ANTICYCLOTOMIC CYCLICITY CONJECTURE

Transcription

1 ANTICYCLOTOMIC CYCLICITY CONJECTURE HARUZO HIDA Abstract. Let F be an iaginary quadratic field. We forulate certain Gorenstein/local coplete intersection property of subrings of the universal deforation ring T of a od p induced representation of a character of Gal(Q/F). These conditions provide a base to prove pseudocyclicity of the Iwasawa odule over Z p-extensions of F. Under ild conditions, we realize this schee and prove anticyclotoic pseudo-cyclicity. Fix a prie p > 3. We have the following conjecture due to Iwasawa (cf. [CPI, No.62 and U3]): Cyclotoic cyclicity conjecture: Let Q /Q be the unique Z p -extension. Let X ± be the Galois group of the axial p-abelian extension everywhere unraified over Q(µ p ) on which coplex conjugation acts by ±1. For an odd character ψ : Gal(Q(µ p )/Q ) µ p 1 (Z p ), define X (ψ) := X Zp[Gal(Q(µ p )/Q )],ψz p (the ψ-eigenspace of X). Then identifying Gal(Q(µ p )/Q) = Z p = µ p 1 Γ and regarding X (ψ) as Z p [[Γ]]-odule naturally, if X (ψ) 0, X (ψ) is pseudo isoorphic to Z p [[Γ]]/(f ψ ) for a power series f ψ prie to pz p [[Γ]]. This conjecture asserts the cyclicity (up to finite error) of X (ψ) as an Iwasawa odule (i.e., having a single generator over the Iwasawa algebra Z p [[Γ]]). Under the assuption that X = 0 (the Kuer Vandiever conjecture), in [CPI, No.48], Iwasawa proved (along with his ain conjecture) pure cyclicity without finite pseudo-null error. The fact p f ψ is a cobination of the vanishing of the µ-invariant of the Kubota Leopoldt p-adic L-function (proven by Ferrero Washington) and the proof of Iwasawa s ain conjecture by Mazur Wiles. There are soe positive results towards this conjecture via Galois deforation theory (e.g. [Ku93], [O03], [Wa15] and [WE15]), relating it to Ribet s proof of the converse of Herbrand s theore, Iwasawa ain conjecture, Sharifi s conjecture, a generalized version of the Kuer Vandiver conjecture (which soeties fails) and a conjecture of Greenberg. Let F be an iaginary quadratic field with discriinant D and integer ring O. Assue that the prie (p) splits into (p) = pp in O with p p. Let F /F be the anti-cyclotoic Z p -extension with Galois group Γ := Gal(F /F); so, cσc = σ 1 for coplex conjugation c and σ Γ = Zp. Take a branch character φ : Gal(Q/F) Q p, fixing an algebraic closure Q p of Q p. Regard it as a finite order idele character φ : F A /F Q p. Most of the tie, we suppose that φ is anticyclotoic; so, φ(x c ) = φ 1 (x). For an anticyclotoic φ, we always find a finite order character ϕ of F A /F such that φ = ϕ for ϕ given by ϕ (x) = ϕ(x)ϕ(x c ) 1 (e.g., [HMI, Lea 5.31]). However, controlling the conductor of ϕ is a difficult task. We write Z p [φ] for the subring of Q p generated by the values of φ over Z p. Consider the anticyclotoic Iwasawa algebra Z p [φ][[γ ]] = li Z p [φ][γ /Γ pn n ]. Let F(φ)/F be the abelian extension cut out by φ (i.e., F(φ) = Q Ker(φ) ). Let Y be the Galois group of the axial p-raified p-abelian extension over the coposite F (φ) := F F(φ). The word: praified eans that it is unraified outside the prie p. Since Gal(F(φ)/F) acts on Y naturally as a factor of Gal(F (φ)/f), we have the φ-eigenspace Y (φ) = Y Zp[Gal(F(φ)/F)],φ Z p (φ), where Z p (φ) is the Z p [φ]-free odule of rank 1 on which Gal(F(φ)/F) acts via φ. Anticyclotoic cyclicity conjecture: Assue φ 1 and that the conductor φ is a product of split pries over Q. If the class nuber of F is prie to p and Y (φ) 0, then the Z p [φ][[γ ]]- odule Y (φ) is pseudo isoorphic to Z p [φ][[γ ]]/(f φ ) as Z p[φ][[γ ]]-odules for an eleent f φ Z p [φ][[γ ]] prie to pz p [φ][[γ ]]. We prove in this paper: Date: January 3, Matheatics Subject Classification. priary 11R23, 11F25, 11F33, 11F80; secondary 11F11, 11G18, 11F27. The author is partially supported by the NSF grant: DMS

2 ANTICYCLOTOMIC CYCLICITY CONJECTURE 2 Theore A: Let the notation be as above. Assue that φ = ϕ for the Teichüller lift ϕ of a odulo p Galois character ϕ of prie-to-p conductor c, and let N = DN F/Q (c). Suppose (h0) p > 3 is prie to N l N (l 1) for prie factors l of N, (h1) c is prie to D, and N F/Q (c) is square-free (so, N is cube-free), (h2) ϕ : Gal(Q/F) F is unraified outside cp with Teichüller lift ϕ, (h3) Np is the conductor of det(ρ) for ρ = Ind Q F ϕ, (h4) ϕ has order at least 3. If the class nuber of F is prie to p, the anticyclotoic cyclicity conjecture holds. The proof of this theore is technical ring theoretic tools applied to a local ring of the big Hecke algebra of tae level N. In this introduction, to give a short outline of our arguent without going into technicality, let us state a typical theore which describe ring-theoretic properties of the Hecke algebra equivalent to pure anti-cyclotoic cyclicity (i.e., without pseudo-null error) of Y (ϕ ). As a base ring, we take a (sufficiently large) coplete discrete valuation ring W C p flat over the p-adic integer ring Z p. Here C p is the p-adic copletion of a fixed algebraic closure Q p of Q p under its nor p noralized so that p p = 1 p. We identify the Iwasawa algebra Λ = W[[Γ]] with the one variable power series ring W[[T]] by Γ γ = (1 p) t = 1 T Λ. Take a Dirichlet character ψ : (Z/NpZ) W, and consider the big ordinary Hecke algebra h (over Λ) of prie-to-p level N and the character ψ whose definition (including its CM coponents) will be recalled in the following section. We just ention here the following three facts (1) h is an algebra flat over Λ interpolating p-ordinary Hecke algebras of level Np r1, of weight k 1 2 and of character ɛψω 1 k for the Teichüller character ω, where ɛ : Z p µ p r (r 0) and k 1 vary. If N is cube-free, h is a reduced algebra [H13, Corollary 1.3]; (2) Each prie P Spec(h) has a unique (continuous) Galois representation ρ P : Gal(Q/Q) GL 2 (κ(p)) for the residue field κ(p) of P; (3) ρ P restricted to Gal(Q p /Q p ) (the p-decoposition group) is isoorphic to an upper triangular representation whose quotient character is unraified. By (2), each local ring T has a od p representation ρ = ρ T : Gal(Q/Q) GL 2 (F) for the residue field F = T/ T. If ρ = Ind Q F ϕ for the reduction ( ) ϕ odulo p of ϕ, we have an involution σ Aut(T/Λ) such that σ ρ P = ρp χ for χ := F/Q. For a subschee Spec(A) Spec(T) stable under σ, we put A ± := {x A σ(x) = ±x}. Then A A is a subring and A is an A -odule. Let Q be a finite set of rational pries in F/Q prie to Np. Let Q be the subset of pries in Q split in F. Write K Q for the ray class field over F of conductor Cp q Q q for C := c cc, and let K Q /F (resp. K C Q ) be the axial p-abelian anticyclotoic sub-extension of K Q /F (resp. the intersection of K Q with the ray class field over F of conductor Cp q Q q). Put H Q = Gal(K Q /F) and C Q = Gal(K C Q /F). When Q is epty, we drop the subscript Q (so, H = H ). Note here H Q = H Q by definition. Moreover the fixed points Spec(T) σ=1 is known to be canonically isoorphic to Spec(W[[H]]), and Y (ϕ ) 0 if and only if σ is non-trivial on T (and hence T W[[H]]; see Corollary 2.5). The ring T is reduced (if N is cube-free), and for the kernel I = T(σ 1)T = Ker(T W[[H]]), the I-span X := I Frac(Λ) in T Λ Frac(Λ) is a ring direct suand X copleentary to Frac(W[[H]]). We write T nc for the iage of T in the ring direct suand X (and call it the non-cm coponent of T). Plainly T nc is stable under σ. Theore B: Let Spec(T) be a connected coponent of Spec(h) associated to the induced Galois representation ρ = Ind Q F ϕ for the reduction ϕ odulo p of ϕ. Suppose (h0 4) as in Theore A. Then if the class nuber of F is prie to p, then, the following four stateents are equivalent: (1) The rings T nc and T nc are both local coplete intersections free of finite rank over Λ. (2) The T nc -ideal I = T(σ 1)T T nc is principal and is generated by a non-zero-divisor θ T = T nc with θ 2 T nc. The eleent θ generates a free T nc -odule T, and T nc = T nc [θ] is free of rank 2 over T nc. (3) The Iwasawa odule Y (ϕ ) is cyclic over W[[Γ ]]. (4) The Iwasawa odule Y (ϕ ω) is cyclic over W[[Γ ]]. Under these equivalent conditions, the ring T is a local coplete intersection.

