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Independence of rational points on twists of a given curve, to appear
 in Compositio Math. arXiv: math.NT/0603557 School of Engineering and Science, International University Bremen, P.O.Box 750561, 28725
"... Abstract. In this paper, we study bounds for the number of rational points on twists C ′ of a fixed curve C over a number field K, under the condition that the group of Krational points on the Jacobian J ′ of C ′ has rank smaller than the genus of C ′. The main result is that with some explicitly g ..."
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Cited by 19 (12 self)
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Abstract. In this paper, we study bounds for the number of rational points on twists C ′ of a fixed curve C over a number field K, under the condition that the group of Krational points on the Jacobian J ′ of C ′ has rank smaller than the genus of C ′. The main result is that with some explicitly given finitely many possible exceptions, we have a bound of the form 2r + c, where r is the rank of J ′ (K) and c is a constant depending on C. For the proof, we use a refinement of the method of ChabautyColeman; the main new ingredient is to use it for an extension field of Kv, where v is a place of bad reduction for C ′. 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.
The arithmetic of Prym varieties in genus 3
 Compositio Math
"... Abstract. Given a curve of genus 3 with an unramified double cover, we give an explicit description of the associated Prym variety. We also describe how an unramified double cover of a nonhyperelliptic genus 3 curve can be mapped into the Jacobian of a curve of genus 2 over its field of definition ..."
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Cited by 3 (1 self)
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Abstract. Given a curve of genus 3 with an unramified double cover, we give an explicit description of the associated Prym variety. We also describe how an unramified double cover of a nonhyperelliptic genus 3 curve can be mapped into the Jacobian of a curve of genus 2 over its field of definition and how this can be used to do Chabauty and BrauerManin type calculations for curves of genus 5 with an unramified involution. As an application, we determine the rational points on a smooth plane quartic with no particular geometric properties and give examples of curves of genus 3 and 5 violating the Hasseprinciple. We also show how these constructions can be used to design smooth plane quartics with specific arithmetic properties. As an example, we give a smooth plane quartic with all 28 bitangents defined over Q(t), By specialization, this also gives examples over Q. 1.
Computing Rational Points on Curves
 In Number theory for the millennium, III (Urbana, IL, 2000), 149–172, A K Peters
, 2000
"... We give a brief introduction to the problem of explicit determination of rational points on curves, indicating some recent ideas that have led to progress. ..."
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Cited by 2 (0 self)
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We give a brief introduction to the problem of explicit determination of rational points on curves, indicating some recent ideas that have led to progress.
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.
UNRAMIFIED COVERS OF GALOIS COVERS OF LOW GENUS CURVES
"... Abstract. Let X → Y be a Galois covering of curves, where the genus of X is ≥ 2 and the genus of Y is ≤ 2. We prove that under certain hypotheses, X has an unramified cover that dominates a hyperelliptic curve; our results apply, for instance, to all tamely superelliptic curves. Combining this with ..."
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Abstract. Let X → Y be a Galois covering of curves, where the genus of X is ≥ 2 and the genus of Y is ≤ 2. We prove that under certain hypotheses, X has an unramified cover that dominates a hyperelliptic curve; our results apply, for instance, to all tamely superelliptic curves. Combining this with a theorem of Bogomolov and Tschinkel shows that X has an unramified cover that dominates y 2 = x 6 − 1, if char k is not 2 or 3. 1.
CYCLES OF COVERS
"... Abstract. We initially consider an example of Flynn and Redmond, which gives an infinite family of curves to which Chabauty’s Theorem is not applicable, and which even resist solution by one application of a certain bielliptic covering technique. In this article, we shall consider a general context, ..."
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Abstract. We initially consider an example of Flynn and Redmond, which gives an infinite family of curves to which Chabauty’s Theorem is not applicable, and which even resist solution by one application of a certain bielliptic covering technique. In this article, we shall consider a general context, of which this family is a special case, and in this general situation we shall prove that repeated application of bielliptic covers always results in a sequence of genus 2 curves which cycle after a finite number of repetitions. We shall also give an example which is resistant to repeated applications of the technique. 1.
RATIONAL POINTS ON CURVES
"... Abstract. This is an extended version of an invited lecture I gave at the Journées Arithmétiques in St. Étienne in July 2009. We discuss the state of the art regarding the problem of finding the set of rational points on a (smooth projective) geometrically integral curve C over Q. The focus is on pr ..."
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Abstract. This is an extended version of an invited lecture I gave at the Journées Arithmétiques in St. Étienne in July 2009. We discuss the state of the art regarding the problem of finding the set of rational points on a (smooth projective) geometrically integral curve C over Q. The focus is on practical aspects of this problem in the case that the genus of C is at least 2, and therefore the set of rational points is finite. 1.