# 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} }

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…

## 10 Citations

Algebraic twists of $\mathrm{GL}_3\times \mathrm{GL}_2$ $L$-functions.

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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,…

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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…

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Averages of coefficients of a class of degree 3 L-functions

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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…

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Subconvexity for $L$-Functions on $\mathrm{GL}_3$ over Number Fields.

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In this paper, over an arbitrary number field, we prove subconvexity bounds for self-dual $\mathrm{GL}_3$ $L$-functions in the $t$-aspect and for self-dual $\mathrm{GL}_3 \times \mathrm{GL}_2$…

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- Mathematics
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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

- MathematicsJournal 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

- MathematicsForum 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

- Mathematics
- 1987

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

- Mathematics
- 2009

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,…