Functiones et Approximatio Commentarii Mathematici

A height inequality for rational points on elliptic curves implied by the abc-conjecture

Ulf Kühn and Jan Steffen Müller

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In this short note we show that the uniform $abc$-conjecture puts strong restrictions on the coordinates of rational points on elliptic curves. For the proof we use a variant of Vojta's height inequality formulated by Mochizuki. As an application, we generalize a result of Silverman on elliptic non-Wieferich primes.

Article information

Funct. Approx. Comment. Math., Volume 52, Number 1 (2015), 127-132.

First available in Project Euclid: 20 March 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11G05: Elliptic curves over global fields [See also 14H52]
Secondary: 11G50: Heights [See also 14G40, 37P30]

elliptic curves abc-conjecture rational points heights Wieferich primes


Kühn, Ulf; Müller, Jan Steffen. A height inequality for rational points on elliptic curves implied by the abc-conjecture. Funct. Approx. Comment. Math. 52 (2015), no. 1, 127--132. doi:10.7169/facm/2015.52.1.10.

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