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CERTAIN GENERALIZED MORDELL CURVES OVER THE RATIONAL NUMBERS ARE POINTLESS

Part of: Curves Arithmetic algebraic geometry Arithmetic problems. Diophantine geometry

Published online by Cambridge University Press:  28 October 2015

NGUYEN NGOC DONG QUAN
NGUYEN NGOC DONG QUAN*
Affiliation:
Department of Mathematics, The University of Texas at Austin, Austin, TX 78712, USA email [email protected]
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Abstract

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A conjecture of Scharaschkin and Skorobogatov states that there is a Brauer–Manin obstruction to the existence of rational points on a smooth geometrically irreducible curve over a number field. In this paper, we verify the Scharaschkin–Skorobogatov conjecture for explicit families of generalized Mordell curves. Our approach uses standard techniques from the Brauer–Manin obstruction and the arithmetic of certain threefolds.

Keywords

Brauer–Manin obstructionrational pointsgeneralized curves

MSC classification

Primary: 14G25: Global ground fields
Secondary: 11G35: Varieties over global fields 14G05: Rational points 14H45: Special curves and curves of low genus
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 100 , Issue 1 , February 2016 , pp. 42 - 64
Copyright
© 2015 Australian Mathematical Publishing Association Inc. 

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