Class groups and local indecomposability for non-CM forms

In the late 1990's, R. Coleman and R. Greenberg (independently) asked for a global property characterizing those $p$-ordinary cuspidal eigenforms whose associated Galois representation becomes decomposable upon restriction to a decomposition group at $p$. It is expected that such $p$-ordinary eigenforms are precisely those with complex multiplication. In this paper, we study Coleman-Greenberg's question using Galois deformation theory. In particular, for $p$-ordinary eigenforms which are congruent to one with complex multiplication, we prove that the conjectured answer follows from the $p$-indivisibility of a certain class group.

1. Introduction 1.1. Overview. As recorded in [GV04, Question 1], R. Greenberg has asked when the 2-dimensional p-adic Galois representation ρ f of Gal(Q/Q) attached to a pordinary cuspidal eigenform f of weight k ≥ 2 has the property of being p-locally split, i.e. its restriction to a decomposition group Gal(Q p /Q p ) at p is isomorphic to the sum of two characters. An equivalent form of this question, which appears to be a very subtle problem in the p-adic theory of modular forms, was independently raised by R. Coleman [Col96, Remark 2, pg. 232]. 1 One easily sees that p-ordinary eigenforms with complex multiplication have this property, and the converse is expected to hold, i.e. (see [Eme97,Conj.  Let Q(f ) ⊂ C be the Hecke field of f . The Galois representation ρ f is valued in GL 2 (E), where E is the completion of Q(f ) at a prime v above p. Serre [Ser68] established (CG) when k = 2 and Q(f ) = Q using Serre-Tate deformation theory. Still in weight 2, Serre's argument was extended independently by Emerton [Eme97] and Ghate [Gha04] provided ρ f is ordinary and p-split for all primes v of Q(f ) above p (we then say that ρ f is totally p-split); the general weight 2 case was recently established by Zhao [Zha14] building on Hida's breakthrough [Hid13]. For weights k > 2, Emerton [Eme97] showed that (CG) follows from a p-adic analogue of Grothendieck's variational Hodge conjecture, provided ρ f is totally p-split. In a different direction, building on modularity lifting results [BT99,Buz03] in weight 1, Ghate-Vatsal [GV04] showed under mild hypotheses that (CG) holds for all but finitely many p-ordinary eigenforms in any single Hida family.
The main result of this paper is Theorem 1.3.1, which gives a sufficient condition for (CG) to hold for all forms in a fixed congruence classf , allowing for any p-adic weight. This condition is that a certain quotient X (later denoted X(ψ − )) of the p-part of the class group of the number field cut out by the associated mod p Galois representationρ f is zero. Such an X can be associated to any congruence class that contains some member with complex multiplication; we impose only mild additional assumptions. We list some examples of vanishing X in §1.8.
Greenberg's pseudo-nullity conjecture [Gre01,Conj. (3.5)] suggests that a certain Iwasawa-theoretic class group X − ∞ (later denoted X − ∞ (ψ − )), which surjects onto X, has finite cardinality. To illustrate the influence of X − ∞ , under an extra assumption, we prove in Theorem 1.4.1 that the finiteness of X − ∞ can be used to produce another proof of the main result of [GV04] for the class ofρ f we consider in this paper.
It is natural to ask whether there exist converse arguments establishing the finiteness of X − ∞ . Thus we give modular characterizations of the vanishing of X − ∞ (Theorem 1.3.4) and its finiteness (Theorem 1.4.4).
1.2. Setup. In order to state question (CG) and the main result of this paper precisely, we introduce the objects of study. Here G F denotes an absolute Galois group of a field F , O F denotes the appropriate standard integer ring of F , and "CM" is short for "complex multiplication." Let p be a prime (later, p ≥ 5).
1.2.1. The question. We fix embeddings of algebraically closed fields Q ֒→ Q q for all primes q, and Q ֒→ C. These embeddings give rise to a choice of q-adic valuation on any algebraic complex number. They also determine a choice of decomposition group G q := Gal(Q q /Q q ) ֒→ G Q and complex conjugation c ∈ G Q . We write I q ⊂ G q for the inertia subgroup. Choose a classical normalized cuspidal Hecke newform f ′ of weight k ≥ 2 and level N ′ ≥ 1. If p ∤ N ′ , let f be a p-stabilization of f ′ of level Γ 0 (p) ∩ Γ 1 (N ′ ); otherwise, let f = f ′ . Thus f is an eigenvector for the U p -operator. Let f = n≥1 a n (f )q n be the q-expansion of f at the cusp ∞, write Q(f )/Q for the subfield of C generated by the coefficients (also the Hecke eigenvalues) a n (f ), and write v = v f for the prime of Q(f ) over p that is distinguished by the embeddings above. We call f p-ordinary when its U p -eigenvalue a p (f ) ∈ C, which is known to be an algebraic integer, is a p-adic unit.
There is attached to f an absolutely irreducible p-adic Galois representation (1.2.1) characterized by the property that (1.2.2) trace ρ f (Frob q ) = a q (f ) for all primes q ∤ N ′ p, where Frob q ∈ G Q is a choice of arithmetic Frobenius element at q. It is known that f is p-ordinary if and only if ρ f | Gp admits a 1-dimensional unramified quotient with Frob p -eigenvalue a p (f ). We call such a representation of G Q , when equipped with the Frob p -eigenvalue, p-ordinary. Similarly, we call a representation ρ of G Q p-locally split when, in addition, ρ| Gp is isomorphic to the direct sum of two characters. We ask the question recorded in §1.1: when k ≥ 2, what property of f determines whether ρ f is p-locally split?
As discussed above, the proposal, denoted (CG), is that such f have CM. While there exist representation-theoretic notions of CM that are arguably more encompassing, we give the simplest equivalent definition: f is called CM when there exists an imaginary quadratic field K/Q such that the attached quadratic Dirichlet character K/Q · satisfies (1.2.3) a n (f ) K/Q n = a n (f ), for almost all n ≥ 1 (the CM condition).
be the associated representation. The Hecke eigenvalues off are determined byρ similarly to (1.2.2). Since f is a p-ordinary eigenform, we know that (1")ρ is odd andρ| Gp admits an unramified quotient with Frob p -eigenvaluē α p := a p (f ).
Our results on (CG) rely on conditions that imply that all Galois representations that give rise toρ arise from Hecke eigenforms, i.e. "R = T." Such R = T-type results are subject to the following assumptions, when p is odd.
1.2.3. The residually CM p-ordinary setting. The following (0)-(4) are the assumptions we work under for the results of this paper.
It follows that there exists an imaginary quadratic field K/Q and a character ψ : G K −→ F × such thatρ ∼ = Ind Q Kψ .
1.3. Results, Part I. Our first main result addresses the representation ρ g : G Q → GL 2 (Q p ) attached to a normalized p-ordinary p-adic eigenform g ∈ Z p [[q]] that has tame level N , arbitrary p-adic weight, and a congruence withf . We refer to whether g has CM by the same definition (1.2.3), which makes sense for any p-adic weight. The theorems in this section are subject to a condition on the following ideal class group. Let ψ − : G K → W × denote the Teichmüller lift ofψ − to the Witt vector ring W = W (F). Let K(ψ − )/K be the finite abelian extension cut out by ψ − , and denote by X(ψ − ) the ψ − -isotypical component of the p-cotorsion of the ideal class group of K(ψ − ). Theorem 1.3.1. Assume (0)-(4) of §1.2. Let g denote a p-ordinary p-adic eigenform of tame level N and arbitrary p-adic weight that is congruent tof . If X(ψ − ) = 0 and ρ g | Gp is split, then g has CM.
We apply the theorem to (CG). Remark 1.3.3. The condition X(ψ − ) = 0 can be ensured analytically in some cases: it is implied by the anti-cyclotomic Katz p-adic L-function L − p (ψ − ) * in §3.2 being a unit (see e.g. [BCG + 19, Cor. 5.2.7]). We also note that the implication (CG) is trivial in the congruence class off unless a different Katz p-adic L-function L − p (ψ − ), also defined in §3.2, is not a unit. Indeed, when L − p (ψ − ) is a unit, any g congruent tof has CM (see Theorem 4.2.2).
In fact, we prove that the vanishing of X(ψ − ) is equivalent to a stronger form of the expected implication (CG). To formulate this, we refer to a modulo p generalized eigenformḡ ′ ∈ F[[q]] whose eigensystem equals that off . We specify these objects in §2.2, also explaining that such aḡ ′ induces a Galois representation where Aḡ′ is a finite-dimensional augmented F-algebra, such that (ρḡ′ mod m Aḡ′ ) ≃ρ and ρḡ′ ≃ρ ⊗ F Aḡ′ .
We also explain that the conditions "p-locally split" and "CM" can be sensibly applied to suchḡ ′ . Theorem 1.3.4. Assume (0)-(4) of §1.2. The following conditions are equivalent.
1.4. Results, Part II. We expect that there are many choices of (K,ψ) such that X(ψ − ) does not vanish, as the results of §1.3 require. The following theorems address the general case. We consider the following Iwasawa-theoretic class group tower over X(ψ − ). Let , and standard arguments about the action of Gal In light of Greenberg's pseudo-nullity conjecture [Gre01, Conj. (3.5)], it is natural to expect that X − ∞ (ψ − ) is finite in cardinality (note that our assumptions rule out trivial zeros). We prove a proportionally weakened version of Theorem 1.3.1 in this case.
Theorem 1.4.1. Assume (0)-(4) of §1.2 and that the class number of K is prime to p. If X − ∞ (ψ − ) has finite cardinality, then there exist at most finitely many ordinary p-adic eigenforms g of tame level N congruent tof such that ρ g | Gp is split and g does not have complex multiplication. In analogy with Theorem 1.3.4, we also can give a modular characterization of the infinitude of X − ∞ (ψ − ). However, a more pleasant criterion applies to a mild generalization 3 for the definition), and is isomorphic to it when p does not divide the class number of K.
Similarly to the mod p case above, to any generalized p-adic eigenform g ′ with eigensystem equal to that of a p-adic eigenform with CM f with coefficient field E/W [1/p], there is associated a Galois representation where A g ′ is a finite-dimensional augmented local E-algebra, such that As before, the conditions of being p-locally split and of being CM can be sensibly applied to ρ g ′ .
(1) X − ∞ (ψ − ) has infinite cardinality. (2) There exists a generalized p-adic eigenform g ′ of tame level N such that: (a) the Hecke eigensystem of g ′ has CM and is congruent tof , (b) g ′ does not have CM, and (c) ρ g ′ | Gp is split. When these conditions are true, then g ′ in (2) may be chosen so that its Hecke span is 2-dimensional, or, equivalently, A g ′ ≃ E[ǫ]/(ǫ 2 ).
