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In this paper we show that for every prime $p\ge 5$ the dimension of the $p$-torsion in the Tate-Shafarevich group of $E/K$ can be arbitrarily large, where $E$ is an elliptic curve defined over a number field $K$, with $[K:\mathbb{Q}]$ bounded by a constant depending only on $p$. From this we deduce that the dimension of the $p$-torsion in the Tate-Shafarevich group of $A/\mathbb{Q}$ can be arbitrarily large, where $A$ is an abelian variety, with $dimA$ bounded by a constant depending only on $p$.

In this paper we give a method for calculating the rank of a general elliptic curve over the field of rational functions in two variables. We reduce this problem to calculating the cohomology of a singular hypersurface in a weighted projective $4$-space. We then give a method for calculating the cohomology of a certain class of singular hypersurfaces, extending work of Dimca for the isolated singularity case.