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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Modularity of residual Galois extensions and the Eisenstein ideal

Authors: Tobias Berger and Krzysztof Klosin
Journal: Trans. Amer. Math. Soc. 372 (2019), 8043-8065
MSC (2010): Primary 11F80; Secondary 11F33, 11R34
Published electronically: June 3, 2019
MathSciNet review: 4029689
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Abstract: For a totally real field $ F$, a finite extension $ \mathbf {F}$ of $ \mathbf {F}_p$, and a Galois character $ \chi : G_F \to \mathbf {F}^{\times }$ unramified away from a finite set of places $ \Sigma \supset \{\mathfrak{p} \mid p\}$,
consider the Bloch-Kato Selmer group $ H:=H^1_{\Sigma }(F, \chi ^{-1})$. The authors previously proved that the number $ d$ of isomorphism classes of (nonsemisimple, reducible) residual representations $ {\overline \rho }$ giving rise to lines in $ H$ which are modular by some $ \rho _f$ (also unramified outside $ \Sigma $) satisfies $ d \geq n:= \dim _{\mathbf {F}} H$. This was proved under the assumption that the order of a congruence module is greater than or equal to that of a divisible Selmer group. We show here that if in addition the relevant local Eisenstein ideal $ J$ is nonprincipal, then $ d >n$. When $ F=\mathbf {Q}$ we prove the desired bounds on the congruence module and the Selmer group. We also formulate a congruence condition implying the nonprincipality of $ J$ that can be checked in practice, allowing us to furnish examples where $ d>n$.

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Additional Information

Tobias Berger
Affiliation: School of Mathematics and Statistics, University of Sheffield, Hicks Building, Hounsfield Road, Sheffield S3 7RH, United Kingdom

Krzysztof Klosin
Affiliation: Queens College, City University of New York, Queens, New York 11367

Received by editor(s): October 17, 2018
Received by editor(s) in revised form: March 3, 2019, and March 6, 2019
Published electronically: June 3, 2019
Additional Notes: The first author’s research was supported by EPSRC Grant #EP/R006563/1.
The second author was supported by Young Investigator Grant #H98230-16-1-0129 from the National Security Agency, by Collaboration for Mathematicians Grant #578231 from the Simons Foundation, and by a PSC–CUNY award jointly funded by the Professional Staff Congress and the City University of New York.
Article copyright: © Copyright 2019 American Mathematical Society