On the $p$-converse of the Kolyvagin–Gross–Zagier theorem

Abstract

Abstract. Let A/Q be an elliptic curve having split multiplicative reduction at an odd prime p. Under some mild technical assumptions, we prove the statement:thus providing a 'p-converse' to a celebrated theorem of Kolyvagin-Gross-Zagier.

On exceptional zeros of Garrett–Hida p-adic L-functions

“…As explained in [5], the main result of [1] can be used to prove cases of Conjecture 2.3 when g and h are theta series associated with certain ray class characters of the same imaginary quadratic field in which p is inert (and P and Q are Heegner points). By combining this with an extension of the height computations carried out in [16,17], the article [4] proves instances of Conjecture 1.1 of [6] in this setting.…”

On the structure of Selmer and Shafarevich–Tate groups of even weight modular forms

“…Conjecture A is the higher weight counterpart of Kolyvagin's conjecture for elliptic curves, which was recently proved in many cases by W. Zhang in [32] (see also the paper [30] by Venerucci). From now on we assume Conjecture A.…”

“…Our goal is to compute the integers r ± in Lemma 8.1 and the integers N i in (30). Unfortunately, in passing from Shafarevich-Tate groups to Selmer groups there is a loss of information and we are not able to compute all the N i as we did for X(K, W p ) in Theorem 7.3.…”

The Iwasawa Main Conjectures for ${\rm GL}_2$ and derivatives of $p$-adic $L$-functions

“…K ]]-adic Gross-Zagier formula for Z ∞ is the following. (see [Ven16,(24)]). The constant term L Hi p,0 (f † φ /K) ∈ I vanishes by Proposition 2.2, and so the factorization (5.10) yields the following equality up to unit in O × φ : (5.11)…”