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Algebraic functional equation for Hida family

Department of Mathematics, Graduate School of Science, Osaka University, Machikaneyama 1-1, Toyonaka, Osaka 5600043, Japan

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Aprameyo Pal

Mathematics Institute, University of Heidelberg, Im Neuenheimer Feld 288, Heidelberg 69120, Germany

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https://doi.org/10.1142/S1793042114500493Cited by:2 (Source: Crossref)

Abstract

We prove a functional equation for the characteristic ideal of the "big" Selmer group 𝒳(𝒯_ℱ/F_cyc) associated to an ordinary Hida family of elliptic modular forms over the cyclotomic ℤ_p extension of a general number field F, under the assumption that there is at least one arithmetic specialization whose Selmer group is torsion over its Iwasawa algebra. For a general number field, the two-variable cyclotomic Iwasawa main conjecture for ordinary Hida family is not proved and this can be thought of as an evidence to the validity of the Iwasawa main conjecture. The central idea of the proof is to prove a variant of the result of Perrin-Riou [Groupes de Selmer et accouplements; cas particulier des courbes elliptiques, Doc. Math.2003 (2003) 725–760, Extra Volume: Kazuya Kato's fiftieth birthday] by constructing a generalized pairing on the individual Selmer groups corresponding to the arithmetic points and make use of the appropriate specialization techniques of Ochiai [Euler system for Galois deformations, Ann. Inst. Fourier (Grenoble)55(1) (2005) 113–146].

Keywords:

AMSC: 11R23, 11F33, 11F80, 11G05, 11G40

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