### Finiteness of Selmer groups and deformation rings, by Chandrashekhar Khare, Ravi Ramakrishna

We prove the finiteness of certain Selmer groups
that can be interpreted as tangent spaces of certain
morphisms of deformation rings of 2 dimensional
mod p representations of the Galois group of the rationals.
We also deduce some consequences for the structure of these
deformation rings that are implied by well-known
conjectures of Fontaine and Mazur.

Chandrashekhar Khare, Ravi Ramakrishna <shekhar@math.tifr.res.in, ravi@math.cornell.edu>