We review some of Kolyvagin's results and conjectures about
elliptic curves, then make a new conjecture that slightly refines
Kolyvagin's conjectures. We introduce a definition of finite index
subgroups W_{p} of E(K), one for each prime p that is inert
in a fixed imaginary quadratic field K. These subgroups generalize
the group Z y_{K} generated by the Heegner point
y_{K} in E(K) in
the case r_{an} = 1. For any curve with r_{an} ≥ 1, we give a
description of W_{p}, which is conditional on truth of the Birch and
SwinnertonDyer conjecture and our conjectural refinement of
Kolyvagin's conjecture. We then deduce the following conditional
theorem, up to an explicit finite set of primes: (a) the set of
indexes [E(K): W_{p}] is finite, and (b) the
subgroups W_{p} with
[E(K): W_{p}] maximal are exactly the subgroups that satisfy a
higherrank generalization of the GrossZagier formula. We also
investigate a higherrank generalization of a conjecture of
GrossZagier.
