Quadratic polynomials represented by norm forms

Tim D Browning; DR Heath-Brown

doi:10.1007/s00039-012-0168-5

Quadratic polynomials represented by norm forms

Tim D Browning, DR Heath-Brown

Research output: Contribution to journal › Article (Academic Journal) › peer-review

17 Citations (Scopus)

Abstract

Let P(t)∈Q[t] be an irreducible quadratic polynomial and suppose that K is a quartic extension of Q containing the roots of P(t). Let N K/Q (x) be a full norm form for the extension K/Q . We show that the variety
P(t)=N K/Q (x)≠0
satisfies the Hasse principle and weak approximation. The proof uses analytic methods.

Original language	English
Pages (from-to)	1124-1190
Number of pages	67
Journal	Geometric and Functional Analysis
Volume	22
Issue number	5
DOIs	https://doi.org/10.1007/s00039-012-0168-5
Publication status	Published - Oct 2012

Keywords

14G05 (11D57, 14G25)

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10.1007/s00039-012-0168-5

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DIOPHANTINE GEOMETRY VIA ANALYTIC NUMBER THEORY
Browning, T. D. (Principal Investigator)
1/09/07 → 1/04/13
Project: Research

Cite this

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title = "Quadratic polynomials represented by norm forms",

abstract = "Let P(t)∈Q[t] be an irreducible quadratic polynomial and suppose that K is a quartic extension of Q containing the roots of P(t). Let N K/Q (x) be a full norm form for the extension K/Q . We show that the variety P(t)=N K/Q (x)≠0 satisfies the Hasse principle and weak approximation. The proof uses analytic methods.",

keywords = "14G05 (11D57, 14G25)",

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T1 - Quadratic polynomials represented by norm forms

AU - Browning, Tim D

AU - Heath-Brown, DR

PY - 2012/10

Y1 - 2012/10

N2 - Let P(t)∈Q[t] be an irreducible quadratic polynomial and suppose that K is a quartic extension of Q containing the roots of P(t). Let N K/Q (x) be a full norm form for the extension K/Q . We show that the variety P(t)=N K/Q (x)≠0 satisfies the Hasse principle and weak approximation. The proof uses analytic methods.

AB - Let P(t)∈Q[t] be an irreducible quadratic polynomial and suppose that K is a quartic extension of Q containing the roots of P(t). Let N K/Q (x) be a full norm form for the extension K/Q . We show that the variety P(t)=N K/Q (x)≠0 satisfies the Hasse principle and weak approximation. The proof uses analytic methods.

KW - 14G05 (11D57, 14G25)

U2 - 10.1007/s00039-012-0168-5

DO - 10.1007/s00039-012-0168-5

M3 - Article (Academic Journal)

SN - 1016-443X

VL - 22

SP - 1124

EP - 1190

JO - Geometric and Functional Analysis

JF - Geometric and Functional Analysis

IS - 5

ER -

Quadratic polynomials represented by norm forms

Abstract

Keywords

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DIOPHANTINE GEOMETRY VIA ANALYTIC NUMBER THEORY

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