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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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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.
The Jacobian and Formal Group of a Curve of Genus 2 over an Arbitrary Ground
 Math. Proc. Cambridge Philos. Soc. 107
, 1990
"... The ability to perform practical computations on particular cases has greatly influenced the theory of elliptic curves. First, it has allowed a rich subbranch of the Mathematics of Computation to develop, devoted to elliptic curves: the search for curves of large rank, large torsion over number fie ..."
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Cited by 19 (3 self)
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The ability to perform practical computations on particular cases has greatly influenced the theory of elliptic curves. First, it has allowed a rich subbranch of the Mathematics of Computation to develop, devoted to elliptic curves: the search for curves of large rank, large torsion over number fields, and — more recently — the application of
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 formal group of the Jacobian of an algebraic curve
 Paci Journal of Mathematics 157
, 1993
"... In this paper we give an explicit construction of the formal group of the Jacobian of an algebraic curve using a basis for the holomorphic differentials on the curve at a rational nonWeierstrass point. We construct the formal group of the Jacobian of the modular curve Xo(l) and using a result of T. ..."
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In this paper we give an explicit construction of the formal group of the Jacobian of an algebraic curve using a basis for the holomorphic differentials on the curve at a rational nonWeierstrass point. We construct the formal group of the Jacobian of the modular curve Xo(l) and using a result of T. Honda, we prove that this formal group is pintegral for all but finitely many p. Introduction. Formal group laws have proven to be very useful tools in many areas of mathematics and computer science. In particular, the formal group of an elliptic curve has been used to great effect in elliptic curve theory (for details see for example Silverman [13]) and the use of the formal group of an abelian variety is pervasive in arithmetic and
LARGE RATIONAL TORSION ON ABELIAN VARIETIES
"... A method of searching for large rational torsion on Abelian varieties is described. A few explicit applications of this method over Q give rational 11 and 13torsion in dimension 2, and rational 29torsion in dimension 4. The search for large rational torsion on elliptic curves goes back to the ear ..."
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A method of searching for large rational torsion on Abelian varieties is described. A few explicit applications of this method over Q give rational 11 and 13torsion in dimension 2, and rational 29torsion in dimension 4. The search for large rational torsion on elliptic curves goes back to the early 1900s, when Levi [6], Billing [1], and others found, for various values of N, the curve X1(N) whose rational points correspond to elliptic curves with rational Ntorsion. These authors found elliptic curves with 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 and 12 torsion over Q, and proved the
The Arithmetic of Hyperelliptic Curves
"... We summarise recent advances in techniques for solving Diophantine problems on hyperelliptic curves; in particular, those for finding the rank of the Jacobian, and the set of rational points on the curve. The constructive theory of hyperelliptic curves has been advanced significantly during the last ..."
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We summarise recent advances in techniques for solving Diophantine problems on hyperelliptic curves; in particular, those for finding the rank of the Jacobian, and the set of rational points on the curve. The constructive theory of hyperelliptic curves has been advanced significantly during the last year. It is intended to give here an indication of the current level of progress, and an outline of the main methods employed. The emphasis in Sections 1 to 4 will be on the group of rational points on the Jacobian of a hyperelliptic curve. Section 5 will concern
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 ..."
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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 descri ..."
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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.
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"... The group law on the Jacobian of a curve of genus 2 E. V. Flynn, Mathematical Institute, University of Oxford An explicit description is given of the group law on the Jacobian of a curve C of genus 2. The Kummer surface provides a useful intermediary stage; bilinear forms relating to the Kummer surf ..."
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The group law on the Jacobian of a curve of genus 2 E. V. Flynn, Mathematical Institute, University of Oxford An explicit description is given of the group law on the Jacobian of a curve C of genus 2. The Kummer surface provides a useful intermediary stage; bilinear forms relating to the Kummer surface imply that the global group law may be given projectively by biquadratic forms defined over the same ring as the coefficients of C. It is not assumed that C has a rational Weierstrass point, and the theory presented applies over an arbitrary ground field. In [8] an explicit embedding in P 15 is described of the Jacobian of a curve of genus 2 over an arbitrary ground field. The defining equations are 72 quadratic forms defined over the ground field ([8], Appendix A). It is the main aim of this paper to describe the biquadratic equations which define the group law on the Jacobian. Although these equations