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arXiv · 2008.02571

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

Francesc Castella·Giada Grossi·Jaehoon Lee·Christopher Skinner
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Abstract

Let $E/\mathbb{Q}$ be an elliptic curve, and $p$ a prime where $E$ has good reduction, and assume that $E$ admits a rational $p$-isogeny. In this paper, we study the anticyclotomic Iwasawa theory of $E$ over an imaginary quadratic field in which $p$ splits, which we relate to the anticyclotomic Iwasawa theory of characters following the method of Greenberg--Vatsal. As a result of our study, we obtain a proof, under mild hypotheses, of Perrin-Riou's Heegner point main conjecture, as well as a $p$-converse to the theorem of Gross--Zagier and Kolyvagin and the $p$-part of the Birch--Swinnerton-Dyer formula in analytic rank $1$ for Eisenstein primes $p$.

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Francesc Castella, Giada Grossi, Jaehoon Lee, Christopher Skinner. 2020-08-06. On the anticyclotomic Iwasawa theory of rational elliptic curves at Eisenstein primes. https://arxiv.org/abs/2008.02571

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