3 ANTICYCLOTOMIC CYCLICITY CONJECTURE 3 The condition p h F could be an analogue of Iwasawa s assuption X = 0, and the cyclotoic cyclicity and anti-cyclotoic cyclicity could be closely related (as pointed out to the author by P. Wake). We actually do not assue p h F in the ain text and prove a result equivalent (under p h F ) to the above theore (see Theore 5.4) replacing the assertions (3) and (4) by suitably odified stateents, and if the stateents for T fails in this general setting, Y (φ) for φ = ϕ, ϕ ω ay not be cyclic over W[[Γ ]]. The fact that f φ in the conjecture is prie to pz p[φ][[γ ]] follows fro the vanishing of the µ-invariant of the anti-cyclotoic Katz p-adic L-function [H10] (and [EAI, Theore 3.37]) and the proof of the ain conjecture by Rubin [Ru88], [Ru91], Tilouine [T89], Mazur [MT90] (and the author [H06]). A slightly stronger and detailed version of Theore B will be proven as Theore 5.4 (and Corollary 2.5). The proof of equivalence of the assertion (4) and the rest of Theore 5.4 relies on a new type of the Taylor Wiles syste arguent proving Theore 4.10 in Section 4 (and on the theory of relative dualizing odules of Grothendieck Hartshorne Kleian recalled in Section 11). The Taylor Wiles syste is ade of the deforation rings R Q of ρ and the corresponding local rings T Q of the Hecke algebras (of level N Q := N q Q q) allowing raification at pries in Q (for a suitably chosen infinite sequence of finite sets Q of pries q with q 1 od p; see Section 4 and [TW95]). Here is a sketch of the proof of the equivalence of (2) (3) in Theore B. For any coutative ring A, we write Frac(A) for the total quotient ring of A (i.e., Frac(A) is the ring of fractions inverting all non-zero-divisors of A). We siply write K for Frac(Λ). As is well known, under (h1), Frac(T) can be decoposed as an algebra direct su Frac(W[[H]]) X in a unique way. Write T nc for the projected iage of T in X. Then we have I T nc, and via the deforation theoretic technique of Mazur Tilouine [MT90] (see also [H16, 6.3.6]), we show that Y (ϕ ) Zp[ϕ ] W is isoorphic to I/I 2 (by an old forula in [H86c, Lea 1.1]). Assue that the class nuber h F of F is prie to p. Then the projection of H to Γ is an isoorphis. By the proof of the anticyclotoic ain conjecture in [T89], [MT90] and [H06], for the Katz p-adic L-function L p (ϕ ) with branch character ϕ giving the characteristic ideal of Y (ϕ ), we have W[[Γ ]]/(L p (ϕ )) = T nc /I (which also shows that the generator of I is a non zero-divisor of T nc ). Since I is principal generated by a non-zero divisor, we have I/I 2 = T nc /I = W[[Γ ]]/(L p (ϕ )), getting the anticyclotoic cyclicity conjecture. If H Γ has non-trivial kernel (which iplies p h F ), Theore 5.4 tells us that Y (ϕ ) Zp[ϕ ] W is not cyclic over W[[Γ ]]. To reach (2) (4) in Theore B, following the techniques of [H98] and [CV03], we construct an involution σ of T (Corollary 2.3). By Taylor Wiles [TW95], T is known to be a local coplete intersection over Λ (so, is Gorenstein over( Λ). ) Adding to the data of the Taylor Wiles syste the F/Q involution σ coing fro the twist by χ =, we argue in the sae way as Taylor and Wiles did. The liit ring R (the syste produced) is a power series ring over Λ with the induced involution σ, and the ring R fixed by involution is proven to be Gorenstein. By the theory of dualizing odules/sheaves for Gorenstein covering X Y (studied by A. Grothendieck [SGA 2.VI V], R. Hartshorne [RDD] and S. Kleian [Kl80]), this is close to the cyclicity of R = {x R σ(x) = x} over R (see Lea 11.4), but we are bit short of proving it. Instead, we prove that the nuber of generator of R over R is actually given by the nuber of generators of Y (ϕ ω) over W[[Γ ]] via a refineent of the original Taylor Wiles arguent. Since T = {x T σ(x) = x} is the surjective iage of R, it is generated over T = {x T σ(x) = x} by a single eleent which is a generator of I, and essentially (4) (2). The Gorenstein-ness of the rings T nc and T nc (i.e., (1)) iplies (2) by Lea 11.4 in the theory of dualizing odules. The identity T nc /(θ) = T nc /(θ 2 ) = W[[H]]/(L p (ϕ )) tells us that T nc and T nc are actually local coplete intersections; so, (2) (1). The sae ring theoretic analysis can be also done for a real quadratic field F, as the conditions (h0 4) do ake sense for real F. We hope to coe back to this proble for real quadratic fields in our future work. An exaple of T Λ given in [H85] is for F = Q[ 3], p = 13 and N = 3. This prie 13 is an irregular prie for Q[ 3] in the sense of [H82] and in the list [H81, (8.11)]. Of course, as easily checked (fro the nuerical values given in [H85]) the equivalent conditions of the theore, and actually (the distinguished factor of) L p (ϕ ) is a linear polynoial in this case. The condition (h4) iplies an assuption for R = T theores of Wiles et al [Wi95] and [TW95]:

4 ANTICYCLOTOMIC CYCLICITY CONJECTURE 4 (W) ρ restricted to Gal(Q/M) for M = Q[ ( 1) (p 1)/2 p] is absolutely irreducible, and the ain reason for us to assue (h4) is the use of the R = T theore for the inial deforation ring R of ρ (see Theore 2.1). The condition (W) is equivalent to the condition that the representation ρ is not of the for Ind Q M ξ for a character ξ : Gal(Q/M) F by Frobenius reciprocity. The iplication: (h4) (W) follows fro [H15, Proposition 5.2]. Actually (W) also follows fro the following condition: (h5) ϕ 1 ϕ raifies at a prie factor l N. Indeed, if ρ = Ind Q K ξ for another quadratic field K F, by [H15, Proposition 5.2 (2)], KF is uniquely deterined degree 4 extension of Q by ρ, and the prie l in (h5) raifies in KF/F as ρ Il = ɛ l δ l for the inertia group I l Gal(Q l /Q l ) with unraified δ l. This is ipossible if K = M as only p raifies in M/Q. Because of this, in near future, we hope to prove Theore A assuing (h5) (i.e., c O) in place of assuing the order of ϕ has at least 3 (though this latter assuption is used for soe other reason than (W) in this paper). Since ρ Gal(Qp /Q p) = ɛ δ with δ unraified, by (h2 3), ɛ has to raify at p (as F/Q is unraified at p), and hence we conclude (Rg) δ ɛ, (R) ϕ in Theores A and B raifies at p. Without (R), the choice of the p-unraified