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Implementing 2Descent for Jacobians of Hyperelliptic Curves
 Acta Arith
, 1999
"... . This paper gives a fairly detailed description of an algorithm that computes (the size of) the 2Selmer group of the Jacobian of a hyperellitptic curve over Q. The curve is assumed to have even genus or to possess a Qrational Weierstra point. 1. Introduction Given some curve C over Q , one w ..."
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. This paper gives a fairly detailed description of an algorithm that computes (the size of) the 2Selmer group of the Jacobian of a hyperellitptic curve over Q. The curve is assumed to have even genus or to possess a Qrational 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 MordellWeil 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...
The Topology of Rational Points
 Experimental Mathematics
, 1992
"... this article is to provoke a discussion concerning the ..."
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Cited by 33 (0 self)
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this article is to provoke a discussion concerning the
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.
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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. 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...
Integral points on punctured abelian surfaces. in Algorithmic number theory
 Lecture Notes in Comput. Sci. 2369
, 2002
"... Abstract. We study the density of integral points on punctured abelian surfaces. Linear growth rates are observed experimentally. 1 ..."
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Abstract. We study the density of integral points on punctured abelian surfaces. Linear growth rates are observed experimentally. 1
THE METHOD OF CHABAUTY AND COLEMAN
"... Abstract. This is an introduction to the method of Chabauty and Coleman, a padic method that attempts to determine the set of rational points on a given curve of genus g ≥ 2. We present the method, give a few examples of its implementation in practice, and discuss its effectiveness. An appendix tre ..."
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Abstract. This is an introduction to the method of Chabauty and Coleman, a padic method that attempts to determine the set of rational points on a given curve of genus g ≥ 2. We present the method, give a few examples of its implementation in practice, and discuss its effectiveness. An appendix treats the case in which the curve has bad reduction. 1. Rational points on curves of genus ≥ 2 We will work over the field Q of rational numbers, although everything we say admits an appropriate generalization to a number field. Let Q be an algebraic closure of Q. For each finite prime p, let Qp be the field of padic numbers (see [Kob84] for the definition). Curves will be assumed to be smooth, projective, and geometrically integral. Let X be a curve over Q of genus g ≥ 2. We suppose that X is presented as the zero set in some P n of an explicit finite set of homogeneous polynomials. We may give instead an equation for a singular (but still geometrically integral) curve in A 2; in this case, it is understood that X is the smooth projective curve birational to this singular curve. Rational points on X can be specified by giving their coordinates. (A little more data may be required if a singular model for X is used.) Let X(Q) be the set of rational points on X. Faltings ’ theorem [Fal83] states that X(Q) is finite. Thus we have the following welldefined problem: Given X of genus ≥ 2 presented as above, compute X(Q). Faltings ’ proof is ineffective in the sense that it does not provide an algorithm for solving this problem, even in principle. In fact, it is not known whether any algorithm is guaranteed to solve the problem. Even the case g = 2 seems hard. Nevertheless there are a few techniques that can be applied: see [Poo02] for a survey. On individual curves these seem to solve the problem often, perhaps even always when used together, though it seems very difficult to prove that they always work. One of the methods used is the method of Chabauty and Coleman.
TWOCOVERINGS OF JACOBIANS OF CURVES OF GENUS TWO
, 905
"... Abstract. Given a curve C of genus 2 defined over a field k of characteristic different from 2, with Jacobian variety J, we show that the twocoverings corresponding to elements of a large subgroup of H1 ` Gal(ks /k), J[2](ks) ´ (containing the Selmer group when k is a global field) can be embedded ..."
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Abstract. Given a curve C of genus 2 defined over a field k of characteristic different from 2, with Jacobian variety J, we show that the twocoverings corresponding to elements of a large subgroup of H1 ` Gal(ks /k), J[2](ks) ´ (containing the Selmer group when k is a global field) can be embedded as intersection of 72 quadrics in P15 k, just as the Jacobian J itself. Moreover, we actually give explicit equations for the models of these twists in the generic case, extending the work of Gordon and Grant which applied only to the case when all Weierstrass points are rational. In addition, we describe elegant equations on the Jacobian itself, and answer a question of Cassels and the first author concerning a map from the Kummer surface in P3 to the desingularized Kummer surface in P5. 1.
HOMOGENEOUS SPACES AND DEGREE 4 DEL PEZZO SURFACES
"... Abstract. It is known that, given a genus 2 curve C: y 2 = f(x), where f(x) is quintic and defined over a field K, of characteristic different from 2, and given a homogeneous space Hδ for complete 2descent on the Jacobian of C, there is a Vδ (which we shall describe), which is a degree 4 del Pezzo ..."
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Abstract. It is known that, given a genus 2 curve C: y 2 = f(x), where f(x) is quintic and defined over a field K, of characteristic different from 2, and given a homogeneous space Hδ for complete 2descent on the Jacobian of C, there is a Vδ (which we shall describe), which is a degree 4 del Pezzo surface defined over K, such that Hδ(K) ̸ = ∅ = ⇒ Vδ(K) ̸ = ∅. We shall prove that every degree 4 del Pezzo surface V, defined over K, arises in this way; furthermore, we shall show explicitly how, given V, to find C and δ such that V = Vδ, up to a linear change in variable defined over K. We shall also apply this relationship to Hürlimann’s example of a degree 4 del Pezzo surface violating the Hasse principle, and derive an explicit parametrised infinite family of genus 2 curves, defined over Q, whose Jacobians have nontrivial members of the ShafarevichTate group. This example will differ from previous examples in the literature by having only two Qrational Weierstrass points. 1.
An Explicit Theory of Heights
"... We consider the problem of explicitly determining the naive height constants for Jacobians of hyperelliptic curves. For genus> 1, it is impractical to apply Hilbert’s Nullstellensatz directly to the defining equations of the duplication law; we indicate how this technical difficulty can be overcome ..."
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We consider the problem of explicitly determining the naive height constants for Jacobians of hyperelliptic curves. For genus> 1, it is impractical to apply Hilbert’s Nullstellensatz directly to the defining equations of the duplication law; we indicate how this technical difficulty can be overcome by use of isogenies. The height constants are computed in detail for the Jacobian of an arbitrary curve of genus 2, and we apply the technique to compute generators of J (Q), the MordellWeil group for a selection of rank 1 examples. There are an increasing number of methods available [2],[6],[7],[11] for performing a Galois 2descent on the Jacobian J of a hyperelliptic curve, giving J (Q)/2J (Q), and hence the rank of J (Q), the MordellWeil group. These methods have collectively found the ranks of 15 Jacobians of curves of genus 2, and one of genus 3; it is likely that minor
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