Bloch-Kato Groups and Iwasawa Theory in Chabauty-Kim

We explain different kinds of Selmer groups, which are subgroups of Galois cohomology, including Bloch-Kato, strict, and Greenberg Selmer groups. We state part of the Bloch-Kato conjectures and describe a bound joint with A. Betts and M. Leonhardt on the number of rational points on a general higher genus curve, conditional on the Bloch-Kato conjectures. Finally, we explain how to use some Iwasawa theory, specifically Kato’s Euler system and a control theorem of Ochiai, to deduce specific cases of Bloch-Kato associated with elliptic curves.