VOL. 86 | 2020 Cyclicity of adjoint Selmer groups and fundamental units
Chapter Author(s) Haruzo Hida
Editor(s) Masato Kurihara, Kenichi Bannai, Tadashi Ochiai, Takeshi Tsuji
Adv. Stud. Pure Math., 2020: 351-411 (2020) DOI: 10.2969/aspm/08610351

Abstract

We study the universal minimal ordinary Galois deformation ρT of an induced representation IndQFφ from a real quadratic field F with values in GL2(T). By Taylor–Wiles, the universal ring T is isomorphic to a local ring of a Hecke algebra. Combining an idea of Cho–Vatsal [CV03] with a modified Taylor–Wiles patching argument in [H17], under mild assumptions, we show that the Pontryagin dual of the adjoint Selmer group of ρT is canonically isomorphic to T/(L) for a non-zero divisor LT which is a generator of the different dT/Λ of T over the weight Iwasawa algebra Λ=W[[T]] inside T. Moreover, defining ε:=(1+T)logp(ε)/logp(1+p) for a fundamental unit ε of the real quadratic field F, we show that the adjoint Selmer group of IndQFΦ for the (minimal) universal character Φ deforming φ is isomorphic to Λ/(ε1) as Λ-modules.

Information

Published: 1 January 2020
First available in Project Euclid: 12 January 2021

Digital Object Identifier: 10.2969/aspm/08610351

Subjects:
Primary: 11F25 , 11F33 , 11F80 , 11R23
Secondary: 11F11 , 11F27 , 11G18

Keywords: adjoint Selmer group , cyclicity , Galois deformation ring , Hecke algebra

Rights: Copyright © 2020 Mathematical Society of Japan

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