1.5. Method of Galois deformation theory. By Hida's influential work [Hid86b], p-ordinary p-adic eigenforms of tame level N with a congruence withf (such as g in the statement of Theorem 1.3.1, for example) are in bijective correspondence with ring homomorphisms T → Q p , where T is the "big" local p-adic Hecke algebra arising from the Hecke action on p-ordinary modular forms of tame level N whose residual Hecke eigensystem is congruent tof . On the other hand, upon assumptions (1") and (2'), there exists a universal p-ordinary deformation ring R ord (constructed by Mazur [Maz89]) parameterizing p-ordinary deformations ofρ. Hida's further result [Hid86a] -that the Galois representations attached to p-ordinary eigenforms interpolate in families -implies that there exists a natural map R ord → T. Under assumptions (1"), (2'), and (3') along with mild local conditions, Diamond [Dia97], following Wiles [Wil95], has shown that this induces an isomorphism R ord ∼ → T. Replacing (1") with (1') so that the expected implication (CG) is not trivial on T, we use a universal Galois deformation ring denoted R spl (constructed by Ghate-Vatsal [GV11]) that parameterizes p-locally split representations of Gal(Q/Q) deformingρ. It follows from the definitions that there is a surjection R ord ։ R spl . Thus, homomorphisms R spl → Q p are in bijection with normalized p-ordinary eigenforms g such that ρ g | Gp is split.
Assuming (0), there exist p-ordinary CM forms congruent tof , resulting in a quotient T ։ T CM , where T CM arises from the Hecke action on these CM forms. The fact that the Galois representations arising from p-ordinary CM eigenforms are p-locally split is reflected in the fact that there exists a surjection R spl ։ T CM fitting in a commutative diagram In terms of this deformation-theoretic picture, our main result is Theorem 5.5.1, which states that the surjection R spl ։ T CM is an isomorphism if and only if X(ψ − ) = 0. Theorem 1.3.1 follows directly from this. The argument for Theorem 1.4.1 is similar, with the addition of commutative algebra arguments set up in §6 and further results on the structure of T reviewed in §4.
Theorem 5.5.1 is deduced from Theorem 5.4.1, which shows that X − ∞ (ψ − ) constitutes the conormal module of Spec(T CM ) ⊂ Spec(R spl ). With this structure of R spl understood, Theorems 1.3.4 and 1.4.4 are applications of R ord ∼ → T and the duality between Hecke algebras and cusp forms.
1.6. A question. One upshot of Theorem 5.5.1 is that (CG) lies somewhat deeper than the simplest possible "big R = T"-type theorem one could hope for, namely, R spl ∼ = T CM . Is there a Hecke algebra that always corresponds to R spl ? What is the module of "p-split" modular forms? We intend to take this up in future work. 1.7. The appendix to this paper. These investigations arose from an attempt to study (CG), for congruence classesρ = Ind Q Kψ as introduced in §1.2.3 above, after restriction of the Galois representations from G Q to G K , using the methods of Wake and the second author [WWE18] to control residually reducible representations. In the process, we realized that some of these arguments amounted to an application of a refined version of Shapiro's lemma to move between deformations of representations of G Q and G K . This is the method that is developed in §5 to prove the key Theorem 5.4.1; in particular, the proof of our results makes no use of the theory of ordinary pseudorepresentations of [WWE18].
Independently and at about the same time as us, Haruzo Hida established similar results to ours by building on [WWE18] as well as his recent work [Hid18a]; see §A.3 for a discussion of the theory of ordinary pseudorepresentations. He has very kindly offered to write his proof of our Theorem 1.3.1 (assuming the class number of K is prime to p) as an appendix to this paper.  [Hid85,pg. 142]), calculated by Maeda or Mestre. They also each satisfy the running assumptions in our paper, because pO K = pp * , ψ − has order at least 3, and ψ is ramified exactly at p. Among these examples, three of them satisfy [K(ψ − ) : K] ≤ 13, so that we found it manageable to calculate K(ψ − ) and its class group using PARI/GP or Magma on a single machine. In each of these three cases, p does not divide the class number of K(ψ − ), so that (ii) is satisfied and Theorem 1.3.1 applies. These examples are The character ω p of G K is the Teichmüller lift of the following characterω p : G K → F × p . Let w := #O × K and letω p : Then, for every multiple a of w, one makes sense of ω a p by taking the (a/w)-th power of the character of G K associated via class field theory to the character To illustrate the example (p, K, ψ) = (13, Q(i), ω 8 p ), we observe that ψ − has order 3 and cuts out the S 3 -extension of Q with minimal polynomial Its class number is 3.
Remark 1.8.1. At the moment, we know of no single example where (ii) fails (which implies that (i) holds), so that the surjection R spl ։ T CM is not an isomorphism and also the conditions of Theorem 1.3.4 are satisfied.
1.9. Acknowledgements. The authors would like to thank Haruzo Hida for helpful discussions on this topic, and for offering to write his results on it as an appendix to this paper. The authors also thank him for providing funding for C.W.E.'s travel to UCLA, where this project was initiated.
The authors also thank Matt Emerton, Ralph Greenberg, Mahesh Kakde, Mark Kisin, Bharath Palvannan, Preston Wake, and Liang Xiao for interesting discussions related to this work, and the anonymous referee for a very careful reading of this paper, whose detailed suggestions helped us improve the exposition of our results.
During the preparation of this work, F.C. was partially supported by NSF grant DMS-1801385; C.W.E. was partially supported by Engineering and Physical Sciences Research Council grant EP/L025485/1; H.H. was partially supported by NSF grant DMS-1464106.
1.10. Notation and conventions. Homomorphisms between profinite topological groups and algebras, and related Galois cohomology modules, are implicitly meant to be continuous.
When F is a number field with a set of places Σ, we let G F,Σ denote the Galois group of F Σ /F , where F Σ is the maximal subextension of Q/F that is ramified only at the places in Σ. Other conventions about Galois groups, such as decomposition groups G q , have been stated in §1.2. We use the case that F = Q and Σ is the set S of places supporting N p∞, thus the Galois group G Q,S . We use G K,S to denote G K,SK , where S K is the set of places of K over S.
When F is either K or Q and T is a G F,S -module, we write C i (O F [1/pN ], T ) for the standard cochain complex of (inhomogeneous) G F,S -cochains valued in T , and H i (O F [1/pN ], T ) for its cohomology. We also use the notation

Ordinary modular forms and Galois representations
In this section, we review background from the theory of p-adic interpolation of p-ordinary modular forms and Galois representations.
2.1. Hida theory. Throughout this paper, we freely refer to the p-adic families of p-ordinary eigenforms constructed by Hida (see [Hid86b,Hid86a]), along with the associated Hecke algebras and big Galois representations. This section summarizes the parts of this theory that we shall apply, following [WWE18,§3] in some of this summary.
We take the dataf ,ρ, and N of §1.2.2 to be fixed in advance.
2.1.1. Ordinary Λ-adic cusp forms and Hecke algebras. For r ≥ 1, let S 2 (Γ 1 (N p r )) ord Zp be the ordinary summand of the Z p -module of cuspidal forms of weight 2 and level N p r with coefficients in Z p . Let the limit being over the natural inclusion maps. Let T ′ be the Z p -algebra generated by the endomorphisms of S ′ Λ given by the Hecke operators (2.1.1) The action of these operators on the modulo p p-stabilized eigenformf gives rise to a maximal ideal of T ′ with residue field F. Let T ′′ denote the completion of T ′ at this maximal ideal. We writeχ for detρ, and χ for the Teichmüller lift ofχ. Using the isomorphism G ab Q ∼ =Ẑ × of class field theory to think of χ as a Dirichlet character on . Likewise, using the projectionẐ × ։ Z × p × (Z/N Z) × , we define the character , that is, we specialize T so that the nebentype on (Z/pN Z) × is constant and equal to χ (as opposed to a non-constant deformation, which is possible when p | φ(N )). Let S Λ := S ′ Λ ⊗ T ′ T; this is the module of p-ordinary Λ-adic cusp forms congruent tof and with nebentype precisely χ, and T the corresponding Hecke algebra. By Hida's control theorem [Hid86b,§3], both T and S Λ are free Λ Q -modules of finite rank, and by [loc. cit., §2] the pairing is a perfect pairing of Λ Q -modules. Consequently, we may view F ∈ S Λ as a Λ-adic where T ′ n = T n for (n, N p) = 1 and, otherwise, T ′ n is the usual polynomial (see e.g. [Shi71,Thm. 3.24]) in the operators of (2.1.1) with coefficients in Z.

Cohomological weights.
We define a p-adic weight to be a characteristic zero height 1 prime P of Λ Q . Any weight arises from a pair of characters (φ k , χ ′ ), In general k is a formal label, but when we start with k ∈ Z, then φ k is the homomorphism φ k (x) := x k−1 . The height 1 prime P = P k,χ ′ ⊂ Λ Q associated to (φ k , χ ′ ) is defined to be the kernel of the factorization of the ring homomorphism By Hida's control theorem, T and S Λ interpolate their classical analogues in cohomological weight. That is, for any p-adic weight (φ k , χ ′ ) with k ∈ Z ≥2 , we recover the module of cusp forms of this weight k and nebentype χ ′ that are congruent tō f via Zp . Similarly, denoting by T k,χ ′ the Hecke algebra generated by the Hecke action on S k,χ ′ , we have a ring isomorphism and the Λ Q -adic duality (2.1.3) specializes modulo P k,χ ′ to thef -congruent part of the classical duality between S k (Γ 1 (N p r ), χ ′ ) and its Hecke algebra.
We will use these consequences of the foregoing theory.
Lemma 2.1.5. There is a bijection between forms in S k,χ ′ ⊗ Λ Q /P k,χ ′ Q p and Λ Qlinear maps T → Q p factoring through T ⊗ Λ Q Λ Q /P k,χ ′ , restricting to a bijection between normalized eigenforms and multiplicative maps.
Lemma 2.1.6. T is reduced.
Proof. This follows from the argument of [Hid15,Lem. 5.4]. Indeed, the nilradical of T k,χ ′ is known to act faithfully on oldforms that are old at levels dividing N according to [Hid86b,Cor. 3.3], and there are no such oldforms in cohomological weight by the assumption that N is the Artin conductor ofρ. Therefore T ⊗ Λ Q Λ Q /P k,χ ′ is reduced for k ∈ Z ≥2 , and since cohomological weights are dense in Spec Λ Q and T is flat over Λ Q , T is reduced.
2.1.3. Associated Galois representations. Hida [Hid86b] proved that the Galois representations ρ f of (1.2.1) associated to p-ordinary cuspidal eigenforms f interpolate along T. Under some assumptions, this interpolation takes on the following particularly strong form. For the statement, we write x f : T → E f ⊂ Q p for the homomorphism associated to a cohomological p-ordinary eigenform f as per Lemma 2.1.5, where E f is the residue field of x f .
Proposition 2.1.7. Upon assumptions (1") and (2') of §1.2, there exists a continuous representation ρ T : G Q −→ GL 2 (T), characterized by the interpolation condition Moreover, ρ T is ramified only at places supporting N p∞ and restricts to G p with form where ν : G p → T × is an unramified character sending an arithmetic Frobenius Frob p to U p and − Q was defined in (2.1.2).