quotient δ of ρ is not uniquely deterined; so, we have two different universal Hecke rings T = T? for? = ϕ or ϕ c insisting δ =? (the proble coing fro the existence of copanion fors giving rise to the sae ρ but a different unraified quotient? assigned). So, Y (ϕ ) Zp[ϕ ] W is not exactly isoorphic to I? /I 2? for the ideal I? T?. Here I? = I as above depends on the choice T = T?. We hope to study this ore coplicated case carefully in a future article. We freely therefore use (Rg) and (R) as we assue (h1 3) and (W) throughout the paper. Here is a brief outline of the paper. In Section 1, we recall the theory of big ordinary Hecke algebras, paying particular attention to the Hecke algebra h Q of auxiliary Q-level used to construct Taylor Wiles systes and its CM coponents W[[H Q ]] as their residue rings. In Section 2, we recall the original R = T theore proven by Taylor Wiles, and in Section 3, we recall soe technical details of the Taylor Wiles arguent and describe the relation of Y (ϕ ω) and R. In Section 4, we prove a sufficient condition for the local intersection property of the subring T of R = T fixed by the involution σ, eploying the ethod of Taylor Wiles adding the datu of the involution. In the following Section 5, we prove a finer version of Theore B, applying the result of Section 4 to a residual representation induced fro an iaginary quadratic field. In Section 6, we show that the inus part of the cotangent space of Q-raified Hecke algebra T Q is generated by T(q) for pries q in Q inert in F/Q. By a Seler group coputation, we show that the nuber r of such inert pries in Q is equal to the iniu nuber of generators of Y (ϕ ) over the Iwasawa algebra. In Section 7, we prove Theore A, introducing ore r q-local involutions acting on T. To have such involutions, the choice of generators given in Section 6 (Theore 6.4) is crucial. These extra involutions fix subring saller than T if r > 1, and studying intricate relations aong eigenspaces of the involutions of Ω T/Λ, we reach the proof of Theore A. In Section 8, we show by a control theore of Rubin that cyclicity (not pseudo cyclicity) of Y (ϕ ) iplies cyclicity of the corresponding Iwasawa odule for any Z p -extension K/F. In Section 9, we study CM irreducible coponents when the class nuber of the CM iaginary quadratic field is divisible by p and shows that the coponent is often far larger than the weight Iwasawa algebra Λ. In Section 10, we explore the close relation of a generator of the ideal I and the adjoint p-adic L-function. In the final section, we gather purely ring theoretic results on Gorenstein local rings and their duality theory used in the proofs of our ain results. Throughout this paper, we write Q (resp. Q p ) for an algebraic closure of Q (resp. Q p ) and fix i p i ebeddings Q p Q C. We write C p for the p-adic copletion of Q p. A nuber field is a subfield of Q by a fixed ebedding. For each local ring A, we write A for the axial ideal of A. For any profinite abelian group G, we write W[G] for its group algebra, and put W[[G]] = li W[G/H] H for H running over all open subgroups of G; so, W[[G]] = W[G] is G is finite.

5 ANTICYCLOTOMIC CYCLICITY CONJECTURE 5 The author would like to thank R. Greenberg and R. Sharifi. Greenberg pointed out the issing hidden assuption (R) of Theore A in the first draft of this paper by counter-exaples presuably exist. Sharifi read the paper carefully and suggested any iproveents. The author also appreciates the coents on the results of the paper ade by P. Wake. Contents 1. Big Hecke algebra 5 2. The R = T theore and an involution of R The Taylor Wiles syste and Taylor Wiles pries A sufficient condition for coplete intersection property for R Proof of Theore B A good choice of generators of R Q over W Proof of Theore A and local involutions Cyclicity for a Z p -extension K /F Degree of CM coponents over the Iwasawa algebra Divisibility of the adjoint p-adic L-function Dualizing odules 48 References Big Hecke algebra We recall the theory of h to the extent we need. We assue that the starting prie-to-p level N is as in (h1); in particular, N is cube-free and its odd part is square-free. We assue that the base discrete valuation ring W flat over Z p is sufficiently large so that its residue field F is equal to T/ T for the axial ideal of the connected coponent Spec(T) (of our interest) in Spec(h). The base ring W ay not be finite over Z p. For exaple, if we deal with Katz p-adic L-functions, the natural ring of definition is the Witt vector ring W(F p ) of an algebraic closure F p (realized in C p ), though the principal ideal generated by a branch of the Katz p-adic L-function descends to an Iwasawa algebra over a finite extension W of Z p (and in this sense, the reader ay assue finiteness over Z p of W just to understand our stateent as it only depends on the ideal in the Iwasawa algebra over W). We consider the following traditional congruence subgroups Γ 0 (Np r ) := {γ = ( a b c d) SL2 (Z) c 0 od Np r }, (1.1) Γ 1 (Np r ) := {γ = ( a b c d) Γ0 (Np r )) d 1 od Np r }. A p-adic analytic faily F of odular fors is defined with respect to the fixed ebedding i p : Q C p. We write α p (α Q) for the p-adic absolute value (with p p = 1/p) induced by i p. Take a Dirichlet character ψ : (Z/Np r Z) W with (p N, r 0), and consider the space of elliptic cusp fors S k1 (Γ 0 (Np r1 ), ψ) with character ψ as defined in [IAT, (3.5.4)]. For our later use, we pick a finite set of pries Q outside Np. We define (1.2) Γ 0 (Q) := {γ = ( a b c d) SL2 (Z) c 0 od q for all q Q}, Γ 1 (Q) := {γ = ( a b c d) Γ0 (Q)) d 1 od q for all q Q}. Let Γ (p) Q quotient of Γ 0 (Q)/Γ 1 (Q) = q Q (Z/qZ). We put be the subgroup of Γ 0(Q) containing Γ 1 (Q) such that Γ 0 (Q)/Γ (p) Q (1.3) Γ Q,r := Γ (p) Q Γ 0(Np r ), is the axial p-abelian and we often write Γ Q for Γ Q,r when r is well understood (ostly when r = 0, 1). Then we put (1.4) Q := (Γ 0 (Np r ) Γ 0 (Q))/Γ Q,r, which is canonically isoorphic to the axial p-abelian quotient of Γ 0 (Q)/Γ 1 (Q) independent of the exponent r. If Q =, we have Γ Q,r = Γ 0 (Np r ), and if q 1 od p for all q Q, we have Γ 1 (N Q p r ) Γ Q,r = Γ 0 (N Q p r ) for N Q := N q Q q.

