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39
Cycles of quadratic polynomials and rational points on a genus 2 curve
, 1996
"... 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), who ..."
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Cited by 45 (13 self)
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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.
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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Cited by 42 (11 self)
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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.
The unipotent Albanese map and Selmer varieties for curves, preprint
, 2005
"... Dedicated to the memory of my teacher Serge Lang. We discuss padic unipotent Albanese maps for curves of positive genus, extending the theory of padic multiple polylogarithms. This construction is then used to relate linear Diophantine conjectures of ‘Birch and SwinnertonDyer type ’ to nonlinear ..."
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Cited by 21 (2 self)
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Dedicated to the memory of my teacher Serge Lang. We discuss padic unipotent Albanese maps for curves of positive genus, extending the theory of padic multiple polylogarithms. This construction is then used to relate linear Diophantine conjectures of ‘Birch and SwinnertonDyer type ’ to nonlinear theorems of FaltingsSiegel type. In a letter to Faltings [14] dated June, 1983, Grothendieck proposed several striking conjectural connections between the arithmetic geometry of ‘anabelian schemes ’ and their fundamental groups, among which one finds issues of considerable interest to classical Diophantine geometers. Here we will trouble the reader with a careful formulation of just one of them. Let F be a number field and f: X→Spec(F) a smooth, compact, hyperbolic curve over F. After the choice of an algebraic closure and a base point y: Spec ( ¯ F)→Spec(F) x: Spec ( ¯ F)→X such that f(x) = y, we get an exact sequence of fundamental groups:
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 15 (2 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...
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 14 (5 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.
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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Cited by 9 (0 self)
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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
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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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.
Minhyong Remark on fundamental groups and effective Diophantine methods for hyperbolic curves. Preprint. Available at mathematics archive
"... Dedicated to the memory of Serge Lang In earlier articles ([8], [9], [10]) attention was drawn to the parallel between the ideas surrounding the wellknown conjecture of Birch and SwinnertonDyer (BSD) for elliptic curves, and the mysterious section conjecture of Grothendieck [6] that concerns hyper ..."
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Cited by 6 (0 self)
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Dedicated to the memory of Serge Lang In earlier articles ([8], [9], [10]) attention was drawn to the parallel between the ideas surrounding the wellknown conjecture of Birch and SwinnertonDyer (BSD) for elliptic curves, and the mysterious section conjecture of Grothendieck [6] that concerns hyperbolic curves. We wish to explain here some preliminary ideas for ‘effective nonabelian descent ’ on hyperbolic curves equipped with at least one rational point. We again follow in an obvious manner the method of descent on elliptic curves and, therefore, rely on conjectures. In fact, the main point is to substitute the section conjecture for the finiteness of the TateShafarevich group. That is to say, the input of the section conjecture is of the form section conjecture ⇒ termination of descent. At a number of different lectures delivered by the author on the topic of fundamental groups and Diophantine geometry, the question was raised about the role of surjectivity in the section conjecture as far as Diophantine applications are concerned. The demonstration of this implication is intended as something of a reply. To start the descent, on the other hand, requires the use of padic Hodge theory and the unipotent
NCOVERS OF HYPERELLIPTIC CURVES
"... Abstract. For a hyperelliptic curve C of genus g with a divisor class of order n = g + 1, we shall consider an associated covering collection of curves Dδ, each of genus g 2. We describe, up to isogeny, the Jacobian of each Dδ via a map from Dδ to C, and two independent maps from Dδ to a curve of ge ..."
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Cited by 5 (1 self)
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Abstract. For a hyperelliptic curve C of genus g with a divisor class of order n = g + 1, we shall consider an associated covering collection of curves Dδ, each of genus g 2. We describe, up to isogeny, the Jacobian of each Dδ via a map from Dδ to C, and two independent maps from Dδ to a curve of genus g(g − 1)/2. For some curves, this allows covering techniques that depend on arithmetic data of number fields of smaller degree than standard 2coverings; we illustrate this by using 3coverings to find all Qrational points on a curve of genus 2 for which 2covering techniques would be impractical. 1. Description of the Jacobian of the Covering Curves We shall consider a hyperelliptic curve of genus g = n − 1 ≥ 1, of the form (1) C: Y 2 = F (X) = G(X) 2 + kH(X) n, where G(X) is of degree n = g + 1 and H(X) is of degree 2, and where G(X), H(X), k are defined over the ring of integers O of a number field K. Here, and elsewhere, we shall adopt the usual convention that C is used to denote the nonsingular curve, even though the equation given in (1) is singular; for the practical purpose of points on C, we can take these to be the affine (X, Y) satisfying (1), together with ∞ +, ∞ − , which will be distinct points on this nonsingular curve. We shall assume that F (X) has nonzero discriminant, which implies that resultant(G(X), H(X)) is also nonzero. Equation (1) is a classical model of a hyperelliptic curve whose Jacobian J has an element of order n defined