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How can we construct abelian Galois extensions of basic number fields?


Author: Barry Mazur
Journal: Bull. Amer. Math. Soc. 48 (2011), 155-209
MSC (2010): Primary 11R04, 18-XX, 20-XX, 23-XX
Published electronically: January 18, 2011
MathSciNet review: 2774089
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Abstract:

Irregular primes--37 being the first such prime--have played a great role in number theory. This article discusses Ken Ribet's construction--for all irregular primes $ p$--of specific abelian, unramified, degree $ p$ extensions of the number fields $ \mathbf{Q}(e^{2\pi i/p})$. These extensions with explicit information about their Galois groups (they are Galois over $ \mathbf{Q}$) were predicted to exist ever since the work of Herbrand in the 1930s. Ribet's method involves a tour through the theory of modular forms; it demonstrates the usefulness of congruences between cuspforms and Eisenstein series, a fact that has inspired, and continues to inspire, much work in number theory.


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Additional Information

Barry Mazur
Affiliation: Department of Mathematics, Harvard University, Cambridge, Massachusetts

DOI: http://dx.doi.org/10.1090/S0273-0979-2011-01326-X
Received by editor(s): September 20, 2009
Received by editor(s) in revised form: January 29, 2010
Published electronically: January 18, 2011
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.