6 ANTICYCLOTOMIC CYCLICITY CONJECTURE 6 Let the ring Z[ψ] C and Z p [ψ] Q p be generated by the values ψ over Z and Z p, respectively. The Hecke algebra over Z[ψ] is the subalgebra of the C-linear endoorphis algebra of S k1 (Γ Q,r, ψ) generated over Z[ψ] by Hecke operators T(n): h = Z[ψ][T(n) n = 1, 2, ] End C (S k1 (Γ Q,r, ψ)), where T(n) is the Hecke operator as in [IAT, 3.5]. We put h Q,k,ψ/W = h k (Γ Q,r, ψ; W) := h Z[ψ] W. Here h k (Γ Q,r, ψ; W) acts on S k1 (Γ Q,r, ψ; W) which is the space of cusp fors defined over W (under the rational structure induced fro the q-expansion at the infinity cusp; see, [MFG, 3.1.8]). More generally for a congruence subgroup Γ containing Γ 1 (Np r ), we write h k (Γ, ψ; W) for the Hecke algebra on Γ with coefficients in W acting on S k1 (Γ, ψ; W). The algebra h k (Γ, ψ; W) can be also realized as W[T(n) n = 1, 2, ] End W (S k1 (Γ, ψ; W)). When we need to indicate that our T(l) is the Hecke operator of a prie factor l of Np r, we write it as U(l), since T(l) acting on a subspace S k1 (Γ 0 (N ), ψ) S k1 (Γ 0 (Np r ), ψ) of level N Np prie to l does not coincide with U(l) on S k1 (Γ 0 (Np r ), ψ). The ordinary part h Q,k,ψ/W h Q,k,ψ/W is the axial ring direct suand on which U(p) is invertible. If Q =, we siply write h k,ψ/w for h,k,ψ/w. We write e for the idepotent of h Q,k,ψ/W, and hence e = li n U(p) n! under the p-adic topology of h Q,k,ψ/W. The idepotent e not only acts on the space of odular fors with coefficients in W but also on the classical space S k1 (Γ Q,r, ψ) (as e descends fro S k1 (Γ Q,r, ψ, Q p ) to S k1 (Γ Q,r, ψ, Q)). We write the iage M ord := e(m) of the idepotent attaching the superscript ord (e.g., Sk1 ord ). Fix a character ψ 0 odulo Np, and assue now ψ 0 ( 1) = 1. Let ω be the odulo p Teichüller character. Recall the ultiplicative group Γ := 1pZ p Z p and its topological generator γ = 1p. Then the Iwasawa algebra Λ = W[[Γ]] = li W[Γ/Γ pn ] is identified with the power series ring n W[[T]] by a W-algebra isoorphis sending γ Γ to t := 1 T. As constructed in [H86a], [H86b] and [GME], we have a unique big ordinary Hecke algebra h Q (of level Γ Q, ). We write h for h. Since Np = DN F/Q (c)p Dp > 4, the algebra h Q is characterized by the following two properties (called Control theores; see [H86a] Theore 3.1, Corollary 3.2 and [H86b, Theore 1.2] for p 5 and [GME, Corollary ] for general p): (C1) h Q is free of finite rank over Λ equipped with T(n) h Q for all 1 n Z prie to Np and U(l) for prie factors l of Np, (C2) if k 1 and ɛ : Z p µ p is a finite order character, h Q /(t ɛ(γ)γ k )h Q = hq,k,ɛψk (γ = 1 p) for ψ k := ψ 0 ω k, sending T(n) to T(n) (and U(l) to U(l) for l Np). Actually a slightly stronger fact than (C1) is known: Lea 1.1. The Hecke algebra h Q is flat over Λ[ Q ] with h Q /A Q h Q = h for the augentation ideal A Q Λ[ Q ]. See [H89, Lea 3.10] and [MFG, Corollary 3.20] for a proof. Hereafter, even if k 0, abusing the notation, we put h Q,k,ɛψk := h Q /(t ɛ(γ)γ k )h Q which acts on p-ordinary p-adic cusp fors of weight k and of Neben character ɛψ k. By the above lea, h Q,k,ɛψk is free of finite rank d over W[ Q ] whose rank over W[ Q ] is equal to rank W h,k,ɛψk (independent of Q). Since N Q is cube-free, by [H13, Corollary 1.3], h Q is reduced. Let Spec(I) be an irreducible coponent of Spec(h Q ). Write a(n) for the iage of T(n) in I (so, a(p) is the iage of U(p)). If a point P of Spec(I)(Q p ) kills (t ɛ(γ)γ k ) with 1 k Z (i.e., P((t ɛ(γ)γ k )) = 0), we call P an arithetic point, and we write ɛ P := ɛ, k(p) := k 1 and p r(p) for the order of ɛ P. If P is arithetic, by (C2), we have a Hecke eigenfor f P S k1 (Γ Q,r(P)1, ɛψ k ) such that its eigenvalue for T(n) is given by a P (n) := P(a(n)) Q for all n. Thus I gives rise to a faily F = {f P arithetic P Spec(I)} of Hecke eigenfors. We define a p-adic analytic faily of slope 0 (with coefficients in I) to be the faily as above of Hecke eigenfors associated to an irreducible coponent Spec(I) Spec(h Q ). We call this faily slope 0 because a P (p) p = 1 for the p-adic absolute value p of Q p (it is also often called an ordinary faily). This faily is said to be analytic because the Hecke eigenvalue a P (n) for T(n) is given by an analytic function a(n) on (the rigid analytic space associated to) the p-profinite foral spectru Spf(I). Identify Spec(I)(Q p ) with

7 ANTICYCLOTOMIC CYCLICITY CONJECTURE 7 Ho W-alg (I, Q p ) so that each eleent a I gives rise to a function a : Spec(I)(Q p ) Q p whose value at (P : I Q p ) Spec(I)(Q p ) is a P := P(a) Q p. Then a is an analytic function of the rigid analytic space associated to Spf(I). Taking a finite covering Spec(Ĩ) of Spec(I) with surjection Spec(Ĩ)(Q p ) Spec(I)(Q p ), abusing slightly the definition, we ay regard the faily F as being indexed by arithetic points of Spec(Ĩ)(Q p), where arithetic points of Spec(Ĩ)(Q p) are ade up of the points above arithetic points of Spec(I)(Q p ). The choice of Ĩ is often the noralization of I or the integral closure of I in a finite extension of the quotient field of I. Each irreducible coponent Spec(I) Spec(h Q ) has a 2-diensional sei-siple (actually absolutely irreducible) continuous representation ρ I of Gal(Q/Q) with coefficients in the quotient field of I (see [H86b]). The representation ρ I restricted to the p-decoposition group D p is reducible with unraified quotient character (e.g., [GME, 4.2]). As is well known now (e.g., [GME, 4.2]), ρ I is unraified outside N Q p and satisfies (Gal) Tr(ρ I (Frob l )) = a(l) (l Np), ρ I ([γ s, Q p ]) ( t s 0 1 ) and ρi ([p, Q p ]) ( 0 a(p) ), where γ s = (1 p) s = ( s n=0 n) p n Z p for s Z p and [x, Q p ] is the local Artin sybol. As for pries in q Q, if q 1 od p and ρ(frob q ) has two distinct eigenvalues, we have (Gal q ) ρ I ([z, Q q ]) ( αq(z) 0 0 β q(z) ) with characters α q and β q of Q q for z Q q, where one of α q and β q is unraified (e.g., [MFG, Theore 3.32 (2)] or [HMI, Theore 3.75]). For each prie ideal P of Spec(I), writing κ(p) for the residue field of P, we also have a sei-siple Galois representation ρ P : Gal(Q/Q) GL 2 (κ(p)) unraified outside N Q p such that Tr(ρ P (Frob l )) is given by a(l) P for all pries l N Q p. If P is the axial ideal I, we write ρ for ρ P which is called the residual representation of ρ I. The residual representation ρ is independent of I as long as Spec(I) belongs