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

Let f be a newform of weight k and level N with trivial nebentypus. Let p 2N be a maximal prime ideal of the coefficient ring of f such that the self-dual twist of the mod-p Galois representation of f is reducible with constituents φ,ψ. Denote a decomposition group over the rational prime p below p by Gₚ. We remove the condition φ|Gₚ ≠ 1, ω from [CGLS22], and generalize their results to newforms of arbitrary weights. 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 Q in this setting. In particular, non-trivial p-torsion is allowed in the Mordell--Weil group. Using Hida families, we prove a Iwasawa main conjecture for newforms of weight $2$ of multiplicative reduction at Eisenstein primes. In the above situations, we also get p-converse theorems 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.