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

TL;DR

This paper advances the understanding of Iwasawa theory for newforms at Eisenstein primes, proving main conjectures and BSD-related results for elliptic curves, especially in cases allowing non-trivial torsion, using Hida families.

Contribution

It generalizes previous results to higher weights and removes certain restrictive conditions, establishing new Iwasawa main conjectures and BSD conjectures for elliptic curves with specific reduction types.

Findings

Proved Iwasawa main conjectures for certain newforms.

Established the $p$-part of the strong BSD conjecture for elliptic curves.

Derived $p$-converse theorems related to Gross--Zagier--Kolyvagin.

Abstract

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 $\phi|_{G_p} \neq \mathbf{1}, \omega$ 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.