to a given connected coponent Spec(T) of Spec(h Q ). Indeed, Tr(ρ P ) od I = Tr(ρ) for any P Spec(T). If P is an arithetic prie, we have det(ρ P ) = ɛ P ψ k ν k p for the p-adic cyclotoic character ν p (regarding ɛ P and ψ k as Galois characters by class field theory). This is the Galois representation associated to the Hecke eigenfor f P (constructed earlier by Shiura and Deligne) if P is arithetic (e.g., [GME, 4.2]). A coponent I is called a CM coponent if there exists a nontrivial character χ : Gal(Q/Q) I such that ρ I = ρi χ. We also say that I has coplex ultiplication if I is a CM coponent. In this case, we call the corresponding faily F a CM faily (or we say that F has coplex ultiplication). If F is a CM faily associated to I with ρ I = ρi χ, then ( χ is ) a quadratic character of Gal(Q/Q) which cuts out an iaginary quadratic field F, i.e., χ = F/Q. Write Ĩ for the integral closure of Λ inside the quotient field of I. The following three conditions are known to be equivalent: ( ) (CM1) F has CM with ρ I = ρi F/Q ( ρ I = Ind Q F λ for a character λ : Gal(Q/F) Ĩ ); (CM2) For all arithetic P of Spec(I)(Q p ), f P is a binary theta series of the nor for of F/Q; (CM3) For soe arithetic P of Spec(I)(Q p ), f P is a binary theta series of the nor for of F/Q. Indeed, (CM1) is equivalent to ρ I = Ind Q λ F for a character λ : Gal(Q/F) Frac(I) unraified outside Np (e.g., [DHI98, Lea 3.2] or [MFG, Lea 2.15]). Since the characteristic polynoial of ρ I (σ) has coefficients in I, its eigenvalues fall in Ĩ; so, the character λ has values in Ĩ (see, [H86c, Corollary 4.2]). Then by (Gal), λ P = P λ : Gal(Q/F) Q p for an arithetic P Spec(Ĩ)(Q p ) is a locally algebraic p-adic character, which is the p-adic avatar of a Hecke character λ P : F A /F C of type A 0 of the quadratic field F /Q. Then by the characterization (Gal) of ρ I, f P is the theta series with q-expansion a λ P(a)q N(a), where a runs over all integral ideals of F. By k(p) 1 (and (Gal)), F has to be an iaginary quadratic field in which p is split (as holoorphic binary theta series of real quadratic field are liited to weight 1 k = 0; cf., [MFM, 4.8]). This shows (CM1) (CM2) (CM3). If (CM2) ( is ) satisfied, we have an identity Tr(ρ I (Frob l )) = a(l) = χ(l)a(l) = Tr(ρ I χ(frob l )) with χ = F/Q for all pries l outside Np. By Chebotarev density, we have Tr(ρ I ) = Tr(ρ I χ), and we get (CM1) fro (CM2) as ρ I is sei-siple. If a coponent Spec(I) contains an arithetic point P with theta series f P as above of F/Q, either I is a CM coponent or otherwise P is in the intersection in Spec(h Q ) of a coponent Spec(I) not

8 ANTICYCLOTOMIC CYCLICITY CONJECTURE 8 having CM by F and another coponent having CM by F (as all failies with CM by F are ade up of theta series of F by the construction of CM coponents in [H86a, 7]). The latter case cannot happen as two distinct coponents never cross at an arithetic point in Spec(h Q ) (i.e., the reduced part of the localization h Q P is étale over Λ P for any arithetic point P Spec(Λ)(Q p ); see [HMI, Proposition 3.78]). Thus (CM3) iplies (CM2). We call a binary theta series of the nor for of an iaginary quadratic field a CM theta series. We describe how to construct residue rings of h Q whose Galois representation is induced fro a quadratic field F (see [LFE, 7.6] and [HMI, 2.5.4]). Here F is either real or iaginary. We write c for the generator of Gal(F/Q) (even if F is real). Let c be the prie-to-p conductor of a character ϕ as in Theore B in the introduction (allowing real F). Put C = c c c. By (h1), c is a square free integral ideal of F with c c c = O (for coplex conjugation c). Since Q is outside N, Q is a finite set of rational pries unraified in F/Q prie to Cp. Let Q be the subset in Q ade up of pries split in F. We choose a prie factor q of q for each q Q (once and for all), and put Q := q Q q. We study soe ray class groups isoorphic to H Q. We put C Q := C q Q q. We siply write C for C. Consider the ray class group Cl(a) (of F) odulo a for an integral ideal a of O, and put (1.5) Cl(cQ p ) = li r Cl(cQ p r ), and Cl(C Q p ) = li r Cl(C Q p r ). On Cl(C Q p ), coplex conjugation c acts as an involution. Let Z Q (resp. Z Q ) be the axial p-profinite subgroup (and hence quotient) of Cl(cQ p ) (resp. Cl(C Q p )). We write Z (resp. Z) for Z (resp. Z ). We have the finite level analogue C Q which is the axial p-profinite subgroup (and hence quotient) of Cl(cQ p). We have a natural ap of (O p O p ) into Cl(C Q p ) = li Cl(C r Q p r ) (with finite kernel). Let Z = Z Q Q /Z c1 Q (the axial quotient on which c acts by 1). We have the projections π : Z Q Z Q and π : Z Q Z Q. Recall p > 3; so, the projection π induces an isoorphis Z 1 c Q = {zz c z Z Q } Z Q. Thus π induces an isoorphis between the p-profinite groups Z Q and Z 1 c Q. Siilarly, π induces π : Z 1 c Q = Z Q. Thus we have for the Galois group H Q as in the introduction (1.6) ι : Z Q = Z Q = H Q by first lifting z Z Q to z Z Q and taking its square root and then project down to π ( z 1/2 ) = z (1 c)/2. Here the second isoorphis Z Q = H Q is by Artin sybol of class field theory. The isoorphis ι identifies the axial torsion free quotients of the two groups Z Q and Z Q which we have written as Γ. This ι also induces W-algebra isoorphis W[[Z Q ]] = W[[Z Q ]] which is again written by ι. Let ϕ be the Teichüller lift of ϕ as in Theore B. Recall N = N F/Q (c)d. Then we have a unique continuous character Φ : Gal(Q/F) W[[Z Q ]] characterized by the following two properties: (1) Φ is unraified outside cq p, (2) Φ(Frob l ) = ϕ(frob l )[l] for each prie l outside Np and Q, where [l] is the projection to Z Q of the class of l in Cl(cQ p ). When F is real, all groups Z Q, Z Q and H Q are finite groups; so, W[[Z Q ]] = W[Z Q ] for exaple. The character Φ is uniquely deterined by the above two properties because of Chebotarev density. We can prove the following result in the sae anner as in [H86c, Corollary 4.2]: Theore 1.2. Suppose that ϕ(frob q ) ϕ(frob q c) for all q Q. Then we have a surjective Λ- algebra hooorphis h Q W[[Z Q ]] such that (1) T(l) Φ(l) Φ(l c ) if l = ll c with l l c and l N Q p; (2) T(l) 0 if l reains prie in F and is prie to N Q p; (3) U(q) Φ(q c ) if q Q ; (4) U(p) Φ(p c ). If F is real, the above hooorphis factors through the weight 1 Hecke algebra h Q /(t p 1)h Q for a sufficiently large 0.

