Deformations of induced Galois representations

Abstract

The object of this article is to study the behavior of certain deformation problems and Hecke rings under base change to a real quadratic field. These questions were the subject of conjectures by Doi, Hida, Ishii (see [DHI98] and [Hid98]); in this paper, we will show that Conjecture 2.2 of [Hid98] holds under a suitable hypothesis, and that the isomorphism predicted by Conjecture 3.8 of [DHI98] holds up to a pseudo-null cokernel. The deformations studied in this paper are also interesting in light of the recent work by Skinner and Wiles (see [SW97] and [SW99]) on the deformation of reducible Galois representations. Namely, the ring R + studied in [Hid98] and [DHI98] turns out to be closely related to a pseudodeformation ring; the explication of this connection is crucial to our results. Furthermore, the central conjecture of [Hid98] turns out to be related to a certain class number condition, which is exactly analogous to the condition that appears in [SW97]. Finally, we would like to point out that the deformations considered here give rise to a number of interesting examples. Specifically, we are able to exhibit Λ-adic representations whose traces generate a nontrivial extension of Λ, and with the property that the specializations to certain arithmetic points of weight one are actually ramified over Spec(Λ). This is in contrast to the well-known theorem of Hida, which asserts that arithmetic points of weight k ≥ 2 are smooth over Spec(Λ). To state the results more precisely, we consider a real quadratic extension F = Q( √ D) of Q, with D > 0. Write σ for a generator of the group ∆ = Gal (F/Q) . We fix a prime p > 2, with (D, p) = 1, such that p = pp in F with p = p. Let Fp denote the maximal algebraic extension of F unramified outside p and infinity. Let H = Gal (Fp/F ), and let G = Gal (Fp/Q). Now consider a character φ : H → F×