Proof. Then the characterization claim follows from the fact that T is flat over Λ Q and reduced by Lemma 2.1.6, as T therefore injects into the product of the E f .
Also, it follows from the above interpolation and the properties of ρ f that the determinant of ρ T is given by In particular, we have equality of F-valued characters of 2.1.4. Complex multiplication in Hida families. When we impose assumption (0) -i.e., thatρ is induced fromψ -there exist classical p-ordinary eigenforms with CM that are congruent tof and have tame level N . In each cohomological weight (φ k , χ ′ ), these form Hecke submodules The action of T on these submodules in cohomological weight results in a quotient This duality along with the control theorem results in a CM-version of the control in cohomological weights (φ k , χ ′ ), We let I CM := ker(T ։ T CM ), and denote by ρ CM the restriction of ρ T to the CM locus: ρ CM := ρ T ⊗ T T CM .
2.2. Non-classical weights and generalized eigenforms. We will have significant interest in both (i) p-ordinary p-adic cusp forms of non-cohomological weight, and (ii) p-ordinary modulo p cusp forms. In both cases, we also need to define generalized eigenforms and their associated Galois representations.
We define p-ordinary cusp forms of non-cohomological weight by interpolation. These are all implicitly "of tame level N ".
(1) A p-adic p-ordinary cusp form of p-adic weight (φ k , χ ′ ) with a congruence withf is an element of S k,χ ′ := S Λ ⊗ Λ Q Λ Q /P k,χ ′ . (2) A p-ordinary p-adic Hecke eigensystem congruent tof is a homomorphism T → Q p , and its weight (φ k , χ ′ ) is determined by the unique height 1 prime P ⊂ Λ Q through which the composite Λ Q → T → Q p factors.
Remark 2.2.2. Note that S k,χ is equal to the module of classical p-ordinary forms, denoted identically, when the weight is cohomological.
The notions of • Hecke eigenform, • generalized Hecke eigenform, and • CM by K (the condition of (1.2.3)) apply to such objects in the same manner as to their classical counterparts. In particular, Lemma 2.1.5 generalizes straightforwardly to any p-adic weight. Thus the eigensystems from Definition 2.2.1(2) are in natural bijection with normalized eigenforms, i.e., "multiplicity one" holds in the presence of (1")-(3').
For the sake of clarity, we specify the meaning of "generalized eigenform". We use the notation (−)[1/p] as shorthand for (−) ⊗ Zp Q p .
Definition 2.2.3. Let g ′ be p-adic p-ordinary cusp form in S k,χ ′ that is congruent tof . Denote by T[1/p]g ′ the T[1/p]-span of g ′ in S k,χ ′ [1/p]. We call g ′ a generalized eigenform when (i) g ′ is not an eigenform, and (ii) soc(T[1/p]g ′ ) is simple as a T[1/p]-module, where soc(T[1/p]g ′ ) denotes the socle of T[1/p]g ′ as a T[1/p]-module. From such a generalized eigenform, we obtain a p-adic p-ordinary eigensystem T → Q p of weight (φ k , χ ′ ) via the T-action on this socle. Denote by E g ′ the subfield of Q p generated by the image of T in End Qp (soc(T[1/p]g ′ )). We also say that the Hecke eigensystem of g ′ is g when g ∈ S k,χ ′ is a eigenform and also is an E g ′ -basis for soc(T[1/p]g ′ ).
We also define the p-ordinary modulo p cusp forms required for Theorem 1.3.4.
Exactly as in the p-adic case, the definition of eigenform, generalized eigenform, and CM by K are identically formulated in S F . Note, however, that the socle of the Hecke span of an element of S F is always simple and even 1-dimensional over F, being spanned byf . Thus every element of S F is a generalized eigenform with Hecke eigensystem preciselyf .
Finally, we require Galois representations associated to generalized eigenforms by the Hecke action on the Hecke span T[1/p]g ′ of g ′ . Thus we have a natural homomorphism T → A g ′ , and the Galois representation ρ g ′ associated to g ′ is given by The definition for ρḡ′ is formulated identically.
Lemma 2.2.6. There is a canonical structure of augmented E g ′ -algebra on A g ′ , compatible with the maps they receive from T. There is an identical statement for a generalized eigenformḡ ′ ∈ S F in place of g ′ .
℘ g ′ at this residue field. As this completion is naturally endowed with the structure of an augmented local Artinian E g ′ -algebra, this gives A g ′ the same kind of structure.
2.3. The ordinary deformation ring. In this section, we recall an minimal ordinary deformation ring and its comparison to a Hecke algebra.
Recall that we have fixedρ as in §1.3, with coefficient field F, and that W = W (F) is the Witt ring of F. Recall also that we denote the semi-simplification ofρ| Gp byχ 1 ⊕χ 2 , whereχ 2 is assumed to be unramified. We use ≃ to represent isomorphisms of representations up to conjugation, while we use = to denote identical homomorphisms into GL 2 . Finally, recall also the notation G Q,S from §1.10.
Let CNL W denote the category of complete Noetherian local W -algebras A with residue field A/m A ∼ = F. Definition 2.3.1 (The minimal ordinary deformation functor, e.g. [DFG04, §3.1]). Let D ord : CNL W → Sets be the functor associating to A the set of strict equivalence classes of homomorphisms ρ A : A is A-free of rank 1. The "strict" equivalence relation is conjugation by an element of 1 + M 2×2 (m A ) ⊂ GL 2 (A). Note also that #ρ(I ℓ ) = p is equivalent toρ(I ℓ ) having unipotent image.
Deformations ρ A ofρ satisfying the conditions defining D ord will be known as p-ordinary of tame level N , or just p-ordinary.
The term "minimal" refers to conditions (iii) and (iv), while "ordinary" refers to condition (ii). These conditions are well-known to be relatively representable on deformation problems, as follows.
Proposition 2.3.2. The conditions (1") and (2') of §1.2 imply that D ord is representable by R ord ∈ CNL W . In this case, there is a universal ordinary deformation Proof. Upon these conditions, the representability of a deformation ring for conditions (i) and (ii) of Definition 2.3.1 is originally due to Mazur [Maz89, §1.7, Prop. 3]. A simplification of the argument for (ii) applies to show that (iv) is relatively representable as well. It is standard that condition (iii) is relatively representable.
Assuming (1")-(3'), and under some mild additional conditions, one may produce a map R ord → T corresponding to the representation ρ T and prove that it is an isomorphism. This was first done in many cases by Wiles [Wil95], followed by generalizations such as those of Diamond [Dia96,Dia97]. Note, however, that some of these generalizations require modifications to R ord or T. We state here only the case we need, where we assume (0)-(4) of §1.2. In this generality, the isomorphism is due to Wiles [Wil95, Thm. 4.8].
Due to assumption (4), there are no ℓ | N of type (iv) in the sense of Definition 2.3.1; they are all of type (iii). While it is implicit in Theorem 2.3.3 that ρ T satisfies condition (iii), it will be useful later to have seen the following verification.
Proof. Because T is reduced (Lemma 2.1.6), by Lemma 2.1.5 it will suffice to prove the result after replacing ρ T by ρ f for an eigenform f with a cohomological weight (k, χ ′ ) of Λ Q .
We have this addendum to Lemma 2.2.6. Lemma 2.3.5. If we let g denote the eigensystem of g ′ , we have Proof. Since the socle of T[1/p]g ′ is one-dimensional over E g ′ but g ′ is not an eigenform, the Hecke action map T → A g ′ cannot factor through the T-algebra map E g ′ → A g ′ that corresponds to the Hecke action on g. Since R ord , and hence T as well (Theorem 2.3.3), is a quotient of the unrestricted deformation ring of ρ, this means that distinct homomorphisms to A g ′ out of T must correspond to non-isomorphic Galois representations.
2.4. The p-locally split deformation ring. The following deformation problem was first considered by Ghate-Vatsal [GV11].
Definition 2.4.1. Let D spl : CNL W → Sets be the subfunctor of D ord associating to A the set of strict equivalence classes of homomorphisms of the form Deformations ρ A ofρ satisfying the conditions defining D spl will be known as p-split.
Proof. We already know that R ord ∼ → T from Theorem 2.3.3. The canonical surjection R ord ։ R spl arises from Proposition 2.4.2. Because ρ CM is induced via Ind Q K (see Proposition 4.1.2) and p splits in K, ρ CM | Gp is p-split. Thus ρ CM induces a surjection R spl ։ T CM . The commutativity of (1.5.1) is clear.

Anti-cyclotomic Iwasawa theory
In this section, we assemble background information about objects of anticyclotomic Iwasawa theory and their relation to Galois cohomology. We will apply the assumptions (0)-(4) of §1.2 and use the charactersψ andψ − defined there.
3.1. Anti-cyclotomic extensions and Iwasawa algebras. Recall that we assume that pO K = pp * splits, with Q ⊂ Q p inducing p. We have G K,S as in §1.10. Our notation mostly follows [Hid15,pg. 636].
Let C be the prime-to-p conductor ofψ − : G K,S → F × , which is equal to c · c c by assumption (4). Then we consider the following abelian quotients of G K,S : Z = the ray class group of K modulo Cp ∞ , Z − = the maximal quotient of Z where complex conjugation acts as −1, Let F ′ be the subfield of F generated by the values ofψ − , and denote by ψ − : . Then ψ − factors through a character on the quotient Z (p) ]. In the following, we let A choice of section s : Γ − K ֒→ Z − p endows Λ − with the structure of an augmented Λ − -algebra. Moreover, it is free of finite rank, receiving a natural isomorphism where H s /K is the finite p-primary unramified extension of K cut out by the be the canonical characters arising from the projection from the group rings (3.1.1), and denote by Λ − − (resp. Λ − − ) the free Λ − -module (resp. Λ − -module) of rank 1 on which G K,S acts via − − (resp. − − ). In particular, the residual character in both cases isψ − : The following extension fields of K are cut out by the characters ψ − , − − , and − − , respectively: 3.2. Anti-cyclotomic Katz p-adic L-functions. We briefly recall Katz's p-adic L-functions attached to K. In this section we write W for the Witt ring W (F p ) of an algebraic closure of F p . For any prime-to-p ideal C ⊂ O K , Hida-Tilouine [HT93], following work of Katz [Kat78] in the case C = 1, produced an element Thm. II.4.14]) characterized by an interpolation property of critical values of the complex L-functions attached to certain Hecke characters of K modulo Cp ∞ . Taking C to be the prime-to-p conductor ofψ − , we shall be concerned with the projection By the Weierstrass preparation theorem, we may and do fix a choice of The same constructions apply when p is replaced by p * (i.e., starting with µ p * ), Altogether we obtain the following avatars of the Katz p-adic L-functions that we will consider: Since we impose condition (4), the following result gives us that the µ-invariants of these p-adic L-functions (when the coefficient ring is a domain) vanish.
