Results 1  10
of
31
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 32 (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 unpubli ..."
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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 This is a review and synthesis of previously known material, both published and unpublished.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.
Finding Rational Points on Bielliptic Genus 2 Curves
"... We discuss a technique for trying to find all rational points on curves of the form Y 2 = f3X 6 + f2X 4 + f1X 2 + f0, where the sextic has nonzero discriminant. This is a bielliptic curve of genus 2. When the rank of the Jacobian is 0 or 1, Chabauty’s Theorem may be applied. However, we shall concen ..."
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Cited by 21 (7 self)
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We discuss a technique for trying to find all rational points on curves of the form Y 2 = f3X 6 + f2X 4 + f1X 2 + f0, where the sextic has nonzero discriminant. This is a bielliptic curve of genus 2. When the rank of the Jacobian is 0 or 1, Chabauty’s Theorem may be applied. However, we shall concentrate on the situation when the rank is at least 2. In this case, we shall derive an associated family of elliptic curves, defined over a number field Q(α). If each of these elliptic curves has rank less than the degree of Q(α) : Q, then we shall describe a Chabautylike technique which may be applied to try to find all the points (x, y) defined over Q(α) on the elliptic curves, for which x ∈ Q. This in turn allows us to find all Qrational points on the original genus 2 curve. We apply this to give a solution to a problem of Diophantus (where the sextic in X is irreducible over Q), which simplifies the recent solution of Wetherell. We also present two examples where the sextic in X is reducible over Q.
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...
The Hasse principle and the BrauerManin obstruction for curves
 Manuscripta Math
, 2004
"... Abstract. We discuss a range of ways, extending existing methods, to demonstrate violations of the Hasse principle on curves. Of particular interest are curves which contain a rational divisor class of degree 1, even though they contain no rational point. For such curves we construct new types of ex ..."
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Cited by 14 (1 self)
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Abstract. We discuss a range of ways, extending existing methods, to demonstrate violations of the Hasse principle on curves. Of particular interest are curves which contain a rational divisor class of degree 1, even though they contain no rational point. For such curves we construct new types of examples of violations of the Hasse principle which are due to the BrauerManin obstruction, subject to the conjecture that the TateShafarevich group of the Jacobian is finite. 1.
Towers of 2covers of hyperelliptic curves
"... Abstract. In this article, we give a way of constructing an unramified Galoiscover of a hyperelliptic curve. The geometric Galoisgroup is an elementary abelian 2group. The construction does not make use of the embedding of the curve in its Jacobian and it readily displays all subcovers. We show th ..."
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Cited by 12 (7 self)
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Abstract. In this article, we give a way of constructing an unramified Galoiscover of a hyperelliptic curve. The geometric Galoisgroup is an elementary abelian 2group. The construction does not make use of the embedding of the curve in its Jacobian and it readily displays all subcovers. We show that the cover we construct is isomorphic to the pullback along the multiplicationby2 map of an embedding of the curve in its Jacobian. We show that the constructed cover has an abundance of elliptic and hyperelliptic subcovers. This makes this cover especially suited for covering techniques employed for determining the rational points on curves. Especially the hyperelliptic subcovers give a chance for applying the method iteratively, thus creating towers of elementary abelian 2covers of hyperelliptic curves. As an application, we determine the rational points on the genus 2 curve arising from the question whether the sum of the first n fourth powers can ever be a square. For this curve, a simple covering step fails, but a second step succeeds. 1.
TWOCOVER DESCENT ON HYPERELLIPTIC CURVES
, 2009
"... We describe an algorithm that determines a set of unramified covers of a given hyperelliptic curve, with the property that any rational point will lift to one of the covers. In particular, if the algorithm returns an empty set, then the hyperelliptic curve has no rational points. This provides a re ..."
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Cited by 10 (6 self)
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We describe an algorithm that determines a set of unramified covers of a given hyperelliptic curve, with the property that any rational point will lift to one of the covers. In particular, if the algorithm returns an empty set, then the hyperelliptic curve has no rational points. This provides a relatively efficiently tested criterion for solvability ofhyperelliptic curves. We also discuss applications of this algorithm to curves ofgenus 1 and to curves with rational points.
On QDerived Polynomials
"... It is known that Qderived univariate polynomials (polynomials defined over Q, with the property that they and all their derivatives have all their roots in Q) can be completely classified subject to two conjectures: that no quartic with four distinct roots is Qderived, and that no quintic with a t ..."
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Cited by 8 (6 self)
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It is known that Qderived univariate polynomials (polynomials defined over Q, with the property that they and all their derivatives have all their roots in Q) can be completely classified subject to two conjectures: that no quartic with four distinct roots is Qderived, and that no quintic with a triple root and two other distinct roots is Qderived. We prove the second of these conjectures.
Covering Collections and a Challenge Problem of Serre
"... We answer a challenge of Serre by showing that every rational point on the projective curve X 4 + Y 4 = 17Z 4 is of the form (±1, ±2, 1) or (±2, ±1, 1). Our approach builds on recent ideas from both Nils Bruin and the authors on the application of covering collections and Chabauty arguments to curve ..."
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Cited by 8 (6 self)
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We answer a challenge of Serre by showing that every rational point on the projective curve X 4 + Y 4 = 17Z 4 is of the form (±1, ±2, 1) or (±2, ±1, 1). Our approach builds on recent ideas from both Nils Bruin and the authors on the application of covering collections and Chabauty arguments to curves of high rank. This is the only value of c ≤ 81 for which the Fermat quartic X 4 +Y 4 = cZ 4 cannot be solved trivially, either by local considerations or maps to elliptic curves of rank 0, and it seems likely that our approach should give a method of attack for other nontrivial values of c.
Application of covering techniques to families of curves
 J. Number Theory
"... Abstract. Much success in finding rational points on curves has been obtained by using Chabauty’s Theorem, which applies when the genus of a curve is greater than the rank of the MordellWeil group of the Jacobian. When Chabauty’s Theorem does not directly apply to a curve C, a recent modification h ..."
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Cited by 5 (2 self)
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Abstract. Much success in finding rational points on curves has been obtained by using Chabauty’s Theorem, which applies when the genus of a curve is greater than the rank of the MordellWeil group of the Jacobian. When Chabauty’s Theorem does not directly apply to a curve C, a recent modification has been to cover the rational points on C by those on a covering collection of curves Di, obtained by pullbacks along an isogeny to the Jacobian; one then hopes that Chabauty’s Theorem applies to each Di. So far, this latter technique has been applied to isolated examples. We apply, for the first time, certain covering techniques to infinite families of curves. We find an infinite family of curves to which Chabauty’s Theorem is not applicable, but which can be solved using bielliptic covers, and other infinite families of curves which even resist solution by bielliptic covers. A fringe benefit is an infinite family of Abelian surfaces with nontrivial elements of the TateShafarevich group killed by a bielliptic isogeny. 1.