. This paper gives a fairly detailed description of an algorithm that computes (the size of) the 2-Selmer group of the Jacobian of a hyperellitptic curve over Q. The curve is assumed to have even genus or to possess a Q-rational Weierstra point. 1. Introduction Given some curve C over Q , one would like to determine as much as possible of its arithmetical properties. One of the more important invariants is the MordellWeil rank of its Jacobian J , i.e., the free abelian rank of J(Q ) (finite by the Mordell-Weil Theorem). There is no algorithm so far that provably determines this rank, but it is possible (at least in theory) to bound it from above by computing the size of a suitable Selmer group. It is also fairly easy to find lower bounds by looking for independent rational points on the Jacobian. (It can be difficult, however, to find the right number of independent points, when some of the generators are large.) With some luck, both bounds coincide, and the rank is determined. In...

We construct examples of families of curves of genus 2 or 3 over Q whose Jacobians split completely and have various large rational torsion subgroups. For example, the rational points on a certain elliptic surface over P of positive rank parameterize a family of genus-2 curves over Q whose Jacobians each have 128 rational torsion points. Also, we find the genus-3 curve ) = 0, whose Jacobian has 864 rational torsion points.

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 2-descent 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 5-cycles, and show that there exist Gal(Q/Q)-stable N-cycles 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.

Abstract. This is the second in a series of papers where we combine the classical approach to exponential Diophantine equations (linear forms in logarithms, Thue equations, etc.) with a modular approach based on some of the ideas of the proof of Fermat’s Last Theorem. In this paper we use a general and powerful new lower bound for linear forms in three logarithms, together with a combination of classical, elementary and substantially improved modular methods to solve completely the Lebesgue-Nagell equation for D in the range 1 ≤ D ≤ 100. x 2 + D = y n, x, y integers, n ≥ 3, 1.

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 K-rational 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 K-rational 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 K-rational points in the collection is in n-to-1 correspondence with the set of K-rational points on C. If Chabauty is applicable to every curve in the collection, then we can use the covers to bound the number of K-rational 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.

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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...

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.

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 Mordell-Weil group. This last stage is performed by a lattice enlarging procedure. 1.

This thesis is available for Library use on the understanding that it is copy-right material and that no quotation from the thesis may be published without proper acknowledgement. I certify that all the material in this thesis which is not my own work has been clearly identified and that no material is included for which a degree has previously been conferred upon me.