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

authored by
Timo Keller, Mulun Yin

Let \(f\) be a newform of weight \(k\) and level \(N\) with trivial nebentypus. Let \(\mathfrak{p}\nmid 2N\) be a maximal prime ideal of the coefficient ring 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 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 \(\mathbf{Q}\) 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.

Institute of Algebra, Number Theory and Discrete Mathematics
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E-pub ahead of print