a degree one and a degree two L On the Rankin–Selberg problem In this paper, we solve the Rankin--Selberg problem. That is, we break the well known Rankin--Selberg's bound on the error term of the second moment of Fourier coefficients of a$\mathrm{GL}(2)$cusp Subconvexity for$L$-Functions on$\mathrm{GL}_3$over Number Fields. In this paper, over an arbitrary number field, we prove subconvexity bounds for self-dual$\mathrm{GL}_3L$-functions in the$t$-aspect and for self-dual$\mathrm{GL}_3 \times \mathrm{GL}_2$Analytic twists of$\rm GL_2\times\rm GL_2$automorphic forms • Mathematics • 2019 Let f and g be holomorphic or Maass cusp forms for SL2(Z) with normalized Fourier coefficients λf (n) and λg(n), respectively. In this paper, we prove nontrivial estimates for the sum ∞ ## References SHOWING 1-10 OF 40 REFERENCES Subconvexity for$GL(3)\times GL(2)L$-functions in$t$-aspect • R. Munshi • Mathematics Journal of the European Mathematical Society • 2021 Let$\pi$be a Hecke-Maass cusp form for$SL(3,\mathbb Z)$and$f$be a holomorphic (or Maass) Hecke form for$SL(2,\mathbb{Z})$. In this paper we prove the following subconvex bound $$Non-linear additive twist of Fourier coefficients of GL(3) Maass forms • Mathematics • 2019 Let \lambda_{\pi}(1,n) be the Fourier coefficients of the Hecke-Maass cusp form \pi for SL(3,\mathbb{Z}). The aim of this article is to get a non trivial bound on the sum which is non-linear Bounds for GL3 L-functions in depth aspect • Mathematics Forum Mathematicum • 2018 Abstract Let f be a Hecke–Maass cusp form for SL 3 ⁢ ( ℤ ) {\mathrm{SL}_{3}(\mathbb{Z})} and χ a primitive Dirichlet character of prime power conductor 𝔮 = p κ {\mathfrak{q}=p^{\kappa}} , with p Hybrid subconvexity bounds for$$L \left( \frac{1}{2}, \hbox {Sym}^2 f \otimes g\right) $$L12,Sym2f⊗g • Mathematics • 2016 Fix an integer$$\kappa \geqslant 2$$κ⩾2. Let P be prime and let$$k> \kappa $$k>κ be an even integer. For f a holomorphic cusp form of weight k and full level and g a primitive holomorphic cusp form Algebraic twists of \mathrm{GL}_3\times \mathrm{GL}_2 L-functions. • Mathematics • 2019 We prove that the coefficients of a \mathrm{GL}_3\times \mathrm{GL}_2 Rankin--Selberg L-function do not correlate with a wide class of trace functions of small conductor modulo primes, On exponential sums involving the Ramanujan function AbstractLet τ(n) be the arithmetical function of Ramanujan, α any real number, and x≥2. The uniform estimate$$\mathop \Sigma \limits_{n \leqslant x} \tau (n)e(n\alpha ) \ll x^6 \log x$\$ is a
Automorphic distributions, L-functions, and Voronoi summation for GL(3)
• Mathematics
• 2004
This paper is third in a series of three, following "Summation Formulas, from Poisson and Voronoi to the Present" (math.NT/0304187) and "Distributions and Analytic Continuation of Dirichlet Series"
Automorphic Forms and L-Functions for the Group Gl(n, R)
• Mathematics
• 2006
Introduction 1. Discrete group actions 2. Invariant differential operators 3. Automorphic forms and L-functions for SL(2,Z) 4. Existence of Maass forms 5. Maass forms and Whittaker functions for
ON SUMS INVOLVING COEFFICIENTS OF AUTOMORPHIC L-FUNCTIONS
Let L(s, π) be the automorphic L-function associated to an automorphic irreducible cuspidal representation π of GL m over Q, and let α π (n) be the nth coefficient in its Dirichlet series expansion.
On the structure of the Selberg class, VI: non-linear twists
• Mathematics
• 2005
of functions F (s) of degree 1 in the extended Selberg class S ]. Precisely, denoting by qF and θF respectively the conductor and the shift of F (s) (see below for definitions) and writing nα = qFα,