Let (A, λ) be a principally polarized abelian variety defined over a global field k, and let ∐ ∐(A) be its Shafarevich-Tate group. Let ∐ ∐(A) nd denote the quotient of ∐ ∐(A) by its maximal divisible subgroup. Cassels and Tate constructed a nondegenerate pairingIf A is an elliptic curve, then by a result of Cassels the pairing is alternating. But in general it is only antisymmetric.Using some new but equivalent definitions of the pairing, we derive general criteria deciding whether it is alternating and whether there exists some alternating nondegenerate pairing on ∐ ∐(A) nd . These criteria are expressed in terms of an element c ∈ ∐ ∐(A) nd that is canonically associated to the polarization λ. In the case that A is the Jacobian of some curve, a down-to-earth version of the result allows us to determine effectively whether #∐ ∐(A) (if finite) is a square or twice a square. We then apply this to prove that a positive proportion (in some precise sense) of all hyperelliptic curves of even genus g ≥ 2 over Q have a Jacobian with nonsquare #∐ ∐ (if finite). For example, it appears that this density is about 13% for curves of genus 2. The proof makes use of a general result relating global and local densities; this result can be applied in other situations.

Let X be a smooth quasiprojective subscheme of P n of dimension m ≥ 0 over F q . Then there exist homogeneous polynomials f over F q for which the intersection of X and the hypersurface f = 0 is smooth. In fact, the set of such f has a positive density, equal to ζ X (m + 1) −1 , where ζ X (s) = Z X (q −s ) is the zeta function of X. An analogue for regular quasiprojective schemes over Z is proved, assuming the abc conjecture and another conjecture.

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 X 1 (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.

We classify the graphs that can occur as the graph of rational preperiodic points of a quadratic polynomial over Q, assuming the conjecture that it is impossible to have rational points of period 4 or higher. In particular, we show under this assumption that the number of preperiodic points is at most 9. Elliptic curves of small conductor and the genus 2 modular curves X 1 (13), X 1 (16), and X 1 (18) all arise as curves classifying quadratic polynomials with various combinations of preperiodic points. To complete the classification, we compute the rational points on a non-modular genus 2 curve by performing a 2-descent on its Jacobian and afterwards applying a variant of the method of Chabauty and Coleman.

Abstract. We construct examples of families of curves of genus 2 or 3 over Q whose Jacobians split completely and have various large rational torsion subgroups. For example, the rational points on a certain elliptic surface over P 1 of positive rank parameterize a family of genus-2 curves over Q whose Jacobians each have 128 rational torsion points. Also, we find the genus-3 curvewhose Jacobian has 864 rational torsion points.