] for some n, where µ p n denotes a p n -th root of unity.
3.3. Anti-cyclotomic Iwasawa class groups. Consider the following metabelian field extensions of K: We have Iwasawa modules coming from Galois groups of these extensions, along with the following integral units in these fields: , and of all of these are known to be finitely generated. They are related by isomorphisms Class field theory then yields the "fundamental" exact sequence of Λ − -modules We apply the main conjecture toward the control of X − ∞ (ψ − ). Proposition 3.3.3. The following equivalences hold.
I have a non-trivial common prime factor P ⊂ I of characteristic zero.
Proof. The equivalences of (i) (and the leftmost equivalence of Proposition 3.3.2) follow from Nakayama's lemma. For example, to the L-functions in the statement of (i) follows from Proposition 3.3.2 and its variant for the module Y − ∞ (ψ − ) * obtained by swapping the roles of p and p * . To . Because X is a quotient of Y, and we know from Proposition 3.2.2 that the µ-invariant of Y is zero, Lemma 3.3.4 below implies that X has a non-zero p-torsion-free quotient.
Therefore X [1/p] is a non-zero Λ[1/p]-module. By examining a choice of presentation (3.1.2), we see that is a regular ring. Therefore X [1/p] is supported at some maximal ideal of ( Λ/I)[1/p] for some choice of irreducible component Spec(I) ⊂ Spec( Λ − ). Since we know that X is a quotient of both Y and Y * (whose characteristic ideals on each I are associated to L − p (ψ − ) I by Proposition 3.3.2), this means that Char I (Y I ) and Char I (Y * I ) have a common factor. By Proposition 3.3.2 this is a common factor of Proof. Because Z is finitely generated and p-power torsion, there exists some t ∈ Z ≥1 such that p t · Z = 0. Because of the surjections ·p s : Z/p ։ p s Z/p s+1 Z, the infinitude of Z implies that Z/p is infinite. Because Λ − /p is generated over Λ − /p by adjoining finitely many nilpotents (via a choice of presentation (3.1.2)), the same argument implies that Z/p is infinite. As Z is supported on Spec(Λ − /p) ⊂ Spec(Λ − ), this means that the µ-invariant of Z as a Λ − -module is positive.
3.4. Galois cohomology with support, and duality. In this section, we compute some Galois cohomology groups often known as "Iwasawa cohomology," relating them to the Iwasawa-theoretic objects defined in §3.3. We follow the approach of [WWE18,§6] and parts of [WWE17, §2], using the notation for Galois cohomology established in §1.10.
We will make use of the modules Λ − − , Λ − − equipped with the canonical characters defined in (3.1.3), and respectively denote by the same underlying modules equipped with the inverse of those characters. Let S ′ ⊂ S K denote some subset of places of K. We will study the cohomology , which is defined to be the cohomology of the cone of the morphism of complexes This gives rise to the standard long exact sequence in cohomology, whose terms in a single degree are We see that we have H i The following module-theoretic version of global Tate duality will be useful.
Proposition 3.4.2. Let T a free module of finite rank over a complete local Noetherian Z p -algebra R that is Gorenstein. Equip T with an R-linear action of G K,S . Let V denote a finitely generated R-module (with a trivial G K,S -action). Then there is a spectral sequence where T * denotes the R-linear dual module with the contragredient G K,S -action.
Proof. This follows directly from [WWE17, Prop. 2.2.1] when R is regular and S ′ ∈ {S K , ∅}. We explain how to adapt the proof of loc. cit. to prove this proposition. The generalization to an arbitrary subset S ′ ⊂ S follows from the fact that classical Poitou-Tate duality (i.e. for T a finite abelian group and T * its Pontryagin dual) holds for an arbitrary S ′ ⊂ S. For this, see e.g. [ , which is an expression of this duality in the derived category of R-modules. In this setting, T may be a bounded complex and T * is a bounded complex representing RHom R (T, ω R ), where ω R is a dualizing complex for R. In our statement, R is assumed to be Gorenstein (thus one may let ω R be R[0]) and T is R-free, so we may use the standard R-linear dual module T * . The , is canonically isomorphic to the fundamental exact sequence (3.3.1). In particular, we have isomorphisms The proof technique is similar to that of [WWE18,§6], which applies when Q is replaced by K.
Lemma 3.5.4. There are canonical isomorphisms The isomorphism (3.5.3) follows just as in the proof of [WWE18, Cor. 6.3.1]. Namely, because ψ − is non-trivial at all primes of K dividing N , and is clearly not congruent modulo p to Z p (1), taking the ψ − -component of the long exact sequence appearing in the statement of [WWE18, Cor. 6.1.3] results in the desired isomorphism.
Similarly, we have the Kummer isomorphism with respect to which the natural maps H 1 1)) = 0. By local Tate duality ("derived" as in Proposition 3.4.2, which can be applied with R = Λ − since this ring is a complete intersection, given its presentation (3.1.2)), the vanishing of H 2 (K p , Λ − # (1)) follows from the fact that H 0 (K p , Λ − − /I) = 0 for all ideals I ⊂ Λ − . It remains to establish (3.5.2) compatibly with the isomorphisms we have already drawn. Using the proof of [Lim12, Prop. 5.3.3(b)] (which is written for S ′ = S K , but applies to any choice of S ′ , such as S ′ = {p}), we find that Because ψ − has order prime to p and is non-constant on G q for all primes q of K r dividing N p * , we deduce Because taking the ψ − -part kills the contribution of the cokernel of , we know that A r,m is canonically isomorphic to the group of ψ − -equivariant homomorphisms from the absolute Galois group of We observe that H 1 (K q ,ψ − ) = 0 for q | N follows from assumption (4); likewise, ker(H 1 (K p * ,ψ − ) → H 1 (K unr p * ,ψ − )) = 0 follows from assumption (2). It follows that triviality of an element of A 1,1 = H 1 (O K [1/N p],ψ − ) at the decomposition group at q | N p * is equivalent to being trivial on the inertia group at q. It is straightforward to generalize this conclusion to general K r and m ≥ 1 from this base case (K 1 = K and m = 1), as K r /K is ramified only at p. By definition of . Applying this isomorphism to the limits over m and r above, we deduce (3.5.2).
To complete the proof of Proposition 3.5.1, it remains to check that the connecting map in (3.4.1) is compatible with the map U − ∞ (ψ − ) → Y − ∞ (ψ − ) coming from the Artin symbol, and that the map from H 2 (p) to H 2 in (3.4.1) is compatible with . This is standard, so we omit it.
4.1. CM Hecke algebras and associated Galois representations. The point of this section is to study the structure of the CM Hecke algebra T CM , a quotient of T which we defined in §2.1.4. This will mainly be applied in §6. We do this by understanding the relation of T CM to Galois representations. Recall that Spec(T CM ) ⊂ Spec(T) is the minimal closed subscheme containing all of the irreducible components of T with CM by K, and ρ CM = ρ T ⊗ T T CM denotes the restriction of ρ T to this CM locus. Recall that c ⊂ O K denotes the prime-to-p Artin conductor of ψ : G K,S → W × .
We will also use the notation for anti-cyclotomic Iwasawa theory established at the beginning of §3.1. We add to it the following definitions. Let K cp ∞ denote the ray class field of K modulo cp ∞ , with ray class group Z. Let Z p denote the maximal pro-p quotient of Z, which is also naturally a direct factor. Also let Γ p K ≃ Z p be the maximal torsion-free quotient of Z p .
We see that ψ factors through a character on the quotient Z (p) := Z/Z p , resulting in a projection sending a group-like element (z p , z (p) ) ∈ Z to ψ(z (p) )z p ∈ W the natural characters arising from projection G K,S ։ Z and π ψ (resp. also via Λ ։ Λ). Each of Λ and Λ are complete local Noetherian W -algebras with residue field F, and these two characters are residually equal toψ. Similarly to Definition 2.3.1, a deformation ψ A ofψ to A ∈ CNL W is called minimal at a prime q of K if reduction modulo m A induces an isomorphism ψ A (I q ) ∼ →ψ(I q ). It is standard (see e.g. [Maz89, §1.4]) that Λ with − is a universal deformation ofψ as follows.
In particular, T CM is a reduced complete intersection.
Proof. As pointed out in the proof of [Hid15, Prop. 5.7(2)], since we are working in the minimal case (the tame level of our forms is equal to the prime-to-p conductor ofρ) this claim follows immediately from Lemma 4.1.1 as long asρ is induced only from K among all quadratic fields. By Proposition 5.2(2) in loc. cit., assumption (3) of §1.2 implies this.
There is a notion of a Zariski-closed maximal induced locus for Ind Q K in Spec R, where R ∈ CNL W supports a Galois representation ρ R : G Q,S → GL 2 (R) deforminḡ ρ = Ind Q Kψ . (See, for example, [DW18].)  Proof. The first statement follows from (2.1.9), as Proposition 4.1.2 tells us that A presentation of Λ Q as a power series ring W [[t]] arises from t → γ Q −1, where γ is any element of I p that projects to a generator of the Galois group of the maximal cyclotomic Z p -extension of Q. From the presentation of Λ given above, and the equality (4.1.5), we see that Λ Q → Λ is an isomorphism if and only if γ − 1 ∈ Λ Q maps to a power series generator of Λ if and only if γ maps to a generator of Z p . This is the case if and only if I p ∼ → I p ⊂ G K,S surjects onto Z p , which is equivalent to p ∤ h K .

4.2.
Congruence module of the CM locus. We recall Hida's determination of the characteristic ideal of the congruence module of the CM locus Spec(T CM ) ⊂ Spec(T).
For this, and for the further study of non-induced deformations of induced representations in §5, we identify how anti-cyclotomic objects over Λ − W ′ set up in §3 (like L − p (ψ − )) are presented over Λ.
Notation. In §3 only, we denoted Λ − W ′ , Λ − W ′ without the subscript. Elsewhere, the relationship between the two notations is as in (4.2.1). We mildly abuse notation by continuing to use − − (resp. − − ) for the base change of this character (as defined in §3.
They are induced by the canonical isomorphism ι :  Moreover, we have the following commutative diagram with exact rows and columns: Proof. This is shown in [Hid15, Thm. 7.2], building on the proof originating from [MT90] of the anti-cyclotomic main conjecture (Proposition 3.3.2). There we find the additional assumption thatψ is ramified at p and p ∤ φ(N ). However, the first assumption is used only in order to apply [Hid15, Thm. 7.1] and ensure that T is a Gorenstein ring. In our setting, this follows from Theorem 2.3.3. The assumption p ∤ φ(N ) is used to rule out the failure of minimality of CM families, but our assumptions guarantee minimality.