9 ANTICYCLOTOMIC CYCLICITY CONJECTURE 9 The last point of the orphis factoring through the weight 1 Hecke algebra is because theta series of a real quadratic field are liited to weight 1. Note that out of a Hecke eigenfor f(z) S k1 (Γ 0 (N Q p r ), φ) with f T(q) = a q f for q Q and two roots α, β of X 2 a q Xφ(q)q k = 0, we can create two Hecke eigenfors f α = f(z) βf(qz) and f β = f(z) αf(qz) of level N Q q with f x U(q) = xf x for x = α, β. This tells us that if we choose a set Σ := {α q q Q } of od p eigenvalues of ρ(frob q ) for q Q := Q Q, we have a unique local ring T Q of h Q and a surjective algebra hooorphis T Q W[[Z Q ]] factoring through h Q W[[Z Q ]] such that U(q) od T Q = α q for all q Q. For q Q, if f is a theta series of F, we have a q = 0; so, the residual class (odulo T Q) of α and β in Z p [α, β] Q p are distinct (because of p > 2). Therefore if we change Σ, the local ring T Q will be changed accordingly. We record this fact as Corollary 1.3. Suppose that ϕ(frob q ) ϕ(frob q c) for all q Q and that W is sufficiently large so that we can choose a set Σ = {α q F q Q } of od p eigenvalues of ρ(frob q ) for q Q = Q Q in the residue field F of W. Then we have a unique local ring T Q of h Q such that we have a surjective Λ-algebra hooorphis T Q W[[Z Q ]] characterized by the following conditions: (1) T(l) Φ(l) Φ(l c ) if l = ll c with l l c and l N Q p; (2) T(l) 0 if l reains prie in F and is prie to N Q p; (3) U(q) Φ(q c ) if q Q ; (4) U(q) ±Φ(q) if q Q, where the sign is deterined by ±Φ(q) od T Q = α q ; (5) U(p) Φ(p c ). If F is real, the above hooorphis factors through the weight 1 Hecke algebra T Q /(t p 1)T Q for a sufficiently large 0. We will later show that the quotient T Q W[[Z Q ]] constructed above is the axial quotient such that the corresponding Galois representation is induced fro F under (h0 4) (see Proposition 2.6). Hereafter, ore generally, fixing an integer k 0 and the set Σ = {α q F q Q }, we put (1.7) T Q = T Q /(t γ k )T Q. The choice of q Q can be also considered to be the choice Σ := {ϕ(frob q c) F : q Q } of the eigenvalue of U(q). Thus the local rings T Q and T Q are considered to be defined with respect to the choice Σ = Σ Σ of one of the od p eigenvalues of U(q) for each q Q. In other words, T Q is a local factor of h Q,k,ψk with the prescribed od p eigenvalues Σ of U(q) for q Q. Note that T Q is classical if k 1 but otherwise, it is defined purely p-adically. In the above corollary, we took k = 0 when F is real. Assue that F is iaginary. In this case, we need later a rapid growth assertion of the group H Q and the group ring W[[H Q ]] if we vary Q suitably. This growth result we describe now. We fix a positive integer r and choose an infinite set Q = {Q = 1, 2,...} of r -sets Q of pries q of O such that N(q) 1 od p. We assue that Q is ade of pries split in F/Q outside cp and that q q Z induces a bijection between Q and Q := {q Z q Q }. We regard Q as a set of rational pries. We write Q soeties for the product q Q q. Then the inclusion Z O induces a natural isoorphis q Q (Z/qZ) = (O/Q ). We identify the two groups by this isoorphis, and write Q for the p-sylow subgroup of this group. Then Q is the product over q Q of the p-sylow subgroup q = q of (O/q) = (Z/qZ). For the ray class group Cl(cQ p n ), we have a natural exact sequence of abelian groups (O/Q ) i Cl(cQ p n ) Cl(cp n ) 1, which induces the exact sequence of its axial p-abelian quotients: 1 Q Cl(cQ p n ) p Cl(cp n ) p 1, since the order of the finite group Ker(i) is prie to p (as p > 3). Passing to the projective liit with respect to n, we have an exact sequence of copact odules (1.8) 1 Q Z Q Z 1. We consider the group algebra W[[Z Q ]] which is an algebra over W[ Q ]. We choose a generator δ q of the cyclic group q and put n to be the quotient of Q by the subgroup generated by

10 ANTICYCLOTOMIC CYCLICITY CONJECTURE 10 {δq pn } q Q for 0 < n ; thus, n = (Z/p n Z) r. This include the ordering Q = {q 1,..., q r } so that the above isoorphis sends qj / δq pn j to the j-th factor Z/p n Z. In this way, we fix the identification of n with (Z/p n Z) r for all n and once and for all. Thus, writing W n := W/p n W, we get a projective syste {W n [ n ] = W n [(Z/p n Z) r ]} n>0 sending (Z/p n Z) r x (x od p n ) (Z/p n Z) r for all n. We then have W[[S 1,..., S r ]] = li n W n [ n ] sending s j = 1 S j to the iage of δ qj in n for all j, q j Q and n. Assuing that F has class nuber prie to p, the natural isoorphis Z p = O p induces a group orphis Z p Cl(cp ), which induces an isoorphis Γ = 1 pz p = Z. Then we can canonically split exact sequence (1.8) so that Z Q = Q Γ, aking the following diagra coutative for all n > n with n: W n [[Γ]][ n ] = W[[Z Q ]]/A n W n [[Z ]] onto π n n W n [[Γ]][ n] = W[[Z Q ]]/A n W n [[Z ]], where A n := (p n, s pn j 1) j=1,2,...,r as an ideal of W[[S 1,..., S r ]]. In this way, we get a (bit artificial) projective syste n π {W[[Z Q ]]/A n n W[[Z Q ]]/A n } n >n. By this ap, W[[Z Q ]]/A n is naturally a Λ-algebra via the canonical splitting Z Q = Q Z, and hence a Λ[[S 1,..., S r ]]-algebra. Since