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11
Cycles of quadratic polynomials and rational points on a genus 2 curve
, 1996
"... Abstract. It has been conjectured that for N sufficiently large, there are no quadratic polynomials in Q[z] with rational periodic points of period N. Morton proved there were none with N = 4, by showing that the genus 2 algebraic curve that classifies periodic points of period 4 is birational to X1 ..."
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Cited by 36 (13 self)
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Abstract. It has been conjectured that for N sufficiently large, there are no quadratic polynomials in Q[z] with rational periodic points of period N. Morton proved there were none with N = 4, by showing that the genus 2 algebraic curve that classifies periodic points of period 4 is birational to X1(16), whose rational points had been previously computed. We prove there are none with N = 5. Here the relevant curve has genus 14, but it has a genus 2 quotient, whose rational points we compute by performing a 2descent on its Jacobian and applying a refinement of the method of Chabauty and Coleman. We hope that our computation will serve as a model for others who need to compute rational points on hyperelliptic curves. We also describe the three possible Gal(Q/Q)stable 5cycles, and show that there exist Gal(Q/Q)stable Ncycles for infinitely many N. Furthermore, we answer a question of Morton by showing that the genus 14 curve and its quotient are not modular. Finally, we mention some partial results for N = 6. 1.
Bounding the Number of Rational Points on Certain Curves of High Rank
, 1997
"... Let K be a number eld and let C be a curve of genus g > 1 dened over K. In this dissertation we describe techniques for bounding the number of Krational points on C. In Chapter I we discuss Chabauty techniques. This is a review and synthesis of previously known material, both published and unpu ..."
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Cited by 25 (2 self)
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Let K be a number eld and let C be a curve of genus g > 1 dened over K. In this dissertation we describe techniques for bounding the number of Krational points on C. In Chapter I we discuss Chabauty techniques. This is a review and synthesis of previously known material, both published and unpublished. We have tried to eliminate unnecessary restrictions, such as assumptions of good reduction or the existence of a known rational point on the curve. We have also attempted to clearly state the circumstances under which Chabauty techniques can be applied. Our primary goal is to provide a exible and powerful tool for computing on specic curves. In Chapter II we develop a technique which, given a Krational isogeny to the Jacobian of C, produces a positive integer n and a collection of covers of C with the property that the set of Krational points in the collection is in nto1 correspondence with the set of Krational points on C. If Chabauty is applicable to every curve in the collection, then we can use the covers to bound the number of Krational points on C. The examples in Chapters I and II are taken from problem VI.17 in the Arabic text of the Arithmetica. Chapter III is devoted to the background calculations for this problem. When we assemble the pieces, we discover that the solution given by Diophantus is the only positive rational solution to this problem. Contents 1. Preface 4 Chapter 1. Chabauty bounds 5 1.
Computational Aspects of Curves of Genus at Least 2
 Algorithmic number theory. 5th international symposium. ANTSII
, 1996
"... . This survey discusses algorithms and explicit calculations for curves of genus at least 2 and their Jacobians, mainly over number fields and finite fields. Miscellaneous examples and a list of possible future projects are given at the end. 1. Introduction An enormous number of people have per ..."
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Cited by 14 (3 self)
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. This survey discusses algorithms and explicit calculations for curves of genus at least 2 and their Jacobians, mainly over number fields and finite fields. Miscellaneous examples and a list of possible future projects are given at the end. 1. Introduction An enormous number of people have performed an enormous number of computations on elliptic curves, as one can see from even a perfunctory glance at [29]. A few years ago, the same could not be said for curves of higher genus, even though the theory of such curves had been developed in detail. Now, however, polynomialtime algorithms and sometimes actual programs are available for solving a wide variety of problems associated with such curves. The genus 2 case especially is becoming accessible: in light of recent work, it seems reasonable to expect that within a few years, packages will be available for doing genus 2 computations analogous to the elliptic curve computations that are currently possible in PARI, MAGMA, SIMATH, apec...
Canonical heights on the Jacobians of curves of genus 2 and the infinite descent
 Acta Arith
, 1997
"... Abstract. We give an algorithm to compute the canonical height on a Jacobian of a curve of genus 2. The computations involve only working with the Kummer surface and so lengthy computations with divisors in the Jacobian are avoided. We use this height algorithm to give an algorithm to perform the “i ..."
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Cited by 11 (3 self)
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Abstract. We give an algorithm to compute the canonical height on a Jacobian of a curve of genus 2. The computations involve only working with the Kummer surface and so lengthy computations with divisors in the Jacobian are avoided. We use this height algorithm to give an algorithm to perform the “infinite descent ” stage of computing the MordellWeil group. This last stage is performed by a lattice enlarging procedure. 1.
When Newton met Diophantus: A study of rationalderived polynomials and their extension to quadratic fields
, 1999
"... We consider the problem of classifying all univariate polynomials, defined over a domain k, with the property that they and all their derivatives have all their roots in k. This leads to a number of interesting subproblems such as finding krational points on a curve of genus 1 and rational points ..."
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Cited by 4 (0 self)
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We consider the problem of classifying all univariate polynomials, defined over a domain k, with the property that they and all their derivatives have all their roots in k. This leads to a number of interesting subproblems such as finding krational points on a curve of genus 1 and rational points on a curve of genus 2. Keywords : polynomial, derivative, diophantine, elliptic, jacobian. 1
Integral points on punctured abelian surfaces. in Algorithmic number theory
 Lecture Notes in Comput. Sci. 2369
, 2002
"... Abstract. We study the density of integral points on punctured abelian surfaces. Linear growth rates are observed experimentally. 1 ..."
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Cited by 2 (2 self)
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Abstract. We study the density of integral points on punctured abelian surfaces. Linear growth rates are observed experimentally. 1
Nontrivial X in the Jacobian of an infinite family of curves of genus 2
 JOURNAL DE THÉORIE DES NOMBRES DE BORDEAUX
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Solving Diophantine Problems on Curves via Descent on the Jacobian
"... The theory of Jacobians of curves has largely been developed in a vacuum, with little computational counterpart to the abstract theory. A recent development has been the explicit construction of Jacobians & formal groups, and workable methods of descent [6],[7] to find the rank. We suggest that ..."
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The theory of Jacobians of curves has largely been developed in a vacuum, with little computational counterpart to the abstract theory. A recent development has been the explicit construction of Jacobians & formal groups, and workable methods of descent [6],[7] to find the rank. We suggest that the following plan will provide a powerful tool for