A quadratic large sieve inequality over number fields

Benoît Louvel,Leo Goldmakher

doi:10.1017/s0305004112000370

Abstract

AbstractWe formulate and prove a large sieve inequality for quadratic characters over a number field. To do this, we introduce the notion of an n-th order Hecke family. We develop the basic theory of these Hecke families, including versions of the Poisson summation formula.

Highlights

In [HB], Heath–Brown proved a large sieve inequality for quadratic characters: λb a bM
We introduce the notion of an n-th order Hecke family
We develop the basic theory of these Hecke families, including versions of the Poisson summation formula

Summary

Introduction

In [HB], Heath–Brown proved a large sieve inequality for quadratic characters: λb a b. (λb) is any sequence of complex numbers, ε > 0, M, N 1, (·/·) is the Jacobi symbol, and the sums are restricted to odd squarefree integers This bound has proved to be extremely useful in applications, and one might wish to generalize it. Property (3) generalizes the following property of the Jacobi symbol: if a and b are positive odd coprime integers, (ab/·) is a Dirichlet character modulo ab if a ≡ b (mod 4) This property will play an essential role in our argument; see (5·2) and Section 6·3. An example of a quadratic Hecke family (the motivating example) was constructed by Fisher and Friedberg in [FF] – see Section 2 for a brief description of their work Their construction is quite natural, and can be readily extended to produce Hecke families of any order; Remark 3 indicates how to modify their construction to produce other Hecke families. This proposition is proved in the final two sections of the paper

The Fisher–Friedberg Hecke family

A Poisson summation formula over number fields

An explicit formula for the norm B3