| Speaker: Victor Flynn Affiliation: University of Oxford Time: 14:00 Monday, 12 November, 2018 Location: 303-257 |
| When one attempts to find the rank of an abelian variety over Q (that is: the number of independent infinite-order generators of the group of rational points on the abelian variety), violations of the Hasse principle contribute to an important group in Arithmetic Geometry: the Tate-Shafarevich group. Much is still unknown about this group: even for elliptic curves, it is not known whether or not it is always finite. It was shown by Cassels in 1964 that Tate-Shafarevich groups of elliptic curves over Q can be arbitrarily large (that is: for each n there exists an elliptic curve over Q with at least n elements in its Tate-Shafarevich group). I shall describe how this may be generalised to absolutely simple abelian varieties of arbitrary dimension. |