For zoom details please contact Pieter Moree (moree@mpim-bonn.mpg.de [3])
We study the capitulation problem for the Tate-Shafarevich group of an elliptic curve E. An n-torsion element of Sha can be represented by a smooth genus one curve C embedded in P^{n-1}, that is a counterexample to the Hasse principle. For n odd, we prove that C admits an L-rational point, where L/Q is a degree n number field, with discriminant bounded by a power of the naive height of E. To do this, we extend the invariant theory of genus one models, developed by Cremona, Fisher and Stoll for n<6, and a part of the theory of ring parametrizations, due to Levi-Delone-Faddeev and Bhargava for n<6, to all n.
Links:
[1] https://www.mpim-bonn.mpg.de/taxonomy/term/39
[2] https://www.mpim-bonn.mpg.de/node/246
[3] mailto:moree@mpim-bonn.mpg.de