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PUBLISHED BY mathematical sciences publishers http://www.mathscipub.org A NON-PROFIT CORPORATION Typeset in LATEX Copyright c 2010 by Mathematical Sciences Publishers ALGEBRA AND NUMBER THEORY 2:8(2008) Integral points on hyperelliptic curves Yann Bugeaud, Maurice Mignotte, Samir Siksek, Michael Stoll and Szabolcs Tengely 2 n Let C V Y D an X C···C a0 be a hyperelliptic curve with the ai rational inte- gers, n ≥ 5, 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.Q/. 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 2 − D 5 − Y D X the genus 2 hyperelliptic models Y Y X X and 2 5 . 1. Introduction Consider the hyperelliptic curve with affine model 2 n n−1 C V Y D an X C an−1 X C···C a0; (1-1) with a0;:::; an rational integers, an 6D 0, n ≥ 5, and the polynomial on the right- hand side irreducible. Let H D maxfja0j;:::; janjg. In one of the earliest appli- cations of his theory of lower bounds for linear forms in logarithms, Baker [1969] showed that any integral point .X; Y / on this affine model satisfies 2 max.jXj; jY j/ ≤ exp exp exp.n10n H/n : Such bounds have been improved considerably by many authors, including Sprin- dzukˇ [1977], Brindza [1984], Schmidt [1992], Poulakis [1991], Bilu [1995], Bu- geaud [1997] and Voutier [1995]. Despite the improvements, the bounds remain astronomical and often involve inexplicit constants. In this paper we explain a new method for explicitly computing the integral points on affine models of hyperelliptic curves (1-1). The method falls into two distinct steps: MSC2000: primary 11G30; secondary 11J86. Keywords: curve, integral point, Jacobian, height, Mordell–Weil group, Baker’s bound, Mordell–Weil sieve. 859 860 Y. Bugeaud, M. Mignotte, S. Siksek, M. Stoll and Sz. Tengely (i) We give a completely explicit upper bound for the size of integral solutions of (1-1). This upper bound combines many refinements found in the papers of Voutier, Bugeaud, and others, together with Matveev’s bounds [2000] for linear forms in logarithms, and a method for bounding the regulators based on a theorem of Landau [1918]. (ii) The bounds obtained in (i), whilst substantially better than bounds given by earlier authors, are still astronomical. We explain a powerful variant of the Mordell–Weil sieve which, combined with the bound obtained in (i), is capa- ble of showing that the known solutions to (1-1) are the only ones. Step (i) requires two assumptions: (a) We assume that we know at least one rational point P0 on C. (b) Let J be the Jacobian of C. We assume that a Mordell–Weil basis for J.Q/ is known. For step (ii) we need assumptions (a), (b) and also: (c) We assume that the canonical height hO V J.Q/ ! R is explicitly computable and that we have explicit bounds for the difference O 0 µ1 ≤ h.D/ − h.D/ ≤ µ1 (1-2) where h is an appropriately normalized logarithmic height on J that allows us to enumerate points P in J.Q/ with h.P/ ≤ B for a given bound B. Assumptions (a)–(c) deserve a comment or two. For many families of curves of higher genus, practical descent strategies are available for estimating the rank of the Mordell–Weil group; see for example [Cassels and Flynn 1996; Poonen and Schaefer 1997; Schaefer 1995; Stoll 2001]. To provably determine the Mordell– Weil group one however needs bounds for the difference between the logarithmic and canonical heights. For Jacobians of curves of genus 2 such bounds have been determined by Stoll [1999; 2002], building on previous work of Flynn and Smart [1997]. At present, no such bounds have been determined for Jacobians of curves of genus ≥3, though work on this is in progress. The assumption about the knowledge of a rational point is a common sense assumption that brings some simplifications to our method, though the method can be modified to cope with the situation where no rational point is known. However, if a search on a curve of genus ≥ 2 reveals no rational points, it is probable that there are none, and the methods of Bruin and Stoll [2008a; 2008b; ≥ 2008] are likely to succeed in proving this. We illustrate the practicality of our approach by proving: Theorem 1.1. The only integral solutions to the equation Y 2 − Y D X 5 − X (1-3) Integral points on hyperelliptic curves 861 are .X; Y / D .−1; 0/; .−1; 1/; .0; 0/; .0; 1/; .1; 0/; .1; 1/; .2; −5/; .2; 6/; .3; −15/; .3; 16/; .30; −4929/; .30; 4930/: Theorem 1.2. The only integral solutions to the equation Y X D (1-4) 2 5 are .X; Y / D .0; 0/; .0; 1/; .1; 0/; .1; 1/; .2; 0/; .2; 1/; .3; 0/; .3; 1/; .4; 0/; .4; 1/; .5; −1/; .5; 2/; .6; −3/; .6; 4/; .7; −6/; .7; 7/; .15; −77/; .15; 78/; .19; −152/; .19; 153/: Equations (1-3) and (1-4) are of historical interest and Section 2 gives a brief outline of their history. For now we merely mention that these two equations are the first two problems on a list of 22 unsolved Diophantine problems, compiled by Evertse and Tijdeman [2007] following a recent workshop on Diophantine equations at Leiden. To appreciate why the innocent-looking (1-3) and (1-4) have resisted previous attempts, let us briefly survey the available methods which apply to hyperelliptic curves and then briefly explain why they fail in these cases. To determine the inte- gral points on the affine model C given by (1-1) there are four available methods: (I) The first is Chabauty’s elegant method which in fact determines all rational points on C in many cases, provided the rank of the Mordell–Weil group of its Jacobian is strictly less than the genus g; see for example [Flynn 1997; Wetherell 1997]. Chabauty’s method fails if the rank of the Mordell–Weil group exceeds the genus. (II) The second method is to use coverings, often combined with a version of Chabauty called Elliptic Curve Chabauty. See [Bruin 1999; 2003; Flynn and Wetherell 1999; 2001]. This approach often requires computations of Mordell–Weil groups over number fields (and does fail if the rank of the Mordell–Weil groups is too large). (III) The third method is to combine Baker’s approach through S-units with the LLL algorithm to obtain all the solutions provided that certain relevant unit groups and class groups can be computed; for a modern treatment, see [Bilu and Hanrot 1998] or [Smart 1998, Section XIV.4]. This strategy often fails in practice as the number fields involved have very high degree. 862 Y. Bugeaud, M. Mignotte, S. Siksek, M. Stoll and Sz. Tengely (IV) The fourth approach is to apply Skolem’s method to the S-unit equations (see [Smart 1998, Section III.2]).