On the anticyclotomic Iwasawa theory of newforms at Eisenstein primes of semistable reduction

Let $f$ be a newform of weight $k=2r$ and level $N$ with trivial nebentypus. Let $\mathfrak{p}\nmid 2N$ be a maximal ideal of the ring of integers of the coefficient field of $f$ such that the self-dual twist of the mod-$\mathfrak{p}$ Galois representation of $f$ is reducible with constituents $\phi ,\psi$. Denote a decomposition group over the rational prime $p$ below $\mathfrak{p}$ by $G_p$. We remove the condition $\phi {\mathrm{\mid }}_{G_p}\mathrm{\ne }\mathbf{1},\omega$ from [CGLS22], and generalize their results to newforms of higher weights $2r$ with $r$ being odd. As a consequence, we prove some Iwasawa Main Conjectures and get the $p$-part of the strong BSD Conjecture for elliptic curves of analytic rank $0$ or $1$ over $\mathbf{Q}$ in this setting. In particular, non-trivial $p$-torsion is allowed in the Mordell–Weil group. Using Hida families, we also prove an Iwasawa Main Conjecture for newforms of weight $2$ of multiplicative reduction at Eisenstein primes. In the above situations, we also get $p$-converse to the theorems of Gross–Zagier–Kolyvagin. The $p$-converse theorems have applications to Goldfeld’s conjecture in certain quadratic twist families of elliptic curves having a $3$-isogeny.