Computation of conormal modules using Shapiro's lemma
In this section, we give an explicit interpretation of the conormal module of the closed CM locus inside the p-ordinary (resp. p-locally split) locus. From this, we deduce the main theorem (Theorem 1.3.1) in §5.5. 5.1. Conormal modules. Assume (0)-(4) of §1.2 in all that follows. We will study the conormal modules of the closed subspaces Remark 5.1.3. Strict equivalence classes within D * ρ amount to conjugacy classes by 1 + M 2 (V ) ⊂ GL 2 ( Λ[V ]), which is why it is non-trivial to take the image in D * .
Proof. Let ρ V represent a strict equivalence class in D * that is the image of a strict equivalence classes in D * ρ . Then ρ V (mod V ) ≃ ρ and det ρ V = − Q . The first condition is equivalent to the map φ ρV : R * → Λ[V ] being induced by ρ V composing with Λ[V ] ։ Λ to produce φ ρ . By examining (2.1.9), we see that the second condition is equivalent to R * → Λ being a Λ Q -algebra homomorphism. Conversely, any strict equivalence class in D * ( Λ[V ]) that satisfies both conditions contains a representative ρ V of a strict equivalence class in D * ρ ( Λ[V ]), and it is clear that such a class is unique.
We also record the relationship between the Hom CM (R * , Λ[V ]), which follows directly from the surjections R ord ։ R spl ։ Λ. Notation. We will write ρ V for a homomorphism . We also mildly abuse terminology by speaking of a deformation ρ V , when really this is the strict equivalence class of ρ V , and refer to ρ V as an element of D * ρ ( Λ[V ]) for D * ρ ∈ {D ord ρ , D spl ρ , Dψ ,ρ }. Next we find these ρ V as elements of an Ext 1 -module.
Lemma 5.1.5. For any finitely generated Λ-module V and R * ∈ {R ord , R spl , Λ}, there exists a Λ-linear injection of ) to the extension class determined by the surjection One may then readily check that the kernel of ρ V ։ ρ is isomorphic to ρ ⊗ Λ V (where V has a trivial G Q,S -action). Then the map to Ext 1 is injective because strict equivalence in D * ρ amounts to conjugation by 1 + M 2 (V ). The fact that this map is Λ-linear is a functorial (in V ) version of the standard fact (see e.g. [Maz89,pg. 399]) that the tangent space of a deformation ring R ρ with residue field k is given, as a k-vector space, to Hom(R, k[ǫ]/ǫ 2 ), and admits a canonical isomorphism of k-vector spaces to Ext 1 k[G Q,S ] (ρ, ρ).

5.2.
Local conditions. Next we address the local conditions that define the deformation problems D ord , D spl , thereby determining the images of the injections of Lemma 5.1.5. We will decompose the condition on the constancy of the determinant of Lemma 5.1.2 into a sum of local inertial conditions. First we address conditions at p. As we have seen, ρ| GK,S ≃ ψ ⊕ ψ c . Because p splits in K (and recall that we have designated p such that G p ∼ → G p ), we also have this decomposition of ρ| Gp . The characters remain distinct after restriction to both G K,S and G p becauseψ| Gp =χ 1 =χ 2 =ψ c | Gp , by the assumptions of §1.2. Therefore, restriction to G K,S induces a canonical map (5.2.1) (where the matrix stands for the direct sum of its entries). For 1 ≤ i, j ≤ 2, write σ p i,j for the projection to the (i, j)-th coordinate of the target of σ p . Likewise, write τ p i,i for the composition of σ p i,i with Lemma 5.2.2. Let V be a finitely generated Λ module.
(1) The ordinary condition and I p -constant determinant condition on the target of σ p are cut out by the kernel of σ p 2,1 ⊕ τ p 1,1 ⊕ τ p 2,2 . (2) The split condition and I p -constant determinant condition on the target of σ p are cut out by the kernel of σ p 2,1 ⊕ σ p 1,2 ⊕ τ p 1,1 ⊕ τ p 2,2 .
Proof. This computation of the ordinary condition amounts to the study of ordinary deformation rings appearing in [Maz89, §1.7, pg. 401], and a straightforward generalization to D spl . We provide more detail, and address the inertial determinant condition. A choice of V -valued cocycles e = a b c d representing a cohomology class in the codomain of σ p may be represented as Next we address the conditions at primes ℓ | N . This is fairly simple, as we have noted that the off-diagonal cohomology is trivial at ℓ in the proof of Lemma 2.3.4. We set up the maps σ ℓ , σ ℓ i,j , and τ ℓ i,j just as for the prime p, above. Lemma 5.2.4. Let ℓ | N be a prime. The condition of minimality at ℓ is cut out by the kernel of τ ℓ 1,1 ⊕ τ ℓ 2,2 . Proof. This condition is part (iii) of Definition 2.3.1. As the codomains of σ ℓ i,j are zero for (i, j) ∈ {(1, 2), (2, 1)}, only the conditions cut out by τ ℓ 1,1 , τ ℓ 2,2 remain. Thus we have determined the image of the injections of Lemma 5.1.5.
Corollary 5.2.5. Let V be a finitely generated Λ-module.
(1) The image of

5.
3. An explicit form of Shapiro's lemma. Because ρ ∼ = Ind Q K − (see Proposition 4.1.2), we can apply Shapiro's lemma to the domain of (5.2.1) to yield that We need to relate this isomorphism to (5.2.1). For this, we develop, in this section, an explicit version of Shapiro's lemma for this particular case. In order to state it, we use the notation (−) c on an extension class as follows, extending the notation for representations of G K established in §1.2.3: When ρ 1 , ρ 2 are representations of G K and e ∈ Ext 1 GK (ρ 2 , ρ 1 ) is an extension class represented by the short exact sequence 0 −→ ρ 1 −→ ρ e −→ ρ 2 −→ 0, then we write e c ∈ Ext 1 GK (ρ c 2 , ρ c 1 ) for the extension class of 0 −→ ρ c 1 −→ ρ c e −→ ρ c 2 −→ 0.
Using the canonical isomorphism between these Ext-groups and group cohomology, we also use the notation (−) c for the map Similarly, choosing matrix-valued representatives for the ρ i and choosing some cocycle a ∈ Z 1 (O K [1/N p], ρ * 2 ⊗ ρ 1 ), we may use the notion of (−) c that applies to homomorphisms: a c (γ) = a(cγc) for γ ∈ G K,S .
We next show that these are compatible.
Lemma 5.3.1. With notation as above, if we write ρ a for the extension of ρ 2 by ρ 1 induced by the cohomology class of a, then the cohomology class of a c corresponds to the extension class of ρ c a . Proof. Using the matrix valued representatives, we can write ρ a as a homomorphism ρ 1 ρ 1 · a ρ 2 and observe that ρ c a is represented by the homomorphism For notational convenience, in the statement of Proposition 5.3.2 we use in place of − .
is injective, and its image is given by Proof. Shapiro's lemma tells us that σ K is injective. Choose e = a b c d in the group of cocycles whose cohomology class lies in the codomain of σ K ; for example, b ∈ Z 1 . This is a function e : G K,S → M 2×2 (V ) that determines the homomorphism ρ e : G K,S → GL 2 ( Λ[V ]) (similar to (5.2.3)) given by It extends to a function on G Q,S = G K,S G K,S c that we denote byρ C e , given bỹ ρ C e : G K,S c ∋ γc → ρ e (γ) · C ∈ GL 2 ( Λ[V ]) (so, in particular,ρ C e (c) = C), where C ∈ GL 2 ( Λ[V ]) has order 2 and satisfies C ≡ 1 1 (mod V ).
We observe that the set of lifts of ρ to Λ[V ] is in bijection with the set of pairs (e, C) such thatρ C e is a homomorphism. We break the determination of the homomorphism condition onρ C e into cases.
, we claim thatρ C e is a homomorphism if and only if a c = d and b c = c, as cocycle functions G K,S → V .
In order to carry out this reduction, we need a bit of additional notation. Write A complement to C ⊂ pgl 2 ⊗ V is 0 w w 0 . Conjugatingρ C e by 1 + 0 1 1 0 w fixes ρ C e (c) = C, fixes a and d, and which maintains the equality b c = c. Altogether, we have calculated that lifts of ρ to Λ[V ] are in bijection with the Λ-module where e and C are defined as The quotient is naturally isomorphic to the claimed image of σ K .
Using the foregoing expression of Shapiro's lemma, we calculate Hom CM (R * , Λ[V ]). Write H p for the p-primary summand of the ideal class group of K.
Proposition 5.3.4. For any finitely generated Λ-module V , there are isomorphisms Proof. We apply throughout the interpretation of Hom CM (R * , Λ[V ]) in Lemma 5.1.2. Thus our goal is to calculate the image of the injections of Lemma 5.1.5, which are determined by Corollary 5.2.5. So it remains is to interpret the conclusion of Corollary 5.2.5 in terms of Proposition 5.3.2. We use the notation of Galois cohomology instead of Ext 1 . For convenience, when v is a rational prime dividing N p and * = ij for i, j ∈ {1, 2}, we use the natural extensions of σ v * and τ v * to the codomain of σ K : these are σ v * , τ v * , where v is the prime over v distinguished by the embeddings of §1.2.1.
We also use the isomorphism of Shapiro's lemma as given by the top row of σ K : The map v|N p τ v 1,1 ⊕ τ v 2,2 factors through the summand H 1 (O K [1/N p], V ) of the codomain of (5.3.5), yielding Using the equivalence a c | Ip = 0 ⇐⇒ a| I p * = 0, we find that these are V -valued homomorphisms factoring through H p . This establishes the final claimed isomorphism, as deformations induced from K are split upon restriction to K. For the first claimed isomorphism, we calculate the ordinary case. Similarly to the previous paragraph, σ p 2,1 factors through the summand H 1 of the codomain of (5.3.5), yielding Let l be a prime of K over N . It follows from the cohomology calculation in the proof of Lemma 2.3.4 that H i (K l ,ψ − ) = 0 for all i ≥ 0. Therefore, the local factors over N of the long exact sequence in cohomology (3.4.1) arising from the cone construction (with S ′ the set of primes of K dividing N p * and T = Λ − # ) are trivial. Likewise, for the local factors over p, we have H 0 (K p ,ψ − ) = H 0 (K p * ,ψ − ) = 0, so there are no local terms in degree zero in this long exact sequence. Also, b c | Gp = 0 if and only if b| G p * = 0. Therefore, the kernel of (5.3.6) is canonically isomorphic has kernel naturally isomorphic to the direct sum of the two kernels above. This gives the first isomorphism. The argument for the second is essentially identical. We replace σ p 2,1 with σ p 1,2 ⊕ σ p 2,1 , which also factors through the summand H 1 of the codomain of (5.3.5). This factorization is Therefore the kernel of σ p 1,2 ⊕ σ p 2,1 ⊕ v|N p τ v 1,1 ⊕ τ v 2,2 is naturally isomorphic to the direct sum of the two kernels from the factorization. Then, (3.4.1) computes this group by the same argument as before, where S ′ is now the set of primes of K dividing N p. Now we can interpret maps out of the conormal modules of the CM locus in the ambient ordinary or split deformation space.
