## Kodai Mathematical Journal

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

### On a rigidity of some modular Galois deformations

#### Abstract

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 $\mathbf{\rho}$.

#### Article information

**Source**

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

**Dates**

First available in Project Euclid: 31 October 2019

**Permanent link to this document**

https://projecteuclid.org/euclid.kmj/1572487231

**Digital Object Identifier**

doi:10.2996/kmj/1572487231

**Mathematical Reviews number (MathSciNet)**

MR4025757

#### Citation

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