Toward a Generalization of the Gross-Zagier Conjecture

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 ⊂ E(K), one for each prime p that is inert in a fixed imaginary quadratic field K. These subgroups generalize the group ZyK generated by the Heegner point yK ∈ 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 satisfy a higher-rank generalization of the Gross-Zagier formula. We also investigate a higher-rank generalization of a conjecture of Gross-Zagier.