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

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 $p$.

arXiv commentFinal version, to appear in Invent. Math

References in corpus (7)

• On the Iwasawa invariants of elliptic curves
• On anticyclotomic mu-invariants of modular forms
• Indivisibility of Heegner points in the multiplicative case
• A proof of Perrin-Riou's Heegner point main conjecture
• Three-isogeny Selmer groups and ranks of abelian varieties in quadratic twist families over a number field
• $p$-adic Gross-Zagier formula at critical slope and a conjecture of Perrin-Riou
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Cited by in corpus (5)

• Genus two curves with full $\sqrt{3}$-level structure and Tate-Shafarevich groups
• Non-vanishing of Kolyvagin systems and Iwasawa theory
• An introduction to $p$-adic $L$-functions
• Anticyclotomic $\mu$-invariants of residually reducible Galois Representations
• Main conjectures for non-CM elliptic curves at good ordinary primes