A Grunwald-Wang type theorem for abelian varieties

Let A be an abelian variety over a number field k. We show that weak approximation holds in the Weil-Ch\^atelet group of A/k but that it may fail when one restricts to the n-torsion subgroup. This failure is however relatively mild; we show that weak approximation holds outside a finite set of primes which is generically empty. This proves a conjecture of Lang and Tate that can be seen as an analog of the Grunwald-Wang theorem in class field theory. The methods apply, for the most part, to arbitrary finite Galois modules and so may be of interest in their own right.Comment: Version 3: minor edits to incorporate suggestions of the refere

Generalized Jacobians and explicit descents

We develop a cohomological description of various explicit descents in terms of generalized Jacobians, generalizing the known description for hyperelliptic curves. Specifically, given an integer $n$ dividing the degree of some reduced effective divisor $\mathfrak{m}$ on a curve $C$, we show that multiplication by $n$ on the generalized Jacobian $J_\frak{m}$ factors through an isogeny $\varphi:A_{\mathfrak{m}} \rightarrow J_{\mathfrak{m}}$ whose kernel is naturally the dual of the Galois module $(\operatorname{Pic}(C_{\overline{k}})/\mathfrak{m})[n]$. By geometric class field theory, this corresponds to an abelian covering of $C_{\overline{k}} := C \times_{\operatorname{Spec}{k}} \operatorname{Spec}(\overline{k})$ of exponent $n$ unramified outside $\mathfrak{m}$. The $n$-coverings of $C$ parameterized by explicit descents are the maximal unramified subcoverings of the $k$-forms of this ramified covering. We present applications of this to the computation of Mordell-Weil groups of Jacobians.Comment: to appear in Math. Com

Potential Sha for abelian varieties

We show that the p-torsion in the Tate-Shafarevich group of any principally polarized abelian variety over a number field is unbounded as one ranges over extensions of degree O(p), the implied constant depending only on the dimension of the abelian variety.Comment: Version 2: improved exposition and corrected various small error

Zero-cycles of degree one on Skorobogatov's bielliptic surface

Skorobogatov constructed a bielliptic surface which is a counterexample to the Hasse principle not explained by the Brauer-Manin obstruction. We show that this surface has a $0$-cycle of degree 1, as predicted by a conjecture of Colliot-Th\'el\`ene

Improved rank bounds from 2-descent on hyperelliptic Jacobians

We describe a qualitative improvement to the algorithms for performing 2-descents to obtain information regarding the Mordell-Weil rank of a hyperelliptic Jacobian. The improvement has been implemented in the Magma Computational Algebra System and as a result, the rank bounds for hyperelliptic Jacobians are now sharper and have the conjectured parity

Relative Brauer groups of torsors of period two

We consider the problem of computing the relative Brauer group of a torsor of period 2 under an elliptic curve E. We show how this problem can be reduced to finding a set of generators for the group of rational points on E. This extends work of Haile and Han to the case of torsors with unequal period and index. Several numerical examples are given.Comment: V2: minor errors corrected; appendix adde