Indivisibility of Heegner points in the multiplicative case

TL;DR

This paper proves the $p$-indivisibility of derived Heegner classes for certain elliptic curves with multiplicative reduction at a prime $p$, confirming a conjecture by Kolyvagin under specific conditions.

Contribution

It extends previous methods to establish $p$-indivisibility of Heegner points in the multiplicative reduction case, broadening the scope of Kolyvagin's conjecture.

Findings

Proves $p$-indivisibility of Heegner classes for multiplicative reduction.

Extends techniques from ordinary to multiplicative reduction cases.

Confirms Kolyvagin's conjecture under new conditions.

Abstract

For certain elliptic curves $E$ over $\mathbb{Q}$ with multiplicative reduction at a prime $p\geq 5$, 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~$p$.