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Computing integral points on genus 2 curves estimating hyperelliptic logarithms

Volume 187 / 2019

Homero R. Gallegos-Ruiz Acta Arithmetica 187 (2019), 329-344 MSC: Primary 11G30; Secondary 11Y50. DOI: 10.4064/aa170315-16-4 Published online: 2 January 2019


Let $C:y^2 = f(x)$ be a hyperelliptic curve with $f(x) \in \mathbb Q[x]$ monic and irreducible over $\mathbb Q$ of degree $n = 5$. Let $J$ be its Jacobian. We give an algorithm to explicitly determine the set of integral points on $C$ provided we know at least one rational point on $C$, a Mordell–Weil basis for $J(\mathbb Q)$ and an upper bound for the height of the integral points. We estimate the hyperelliptic logarithms of the elements on the Mordell–Weil basis to reduce the bound for the height of the integral points to manageable proportions using linear forms. We then find all the integral points on $C$ by a direct search. We illustrate the method by finding all integral points on the rank $5$, genus $2$ curve $y^2 =x^5 + 105x^4 + 4405x^3 + 92295x^2 + 965794x + 4038280$.


  • Homero R. Gallegos-RuizCONACyT fellow
    Unidad Académica de Matemáticas Universidad Autónoma de Zacatecas
    Calzada Solidaridad y Paseo de la Bufa
    Zacatecas, Zacatecas, CP 98000 Mexico

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