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 2-descent 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 5-cycles, and show that there exist Gal(Q/Q)-stable N-cycles 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.

. 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...

Dedicated to the memory of my teacher Serge Lang. We discuss p-adic unipotent Albanese maps for curves of positive genus, extending the theory of p-adic multiple polylogarithms. This construction is then used to relate linear Diophantine conjectures of ‘Birch and Swinnerton-Dyer type ’ to non-linear theorems of Faltings-Siegel 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:

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

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, thesize|X(K) | of the set of K-rational 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

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 non-hyperelliptic 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 Brauer-Manin 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 Hasse-principle. 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.

This paper gives a new proof of Kamienny's Criterion using the method of Coleman and Chabauty. 1.

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.

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 d-th symmetric power of C by C (d). In this paper we adapt the classic Chabauty–Coleman method to study the K-rational 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.

By using the so-called elliptic curve Chabauty method, N. Bruin [1], V. Flynn and J. Wetherell [6] have extended Chabauty's method to some cases where the rank of the Jacobian may not be less than the genus. The main tool in these methods is a theorem of Strassman on p-adic zeros of power series in one variable, and is applicable only if certain Jacobians are of rank less than or equal to 1. In the present paper, we give an explicit generalization of Strassman's theorem to several variables, enabling us to treat cases where the rank is greater than 1. We apply this to find all the rational points on a hyperelliptic curve of rank and genus equal to 4.