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Mathematics > Number Theory

arXiv:2302.05181 (math)
[Submitted on 10 Feb 2023]

Title:Bipartite Euler Systems for certain Galois Representations

Authors:Chandrakant Aribam, Pronay Kumar Karmakar
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Abstract:Let $E/\mathbb{Q}$ be an elliptic curve with ordinary reduction at a prime $p$, and let $K$ be an imaginary quadratic field. The anticyclotomic Iwasawa main conjecture, depending upon the sign of the functional equation of $L(E/K,s)$, predicts the behavior of Selmer group of $E/\mathbb{Q}$ along the anticyclotomic tower of $K$. Some of the crucial ideas of Bertolini and Darmon on this conjecture have been abstracted by Howard into an axiomatic set-up through a notion of Bipartite Euler systems, assuming that $E[p]$ is an irreducible representation of $G_{K}$. We generalize this work by assuming only $(E[p])^{G_K}=0$. We use the results of Howard, Nekovář and Castella \emph{et al}., along with those of Mazur and Rubin on Kolyvagin systems to show one divisibility of the anticyclotomic main conjecture, for both the signs. The other divisibility can be reduced to proving the nonvanishing of sufficiently many $p$-adic $L$-functions attached to a family of congruent modular forms.
Comments: arXiv admin note: text overlap with arXiv:1202.6353 by other authors
Subjects: Number Theory (math.NT)
MSC classes: 11G05, 11G40, 11R23
Cite as: arXiv:2302.05181 [math.NT]
  (or arXiv:2302.05181v1 [math.NT] for this version)
  https://doi.org/10.48550/arXiv.2302.05181
arXiv-issued DOI via DataCite

Submission history

From: Pronay Kumar Karmakar [view email]
[v1] Fri, 10 Feb 2023 11:25:52 UTC (26 KB)
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