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

Abstract

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 representationρ 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.

Big Image of Galois Representations Associated with Finite Slope p-adic Families of Modular Forms

“…2.4] that, under some technical hypotheses, the image of the Galois representation ρ : G Q → GL 2 (I • ) associated with a non-CM ordinary family θ : T → I • contains a congruence subgroup of SL 2 (I • 0 ), where I • 0 is the subring of I • fixed by certain "symmetries" of the representation ρ. In order to study the Galois representation associated with a non-ordinary family we will adapt some of the results in [La16] to this situation. Since the crucial step ([La16, Th.…”

“…The level of a general ordinary family. We recall the main result of [La16]. Denote by T the big ordinary Hecke algebra, which is finite over Λ = Z p [[T ]].…”

“…In her PhD dissertation (see [La16]), J. Lang improved on Hida's Theorem I. Let T be Hida's big ordinary cuspidal Hecke algebra; it is finite and flat over Λ.…”

“…We suppose for simplicity that I is normal. Consider the Λ-algebra automorphisms σ of I for which there exists a finite order character η σ : G Q → I × such that for every prime ℓ not dividing the level, σ • θ(T ℓ ) = η σ (ℓ)θ(T ℓ ) (see [Ri85] and [La16]). These automorphisms form a finite abelian 2-group Γ.…”

“…In this subsection we prove an analogue of [HT15, Lemma 4.5]. It replaces in our approach the use of Pink's Lie algebra theory, which is relied upon in the case of ordinary representations in [Hi15] and [La16]. Let I • 0 be a domain that is finite torsion free over Λ h .…”