#### Vol. 10, No. 1, 2016

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On the image of the Galois representation associated to a non-CM Hida family

### Jaclyn Lang

Vol. 10 (2016), No. 1, 155–194
##### Abstract

Fix a prime $p>2$. Let $\rho :Gal\left(\overline{ℚ}∕ℚ\right)\to {GL}_{2}\left(\mathbb{I}\right)$ be the Galois representation coming from a non-CM irreducible component $\mathbb{I}$ of Hida’s $p$-ordinary Hecke algebra. Assume the residual representation $\stackrel{̄}{\rho }$ is absolutely irreducible. Under a minor technical condition we identify a subring ${\mathbb{I}}_{0}$ of $\mathbb{I}$ containing ${ℤ}_{p}\left[\left[T\right]\right]$ such that the image of $\rho$ is large with respect to ${\mathbb{I}}_{0}$. That is, $Im\phantom{\rule{0.3em}{0ex}}\rho$ contains $ker\left({SL}_{2}\left({\mathbb{I}}_{0}\right)\to {SL}_{2}\left({\mathbb{I}}_{0}∕\mathfrak{a}\right)\right)$ for some nonzero ${\mathbb{I}}_{0}$-ideal $\mathfrak{a}$. This paper builds on recent work of Hida who showed that the image of such a Galois representation is large with respect to ${ℤ}_{p}\left[\left[T\right]\right]$. Our result is an $\mathbb{I}$-adic analogue of the description of the image of the Galois representation attached to a non-CM classical modular form obtained by Ribet and Momose in the 1980s.

##### Keywords
Galois representation, Galois deformation, Hida family
##### Mathematical Subject Classification 2010
Primary: 11F80
Secondary: 11F85, 11F11
##### Milestones
Received: 7 January 2015
Revised: 6 October 2015
Accepted: 27 November 2015
Published: 14 February 2016
##### Authors
 Jaclyn Lang UCLA Mathematics Department University of California, Los Angeles Box 951555 Los Angeles, CA 90095-1555 United States