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

Lang, Jaclyn

doi:10.2140/ant.2016.10.155

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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

DOI: 10.2140/ant.2016.10.155

Abstract

Fix a prime $p > 2$ . Let $ρ : Gal (\bar{ℚ} ∕ ℚ) \to {GL}_{2} (I)$ be the Galois representation coming from a non-CM irreducible component $I$ of Hida’s $p$ -ordinary Hecke algebra. Assume the residual representation $\bar{ρ}$ is absolutely irreducible. Under a minor technical condition we identify a subring $I_{0}$ of $I$ containing $ℤ_{p} [[T]]$ such that the image of $ρ$ is large with respect to $I_{0}$ . That is, $Im ρ$ contains $ker ({SL}_{2} (I_{0}) \to {SL}_{2} (I_{0} ∕ a))$ for some nonzero $I_{0}$ -ideal $a$ . This paper builds on recent work of Hida who showed that the image of such a Galois representation is large with respect to $ℤ_{p} [[T]]$ . Our result is an $I$ -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

Vol. 10, No. 1, 2016