Kolyvagin's conjecture for modular forms

Abstract

Our main result in this article is a proof (under mild technical assumptions) of an analogue for $p$-adic Galois representations attached to a newform $f$ of even weight $k\geq4$ of Kolyvagin's conjecture on the $p$-indivisibility of derived Heegner points on elliptic curves, where $p$ is a prime number that is ordinary for $f$. Our strategy, which is inspired by work of W. Zhang in weight $2$, is based on a variant for modular forms of the congruence method originally introduced by Bertolini-Darmon to prove one divisibility in the anticyclotomic Iwasawa main conjecture for rational elliptic curves. We adapt to higher (even) weight modular forms this approach via congruences, building crucially on results of Wang on the indivisibility of Heegner cycles over Shimura curves. Then we offer an application of our results on Kolyvagin's conjecture to the Tamagawa number conjecture for the motive of $f$ and describe other (standard) consequences on structure theorems for Bloch-Kato-Selmer groups, $p$-parity results and converse theorems for $f$. Since in the present paper we need $p>k+1$, our main theorem and its applications can be viewed as complementary to results obtained by the first and third authors in their article on the Tamagawa number conjecture for modular motives, where Kolyvagin's conjecture was proved (in a completely different way exploiting the arithmetic of Hida families) under the assumption that $k$ is congruent to $2$ modulo $2(p-1)$, which forces $p<k$. In forthcoming work, we will use results contained in this paper to prove (under analogous assumptions) the counterpart for an even weight newform $f$ of Perrin-Riou's Heegner point main conjecture for elliptic curves ("Heegner cycle main conjecture" for $f$).