Z = Γ, we get li W[[Z n Q ]]/A n = Λ[[S1,..., S r ]]. We thus conclude Proposition 1.4. Assue that F is iaginary with class nuber prie to p. Identify H Q with Z Q by (1.6) (whence A n is the ideal of W[[H Q ]]). Then the liit ring li W[[H n Q ]]/A n is isoorphic to Λ[[S 1,..., S r ]]. This follows fro the above arguent, after identifying Z Q with H Q and identifying Λ with W[[Γ ]]. We now explore the case where the class nuber of F is divisible by p. In this case, we again study the set Q of r -sets Q of split pries in F outside N such that N(q) 1 od p with Q := {(q) = q Z q Q } with an ordering. We still have the following exact sequence (1.8): π Q 1 Q Z Q Z 1. Write Z tor for the axial torsion subgroup of Z, and fix a splitting Z = Γ F Z tor with a torsion-free group Γ F. The projection π Q identifies the axial torsion-free quotient of Z Q with Γ F. Write Z Q : Ker(Z,tor Q Γ F) (the axial torsion subgroup of Z Q ). Note that Q. For running over integers with n, the isoorphis classes of the set of cokernels Z Q,tor {Z Q,tor / pn } Q n of pairs of abelian groups is finite. Here Z Q,tor / pn and Z Q Q isoorphic if the following diagra for > is coutative: Q / pn Q Q i, / p n Q Z Q Z Q,tor / pn Q n,tor/ p. Q n,tor / p Q Here i, is induced by sending the generator δ qj pn for Q Q = {q 1,..., q r } to the generator δ q j pn writing Q Q = {q 1,..., q r } according to our choice of ordering. Starting with n = 1, we have an isoorphis class I 1 in {Z Q,tor / p Q } 1 with infinite eleents. Suppose that we have are

11 ANTICYCLOTOMIC CYCLICITY CONJECTURE 11 constructed a sequence I n I n 1 I 1 of isoorphis classes I j in {Z Q,tor / pj Q } j such that Z Q,tor / pj I Q j is sent onto to Z Q,tor / pj 1 in I Q j 1 for all j = 2, 3,..., n. Since I n1 := {Z Q,tor / pn1 (Z Q Q,tor / pn I Q ) n } n1 is an infinite set, we can choose an isoorphis class I n1 I n1 with I n1 =. Thus by induction on n, we find an infinite sequence I n I n 1 I 1 as above. Then we define (n) for each n to be the inial appearing I n. Thus we have a projection πn1,tor n : Z,tor/ pn1 Q Z (n1) Q Q and a projective syste of groups (n1) (n),tor/ pn Q (n) Z,tor/ pn1 Q (n1) Q (n1) Z Q π n n1,tor π n n1 Z Q (n),tor/ pn Q (n) Passing to the liit, we have an exact sequence: / pn1 (n1) Q (n1) Z Q(n) / pn Q (n) Γ F Γ F. 1 liz n Q liz (n),tor/ pn Q Q(n) / pn Γ (n) n Q F 1. (n) Note here the subgroup := li n Q / pn (n) Q (n) = Z r p with W[[ ]] = W[[S 1,..., S r ]] for the variable chosen as in Proposition 1.4 and W[[Z S ]] for Z S := li Z n Q is an algebra (n),tor/ pn Q (n) free of finite rank over W[[ ]]. We write Γ S = Z S /Z S,tor for the axial torsion subgroup Z S,tor of Z S. Choose a splitting of the exact sequence Z S,tor Z S Γ S so that Γ S as a subgroup of Z S contains. Then W[[Z S ]] = W[[Γ S ]][Z S,tor ] = W[[Γ S ]][Z S /Γ S ]]. By splitting the projection Z := li Z Q(n) / pn Γ n Q F, we have a W[[Γ F ]]-algebra structure of W[[Z ]]. (n) Proposition 1.5. Let the notation be as above. Assue that F is iaginary with class nuber divisible by p. Identify H Q with Z Q by (1.6). Then there is a subsequence {Q (n) } n=1,2,... Q such that {W[[H Q ]]/A n } n fors a projective syste of finite rings and that the liit ring (n) li W[[H n Q ]]/A n is isoorphic to the profinite group algebra W[[Γ F Γ S ]][Z /Γ S ], and Γ S (resp. Γ F ) contains (resp. Γ) as a subgroup of finite index. In particular, li W[[H n Q ]]/A n is free of finite rank over Λ[[S 1,..., S r ]] and is a local coplete intersection over Λ. 2. The R = T theore and an involution of R We place ourselves in the setting of Theore B, but we allow any quadratic extension F/Q (which can be real or iaginary). We assue that the residue field of W is given by F = T/ T. For the oent, we only assue (h0 3) for a fixed connected coponent Spec(T) of Spec(h) for h := h and its residual representation ρ of the for Ind Q F ϕ for a Galois character ϕ : Gal(Q/F) F. We fix a weight k 0 and pick a Hecke character ϕ k : Gal(Q/F) W of conductor at ost cp with p-type ki p F for the identity ebedding i p F : F Q p such that ϕ k ϕ od W. Let θ(ϕ k ) S k1 (Γ 0 (Np), ψ k ) for the corresponding theta series. Then ψ k is deterined by ϕ k (i.e., ψ k = χϕ k A νp k regarding ϕ k and ψ k as idele characters; see [HMI, Theore 2.71]). When F is iaginary (that is usually the case), we assue that k 1. Recall the identity ψ k νp k od W = det(ρ) for the p-adic cyclotoic character ν p ; so, ψ 0 is the Teichüller lift of det(ρ). Hereafter, we siply write ψ for ψ 0 = ψ k ω k. Writing c for the prie-to-p conductor of ϕ, by (h3), N F/Q (c)d = N for the discriinant D of F (cf. [GME, Theore 5.1.9]). By (h1), the conductor c is square-free and only divisible by split pries in F/Q. Since ρ = Ind Q F ϕ, for l N, the prie l either splits in F or raified in F. Write l for the prie factor of (l) in F. If (l) splits into ll, we ay assue that the character ϕ raifies at l and is unraified at l, and hence ρ Gal(Ql /Q l) = ϕ l ϕ l. If (l) = l 2 raifies in F, we have ρ Il = 1 χ for the quadratic character ( ) χ = F/Q. Here I l is the inertia subgroup of Gal(Q l /Q l ).

