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The p -part of Tate-Shafarevich groups of elliptic curves can be arbitrarily large

Remke Kloosterman — 2005

Journal de Théorie des Nombres de Bordeaux

In this paper we show that for every prime p 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 : ] 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 / can be arbitrarily large, where A is an abelian variety, with dim A bounded by a constant depending only on p .

Calculating the Mordell-Weil rank of elliptic threefolds and the cohomology of singular hypersurfaces

Klaus HulekRemke Kloosterman — 2011

Annales de l’institut Fourier

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.

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