Heegner Point Kolyvagin System and Iwasawa Main Conjecture

Abstract

In this paper we prove an anticyclotomic Iwasawa main conjecture proposed by Perrin-Riou for Heegner points when the global sign is −1, using a recent work of the author on one divisibility of Iwasawa-Greenberg main conjecture for Rankin-Selberg p-adic L-functions. As a byproduct we also prove the equality for the above mentioned main conjecture under some local conditions, and an improvement of C.Skinner's result on a converse of Gross-Zagier-Kolyvagin theorem.

Perrin-Riou's main conjecture for elliptic curves at supersingular primes

“…Similarly as in [Wan14], Theorem A yields in particular a converse to the Gross-Zagier-Kolyvagin theorem, in the same spirit as the result first obtained by Skinner [Ski14, Thm. B] for ordinary primes (cf.…”

“…Since the Selmer structure F ε introduced above is such that Sel ε,ε (K, T ac ) = H 1 F ε (K, T ac ), the result follows from Lemma 5.14 in the same manner that Proposition 2.2.8 in [How04b] is deduced from [How04b, Lem. 2.2.7] (see also [Wan14,Lem. 3.5]).…”

“…As mentioned above, one of the divisibilities in Perrin-Riou's main conjecture follows from [Ber95] and [How04b]. More recently, the authors established the converse divisibility, leading to a proof of Conjecture 1.1 under mild hypotheses (see [Wan14] and [Cas17]). As for the supersingular setting, our first main result in this paper is the following result on Conjecture 1.2: Theorem A.…”

On the anticyclotomic Iwasawa theory of rational elliptic curves at Eisenstein primes

“…As first observed in [Cas13] and [Wan14] (see also [Cas17,App. ]), when p splits in K, Conjecture B and Conjecture A are essentially equivalent.…”

p‐adic heights of Heegner points and Beilinson–Flach classes

“…As already mentioned, the first result follows easily from the combination of three ingredients: Howard's divisibility in the Heegner point main conjecture, Wan's converse divisibility in the main conjecture for L p (f /K), and an explicit reciprocity law for Heegner points building a bridge between the two. The arguments that follow closely parallel those in [CW16, §4.3], and the interested reader may also wish to consult [Wan14a].…”