The authors prove Perrin-Riou's Heegner point main conjecture for elliptic curves over the rationals under mild hypotheses, and also prove the Iwasawa-Greenberg main conjecture for Bertolini-Darmon-Prasanna p-adic L-functions.
Iwasawa Main Conjecture for Rankin-Selberg p-adic L-functions
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abstract
In this paper we prove that the p-adic L-function that interpolates the Rankin-Selberg product of a general modular form and a CM form of higher weight divides the characteristic ideal of the corresponding Selmer group. This is one divisibility of the Iwasawa main conjecture for the p-adic L-function. We prove this conjecture using the congruences between Klingen Eisensteinseries and cusp forms on the group GU(3; 1), following the strategy of a recent work of C. Skinner and E. Urban. This theorem can be used to deduce a converse of Gross-Zagier-Kolyvagin theorem and the precise BSD formula in the rank one case.
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2019 1verdicts
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A proof of Perrin-Riou's Heegner point main conjecture
The authors prove Perrin-Riou's Heegner point main conjecture for elliptic curves over the rationals under mild hypotheses, and also prove the Iwasawa-Greenberg main conjecture for Bertolini-Darmon-Prasanna p-adic L-functions.