E. V. Flynn

Prolegomena to a middlebrow arithmetic of curves of genus 2

Background and conventions 1. Curves of genus 2 2. Construction of the jacobian 3. The Kummer surface 4. The dual of the Kummer 5. Weddles surface 6. G/2G 7. The jacobian over local fields Formal groups 8. Torsion 9. The isogeny Theory 10. The isogeny Applications 11. Computing the Mordell-Weil group 12. Heights 13. Rational points Chabautys theorem 14. Reducible jacobians 15. The endomorphism ring 16. The desingularized Kummer 17. A neoclassical approach 18. Zukunftsmusik Appendix 1. MAPLE programs Appendix 2. Files available by FTP.

Cycles of quadratic polynomials and rational points on a genus-

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 X

2

_1

curve

(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 Galois-stable 5-cycles, and show that there exist Galois-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.

A flexible method for applying Chabauty's Theorem

A strategy is proposed for applying Chabautys Theoremto hyperelliptic curves of genus > 1. In the genus 2case, it shown how recent developments on the formal group of the Jacobiancan be used to give a flexible and computationallyviable method for applying this strategy. The details are described for a general curveof genus 2, and are then applied to find C(ℚ) fora selection of curves. A fringe benefit is a more explicit proof of a result of Coleman.

The jacobian and formal group of a curve of genus 2 over an arbitrary ground field

An embedding of the Jacobian variety of a curve of genus 2 is given, together with an explicit set of defining equations. A pair of local parameters is chosen, for which the induced formal group is defined over the same ring as the coefficients of . It is not assumed that has a rational Weierstrass point, and the theory presented applies over an arbitrary ground field (of characteristic ╪ 2, 3, or 5).

Towers of 2-covers of hyperelliptic curves

In this article, we give a way of constructing an unramified Galois-cover of a hyperelliptic curve. The geometric Galois-group is an elementary abelian 2-group. The construction does not make use of the embedding of the curve in its Jacobian, and it readily displays all subcovers. We show that the cover we construct is isomorphic to the pullback along the multiplication-by-2 map of an embedding of the curve in its Jacobian. We show that the constructed cover has an abundance of elliptic and hyperelliptic subcovers. This makes this cover especially suited for covering techniques employed for determining the rational points on curves. In particular the hyperelliptic subcovers give a chance for applying the method iteratively, thus creating towers of elementary abelian 2-covers of hyperelliptic curves. As an application, we determine the rational points on the genus 2 curve arising from the question of whether the sum of the first n fourth powers can ever be a square. For this curve, a simple covering step fails, but a second step succeeds.

Large rational torsion on Abelian varieties

A method of searching for large rational torsion on Abelian varieties is described. A few explicit applications of this method over Q give rational 11- and 13-torsion in dimension 2, and rational 29-torsion in dimension 4.

Descent via isogeny on elliptic curves with large rational torsion subgroups

We outline PARI programs which assist with various algorithms related to descent via isogeny on elliptic curves. We describe, in this context, variations of standard inequalities which aid the computation of members of the Tate-Shafarevich group. We apply these techniques to several examples: in one case we use descent via 9-isogeny to determine the rank of an elliptic curve; in another case we find nontrivial members of the 9-part of the Tate-Shafarevich group, and in a further case, nontrivial members of the 13-part of the Tate-Shafarevich group.

The Brauer-Manin Obstruction and Sha[2].

We discuss the Brauer-Manin obstruction on del Pezzo surfaces of degree 4. We outline a detailed algorithm for computing the obstruction and provide associated programs in magma. This is illustrated with the computation of an example with an irreducible cubic factor in the singular locus of the defining pencil of quadrics (in contrast to previous examples, which had at worst quadratic irreducible factors). We exploit the relationship with the Tate-Shafarevich group to give new types of examples of Sha[2], for families of curves of genus 2 of the form y^2 = f(x), where f(x) is a quintic containing an irreducible cubic factor.