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19
Bounding the Number of Rational Points on Certain Curves of High Rank
, 1997
"... 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 Krational points on C. In Chapter I we discuss Chabauty techniques. This is a review and synthesis of previously known material, both published and unpu ..."
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Cited by 25 (2 self)
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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 Krational 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 This is a review and synthesis of previously known material, both published and unpublished.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 Krational 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 Krational points in the collection is in nto1 correspondence with the set of Krational points on C. If Chabauty is applicable to every curve in the collection, then we can use the covers to bound the number of Krational 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.
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 21 (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.
The Hasse principle and the BrauerManin obstruction for curves
 Manuscripta Math
, 2004
"... Abstract. We discuss a range of ways, extending existing methods, to demonstrate violations of the Hasse principle on curves. Of particular interest are curves which contain a rational divisor class of degree 1, even though they contain no rational point. For such curves we construct new types of ex ..."
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Abstract. We discuss a range of ways, extending existing methods, to demonstrate violations of the Hasse principle on curves. Of particular interest are curves which contain a rational divisor class of degree 1, even though they contain no rational point. For such curves we construct new types of examples of violations of the Hasse principle which are due to the BrauerManin obstruction, subject to the conjecture that the TateShafarevich group of the Jacobian is finite. 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...
Application of covering techniques to families of curves
 J. Number Theory
"... Abstract. Much success in finding rational points on curves has been obtained by using Chabauty’s Theorem, which applies when the genus of a curve is greater than the rank of the MordellWeil group of the Jacobian. When Chabauty’s Theorem does not directly apply to a curve C, a recent modification h ..."
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Abstract. Much success in finding rational points on curves has been obtained by using Chabauty’s Theorem, which applies when the genus of a curve is greater than the rank of the MordellWeil group of the Jacobian. When Chabauty’s Theorem does not directly apply to a curve C, a recent modification has been to cover the rational points on C by those on a covering collection of curves Di, obtained by pullbacks along an isogeny to the Jacobian; one then hopes that Chabauty’s Theorem applies to each Di. So far, this latter technique has been applied to isolated examples. We apply, for the first time, certain covering techniques to infinite families of curves. We find an infinite family of curves to which Chabauty’s Theorem is not applicable, but which can be solved using bielliptic covers, and other infinite families of curves which even resist solution by bielliptic covers. A fringe benefit is an infinite family of Abelian surfaces with nontrivial elements of the TateShafarevich group killed by a bielliptic isogeny. 1.
Thue equations and the method of ChabautyColeman
, 2002
"... Let OK be any domain with field of fractions K. LetF(x, y) ∈ OK[x, y] be a homogeneous polynomial of degree n, coprime to y, andassumedto have unit content (i.e., the coefficients of F generate the unit ideal in OK). Assume that gcd(n, char(K)) = 1. Let h ∈ OK and assume that the polynomial hz n − ..."
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Let OK be any domain with field of fractions K. LetF(x, y) ∈ OK[x, y] be a homogeneous polynomial of degree n, coprime to y, andassumedto have unit content (i.e., the coefficients of F generate the unit ideal in OK). Assume that gcd(n, char(K)) = 1. Let h ∈ OK and assume that the polynomial hz n − F(x, y) is irreducible in K[x, y, z]. We denote by X F,h/K the nonsingular complete model of the projective plane curve CF,h/K defined by the equation hz n − F(x, y) = 0. We shall assume in this article that g(X F,h) ≥ 2. When K is a number field, Mordell’s Conjecture (now Faltings ’ Theorem) implies that X F,h(K)  < ∞. Caporaso, Harris, and Mazur ([CHM, 1.1]) have shown that if Lang’s conjecture for varieties of general type is true, then for any number field K, thesizeX(K)  of the set of Krational points of any curve X/K of genus g(X) ≥ 2 can be bounded by a constant depending only on g(X). Prior to the paper [CHM], Mazur and others had asked whether X(K)  can be bounded by a constant depending only on
EXPLICIT CHABAUTY OVER NUMBER FIELDS
, 2009
"... Let C be a smooth projective absolutely irreducible curve of genus g ≥ 2 over a number field K of degree d, and denote its Jacobian by J. Denote the Mordell–Weil rank of J(K) by r. We give an explicit and practical Chabautystyle criterion for showing that a given subset K ⊆ C(K) is in fact equal t ..."
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Cited by 4 (1 self)
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Let C be a smooth projective absolutely irreducible curve of genus g ≥ 2 over a number field K of degree d, and denote its Jacobian by J. Denote the Mordell–Weil rank of J(K) by r. We give an explicit and practical Chabautystyle criterion for showing that a given subset K ⊆ C(K) is in fact equal to C(K). This criterion is likely to be successful if r ≤ d(g − 1). We also show that the only solutions to the equation x 2 + y 3 = z 10 in coprime nonzero integers is (x, y, z) = (±3, −2, ±1). This is achieved by reducing the problem to the determination of Krational points on several genus 2 curves where K = Q or Q ( 3 √ 2), and applying the method of this paper.
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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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.
Chabauty for symmetric powers of curves
 Algebra & Number Theory
"... Abstract. Let C be a smooth projective absolutely irreducible curve of genus g ≥ 2 over a number field K, and denote its Jacobian by J. Let d ≥ 1 be an integer and denote the dth symmetric power of C by C (d). In this paper we adapt the classic Chabauty–Coleman method to study the Krational points ..."
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Abstract. Let C be a smooth projective absolutely irreducible curve of genus g ≥ 2 over a number field K, and denote its Jacobian by J. Let d ≥ 1 be an integer and denote the dth symmetric power of C by C (d). In this paper we adapt the classic Chabauty–Coleman method to study the Krational points of C (d). Suppose that J(K) has Mordell–Weil rank at most g − d. We give an explicit and practical criterion for showing that a given subset L ⊆ C (d) (K) is in fact equal to C (d) (K). 1.