Algebra & Number Theory

Integral points on hyperelliptic curves

Yann Bugeaud, Maurice Mignotte, Samir Siksek, Michael Stoll, and Szabolcs Tengely

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Let C:Y2=anXn++a0 be a hyperelliptic curve with the ai rational integers, n5, and the polynomial on the right-hand side irreducible. Let J be its Jacobian. We give a completely explicit upper bound for the integral points on the model C, provided we know at least one rational point on C and a Mordell–Weil basis for J(). We also explain a powerful refinement of the Mordell–Weil sieve which, combined with the upper bound, is capable of determining all the integral points. Our method is illustrated by determining the integral points on the genus 2 hyperelliptic models Y2Y=X5X and Y2=X5.

Article information

Algebra Number Theory, Volume 2, Number 8 (2008), 859-885.

Received: 28 January 2008
Revised: 2 September 2008
Accepted: 12 September 2008
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11G30: Curves of arbitrary genus or genus = 1 over global fields [See also 14H25]
Secondary: 11J86: Linear forms in logarithms; Baker's method

curve integral point Jacobian height Mordell–Weil group Baker's bound Mordell–Weil sieve


Bugeaud, Yann; Mignotte, Maurice; Siksek, Samir; Stoll, Michael; Tengely, Szabolcs. Integral points on hyperelliptic curves. Algebra Number Theory 2 (2008), no. 8, 859--885. doi:10.2140/ant.2008.2.859.

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