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

TL;DR

This paper formulates and proves a main conjecture for elliptic curves at supersingular primes, extending Perrin-Riou's work and deriving significant new results in $p$-adic number theory and elliptic curve conjectures.

Contribution

It introduces a supersingular analogue of Perrin-Riou's main conjecture and proves it, leading to new $p$-adic formulas and cases of conjectures in elliptic curve theory.

Findings

Proved the supersingular main conjecture under mild hypotheses.

Derived a $ ext{Lambda}$-adic extension of Kobayashi's $p$-adic Gross-Zagier formula.

Established new cases of B.-D. Kim's doubly-signed main conjectures.

Abstract

In 1987, B. Perrin-Riou formulated a Heegner point main conjecture for elliptic curves at primes of ordinary reduction. In this paper, we formulate an analogue of Perrin-Riou's main conjecture for supersingular primes. We then prove this conjecture under mild hypotheses, and deduce from this result a $\Lambda$-adic extension of Kobayashi's $p$-adic Gross-Zagier formula, new cases of B.-D. Kim's doubly-signed main conjectures, and a strengthened version of Skinner's converse to the Gross-Zagier-Kolyvagin theorem for supersingular primes.