On the image of the Galois representation associated to a non-CM Hida family
نویسنده
چکیده
Fix a prime p > 2. Let ρ : Gal(Q/Q) → 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 Zp[[T ]] such that the image of ρ is large with respect to I0. That is, Im ρ contains ker(SL2(I0) → SL2(I0/a)) for some non-zero 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 Zp[[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.
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