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,. Denote a decomposition group over the rational prime p below p by Gₚ. We remove the condition |₆䂹 1, 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.