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Hilbert Modular Forms and Iwasawa Theory$
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Haruzo Hida

Print publication date: 2006

Print ISBN-13: 9780198571025

Published to Oxford Scholarship Online: September 2007

DOI: 10.1093/acprof:oso/9780198571025.001.0001

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HECKE ALGEBRAS AS GALOIS DEFORMATION RINGS

HECKE ALGEBRAS AS GALOIS DEFORMATION RINGS

Chapter:
(p.162) 3 HECKE ALGEBRAS AS GALOIS DEFORMATION RINGS
Source:
Hilbert Modular Forms and Iwasawa Theory
Author(s):

Haruzo Hida

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780198571025.003.0003

The deformation theoretic techniques of Wiles-Taylor were introduced for elliptic modular forms in the introductory Chapter 1, and are generalized to Hilbert modular forms (following Fujiwara's treatment) in this chapter. In particular, Fujiwara's ‘R=T’ theorem (the identification of the Galois deformation ring and the corresponding Hecke algebra) is proven in the minimal case. In addition to the Taylor-Wiles methods, an explicit formula of the L-invariant (of the adjoint L-functions) as well as an integral solution to Eichler's basis problem are presented for Hilbert modular forms.

Keywords:   R=T theorem, Taylor-Wiles system, Adjoint Selmer group, near ordinarity

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