Abstract. For every prime power p n with p = 2 or 3 and n ≥ 2 we give an example of an elliptic curve over Q containing a rational point which is locally divisible by p n but is not divisible by p n . For these same prime powers we construct examples showing that the analogous local-global principle for divisibility in the Weil-Châtelet group can also fail.

Abstract. For every prime p we give infinitely many examples of torsors under abelian varieties over Q that are locally trivial but not divisible by p in the Weil-Châtelet group. We also give an example of a locally trivial torsor under an elliptic curve over Q which is not divisible by 4 in the Weil-Châtelet group. This gives a negative answer to a question of Cassels.

Abstract. We construct unramified central simple algebras representing 2-torsion classes in the Brauer group of a hyperelliptic curve, and show that every 2-torsion class can be constructed this way when the curve has a rational Weierstrass point or when the base field is C 1 . In general, we show that a large (but in general proper) subgroup of the 2-torsion classes are given by the construction. Examples demonstrating applications to the arithmetic of hyperelliptic curves defined over number fields are given.

Let p be a prime and let C be a genus one curve over a number field k representing an element of order dividing p in the Shafarevich-Tate group of its Jacobian. We describe an algorithm which computes the set of D in the Shafarevich-Tate group such that pD = C and obtains explicit models for these D as curves in projective space. This leads to a practical algorithm for performing explicit 9-descents on elliptic curves over Q.

Let ϕ : E → E ′ be an isogeny of prime degree ℓ between elliptic curves defined over a number field. We describe how to perform ϕ-descents on the nontrivial elements in the Shafarevich-Tate group of E ′ which are killed by the dual isogeny ϕ ′ . This makes computation of ℓ-Selmer groups of elliptic curves admitting an ℓ-isogeny over Q feasible for ℓ = 5, 7 in cases where a ϕ-descent on E is insufficient and a full ℓ-descent would be infeasible. As an application we complete the verification of the full Birch and Swinnerton-Dyer conjectural formula for all elliptic curves over Q of rank zero or one and conductor less than 5000.