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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 Wp of E(K), one for each prime p that is inert
in a fixed imaginary quadratic field K. These subgroups generalize
the group Z yK generated by the Heegner point
yK in E(K) in
the case ran = 1. For any curve with ran ≥ 1, we give a
description of Wp, which is conditional on truth of the Birch and
Swinnerton-Dyer 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): Wp] is finite, and (b) the
subgroups Wp with
[E(K): Wp] maximal are exactly the subgroups that satisfy a
higher-rank generalization of the Gross-Zagier formula. We also
investigate a higher-rank generalization of a conjecture of
Gross-Zagier.
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