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We describe a qualitative improvement to the algorithms for performing [Formula: see text]-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.

Suppose X is a torsor under an abelian variety A over a number field. We show that any adelic point of X that is orthogonal to the algebraic Brauer group of X is orthogonal to the whole Brauer group of X. We also show that if there is a Brauer–Manin obstruction to the existence of rational points on X, then there is already an obstruction coming from the locally constant Brauer classes. These results had previously been established under the assumption that A has finite Tate–Shafarevich group. Our results are unconditional.

A pair of symmetric bilinear forms A and B determine a binary form $f(x,y) = disc(Ax-By)$. We prove that the question of whether a given binary form can be written in this way as a discriminant form generically satisfies a local-global principle and deduce from this that most binary forms over $\mathbf{Q}$ are discriminant forms. This is related to the arithmetic of the hyperelliptic curve $z^2 = f(x,y)$. Analogous results for non-hyperelliptic curves are also given., Comment: v3: updated references

For a curve over a global field we consider for which integers d the d-primary part of the Brauer group can obstruct the existence of rational points. We give examples showing it is possible that there is a d-primary obstruction for infinitely many pairwise coprime d, and also an example where the odd part of the Brauer group does not obstruct although there is a Brauer–Manin obstruction. These examples demonstrate that a slightly stronger form of a conjecture of Poonen is false, despite being supported by the same heuristic.

Schenkel proved that the automorphism group of a flat Minkowski plane is a Lie group of dimension at most 6 and described planes whose automorphism group has dimension at least 4 or one of whose kernels has dimension 3. We extend these results to the case of toroidal circle planes.

Let X be a smooth double cover of a geometrically ruled surface defined over a separably closed field of characteristic different from 2. The main result of this paper is a finite presentation of the 2-torsion in the Brauer group of X with generators given by central simple algebras over the function field of X and relations coming from the N\'eron-Severi group of X. In particular, the result gives a central simple algebra representative for the unique nontrivial Brauer class on any Enriques surface. An example demonstrating the applications to the study of rational points is given., Comment: v3: Sections 4-6 expanded/rewritten

Let X be a smooth variety over a number field k embedded as a degree d subvariety of $\mathbb{P}^n$ and suppose that X is a counterexample to the Hasse principle explained by the Brauer-Manin obstruction. We consider the question of whether the obstruction is given by the d-primary subgroup of the Brauer group, which would have both theoretic and algorithmic implications. We prove that this question has a positive answer in the case of torsors under abelian varieties, Kummer surfaces and (conditional on finiteness of Tate-Shafarevich groups) bielliptic surfaces. For Kummer varieties we show that the obstruction is already given by the 2-primary torsion. We construct a conic bundle over an elliptic curve that shows that, in general, the answer is no., v3: minor corrections and updated references

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-Thelene.

Given a curve X of the form y^p = h(x) over a number field, one can use descents to obtain explicit bounds on the Mordell-Weil rank of the Jacobian or to prove that the curve has no rational points. We show how, having performed such a descent, one can easily obtain additional information which may rule out the existence of rational divisors on X of degree prime to p. This can yield sharper bounds on the Mordell-Weil rank by demonstrating the existence of nontrivial elements in the Shafarevich-Tate group. As an example we compute the Mordell-Weil rank of the Jacobian of a genus 4 curve over Q by determining that the 3-primary part of the Shafarevich-Tate group is isomorphic to Z/3 x Z/3., Comment: v2: Example 7.5 (version 1) was incorrect and has been removed

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 9-descents on elliptic curves over the rationals., Comment: 45 pages

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-Ch\^atelet 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-Ch\^atelet group. This gives a negative answer to a question of Cassels., Comment: v2: updated/corrected bibliography

We extend existing results characterizing Weil-Ch\^atelet divisibility of locally trivial torsors over number fields to global fields of positive characteristic. Building on work of Gonz\'alez-Avil\'es and Tan, we characterize when local-global divisibility holds in such contexts, providing examples showing that these results are optimal. We give an example of an elliptic curve over a global field of characteristic $2$ containing a rational point which is locally divisible by $8$, but is not divisible by $8$ as well as examples showing that the analogous local-global principle for divisibility in the Weil-Ch\^atelet group can also fail., Comment: v2: incorporates helpful feedback from the community

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. Comp

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 errors

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 added

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., 25 pages. This was part I of arXiv:1306.3251v1; part II will be posted as arXiv:1306.3251v2

For every prime power p^n with p = 2 or 3 and n > 1 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-Ch��telet group can also fail., Magma script embedded in source file

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 referee

Let h be a p-isogeny of elliptic curves. We describe how to perform h-descents on the nontrivial elements in the Shafarevich-Tate group which are killed by the dual isogeny h'. This makes computation of p-Selmer groups of elliptic curves admitting a p-isogeny over Q feasible for p = 5,7 in cases where an isogeny descent is insufficient and a full p-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., Comment: version 2: Definition 2.1 has been corrected; other minor edits and corrections