Number Theory seminar: Anna Arnth-Jensen

Title: Exhibiting non-trivial elements of the Tate-Shafarevich group for the Jacobian of a hyperelliptic curve

Speaker: Anna Arnth-Jensen

Abstract: By the Mordell-Weil theorem, the set of K-rational points on the Jacobian of a hyperelliptic curve over a number field K is a finitely generated abelian group. Via descent methods, one may hope to determine the Mordell-Weil group completely. However, the Tate-Shafarevich group constitutes an obstruction to the successful application of this methodology, since there is no known algorithm for computing this group.
In this talk I will show how non-trivial elements of the Tate-Shafarevich group for the Jacobian of a hyperelliptic curve may be exhibited by exploiting information from descents on isogenous abelian varieties. I will present some explicit examples of infinite families of higher genus curves whose Jacobians have a non-trivial Tate-Shafarevich group.