Indivisibility of Heegner points in the multiplicative case

Abstract

For certain elliptic curves $E$ over $\mathbb{Q}$ with multiplicative reduction at a prime $p\ge 5$, we prove the $p$-indivisibility of the derived Heegner classes defined with respect to an imaginary quadratic field $K$, as conjectured by Kolyvagin. The conditions on $E$ include that $E[p]$ be irreducible and not finite at $p$ and that $p$ split in the imaginary quadratic field $K$, along with certain $p$-indivisibility conditions on various Tamagawa factors. The proof extends the arguments of the second author for the case where $E$ has good ordinary reduction at~$p$.

References in corpus (2)

• On isomorphisms between deformation rings and Hecke rings
• A majority of elliptic curves over $\mathbb{Q}$ satisfy the Birch and Swinnerton-Dyer conjecture

Cited by in corpus (12)

• Iwasawa Main Conjecture for Supersingular Elliptic Curves and BSD conjecture
• A proof of Perrin-Riou's Heegner point main conjecture
• A majority of elliptic curves over $\mathbb{Q}$ satisfy the Birch and Swinnerton-Dyer conjecture
• On the anticyclotomic Iwasawa theory of rational elliptic curves at Eisenstein primes
• The Birch and Swinnerton-Dyer Formula for Elliptic Curves of Analytic Rank One
• The Iwasawa Main Conjecture for elliptic curves at odd supersingular primes
• On the Iwasawa main conjectures for modular forms at non-ordinary primes
• Iwasawa Main Conjecture for Non-Ordinary Modular Forms
• Codimension two cycles in Iwasawa theory and elliptic curves with supersingular reduction
• On Howard's main conjecture and the Heegner point Kolyvagin system
• Beilinson-Kato and Beilinson-Flach elements, Coleman-Rubin-Stark classes, Heegner points and the Perrin-Riou Conjecture
• Kolyvagin's Conjecture and patched Euler systems in anticyclotomic Iwasawa theory