# Bounds for automorphic L-functions

@inproceedings{Duke2005BoundsFA, title={Bounds for automorphic L-functions}, author={William Duke and John B. Friedlander and Henryk Iwaniec}, year={2005} }

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 applications following therefrom. The proof of Burgess appeals to the Riemann Hypothesis for curves established by Weil. Another method to break the convexity barrier was given recently in [F-I] . This method, as well as being more elementary, combines well with the methods developed in the series [D-I2] to…

## 140 Citations

### Shifted convolution sums and subconvexity bounds for automorphic L-functions

- Mathematics
- 2004

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…

### Subconvexity bounds for automorphic L-functions

- MathematicsJournal of the Institute of Mathematics of Jussieu
- 2009

Abstract We break the convexity bound in the t-aspect for L-functions attached to cusp forms f for GL2(k) over arbitrary number fields k. The argument uses asymptotics with error term with a power…

### Subconvexity bounds in depth-aspect for automorphic L-functions on GL(2)

- Mathematics
- 2009

From a spectral identity we obtain asymptotics with error term for the second integral moments of families of automorphic L-functions for GL(2) over an arbitrary number field according to twists by…

### Subconvex Bounds for Automorphic L-functions and Applications

- Mathematics
- 2011

This work presents subconvex bounds in the q-aspect for automorphic L-functions of GL2×GL1, GL2, GL2×GL2 type over Q and some of their consequences. The results were published earlier in [BlHa08a,…

### DEPENDENCY ON THE GROUP IN AUTOMORPHIC SOBOLEV INEQUALITIES

- Mathematics
- 2008

In [1] and [2], Bernstein and Reznikov have introduced a new way of estimating the coefficients in the spectral expansion of φ2, where φ is a Maass cusp of norm 1 on a quotient Y = Γ\H of the…

### 3 0 Se p 20 03 Riemann ’ s Zeta Function and Beyond

- Mathematics
- 2004

In recent years L-functions and their analytic properties have assumed a central role in number theory and automorphic forms. In this expository article, we describe the two major methods for proving…

### New bounds for automorphic L-functions

- Mathematics
- 2003

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…

### UNIFORM SUBCONVEXITY BOUNDS FOR GL(3)×GL(2) L-FUNCTIONS

- Mathematics
- 2021

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…

### Twisted Hilbert modular L-functions and spectral theory

- Mathematics
- 2014

These are notes for four lectures given at the 2010 CIMPA Research School "Automorphic Forms and L-functions" in Weihai, China. The lectures focused on a Burgess-like subconvex bound for twisted…

### Equidistribution on the modular surface and L-functions

- Mathematics
- 2011

Abstract. These are notes for two lectures given at the 2007 summer school “Homogeneous Flows, Moduli Spaces and Arithmetic” in Pisa, Italy. The first lecture introduces Heegner points and closed…

## References

SHOWING 1-9 OF 9 REFERENCES

### Hyperbolic distribution problems and half-integral weight Maass forms

- Mathematics
- 1988

(Actually n ~ is replaced by d(n)log ~ 2n where d(n) is the divisor function.) A striking application of(1.2) is to give the uniform distribution of certain lattice points in Z 3 on a sphere centered…

### Estimates for coefficients of L -functions. III

- Mathematics
- 1994

In this sequence of papers we investigate Dirichlet series
$${\rm A}(s,X) = \sum\limits_1^\infty {{a_n}} X(n){n^{ - 3}}$$
(1)
having Euler products and compatible functional equations with…

### Introduction to the arithmetic theory of automorphic functions

- Mathematics
- 1971

* uschian groups of the first kind * Automorphic forms and functions * Hecke operators and the zeta-functions associated with modular forms * Elliptic curves * Abelian extensions of imaginary…

### Representation of integers by positive ternary quadratic forms and equidistribution of lattice points on ellipsoids

- Mathematics
- 1990

It is a classical problem to find an asymptotic formula for the number of integral points in a region on the ellipsoid q(x l . . . . . x r ) = n as n--+ oo where q is a positive definite integral…

### Some Applications of Modular Forms

- Mathematics
- 1990

Introduction 1. Modular forms 2. Invariant means on L (Sn) 3. Ramanujan graphs 4. Bounds for Fourier coefficients of 1/2-integral weight Bibliogrpahy Index.

### Bilinear forms in the Fourier coefficients of half-integral weight cusp forms and sums over primes

- Mathematics
- 1990