We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
We show that, for any prime p, there exist absolutely simple abelian varieties over 
$\mathbb {Q}$ with arbitrarily large p-torsion in their Tate-Shafarevich groups. To prove this, we construct explicit 
$\mu _p$-covers of Jacobians of curves of the form 
$y^p = x(x-1)(x-a)$ which violate the Hasse principle. In the appendix, Tom Fisher explains how to interpret our proof in terms of a Cassels-Tate pairing.
$7$ in the Tate–Shafarevich group of an elliptic curve
We study the elliptic curves in Cremona’s tables that are predicted by the Birch–Swinnerton-Dyer conjecture to have elements of order 
$7$ in their Tate–Shafarevich group. We show that in many cases these elements are visible in an abelian surface or abelian 3-fold.
We use an invariant-theoretic method to compute certain twists of the modular curves 
$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}X(n)$ for 
$n=7,11$. Searching for rational points on these twists enables us to find non-trivial pairs of 
$n$-congruent elliptic curves over 
${\mathbb{Q}}$, that is, pairs of non-isogenous elliptic curves over 
${\mathbb{Q}}$ whose 
$n$-torsion subgroups are isomorphic as Galois modules. We also find a non-trivial pair of 11-congruent elliptic curves over 
${\mathbb{Q}}(T)$, and hence give an explicit infinite family of non-trivial pairs of 11-congruent elliptic curves over 
${\mathbb{Q}}$.
$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}6$-coverings of elliptic curves
In this paper we give a new formula for adding 
$2$-coverings and 
$3$-coverings of elliptic curves that avoids the need for any field extensions. We show that the 
$6$-coverings obtained can be represented by pairs of cubic forms. We then prove a theorem on the existence of such models with integer coefficients and the same discriminant as a minimal model for the Jacobian elliptic curve. This work has applications to finding rational points of large height on elliptic curves.
