Related papers: Visualizing elements of Sha[3] in genus 2 jacobian…

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…

Let $A$ be an abelian variety over a number field $F$ and let $p$ be a prime. Cohen-Lenstra-Delaunay-style heuristics predict that the Tate-Shafarevich group of $A_s$ should contain an element of order $p$ for a positive proportion of…

For an abelian variety A over a number field k we discuss the maximal divisibile subgroup of H^1(k,A) and its intersection with the subgroup Sha(A/k). The results are most complete for elliptic curves over Q.

We give parametrisation of curves C of genus 2 with a maximal isotropic (ZZ/3)^2 in J[3], where J is the Jacobian variety of C, and develop the theory required to perform descent via (3,3)-isogeny. We apply this to several examples, where…

Let $C$ be a $4$-cover of an elliptic curve $E$, written as a quadric intersection in $\mathbb{P}^3$. 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…

Given a genus two curve $X: y^2 = x^5 + a x^3 + b x^2 + c x + d$, we give an explicit parametrization of all other such curves $Y$ with a specified symplectic isomorphism on three-torsion of Jacobians $\mbox{Jac}(X)[3] \cong…

Given a smooth geometrically connected curve $C$ over a field $k$ and a smooth commutative group scheme $G$ of finite type over the function field $K$ of $C$ we study the Tate--Shafarevich groups given by elements of $H^1(K,G)$ locally…

For each prime $p$, we show that there exist geometrically simple abelian varieties $A/\mathbb Q$ with non-trivial $p$-torsion in their Tate-Shafarevich groups. Specifically, for any prime $N\equiv 1 \pmod{p}$, let $A_f$ be an optimal…

We consider the Kolyvagin cohomology classes associated to an elliptic curve $E$ defined over $\mathbb{Q}$ from a computational point of view. We explain how to go from a model of a class as an element of…

We give a formula relating the order of the Brauer group of a surface fibered over a curve over a finite field to the order of the Tate-Shafarevich group of the Jacobian of the generic fiber. The formula implies that the Brauer group of a…

For any number field not containing $\QQ(i),$ we give an explicit construction to prove that there exists an elliptic curve defined over this field such that its Shafarevich-Tate group is nontrivial.

We show that there exist genus one curves of every index over the rational numbers, answering affirmatively a question of Lang and Tate. The proof is "elementary" in the sense that it does not assume the finiteness of any Shafarevich-Tate…

The Shafarevich-Tate group $W (\mathscr{A})$ measures the failure of the Hasse principle for an abelian variety $\mathscr{A}$. Using a correspondence between the abelian varieties and the higher dimensional non-commutative tori, we prove…

We determine the limiting distribution of the normalized Euler factors of an abelian surface A defined over a number field k when A is isogenous to the square of an elliptic curve defined over k with complex multiplication. As an…

Let A be an abelian surface over a fixed number field. If A is principally polarised, then it is known that the order of the Tate-Shafarevich group of A must, if finite, be a square or twice a square. The situation for A not principally…

We will prove that given a genus-2 fibration $f: X \rightarrow C$ on a smooth projective surface $X$ such that $b_1(X)=b_1(C)+2$, the fundamental group of $X$ is almost isomorphic to $\pi_1(C) \times \pi_1(E)$, where $E$ is an elliptic…

In an earlier paper we generalised the notion of the Tate-Shafarevich group of an elliptic K3 surface to the Tate-Shafarevich group of a polarised K3 surface. In the present note, we complement the result by proving that the…

Let (A,\lambda) be a principally polarized abelian variety defined over a global field k, and let \Sha(A) be its Shafarevich-Tate group. Let \Sha(A)_\nd denote the quotient of \Sha(A) by its maximal divisible subgroup. Cassels and Tate…

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…

In this paper we generalize an argument of Neukirch from birational anabelian geometry to the case of arithmetic curves. In contrast to the function field case, it seems to be more complicate to describe the position of decomposition groups…