Results 1  10
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37
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 34 (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 This is a review and synthesis of previously known material, both published and unpublished.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.
The MordellWeil sieve: Proving nonexistence of rational points on curves
"... Abstract. We discuss the MordellWeil sieve as a general technique for proving results concerning rational points on a given curve. In the special case of curves of genus 2, we describe quite explicitly how the relevant local information can be obtained if one does not want to restrict to mod p info ..."
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Cited by 15 (11 self)
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Abstract. We discuss the MordellWeil sieve as a general technique for proving results concerning rational points on a given curve. In the special case of curves of genus 2, we describe quite explicitly how the relevant local information can be obtained if one does not want to restrict to mod p information at primes of good reduction. We describe our implementation of the MordellWeil sieve algorithm and discuss its efficiency. 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...
TWISTS OF X(7) AND PRIMITIVE SOLUTIONS TO x 2 + y 3 = z 7
"... Abstract. We find the primitive integer solutions to x 2 + y 3 = z 7. A nonabelian descent argument involving the simple group of order 168 reduces the problem to the determination of the set of rational points on a finite set of twists of the Klein quartic curve X. To restrict the set of relevant t ..."
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Cited by 14 (9 self)
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Abstract. We find the primitive integer solutions to x 2 + y 3 = z 7. A nonabelian descent argument involving the simple group of order 168 reduces the problem to the determination of the set of rational points on a finite set of twists of the Klein quartic curve X. To restrict the set of relevant twists, we exploit the isomorphism between X and the modular curve X(7), and use modularity of elliptic curves and level lowering. This leaves 10 genus3 curves, whose rational points are found by a combination of methods. 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.
Rational 6cycles under iteration of quadratic polynomials
 London Math. Soc. J. Comput. Math
"... Abstract. We present a proof, which is conditional on the Birch and SwinnertonDyer Conjecture for a specific abelian variety, that there do not exist rational numbers x and c such that x has exact period N = 6 under the iteration x ↦ → x2 + c. This extends earlier results by Morton for N = 4 and Fl ..."
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Cited by 9 (3 self)
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Abstract. We present a proof, which is conditional on the Birch and SwinnertonDyer Conjecture for a specific abelian variety, that there do not exist rational numbers x and c such that x has exact period N = 6 under the iteration x ↦ → x2 + c. This extends earlier results by Morton for N = 4 and Flynn, Poonen and Schaefer for N = 5. 1.
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.