Corollary 5.3.7. For any finitely generated Λ-module V , we have canonical isomorphisms Proof. We claim that the injections

5.4.
Interpretation as class groups. We arrive at the identification of the conormal modules. We apply the map δ of (4.2.1), usually restricting it from its domain as functors on finitely generated Λ-modules. Because both J/J 2 and Λ are finitely generated as Λ-modules, Yoneda's lemma implies the result (i). The proof of (ii) is essentially the same. Because H i (O K [1/N p], Λ − # (1)) = 0 for i > 2, the duality spectral sequence of Proposition 3.4.2 yields By Proposition 3.5.1, we can replace H 2 The rest of the proof proceeds as in the proof of (i). Proof of Theorems 1.3.4 and 1.4.4. It follows from Proposition 2.4.2 that the plocally split condition is well-defined on the Galois representations associated to generalized eigenforms g ′ ,ḡ ′ , even though their coefficient rings are not domains. Thus condition (c) of the theorems is equivalent to the map T → Aḡ′ (resp. T → A g ′ ) factoring through T ։ R spl .
Similarly, as we have noted that the CM condition is well-defined on generalized eigenforms in §2.2, the "not CM" condition (b) of both theorems is equivalent to the map T → Aḡ′ (resp. T → A g ′ ) not factoring through T ։ T CM .
Case of Theorem 1.4.4. Assume that X − ∞ (ψ − ) is infinite, which is equivalent to Then as a Λ Q -module (where this module structure arises from β : Λ Q → Λ discussed in Lemma 4.1.4), X has support on some height 1 prime P ⊂ Λ Q . By Proposition 3.3.3(ii), P has characteristic zero; hence P = P k,χ ′ for some p-adic weight (k, χ ′ ).
Let E = E k,χ ′ denote the residue field of P k,χ ′ , which is a finite extension of Q p . We now consider the surjection with square-zero kernel By Theorem 5.4.1, its kernel surjects onto X ⊗ Λ Q E, which is non-zero. Because T CM ⊗ Λ Q E is a finite product of finite extension fields over E, it has some factor E x = (T CM ⊗ Λ Q E)/m x with the following property: letting m ′ x be the kernel of the surjection from ( be the corresponding square-zero extension of E x . Then we may factor We now recall the discussion of generalized eigenforms and their attached Galois representations from §2.2. The composite T ։ R spl ։ A x corresponds (via the duality of Lemma 2.1.5) to a p-adic p-ordinary generalized eigenform g ′ of p-adic weight (k, χ ′ ) with eigensystem corresponding to the composite T ։ A x ։ E x . The corresponding Galois representation ρ g ′ : G Q,S → GL 2 (A x ) arising as ρ g ′ := ρ T ⊗ T A x has the following properties: (a) The eigensystem induced by T → E x has CM and is congruent tof , because it factors through T ։ T CM . (b) g ′ does not have CM, because T → A x cannot factor through T CM : indeed, by Theorem 5.4.1, if it did factor, then X must vanish when projected to A x . But T → A x has been constructed so that it does not have this property.
These are the properties (a), (b), and (c) of Theorem 1.4.4. We have also arranged for A x ≃ E x [ǫ]/(ǫ 2 ), as claimed.
For the converse, note that if g ′ inducing T → A g ′ arises from the action on a generalized eigenform with properties (a), (b), and (c), then (a) implies that the composite map T → A g ′ → E g ′ ∼ = A g ′ /m g ′ to the residue field of A g ′ amounts to an eigensystem that has CM, (b) implies that this map does not factor through T ։ T CM , and (c) implies that this map does factor through T ։ R spl .
Consider the image A ⊂ A g ′ of R spl , which is a local ring that is not a field (by (a) and (b)). Writing m A ⊂ A for its maximal ideal, we consider the induced map R spl ։ A/m 2 A . Its restriction to J s factors through J s /(J s ) 2 , and (b) implies that its image is non-zero. Since this image is a Z p -submodule of a Q p -vector space, we deduce from Theorem 5.4.1 that X − ∞ (ψ − ) is infinite.
Case of Theorem 1.3.4. The proof of this case is essentially the same. The only difference is that F plays the role of both E and E x , while T CM ⊗ Λ Q F is an Artinian local F-algebra. Then the surjection of Artinian local algebras R spl ⊗ Λ Q F ։ T CM ⊗ Λ Q F induces a surjection of the square-zero extension quotients. By Theorem 5.4.1 and by letting V = F in Proposition 5.3.4, this surjection is (in the notation of Proposition 5.3.4). It is straightforward to deduce the result from here, using arguments analogous to the case of Theorem 1.4.4.

Commutative algebra
In this section, we set up a proposition from commutative algebra and deduce Theorem 1.4.1.
6.1. A proposition using the resultant. The following lemma summarizes the theory of the resultant that we will require. Lemma 6.1.1. Let R be a domain, and let F (y), G(y) ∈ R[y] be polynomials. There is a resultant π ∈ R of F (y) and G(y) with the following properties.
(1) π = 0 if and only if F (y) and G(y) have a non-constant common factor.
In the following proposition, we refer to the generic rank of a module M over a domain R. This is defined to be the Frac(R)-dimension of M ⊗ R Frac(R).
Proposition 6.1.2. Let R be a complete Noetherian regular local ring. Let S be an augmented reduced local R-algebra that is finitely generated and torsion-free as an R-module. Let T be an augmented local R-algebra quotient of S, and denote by K the kernel of T ։ R.
Assume that K/K 2 is supported in codimension at least 2 as an R-module. Then T has generic rank equal to 1.
Proof. For this proof, given an augmented R-algebra R ֒→ A ։ R, we denote by A c the R-module complement to the summand R ⊂ A determined by the augmented R-algebra structure. That is, we have a canonical isomorphism of R-modules A ∼ = R ⊕ A c . We note that A has generic rank 1 if and only if A c is R-torsion; we will implicitly use this equivalence in this proof.
Denote by J the kernel of S ։ R, and choose a minimal set G of generators for the ideal J, which is also a minimal set of generators for S as an R-algebra. Choose an element y ∈ G and write S ′ y ⊂ S, T ′ y ⊂ T for the sub-R-algebras generated by y. We observe that S ′ y → T ′ y is a morphism of augmented R-algebras. We claim that it suffices to prove that T ′ y has generic rank 1 for all y ∈ G. Indeed, consider these product algebras with an augmented R G -algebra structure where the additional rightmost arrow is the diagonal projection homomorphism. We also have a natural map y∈G T ′ y ։ T lying over the diagonal projection, inducing a surjection of R-modules Thus we observe that T has generic rank 1 if and only if T ′ y has generic rank 1 for all y ∈ G.
Now we have J = (y). Note that J/J 2 is a torsion R-module generated by y (mod J 2 ). Indeed, if this were not the case, let m ≥ 2 be minimal such that is a monic polynomial of minimal degree satisfied by y, then y m | P (y) because J i /J i+1 is free of rank one for i < m. Thus y · P (y) is a nilpotent element of S, contradicting our assumption that S is reduced. Observe that J/J 2 is a cyclic R-module, generated by y, and isomorphic as an R-module to J/J 2 ∼ −→ R (F 1 (0), . . . , F n (0)) .
We claim that there exist a pair of polynomials F (y), G(y) in the set {F 1 (y), . . . , F n (y), G 1 (y), . . . , G r (y)} such that R/(F (0), G(0)) is supported in codimension 2. This follows directly from the assumption that K/K 2 is supported in codimension 2. We note that R[y] (y · F (y)) , R[y] (y · F (y), y · G(y)) are naturally augmented local R-algebras with augmentation ideal generated by y, and with a surjective augmented R-algebra map to S and T , respectively. Therefore, it suffices to replace S and T with these algebras. Indeed, having done this, we observe that J/J 2 is torsion and K/K 2 is supported in codimension 2. We define , the quotient of T by (F (y)), but note that T ′ is not an augmented R-algebra.
Because the kernel of T ։ T ′ is a cyclic R-module (generated by F (y)), and we know that T has generic rank at least 1, it will suffice to show that T ′ is a torsion R-module.
Thus we want to show that π = 0. By Lemma 6. 1.1(1), it suffices to prove that F (y) and y · G(y) do not have any non-constant common factors. Assume, for the sake of contradiction, that there exists such a divisor H(y) ∈ R[y]. We may assume that H(y) is irreducible and monic, since both F (y) and y · G(y) are monic. We see that H(y) = y, because F (0) = 0. Next, note that H(0) is not a unit in R, because if H(y) | F (y) with quotient Q(y), then S ∼ = R[y]/(y · H(y) · Q(y)) would not be a local ring (consider S/m R S). Then H(0) | F (0) and H(0) | G(0). This contradicts the fact that R/(F (0), G(0)) is finite, as it surjects onto the non-finite R/(H(0)).

6.2.
Proof of Theorem 1.4.1. We will apply Proposition 6.1.2 to R spl in order to prove Theorem 1.4.1.
Proof. We see that the conclusion of the lemma will follow from verifying that the assumptions of Proposition 6.1.2 about (R, S, T, K) are satisfied by where the augmented Λ Q -algebra structure of R ord ∼ = T is understood to be defined by the ideal J ∼ = I CM .
Recall from Lemma 4.1.4 the sequence of homomorphisms There, we see that these induce isomorphisms Λ Q ∼ → Λ ∼ → Λ if and only if p ∤ h K . Thus we apply the assumption p ∤ h K and identify Λ Q ∼ → T CM ∼ = Λ, treating T ։ R spl as a morphism of augmented Λ Q -algebras.
All of the assumptions of Proposition 6.1.2, except the one that J s /(J s ) 2 is supported in codimension at least 2, are satisfied by the properties of T checked in §2, especially Lemma 2.1.6. We will show that the remaining property follows from the assumption that X − ∞ (ψ − ) is finite in cardinality. For R = Λ Q , an R-module is supported in codimension 2 if and only if it has finite cardinality. By Theorem 5.4.1, there is an isomorphism Then the tensor product operation ⊗ Λ − W ′ Λ preserves the finite cardinality property of these modules.
Proof of Theorem 1.4.1. By Lemma 6.2.1, we know that the assumptions of Theorem 1.4.1 imply that R spl has generic rank 1 as a Λ Q -module.
Because the locus Spec(T CM ) ⊂ Spec(T) parameterizes exactly the CM p-adic eigenforms congruent tof , it follows from the constructions of §4.2 that the map x g : T → Q p of Lemma 2.1.5 corresponding to a p-adic eigenform g (congruent tō f ) factors through T ։ T nCM if g does not have CM. We also know that ρ g is p-locally split if and only if x g factors through T ։ R spl . Thus it will suffice to show that R sn := T nCM ⊗ T R spl is torsion as a Λ Q -module.
