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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Modularity of residual Galois extensions and the Eisenstein ideal
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by Tobias Berger and Krzysztof Klosin PDF
Trans. Amer. Math. Soc. 372 (2019), 8043-8065 Request permission


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
  • MR Author ID: 830077
  • Krzysztof Klosin
  • Affiliation: Queens College, City University of New York, Queens, New York 11367
  • MR Author ID: 842947
  • Email:;
  • 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.
  • © Copyright 2019 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 372 (2019), 8043-8065
  • MSC (2010): Primary 11F80; Secondary 11F33, 11R34
  • DOI:
  • MathSciNet review: 4029689