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Capitulation discriminants of genus one curves

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Lazar Radicevic
Mit, 19/01/2022 - 14:30 - 15:30
Parent event: 
Number theory lunch seminar

For zoom details please contact Pieter Moree (

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.

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