Kodai Mathematical Journal

On a rigidity of some modular Galois deformations

Yuichi Shimada

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Let $F$ be a totally real field and $\mathbf{\rho} = (\rho_{\lambda})_{\lambda}$ be a compatible system of two dimensional $\lambda$-adic representations of the Galois group of $F$. We assume that $\mathbf{\rho}$ has a residually modular $\lambda$-adic realization for some $\lambda$. In this paper, we consider local behaviors of modular deformations of $\lambda$-adic realizations of $\mathbf{\rho}$ at unramified primes. In order to control local deformations at specified unramified primes, we construct certain Hecke modules. Applying Kisin's Taylor-Wiles system, we obtain an $R = T$ type result supplemented with local conditions at specified unramified primes. As a consequence, we shall show a potential rigidity of some modular deformations of infinitely many $\lambda$-adic realizations of $\mathbf{\rho}$.

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Kodai Math. J., Volume 42, Number 3 (2019), 526-565.

First available in Project Euclid: 31 October 2019

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Shimada, Yuichi. On a rigidity of some modular Galois deformations. Kodai Math. J. 42 (2019), no. 3, 526--565. doi:10.2996/kmj/1572487231. https://projecteuclid.org/euclid.kmj/1572487231

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