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

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