On Howard's main conjecture and the Heegner point Kolyvagin system

Abstract

We upgrade Howard's divisibility toward Perrin-Riou's Heegner point Main Conjecture to an equality under some mild conditions. As in the recent result of Burungale, Castella and Kim, we do this by exploiting Wei Zhang's proof of the Kolyvagin conjecture but, unlike their result, we do not restrict ourselves to the case of analytic rank 1 over K. The main ingredient for this is an improvement of Howard's Kolyvagin system formalism. As another consequence of this improvement, we establish the equivalence between this main conjecture and the primitivity of the Kolyvagin system in certain cases, by also exploiting a explicit reciprocity law for Heegner points.