12 ANTICYCLOTOMIC CYCLICITY CONJECTURE 12 Write CL W for the category of p-profinite local W-algebras with residue field F := W/ W whose orphiss are local W-algebra hooorphiss. Let Q (Np) Q be the axial extension of Q unraified outside Np. Consider the following deforation functor D : CL W SETS given by D(A) = D (A) := {ρ : Gal(Q (Np) /Q) GL 2 (A) : a representation satisfying (D1 4)}/ =. Here are the conditions (D1 4): (D1) ρ od A = ρ (i.e., there exists a GL2 (F) such that aρ(σ)a 1 = (ρ od A ) for all σ Gal(Q/Q)). (D2) ρ Gal(Qp /Q p) = ( 0 ɛ δ ) with δ unraified. (D3) det(ρ) Il is equal to ι A ψ l for the l-part ψ l of ψ for each prie l N, where ι A : W A is the orphis giving W-algebra structure on A and ψ l = ψ Il regarding ψ as a Galois character by class field theory. (D4) det(ρ) Ip ψ Ip od A (which is equivalent to ɛ Ip ψ Ip od A ). If we want to allow raification at pries in a finite set Q of pries outside Np, we write Q (QNp) for the axial extension of Q unraified outside Q {l N p} { }. Consider the following functor D Q (A) := {ρ : Gal(Q (QNp) /Q) GL 2 (A) : a representation satisfying (D1 4) and (UQ)}/ =, where (UQ) det ρ is unraified at all q Q. We ay also ipose another condition if necessary: (det) det(ρ) = ι A ν k pψ k for the p-adic cyclotoic character ν p, and consider the functor D Q,k,ψk (A) := {ρ : Gal(Q (QNp) /Q) GL 2 (A) : a representation satisfying (D1 4) and (det)}/ =. The condition (det) iplies that if deforation is odular and satisfies (D1 4), then it is associated to a weight k 1 cusp for of Neben character ψ k ; strictly speaking, if k = 0 (i.e., F is real), we allow non-classical p-ordinary p-adic cusp fors. We often write siply D k,ψk for D,k,ψk when Q is epty. For each prie q, we write D q Q,k,ψ k for the deforation functor of ρ Gal(Qq /Q satisfying q) the local condition (D2 4) which applies to q. By our choice of ρ = Ind Q F ϕ, we have ρ Gal(Q q /Q) = ( ɛq 0 0 δ q ) for two local characters ɛ q, δ q for all q Q. If δ ɛ (i.e., (Rg)) and ɛ q (Frob q ) δ q (Frob q ) for all q Q, D, D Q, D k,ψk and D Q,k,ψk are representable by universal objects (R, ρ) = (R, ρ ), (R Q, ρ Q ), (R, ρ ) and (R Q, ρ Q ), respectively (see [MFG, Proposition 3.30] or [HMI, Theore 1.46 and page 186]). Here is a brief outline of how to show the representability of D. It is easy to check the deforation functor D ord only iposing (D1 2) is representable by a W-algebra R ord. The condition (D4) is actually redundant as it follows fro the universality of the Teichüller lift and the conditions (D1 2). Since N is the prie-to-p conductor of det ρ and p is unraified in F/Q (h2 3), if l is a prie factor of N, writing ρ ss I l for its sei-siplification of ρ over I l, we see fro (h0) that (ρ Il ) ss = ɛ l δ l for two characters ɛ l and δ l (of order prie to p) with δ l unraified and ɛ l ψ Il od A. Thus by the character ɛ N := l N ɛ l of I N = l N I l, A is canonically an algebra over the group algebra W[I N ]. Then R is given by the axial residue ring of R ord on which I N acts by ψ 1,N = l N ψ I l ; so, R = R ord W[IN],ψ 1,N W, where the tensor product is taken over the algebra hooorphis W[I N ] W induced by the character ψ 1,N. Since ρ is an induced representation, ρ Il is sei-siple and ρ Il = ɛ l δ l with ɛ l = ɛ l od A. Siilarly one can show the representability of D Q and D Q,k,ψk. Let T be the local ring of h = h as in Theore B whose residual representation is ρ = Ind Q F ϕ. The ring T is uniquely deterined by (h1 3) as the unraified quotient of ρ at each l Np is unique. Without assuing (h1 3), to have a universal ring and to have uniquely defined T, we need to specify in the deforation proble the unraified quotient character and for T, the residue class of U(l)-eigenvalue (because of the existence of copanion fors). Since ρ is irreducible, by the technique of pseudo-representation, we have a unique representation ρ T : Gal(Q (Np) /Q) GL 2 (T)

13 ANTICYCLOTOMIC CYCLICITY CONJECTURE 13 up to isoorphiss such that Tr(ρ T (Frob l )) = T(l) T for all prie l Np (e.g., [HMI, Proposition 3.49]). This representation is a deforation of ρ in D (T). Thus by universality, we have projections π : R = R T. such that π ρ = ρ T. Here is the R = T theore of Taylor, Wiles ét al: Theore 2.1. Assue (Rg) and (h0 3) with either (h4) or (h5). Then the orphis π : R T is an isoorphis, and T is a local coplete intersection over Λ. See [Wi95, Theore 3.3] and [DFG04] for a proof (see also [HMI, 3.2] or [MFG, Theore 3.31] for details of how to lift the results in [Wi95] to the (bigger) ordinary deforation ring with varying deterinant character). These references require the assuption (W) which is absolute irreducibility of ρ Gal(Q/M) for M = Q[ p ] with p := ( 1) (p 1)/2 p. Note that (W) follows fro either (h4) or (h5), as entioned in the introduction. To eliinate the assuption (h0), we need to ipose in addition to (D3) that H 0 (I l, ρ) = A for prie factors l of N with l 1 od p to have the identity R = T (or work with Γ 1 (l)-level Hecke algebra), which not only coplicates the setting but also the identification of T/I = W[[H]] (for I in Theore B) could fail if (h0) fails (so, we always assue (h0); see Lea 2.4). We will recall the proof of Theore 2.1 in the following Section 4 to good extent in order to facilitate a base for a finer version we study there. Perhaps the following fact is well known (e.g., [Ru91, Theore 5.3]): Corollary 2.2. Assue (h0 4) and that F is an iaginary quadratic field of class nuber prie to p. Then Y (ϕ ) has hoological diension 1 (so, it does not have any pseudo-null subodule non-null). Thus if Y (ϕ ) is pseudo isoorphic to a cyclic Z p [ϕ ][[Γ ]]-odule Z p [ϕ ][[Γ ]]/(f ϕ ) with f ϕ Z p [ϕ ][[Γ ]], it has an injection into the cyclic odule with finite cokernel. Proof. Write the presentation of R = T as R = Λ[[T 1,..., T r ]]/(S 1,..., S r ) for a regular sequence (S 1,..., S r ) of Λ[[T 1,..., T r ]]. Then by the fundaental exact sequence of differentials (e.g., [CRT, Theore 25.2] and [HMI, page 370]), we get the following exact sequence 0 i RdS i = (S 1,..., S r )/(S 1,..., S r ) 2 i RdT i Ω R/Λ 0. Since the class nuber of F is prie to p, the CM coponent W[[H]] of T = R is isoorphihc to Λ; so, tensoring Λ over R, we get another exact sequence: 0 i ΛdS i i ΛdT i Ω R/Λ R Λ 0. By a theore of Mazur (cf. [MT90], [HT94, 3.3.7] and [H16, 6.3.6]), under (R) (which follows fro (h2 3)) and (h0), we have Ω R/Λ R Λ = Y (ϕ ) Zp[ϕ ] W. Thus we get a Λ-free resolution of length 2 of the Iwasawa odule, and hence it has hoological diension 1. Suppose that we have a pseudo-isoorphis i : Y (ϕ ) Z p [ϕ ][[Γ ]]/(f ϕ ). Then i is an injection as Y (ϕ ) does not have any pseudo-null subodule non-null, and Coker(i) is finite. Since ρ = Ind Q F ϕ, for χ = ( F/Q ), ρ χ = ρ. By ordinarity, p splits in F; so, χ is trivial on Gal(Q l /Q l ) for prie factors of pn F/Q (c) and raified quadratic on Gal(Q l /Q l ) for l D. Thus ρ ρ χ is an autoorphis of the functor D Q and D Q,k,ψk, and ρ ρ χ induces autoorphiss σ Q of R Q and R Q. We identify R and T now by Theore 2.1; in particular, we have an autoorphis σ = σ Aut(T) as above. We could think about h /W0 defined over a saller coplete discrete valuation ring W 0 W (the sallest possible ring is Z p [ψ]). After extending scalar fro W 0 to W, we get an involution. We ay assue that W = W(F) (the Witt vector ring of F = T/ T ). Since σ fixes W as it is an identity on F, we know that σ preserves T before extending scalar to W. Thus we get Corollary 2.3. Assue (h0 4). Then for a coplete discrete valuation ring W 0 flat over Z p [ψ], we have an involution σ Aut(T /W0 ) with σ ρ T = ρt χ. We write T for the subring of T fixed by the involution in Corollary 2.3. More generally, for any odule X on which the involution σ acts, we put X ± = X ± = {x X σ(x) = ±x}. In particular, we have T ± := {x T x σ = ±x}.