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 L∈T 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/(⟨ε⟩−1) as Λ-modules.
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Digital Object Identifier: 10.2969/aspm/08610351