Publication details
On the anticyclotomic Iwasawa theory of newforms at Eisenstein primes of semistable reduction
 authored by
 Timo Keller, Mulun Yin
 Abstract
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 selfdual 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, nontrivial \(p\)torsion is allowed in the MordellWeil 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 GrossZagierKolyvagin. The \(p\)converse theorems have applications to Goldfeld's conjecture in certain quadratic twist families of elliptic curves having a \(3\)isogeny.
 Organisation(s)

Institute of Algebra, Number Theory and Discrete Mathematics
 Type
 Preprint
 Publication date
 20.02.2024
 Publication status
 Epub ahead of print