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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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Hilbert Modular Forms and Iwasawa Theory

Haruzo Hida

Oxford University Press

This chapter proves the torsion of the anticyclotomic Iwasawa module of a (p-ordinary) CM field, and presents an explicit formula of the L-invariant of the CM field, which is a natural generalization of the formula by Ferrero-Greenberg and Gross-Koblitz from the 1970s for imaginary quadratic fields. These results are proven through the comparison theorem (the ‘R=T’ theorem) between the Iwasawa-theoretic version of the universal deformation ring and the universal nearly p-ordinary Hecke algebra over the (infinite) cyclotomic Iwasawa tower. These combined results enable us to compute the adjoint L-invariant of a CM theta family in terms of the U(p)-eigenvalue of the theta family.

Keywords:   nearly ordinary Hecke algebra, adjoint Selmer group, anticyclotomic Iwasawa module, CM fields

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