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

Fix a prime p > 2. Let ρ : Gal( ¯) GL2(I) be the Galois representation coming from a non-CM irreducible component I of Hida’s p-ordinary Hecke algebra. Assume the residual representation ρ̄ is absolutely irreducible. Under a minor technical condition we identify a subring I0 of I containing p[[T]] such that the image of ρ is large with respect to I0. That is, Imρ contains ker(SL2(I0) SL2(I0a)) for some nonzero I0-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 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.

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