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2015•Journal of the Australian Mathematical SocietyOpen access

CERTAIN GENERALIZED MORDELL CURVES OVER THE RATIONAL NUMBERS ARE POINTLESS

Nguyen Ngoc Dong Quan

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Abstract

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.

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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.

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Available abstract

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.

Key concepts: Mathematics, Conjecture, Pure mathematics, Algebraic number field, Field (mathematics), Rational number, Algebra over a field

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CERTAIN GENERALIZED MORDELL CURVES OVER THE RATIONAL NUMBERS ARE POINTLESS — Research Paper | ScholarLens