Since we have already deduced that R spl has generic rank 1, it suffices to show that the kernel of R spl ։ R sn has generic rank 1. In view of Theorem 4.2.2, we want to show that the kernel I nCM ⊂ T of T ։ T nCM , injects into R spl under T ։ R spl . But this follows from the same theorem, as we see there that I nCM injects under the composite quotient map T ։ R spl ։ T CM ∼ = Λ Q , with torsion cokernel.  (4), and the finiteness of X − ∞ (ψ − ). There, the authors use the fact that the ideal of ( * ) ⊂ T generated by the image of G p under the " * " of (2.1.8) cuts out the quotient T ։ R spl . Our method hinges on the study of maximal square-zero augmented T CM -algebra quotients of T (resp. R spl ) over Λ Q . We found in Theorem 5.4.1 that this maximal quotient is . So our method relies on detecting " * " in the conormal module

Appendix. Local indecomposability via a presentation of the Hecke algebra by Haruzo Hida
A.1. Summary. Let p ≥ 5 be a prime. In this appendix, we give a proof of Greenberg's conjecture ((CG) in the main text) of local indecomposability of a non-CM residually CM Galois representation based on the presentation of the universal ring given in [Hid18a] (so, the proof is different from the one given in the main text). We impose an extra assumption (H3-4) in addition to the set of the assumptions made in the main text (we list our set of assumptions as (H0-4) below). We use the notation introduced in the main text. For each Galois representation ρ of G K , we write K(ρ) = Q Ker(ρ) for the splitting field of ρ. We fix an algebraic closure F Assuming T = Λ, the minimal presentation we found in [Hid18a] has the following form: Here Let P be a prime factor of p in K(ρ) (the splitting field of ρ). Write the image of U (p) in T as u. Writing the local Artin symbol [x, K p ] (identifying K p = Q p ), for the residual degree f of P, the semi-simplification of ρ T ([p, Theorem A.1.2. Let the notation be as above. Assume (H0-4) and σ = id on T. Let I p be the wild p-inertia subgroup of Gal(K(ρ T )/Q) for the splitting field K(ρ T ) of ρ T . Then we have a decomposition I p = U ⋊ Gal(Q ∞ /Q) for the Z pextension Q ∞ /Q, where U is an abelian group mapped by ρ T into the unipotent radical of a Borel subgroup in GL 2 (T) whose logarithmic image u = Lie(U) (in the nilpotent Lie Λ-algebra T) is equal to Θ · Λ 1 . In short, we have an isomorphism This theorem supplies us with a very explicit unipotent element ( 1 Θ 0 1 ) in the image of ρ T with (ΘT ⊗ W W) ∩ Λ W = (L − p (ψ − )); therefore, we can answer the question of Greenberg: Corollary A.1.3. Assume (H0-4) and σ = id on T. For all prime divisors P ∈ Spec(T nCM ) with associated Galois representation ρ P , the following conditions are equivalent: (1) the Galois representation ρ P is completely reducible over the inertia group I p at p, (2) P ∈ Spec(T nCM ) ∩ Spec(T CM ), (3) P |(L − p (ψ − )Λ W ∩ Λ). As described in the main text, from [Eme97] and [Gha05,Prop. 11], the above corollary implies: Corollary A.1.4 (Coleman's question). Assume (H0-4). For every classical modular form f of weight k ≥ 2 and of level N with residual representation ρ, write g for the p-critical stabilization of the primitive form associated to f . Then g is in the image of (q d dq ) k−1 if and only if f has complex multiplication. A.2. Presentation of a Galois deformation ring. For a set Q of Taylor-Wiles primes satisfying the conditions (Q0-10) in [Hid18a, § §3-4], we write K(ρ) (pQ) for the maximal p-profinite extension of K(ρ) unramified outside {p} ⊔ Q. We simply write K(ρ) (p) for K(ρ) (pQ) if Q = ∅. Let G Q := Gal(K(ρ) (pQ) /Q) and H Q := Gal(K(ρ) (pQ) /K) with G = G ∅ and H = H ∅ . We first note that G Q = Gal(K(ρ) (pQ) /K(ρ))⋊Gal(K(ρ)/Q) and H Q = Gal(K(ρ) (pQ) /K(ρ))⋊Gal(K(ρ)/K) as p > 2 and p ∤ [K(ρ) : Q]. We fix such a decomposition; so, Gal(K(ρ)/Q) ∼ = ∆ G for a subgroup ∆ G of Gal(K(ρ) (p) /Q). Write ∆ ⊂ ∆ G for the subgroup isomorphic to Gal(K(ρ)/K); so, [∆ G : ∆] = 2.
Let N = DN K/Q (c ′ ). Let h Q be the big Hecke algebra described in [Hid18b,§1] for each Q. We have a local ring T Q of h Q whose residual representation is isomorphic to ρ. Let ρ Q : G Q → GL 2 (T Q ) be the Galois representation of T Q such that Tr(ρ Q (Frob l )) for primes l outside {l|N p} ⊔ Q is given by the image in T Q of the Hecke operator T (l). On T Q , we have an involution σ with the property that It is known that T Q and T Q CM are reduced algebras finite flat over Λ. Further we have an algebra decomposition In the above notation, if Q = ∅, we remove the superscript or subscript Q from the notation. If σ is the identity on T, we have Strictly speaking, the patching argument is given in [Hid18a] under the following extra assumptions: (h2) N := DN K/Q (c ′ ) for an O-ideal c ′ prime to D with square-free N K/Q (c ′ ) (so, N is cube-free), (h3) p is prime to N l|N (l − 1) for prime factors l of N .
Here is the reason why we can remove these two assumptions: We studied the minimal deformation problem in [Hid18a] over the absolute Galois group G Q , but as was explained in [DFG04,pg. 717], under the condition that p ∤ |ρ(I l )| (which holds in our case), all minimal deformations factor through G, and considering the deformation problem over {G Q } Q for appropriate sets Q of Taylor-Wiles primes satisfying [Hid18b, §3 (Q0-8)], every argument in the proof of [Hid18a, Thm. 5.4] goes through for the above choice of T Q (as easily checked), and thus we obtain the theorem. Indeed, we used (h3) in [Hid18a] just because the universal minimal ordinary Galois representation of prime-to-p conductor N (considered in [Hid18a]) factors through G; so, just imposing deformations to factor through G the arguments simply work; so, we do not need to assume (h3). The condition (h2) is assumed to guarantee the big Hecke algebra is reduced, but again, all deformations over G has prime-to-p conductor equal to N which is equal to the prime-to-p conductor of its determinant (the Neben character). Then, by the theory of new forms, the Hecke algebra is reduced if its tame character has conductor equal to the tame level; so, we do not need (h2).
Since σ acts trivially on T CM = T/(Θ), writing ρ := (ρ T mod (Θ)), we find ρ ∼ = ρ ⊗ χ for χ = K/Q . Note that ρ is a minimal deformation of ρ; so, it factors through G. Thus by [DHI98,Lem. 3.2] applied to G = G and H = H (under the notation of the lemma), we find ρ ∼ = Ind Q K Ψ for a character Ψ : H → T CM,× unramified outside c ′ p deforming ψ. Let Γ p be the Galois group over K(ρ) of the maximal p-abelian extension of K inside K(ρ) (p) unramified outside p. By We identify the two rings. Since p ∤ [K(ρ) : K], there exists a class field K(p)/K in K(ρ) (p) with Gal(K(p)/K) ∼ = Γ p by Artin symbol. Define a character Φ : . Then Φ factors through H. Since (Λ, Φ) for the character Φ : H → Λ × is a universal pair for the deformation problem of ψ unramified outside pc ′ over the group H, we have a canonical surjective algebra homomorphism Λ ։ T CM inducing Ψ. By the same argument which proves [Hid18a, Cor. 2.5], this is an isomorphism. We record this fact as Recall G = Gal(K(ρ) (p) /Q) and H = Gal(K(ρ) (p) /K). Let ρ A : G K → GL 2 (A) be a minimal p-ordinary deformation of ρ for a p-profinite local W -algebra A with residue field F. The representation ρ A factors through G by minimality (so, hereafter, we consider the deformation problem over G). By p-ordinarity, we have where m A is the maximal ideal of the local ring A. This gives rise to an exact sequence ǫ A ֒→ ρ A ։ δ A . Realize sl 2 (A) inside the A-linear endomorphism algebra End A (ρ A ), and write F + (ρ A ) the subspace of {T ∈ sl 2 (A)|T (ǫ) = 0} = Hom A (δ A , ǫ A ) on which Ad(ρ A ) acts by the character ǫ A /δ A (the upper nilpotent Lie subalgebra if ρ A | Gp has upper triangular form as above). Write Ad(ρ A ) * for the Galois module Ad(ρ A ) ⊗ A A ∨ for the Pontryagin dual A ∨ of A, where G Q acts on the factor Ad(ρ A ). Similarly we put F + (ρ A ) * := F + (ρ A ) ⊗ A A ∨ which is a p-local Galois module. Then we define (A.3) for the product of restriction maps to the inertia group I l ⊂ G of l. In the Galois e i Ee i = A and put B = e 1 Ee 2 and C = e 2 Ee 1 . Then a generalized matrix algebra over A is a pair of an associative A-algebra E and E. It is isomorphic to A ⊕ B ⊕ C ⊕ A as A-modules; so, we write instead (E, E) = ( A B C A ) which we call a GMA structure. There is an A-linear map B ⊗ A C → A such that the multiplication in E is given by 2-by-2 matrix product. In this case, A is called the scalar subring of (E, E) and (E, E) is called an A-GMA. A Cayley-Hamilton representation with coefficients in A and residual representation ψ 0 0 ψ c (with this order ψ at the top) is a homomorphism ρ : H → E × , such that (E, E) is an A-GMA, and such that in matrix coordinates, ρ is given by σ → with (ρ 11 (σ) mod m A ) = ψ(σ), (ρ 22 (σ) mod m A ) = ψ c (σ), and ρ 12 (σ)ρ 21 (σ) ≡ 0 mod m A . For a given ρ, if we change the set E of idempotents, the matrix expression changes; so, we added the superscript E to the matrix entries ρ E ij to indicate its dependence on E. If the input of E is clear from the context, we omit the superscript E.
In H, we have two conjugacy classes of the p-decomposition groups depending on prime factors of p in K. Fix a decomposition subgroup D p ⊂ H for p and put D p * for p * . We define p-ordinarity (resp. p * -ordinarity) of ρ to have E (resp. E * ) such that ρ E 12 (σ) = 0 for all σ ∈ D p and ρ E 22 (I p ) = 1 (resp. ρ E * 21 (σ) = 0 for all σ ∈ D p * and ρ E * 11 (I p * ) = 1). We say ρ is ordinary if it is p and p * -ordinary at the same time. This definition does not depends on the choice of D p and D p * . For example, if we replace D p by σD p σ −1 , (E, ρ(σ)Eρ(σ) −1 ) satisfies the required conditions.
