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# On the Dihedral Main Conjectures of Iwasawa Theory for Hilbert Modular Eigenforms

Published:2012-04-19
Printed: Apr 2013
• Jeanine Van Order,
Section de Mathématiques, Ecole Polytechnique Fédérale de Lausanne, Lausanne 1015, Switzerland
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## Abstract

We construct a bipartite Euler system in the sense of Howard for Hilbert modular eigenforms of parallel weight two over totally real fields, generalizing works of Bertolini-Darmon, Longo, Nekovar, Pollack-Weston and others. The construction has direct applications to Iwasawa main conjectures. For instance, it implies in many cases one divisibility of the associated dihedral or anticyclotomic main conjecture, at the same time reducing the other divisibility to a certain nonvanishing criterion for the associated $p$-adic $L$-functions. It also has applications to cyclotomic main conjectures for Hilbert modular forms over CM fields via the technique of Skinner and Urban.
 Keywords: Iwasawa theory, Hilbert modular forms, abelian varieties
 MSC Classifications: 11G10 - Abelian varieties of dimension $> 1$ [See also 14Kxx] 11G18 - Arithmetic aspects of modular and Shimura varieties [See also 14G35] 11G40 - $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture [See also 14G10]

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