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Integer points on algebraic curves with exceptional units

Part of: Diophantine equations Arithmetic algebraic geometry

Published online by Cambridge University Press:  09 April 2009

Dimitrios Poulakis
Dimitrios Poulakis
Affiliation:
Department of Mathematics Aristotle University of Thessaloniki54006 Thessaloniki, Greece
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Abstract

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Let F(X, Y) be an absolutely irreducible polynomial with coefficients in an algebraic number field K. Denote by C the algebraic curve defined by the equation F(X, Y) = 0 and by K[C] the ring of regular functions on Cover K. Assume that there is a unit ϕ in K[C] − K such that 1 − ϕ is also a unit. Then we establish an explicit upper bound for the size of integral solutions of the equation F(X, Y) = 0, defined over K. Using this result we establish improved explicit upper bounds on the size of integral solutions to the equations defining non-singular affine curves of genus zero, with at least three points at ‘infinity’, the elliptic equations and a class of equations containing the Thue curves.

MSC classification

Secondary: 11D41: Higher degree equations; Fermat's equation 11G30: Curves of arbitrary genus or genus $ne 1$ over global fields
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 63 , Issue 2 , October 1997 , pp. 145 - 164
Copyright
Copyright © Australian Mathematical Society 1997

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