The fine Tate–Shafarevich group

Abstract

Abstract Within the Tate–Shafarevich group of an elliptic curve E defined over a number field K , there is a canonical subgroup defined by imposing stronger conditions at the places above a given prime p . This group appears naturally in the Iwasawa theory for E . We propose a study of what one can say about the relation to the full Tate–Shafarevich group. Some numerical examples are included, as well as a few conjectures.