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23
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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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...
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 7 (1 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:
Rational points on certain families of curves of genus at least 2
 Proc. London Math. Soc
, 1987
"... A conjecture of Mordell, recently proven by Faltings [7], states that a curve of genus at least 2 has only finitely many rational points. Faltings ' proof is not effective, although a careful reworking of his proof, combined with some further ideas of Faltings, Mumford, Parshin, and Raynaud, allows ..."
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Cited by 4 (0 self)
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A conjecture of Mordell, recently proven by Faltings [7], states that a curve of genus at least 2 has only finitely many rational points. Faltings ' proof is not effective, although a careful reworking of his proof, combined with some further ideas of Faltings, Mumford, Parshin, and Raynaud, allows one to give an upper bound for the number of rational points. (See [19, XI, §2].) Unfortunately, the resulting bound depends in quite a nasty manner on the set of primes at which the curve has bad reduction. Prior to Faltings ' proof of Mordell's conjecture, there were two methods which in certain rather restrictive cases could be used to prove finiteness of the number of rational points. The first was due to Chabauty [3], and the second to Dem'janenko [5], generalized by Manin [12]. Recently, Coleman [4] has analysed Chabauty's method and used it to give relatively small upper bounds in those cases where it can be applied. For example, he proves that if C/Q is a smooth curve of genus g 3 = 2, and if the Jacobian variety J of C satisfies rank/(Q)<g, then
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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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 3 (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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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 motivic fundamental group of P 1 \ {0, 1, ∞} and the theorem of Siegel
, 2008
"... One of the strong motivations for studying the arithmetic fundamental groups of algebraic varieties comes from the hope that they will provide grouptheoretic access to Diophantine geometry. This is most clearly expressed by the socalled ‘section conjecture ’ of Grothendieck: Let F be a number fiel ..."
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One of the strong motivations for studying the arithmetic fundamental groups of algebraic varieties comes from the hope that they will provide grouptheoretic access to Diophantine geometry. This is most clearly expressed by the socalled ‘section conjecture ’ of Grothendieck: Let F be a number field and let X→Spec(F) be a smooth hyperbolic curve. After some choice of base point, we get an exact sequence 0→ˆπ1 ( ¯ X)→ˆπ1(X) p → Γ→0 where Γ is the Galois group of F and ˆπ1 refers to the profinite fundamental group. The section conjecture states that there is a onetoone correspondence between conjugacy classes of splittings of p and the geometric sections of the morphism X→Spec(F). There is apparently a compactness argument that yields a direct implication from the section conjecture to the theorems of Siegel and Faltings. The purpose of this paper is to illustrate a somewhat different methodology for deriving Diophantine consequences from a study of the fundamental group. To this end, we give a π1 proof of the theorem of Siegel on the finiteness of integral points for the thricepunctured projective line. The notion of a ‘π1 proof ’ may not be entirely welldefined, but it is hoped that the techniques employed
Kamienny's Criterion And The Method Of Coleman And Chabauty
, 1999
"... This paper gives a new proof of Kamienny's Criterion using the method of Coleman and Chabauty. 1. ..."
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This paper gives a new proof of Kamienny's Criterion using the method of Coleman and Chabauty. 1.