A Burgess-like subconvex bound for twisted L-functions

  title={A Burgess-like subconvex bound for twisted L-functions},
  author={Valentin Blomer and Gergely Harcos and Philippe Michel and Zhouhang Mao},
Abstract Let g be a cuspidal newform (holomorphic or Maass) of arbitrary level and nebentypus, χ a primitive character of conductor q, and s a point on the critical line ℜs = ½. It is proved that , where ε > 0 is arbitrary and θ = is the current known approximation towards the Ramanujan–Petersson conjecture (which would allow θ = 0); moreover, the dependence on s and all the parameters of g is polynomial. This result is an analog of Burgess' classical subconvex bound for Dirichlet L-functions… 

Level aspect subconvexity for twisted L-functions

Hybrid bounds for twisted L-functions

Abstract The aim of this paper is to derive bounds on the critical line ℜs = 1/2 for L-functions attached to twists f ⊗ χ of a primitive cusp form f of level N and a primitive character modulo q that

Rankin-Selberg L-functions on the critical line

Let f and g be two primitive (holomorphic or Maass) cusp forms of arbitrary level, character and infinity parameter by which we mean the weight in the holomorphic case and the spectral parameter in


Let F be a number field, π an irreducible cuspidal representation of GL2(AF ) with unitary central character, and χ a Hecke character of analytic conductor Q. Then L(1/2, π⊗χ) Q 2 − 8 (1−2θ)+ , where

The Second Moment of Rankin-Selberg L-functions, Hybrid Subconvexity Bounds, and Related Topics

In this thesis, we study three problems related to subconvexity bounds of RankinSelberg L-functions. Let M,N be two coprime square-free integers. Let f be either a holomorphic or a Maaß cusp form of

On an analogue of Titchmarsh’s divisor problem for holomorphic cusp forms

the estimation of (1.1) for ξ = 1 is related to the location of the zeros of ζ, which is the classical approach to the prime number theorem, and for multiplicative ξ in general the problem can be

Non-split sums of coefficients of GL(2)-automorphic forms

Given a cuspidal automorphic form π on GL2, we study smoothed sums of the form $$\sum\nolimits_n {{a_\pi }({n^2} + d)V({n \over x})} $$. The error term we get is sharp in that it is uniform in both d

Weyl-type bounds for twisted $GL(2)$ short character sums

Let f be a Hecke-Maass or holomorphic primitive cusp form for SL(2,Z) with normalized Fourier coefficients λf (n). Let χ be a primitive Dirichlet character of modulus p, where p is a prime number. In

A generalized cubic moment and the Petersson formula for newforms

Using a cubic moment, we prove a Weyl-type subconvexity bound for the quadratic twists of a holomorphic newform of square-free level, trivial nebentypus, and arbitrary even weight. This generalizes



Bounds for automorphic L–functions. III

We continue our study of GL2 L–functions with the aim of providing upper bounds for their order of magnitude. As is familiar it suffices to provide such bounds on the critical line and, both for the

New bounds for automorphic L-functions

This dissertation contributes to the analytic theory of automorphic L-functions. We prove an approximate functional equation for the central value of the L-series attached to an irreducible

Bounds for automorphic L-functions

on the line Re s = 2 x, the implied constant depending on s. This classical estimate resisted improvement for many years until Burgess I-B] reduced the exponent from 88 to ~ , many important

Shifted convolution sums and subconvexity bounds for automorphic L-functions

The behavior of L-functions in the critical strip has received a lot of attention from the first proof of the prime number theorem up to now. In fact, the deeper arithmetic information of the

The subconvexity problem for Rankin–Selberg L-functions and equidistribution of Heegner points. II

We prove a general subconvex bound in the level aspect for Rankin–Selberg L-functions associated with two primitive holomorphic or Maass cusp forms over Q. We use this bound to establish the

Fourier coefficients of modular forms of half-integral weight

It is known that if the Fourier coefficients a(n)(n ≥ 1) of an elliptic modular form of even integral weight k ≥ 2 on the Hecke congruence subgroup Γ0(N)(N ∈ N) satisfy the bound a(n) ≪f nc for all n

An additive problem in the Fourier coefficients of cusp forms

Abstract. We establish an estimate on sums of shifted products of Fourier coefficients coming from holomorphic or Maass cusp forms of arbitrary level and nebentypus. These sums are analogous to the

Hybrid bounds for Dirichlet's L-function

  • M. HuxleyN. Watt
  • Mathematics
    Mathematical Proceedings of the Cambridge Philosophical Society
  • 2000
This is a paper about upper bounds for Dirichlet's L-function, L(s, χ), on its critical line (s + s¯ = 1). It is to be assumed throughout that, unless otherwise stated, the Dirichlet character, χ, is


Let ∧ : GLn(C) −→ GLN (C), where N = n(n−1) 2 , be the map given by the exterior square. Then Langlands’ functoriality predicts that there is a map from cuspidal representations of GLn to automorphic

Cuspidality of symmetric powers with applications

The purpose of this paper is to prove that the symmetric fourth power of a cusp form on GL(2), whose existence was proved earlier by the first author, is cuspidal unless the corresponding automorphic