If (E, E) can be embedded into the matrix algebra M 2 ( A) for a complete local W -algebra A with residue field F containing A, the Cayley-Hamilton representation ρ : H → E × can be regarded as a representation into GL 2 ( A). Since ρ = Ind Q K ψ is irreducible over G, we may have an extension ρ of the GMA representation ρ to G. If an extension ρ exists, the extension is a usual representation into GL 2 ( A). As usual, we call ρ p-ordinary if ρ| Gp ∼ = ( ǫ * 0 δ ) with unramified δ ≡ ψ c mod m A . The ordering of the residual representation ψ 0 0 ψ c (with this order ψ at the top) is fixed; so, plainly, to have compatibility of ordinarity of ρ over H and Q-ordinarity of ρ (and to preserve residual order of the characters ψ and ψ c ), we need to define p * -ordinarity to have a set of idempotent E * so that ρ E * | D * p in the lower triangular form. Indeed, if ρ(c) = ( 0 1 1 0 ), ρ is p-ordinary for E if and only if ρ is p * -ordinary for the same E by choosing D p * = cD p c −1 . As we describe in the following proposition, this phenomenon occurs if we take ρ := ρ T | H for A = T + and A = T. Details of the deformation theory of ρ in the category of representations over G and in the category of Cayley-Hamilton representations over H will be discussed in a forthcoming paper [Hid19]. Proposition A.3.1. The Galois representation ρ = ρ T | H associated to T restricted to H is an ordinary Cayley-Hamilton representation with values in the following T + -GMA given by Θb ⊗ Θc → θbc for θ = Θ 2 (the product in T).
is the eigenspace under the conjugation action of ρ T (τ ) with eigenvalue ψ − (τ ) (resp. ψ − (τ ) −1 ). Thus our expression of ρ T | H is associated to (E, e 1 , e 2 ). By ordinarity of ρ T on G p (inducing D p ), we see ρ T | H is p-ordinary. Plainly c ∈ G interchanges e 1 and e 2 ; i.e., ρ T (c)e 1 ρ T (c) = e 2 . Thus over D p * = cD p c, we conclude ρ T | H with values in (E, E) is also p * -ordinary. Since the residual representation is exactly ψ 0 0 ψ c (with this order ψ at the top), the choice of (e 2 , e 1 ) is impossible violating the residual order of the characters (the definition of p * -ordinarity is lower triangular on D p * to accommodate to preserve this residual order). Therefore we need to choose E = (e 1 , e 2 ) for p * -ordinary.
Under the normalization as above, we may and do assume that ρ T (c) = ( 0 1 1 0 ).
A.4. Local Iwasawa theory. Let k/Q p (inside Q p ) is a Galois extension with p ∤ [k : Q p ]. Write F/k for the cyclotomic Z p -extension inside Q p . Let Γ := Gal(F/k) = γ Zp and put Γ n = Γ p n . Set F n := F Γn with p-adic integer ring o n . Let L (resp. L n ) be the maximal abelian p-extension of F (resp. F n ). Write X n := Gal(L n /k n ) and X := Gal(L/F ). We have Gal(F/Q p ) = Gal(F/Q p ) ⋉ X. The exact sequence 1 → X → Gal(L/k) → Γ → 1 is split just by lifting γ to an element γ ∈ Gal(L/k) taking splitting image γ Zp . Therefore the commutator subgroup of Gal(L/k n ) is given by (γ p n − 1)X, and we have the corresponding exact sequence at each level n: 1 → X/(γ p n − 1)X → Gal(L n F/F ) → Γ n → 1.
Let k ∞ /k be the unramified Z p -extension inside Q p with its n-th layer k n , and put F n = F k n . Let L (resp. L n ) be the maximal abelian p-extension of F ∞ (resp. F n ). Set X := Gal(L/F ∞ ). Pick a lift φ ∈ Gal(L/k) of the Frobenius element [p, Q p ] f (for the residual degree f of k/Q p ) generating Gal(k ∞ F/k) and a lift γ ∈ Gal(L/k) of the generator γ of Gal(kQ p,∞ /k 0 ) = Γ. The commutator τ := [φ, γ], acts on X by conjugation, and (τ − 1)x := [τ, x] = τ xτ −1 x −1 for x ∈ X is uniquely determined independent of the choice of γ and φ. Define L ′ ⊂ L and L ′ n ⊂ L n by the fixed field of (τ −1)X (i.e., the fixed field of τ ), which is independent of the choice of γ and φ. Let X ′ = Gal(L ′ /F ∞ ) and X ′ n = Gal(L ′ n /F n ). Proposition A.4.1. Let the notation and the assumptions be as above.
(1) We have a canonical decomposition as Gal(F/Q p )-modules. Here κ is the residue field of the subalgebra of M dim(η) (Z p ) generated by the values of η over Z p , ω is the Teichmüller character and σ ∈ Gal(F/Q p ) acts on W (κ) via η regarded as having values in W (κ) × .
(2) The restriction map X ′ → X induces an isomorphism of X ′ /(φ−1)X ′ onto the augmentation ideal of Z p [[Gal(F/Q)]] ⊂ X. Note that the subalgebra of M dim(η) (Z p ) generated by the values of η over Z p is isomorphic to the Witt vector ring W (κ) with coefficients in its residue field κ.
Proof. We first prove the assertion (1). ]-free quotient of X. Since Gal(k/Q p ) has order prime to p, Gal(K/Q p ) ∼ = Gal(k/Q p ) ⋉ Γ, and its action on Y is determined by its action on Y 0 = Y /(γ − 1)Y .
Z p as Z p [Gal(k n /Q p )]-modules. By this diagram and L ′ n ⊃ k ∞ , we still have Gal(L ′ n /k n ) = X ′ n ⊕ Gal(k ∞ /k n ) with Gal(k ∞ /k n ) ∼ = Z p . By the same argument as in the case proving (1), if µ p (k) = µ p (Q p ), we have X n ∼ = Y n ⊕ Z p (1) as Z p [[Gal(k n Q p,∞ /Q p )]]-modules for a unique direct summand Y n . On Z p (1), φ acts trivially (as ν p ([p, Q p ]) = 1 for the p-adic cyclotomic character ν p ); so, [ γ, φ] acts trivially on the factor Z p (1). Hence we still have the decomposition X ′ n = Y ′ n ⊕ Z p (1). The restriction from X ′ m → X ′ n for m > n induces on Z p (1) multiplication by p m−n as φ = [p, Q p ] f acts trivially on µ p ∞ (Q p ). Thus passing to the limit, the factor Z p (1) disappears. Therefore, by Kummer theory, Coker(X ′ Res − − → X) = Z p ⊕ Z p (1) if µ p (k) = µ p (Q p ) and otherwise Z p ; so, by definition, the restriction map Y ′ m → Y ′ n is onto, and its image passing to the limit is the augmentation ideal of Z p [[Gal(F/Q p )]] (as we lose the augmentation quotient Z p which corresponds to the factor Z p in Gal(L ′ n /k n )). Since Ker(X ′ → X) is plainly (φ − 1)X ′ , we find that X ′ /(φ − 1)X ′ is isomorphic to the augmentation ideal of Pick a prime ℘ of K(ρ) above p. Let I p (resp. I p * , D p ) be the p-inertia (resp. p * -inertia, p-decomposition) subgroup of Gal(K(ρ)/K(ρ)) corresponding to ℘ and ℘ c . Regard [p, Q p ] f ∈ D p for the residual degree f of P = ℘ ∩ K(ρ), and recall ϕ ′ := ρ([p, a]] ⊂ T for a = u 2f − 1, and recall t = 1 + T . We restate Theorem A.1.2 in the introduction in the following way: Theorem A.5.1. Let the notation be as above. Suppose (H0-4). Then we can choose conjugacy classes of I p and I p * in G and a generator Θ of the σ-different I = T(σ − 1)T with Θ σ = −Θ so that we have ρ(I p ) = ( a b 0 1 ) a ∈ t Zp , b ∈ ΘΛ 1 ⊂ E × and ρ nCM (I p ) = ( a b 0 1 ) a ∈ t Zp , b ∈ ΘΛ 1 ⊂ E nCM,× and ρ(I p * ) = Jρ(I p )J −1 , where J = ( 0 1 1 0 ) for ρ = ρ T and ρ T nCM . Here t Zp ⊂ Λ is embedded in E and E nCM by the structure homomorphism.
We can conjugate ρ by ( a 0 0 1 ) for any a ∈ T × , and by doing this, Θ will be replaced by aΘ; so, actually, we can always assume that for any choice of the generator Θ with Θ σ = −Θ of the ideal (Θ), we can arrange ρ(I p ) (and ρ(I p * )) as in the corollary.
By [Hid15,Lem. 1.4], this extension is split by the action of ∆ for U being an eigenspace on which ∆ acts by ψ − ; so, we may assume to have a section s : T ֒→ I identifying T with ( a 0 0 1 ) a ∈ T . Replacing ϕ ′ by an element ϕ ∈ ϕ ′ U, we may assume that ϕ = u −f 0 0 u f commuting with t Zp 0 0 1 = Gal(Q p,∞ K(ρ)/K(ρ)). Take φ ∈ D p such that ρ(φ) = ϕ and γ ∈ D p with ρ( γ) = ( t 0 0 1 ). For the commutator [φ, γ], we have ρ([φ, γ]) = 1 (i.e. it acts on K(ρ) P trivially; the requirement for the validity of Proposition A. 4.1 (3)). The module U is a Λ 1 -module by the adjoint action of T · ϕ Zp . Since ρ CM | I has kernel U, we see that I = ρ(I p ) ∼ = ρ nCM (I p ); so, we only need to prove the assertion for ρ. If Write P|p for the prime factor in K(ψ − ) corresponding to I p . We apply Proposition A.4.1 to the P-adic completion k of K(ψ − ), its cyclotomic Z p -extension F and the composite F ∞ of F and the unramified Z p -extension of k. Thus U is made of unipotent matrices, and writing Since H(Φ − ) := Gal(K(ρ) (p) /K(Φ − )) only ramifies at p, u is unramified at c ′ c ′ c . Since I p * is lower triangular contained in JM (T)J −1 , u is unramified everywhere. Let N ∞ ⊂ K(ρ) (p) be the fixed field by Ker(u : Gal(K(ρ) (p) /K(Φ − )) → T − /m nCM + T − ) and put X := Gal(N ∞ /K(Φ − )). Then N ∞ /K(Φ − ) is an everywhere unramified pabelian extension. Since K(Φ − )/K(ψ − ) is a fully p-ramified Z p -extension generated by an element γ, we find X/(γ − 1)X is a Galois group of an everywhere unramified p-abelian extension of K(ψ − ), which is non-trivial by our assumption. Since p ∤ h K(ψ − ) , this is a contradiction. Thus the T + -span of u(I 1 ) is F; so, the T + -span of u(I 1 ) is equal to T − by Nakayama's lemma. Thus T + u(I 1 ) ≡ 0 mod m nCM + T − ; so, we may assume that Θ ∈ u(I 1 ).