Some quartic curves with no points in any cubic field

Andrew Bremner

doi:10.1112/plms/s3-52.2.193

Some quartic curves with no points in any cubic field

Andrew Bremner

Mathematical and Statistical Sciences, School of (SoMSS)

Research output: Contribution to journal › Article › peer-review

14 Scopus citations

Abstract

A sufficient condition is given such that the curves Γ_D: x⁴ + y⁴ = Dz⁴ have no points in any odd-degree (greater than 1) extension field of the rationals. The condition is in terms of the rational rank of an elliptic curve in the Jacobian of Γ_D. Various examples are given.

Original language	English (US)
Pages (from-to)	193-214
Number of pages	22
Journal	Proceedings of the London Mathematical Society
Volume	s3-52
Issue number	2
DOIs	https://doi.org/10.1112/plms/s3-52.2.193
State	Published - Mar 1986

ASJC Scopus subject areas

General Mathematics

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10.1112/plms/s3-52.2.193

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title = "Some quartic curves with no points in any cubic field",

abstract = "A sufficient condition is given such that the curves ΓD: x4 + y4 = Dz4 have no points in any odd-degree (greater than 1) extension field of the rationals. The condition is in terms of the rational rank of an elliptic curve in the Jacobian of ΓD. Various examples are given.",

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AB - A sufficient condition is given such that the curves ΓD: x4 + y4 = Dz4 have no points in any odd-degree (greater than 1) extension field of the rationals. The condition is in terms of the rational rank of an elliptic curve in the Jacobian of ΓD. Various examples are given.

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Some quartic curves with no points in any cubic field

Abstract

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