  It is often the case that a Selmer group of an abelian variety and a group
related to an ideal class group can both be naturally embedded into the same
cohomology group. One hopes to compute one from the other by finding how close
each is to their intersection. In this paper we compute the two groups and
their intersection explicitly in the local case and put together the local
information to get sharp upper bounds in the global case. The techniques in
this paper can be used for arbitrary abelian varieties, isogenies and number
fields assuming a frequently occurring condition. Several examples are worked
out for the Jacobians of elliptic and hyperelliptic curves.
