Visualising Sha[2] in Abelian Surfaces

Given an elliptic curve E1 over a number field and an element s in its 2-Selmer group, we give two different ways to construct infinitely many Abelian surfaces A such that the homogeneous space representing s occurs as a fibre of A over another elliptic curve E2. We show that by comparing the 2-Selmer groups of E1, E2 and A, we can obtain information about Sha(E1/K)[2] and we give examples where we use this to obtain a sharp bound on the Mordell-Weil rank of an elliptic curve. As a tool, we give a precise description of the m-Selmer group of an Abelian surface A that is m-isogenous to a product of elliptic curves E1 x E2. One of the constructions can be applied iteratively to obtain information about Sha(E1/K)[2^n]. We give an example where we use this iterated application to exhibit an element of order 4 in Sha(E1/Q).Comment: 17 page

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

Mazur’s rational torsion result for pointless genus one curves:Examples

This note reformulates Mazur’s result on the possible orders of rational torsion points on elliptic curves over Q in a way that makes sense for arbitrary genus one curves, regardless whether or not the curve contains a rational point. The main result is that explicit examples are provided of ‘pointless’ genus one curves over Q corresponding to the torsion orders 7, 8, 9, 10, 12 (and hence, all possibilities) occurring in Mazur’s theorem. In fact three distinct methods are proposed for constructing such examples, each involving different in our opinion quite nice ideas from the arithmetic of elliptic curves or from algebraic geometry

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

Visibility of 4-covers of elliptic curves

Let C be a 4-cover of an elliptic curve E, written as a quadric intersection in P^3. Let E' be another elliptic curve with 4-torsion isomorphic to that of E. We show how to write down the 4-cover C' of E' with the property that C and C' are represented by the same cohomology class on the 4-torsion. In fact we give equations for C' as a curve of degree 8 in P^5. We also study the K3-surfaces fibred by the curves C' as we vary E'. In particular we show how to write down models for these surfaces as complete intersections of quadrics in P^5 with exactly 16 singular points. This allows us to give examples of elliptic curves over Q that have elements of order 4 in their Tate-Shafarevich group that are not visible in a principally polarized abelian surface

Relative Brauer groups of torsors of period two

We consider the problem of computing the relative Brauer group of a torsor of period two under an elliptic curve. We show how this problem can be reduced to finding a set of generators for the group of rational points on the elliptic curve. This extends work of Haile and Han to the case of torsors with unequal period and index. Our results also apply to torsors under higher dimensional abelian varieties. Several numerical examples are given

Explicit n-descent on elliptic curves. III. Algorithms

This is the third in a series of papers in which we study the n-Selmer group of an elliptic curve, with the aim of representing its elements as curves of degree n in P^{n-1}. The methods we describe are practical in the case n=3 for elliptic curves over the rationals, and have been implemented in Magma. One important ingredient of our work is an algorithm for trivialising central simple algebras. This is of independent interest: for example, it could be used for parametrising Brauer-Severi surfaces.Comment: 43 pages, comes with a file containing Magma code for the computations used for the examples. v2: some small edit