Results 1 
9 of
9
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 ..."
Abstract

Cited by 32 (13 self)
 Add to MetaCart
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.
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 ..."
Abstract

Cited by 14 (3 self)
 Add to MetaCart
. 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 ..."
Abstract

Cited by 8 (3 self)
 Add to MetaCart
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.
Computing torsion points on curves
 Experimental Math
"... Let X be a curve of genus g> 2 over a field k of characteristic 1. Introduction zero Let X ^ ^ A be an Albanese map associated to a point Po 2. Notation on X. The ManinMumford conjecture, first proved by Raynaud, ..."
Abstract

Cited by 6 (0 self)
 Add to MetaCart
Let X be a curve of genus g> 2 over a field k of characteristic 1. Introduction zero Let X ^ ^ A be an Albanese map associated to a point Po 2. Notation on X. The ManinMumford conjecture, first proved by Raynaud,
Descent via Isogeny in Dimension 2
"... A technique of descent via 4isogeny is developed on the Jacobian of a curve of genus 2 of the form: Y 2 = q1(X)q2(X)q3(X), where each qi(X) is a quadratic defined over Q. The technique offers a realistic prospect of calculating rank tables of MordellWeil groups in higher dimension. A selection of ..."
Abstract
 Add to MetaCart
A technique of descent via 4isogeny is developed on the Jacobian of a curve of genus 2 of the form: Y 2 = q1(X)q2(X)q3(X), where each qi(X) is a quadratic defined over Q. The technique offers a realistic prospect of calculating rank tables of MordellWeil groups in higher dimension. A selection of worked examples is included as illustration. The study of curves of genus 2 and their Jacobians is rapidly becoming more constructive in nature. An explicit embedding of the Jacobian variety has been described in P9 for the case when there is a rational Weierstrass point [10], and in P15 for the general situation [7]. The defining equations have been determined in a manner which preserves
A Flexible Method for Applying Chabauty’s Theorem
"... A strategy is proposed for applying Chabauty’s Theorem to hyperelliptic curves of genus> 1. In the genus 2 case, it shown how recent developments on the formal group of the Jacobian can be used to give a flexible and computationally viable method for applying this strategy. The details are described ..."
Abstract
 Add to MetaCart
A strategy is proposed for applying Chabauty’s Theorem to hyperelliptic curves of genus> 1. In the genus 2 case, it shown how recent developments on the formal group of the Jacobian can be used to give a flexible and computationally viable method for applying this strategy. The details are described for a general curve of genus 2, and are then applied to find C(Q) for a selection of curves. A fringe benefit is a more explicit proof of a result of Coleman. We recall the following result of Chabauty [4], which gives a way of deducing information about the Krational points on a curve from its Jacobian. Proposition 0.1. Let C be a curve of genus g defined over a number field K, whose Jacobian has MordellWeil rank � g − 1. Then C has only finitely many Krational points.
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 the ..."
Abstract
 Add to MetaCart
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
E.V. Flynn On a Theorem of Coleman 1 On a Theorem of Coleman
"... A simplified method of descent via isogeny is given for Jacobians of curves of genus 2. This method is then used to give applications of a theorem of Coleman for computing all the rational points on certain curves of genus 2. 0 ..."
Abstract
 Add to MetaCart
A simplified method of descent via isogeny is given for Jacobians of curves of genus 2. This method is then used to give applications of a theorem of Coleman for computing all the rational points on certain curves of genus 2. 0
THE COMPLETE CLASSIFICATION OF RATIONAL PREPERIODIC POINTS OF QUADRATIC POLYNOMIALS OVER Q: A REFINED CONJECTURE
, 1995
"... Abstract. We classify the graphs that can occur as the graph of rational preperiodic points of a quadratic polynomial over Q, assuming the conjecture that it is impossible to have rational points of period 4 or higher. In particular, we show under this assumption that the number of preperiodic point ..."
Abstract
 Add to MetaCart
Abstract. We classify the graphs that can occur as the graph of rational preperiodic points of a quadratic polynomial over Q, assuming the conjecture that it is impossible to have rational points of period 4 or higher. In particular, we show under this assumption that the number of preperiodic points is at most 9. Elliptic curves of small conductor and the genus 2 modular curves X1(13), X1(16), and X1(18) all arise as curves classifying quadratic polynomials with various combinations of preperiodic points. To complete the classification, we compute the rational points on a nonmodular genus 2 curve by performing a 2descent on its Jacobian and afterwards applying a variant of the method of Chabauty and Coleman. 1.