  In an earlier article we proved the existence of a canonical Kolyvagin
derivative homomorphism between the modules of Euler and Kolyvagin systems (in
any given rank) that are associated to $p$-adic representations over number
fields. We now explain how the existence of such a homomorphism leads to new
results on the structure of the Selmer modules of Galois representations over
Gorenstein orders and to a strategy for verifying (refinements of) the Tamagawa
number conjecture of Bloch and Kato. We describe concrete applications relating
to the multiplicative group over arbitrary number fields and to elliptic curves
over abelian extensions of the rational numbers.
