MODULAR IWASAWA THEORY

HARUZO HIDA

Contents

1. Introduction 2 1.1. Good old Iwasawa theory 2 1.2. Galois theoretic interpretation of the class group 4 1.3. The Iwasawa algebra as a deformation ring 6 2. Selmer Groups 7 2.1. Ordinary Galois representations 7 2.2. Greenberg’s Selmer groups 8 2.3. Selmer groups with general coefficients 8 3. Deformation Rings and Adjoint Square Selmer Groups 10 3.1. Nearly ordinary deformation rings 10 3.2. Adjoint square Selmer groups and differentials 11 4. Iwasawa Theory for Deformation Rings 18 4.1. Galois action on deformation rings 18 4.2. Control of adjoint square Selmer groups 19 4.3. Λ-Adic forms 23 5. L-Invariants of Adjoint Square Galois Representations 24 5.1. Balanced Selmer groups 26 5.2. Greenberg’s L-invariant 28 5.3. Proof of the theorem 28 References 30

Date: February 22, 2004.

The author would like to thank Professors Chunlai Zhao and King-Fai Lai for their invitation to Peking university where a series of lectures described by these notes is given. The author is partially supported by the NSF grant: DMS 0244401.

1. Introduction

We fix a prime p > 2, an algebraic closure \overline{ℚ} of ℚ and \overline{ℚ}_{p} of ℚ_{p} and field embeddings i_{p} : \overline{ℚ} ↪ \overline{ℚ}_{p} and i_{∞} : \overline{ℚ} ↪ ℂ.

1.1. Good old Iwasawa theory. We regard μ_{N} = Spec(ℤ[X]/((1 + X)^{N} − 1)) as a group scheme whose A points μ_{N}(A) is made of N-th roots of unity in A. We put μ_{p^{∞}} = ⋃_{n} μ_{p^{n}}. If A is a subring of B = \overline{ℚ}_{p} or ℂ, we often write A[μ_{p^{n}}] for the extension of A generated by μ_{p^{n}}(B). We have N : Gal(ℚ[μ_{p^{∞}}]/ℚ) ≅ ℤ_{p}^{×} by ζ^{σ} = ζ^{N (σ)}, and for the unique subgroup μ_{p−1} ⊂ ℤ_{p}^{×}, we define ℚ_{∞} by the subfield of ℚ[μ_{p^{∞}}] fixed by N^{−1}(μ_{p−1}(ℤ_{p})). Then Gal(ℚ_{∞}/ℚ) ≅ Γ = 1 + pℤ_{p}. Note that Γ = (1 + p)^{ℤ_{p}} ≅ ℤ_{p} by (1 + p)^{s} = ∑_{n=0}^{∞} \binom{s}{n}p^{n} ∈ Γ, and ℚ_{∞} is called the cyclotomic ℤ_{p}-extension over ℚ. For a given number field M, the composite M_{∞} = ℚ_{∞}M is called the cyclotomic ℤ_{p}-extension of M.

In the late 1950’s, Iwasawa started studying the arithmetic of ℤ_{p}-extensions (particularly the cyclotomic ones) and created many important profinite modules X with a continuous action of Γ, which are often called Iwasawa modules, and it turned out to be a great success, bringing us new knowledge of arithmetic of ℤ_{p}-extensions (see [CPI]). The book [ICF] by Washington is a good introduction of Iwasawa’s theory. For any complete valuation ring W finite flat over ℤ_{p} with residue field 𝔽, the completed group algebra

W [[Γ]] = \underleftarrow{lim}_{n}W [Γ/Γ^{p^{n}}] = \underleftarrow{lim}_{n}W [x]/((1 + x)^{p^{n}} − 1) ≅ W [[x]]

therefore acts on X (if X is W -module in addition to being a Γ-module). We write Λ = Λ_{W} for W [[Γ]], which is a local ring with maximal ideal 𝔪_{Λ} and Λ/𝔪_{Λ} = 𝔽.

Let us discuss briefly a proto-typical example of Iwasawa modules. In this section, we assume W = ℤ_{p}. For simplicity, we assume only in this section that M/ℚ is at most tamely ramified at p. Since p fully ramifies in ℚ_{∞}/ℚ, M and ℚ_{∞} are linearly disjoint; so, we have Gal(M_{∞}/M) ≅ Γ, canonically, by restricting σ ∈ Gal(M_{∞}/M) to ℚ_{∞}. The ℤ_{p}-extension M_{∞}/M has layers of intermediate fields M_{n} fixed by Γ_{n} = Γ^{p^{n}}. Let C_{n} be the p-Sylow subgroup of the class group of M_{n}. By Galois conjugation, Gal(M_{n}/M) = Γ/Γ_{n} ≅ ℤ/p^{n}ℤ acts on C_{n}. The norm map N_{m,n} : C_{m} → C_{n} (m > n) is compatible with the Galois action, and we can form the projective limit C_{∞} = \underleftarrow{lim}_{n}C_{n} with respect to the norm map. Since C_{n} is a finite p-group, C_{∞} is a p-profinite (compact) group with a continuous Γ-action. Thus we get an Iwasawa module C_{∞}, which is a module over Λ.

By class field theory, C_{n} is isomorphic to the Galois group Gal(H_{n}/M_{n}) of p-Hilbert class field H_{n} by the Artin reciprocity map. Since H_{m} ⊃ H_{n} (m > n), we have the restriction map Res_{m,n} : Gal(H_{m}/M_{m}) → Gal(H_{n}/M_{n}) which is surjective. By class field theory, we have the following commutative diagram:

Gal(H_{m}/M_{m}) ⟶^{∼} C_{m}
Res^{m,n}↓  ↓^{N_{m,n}}
Gal(H_{m}/M_{m}) ⟶^{∼} C_{m}.

Thus C_{∞} is canonically isomorphic to X = Gal(H_{∞}/M_{∞}), where H_{∞}/M_{∞} is the maximal unramified p-abelian extension. The action of σ ∈ Gal(M_{m}/M) on Gal(H_{m}/M_{m}) is given by taking an extension σ̃ of σ to H_{m}/M and conjugating τ ∈ Gal(H_{m}/M_{m}) by σ̃: τ ↦ σ̃τσ^{−1}. Since Gal(H_{m}/M_{m}) is commutative, this action of σ is independent of the choice of the extension σ̃. Since H_{n}M_{m} is a maximal abelian extension of M_{n} inside H_{m}, the Galois group Gal(H_{m}/H_{n}M_{m}) is the derived group of Gal(H_{m}/M_{n}). Writing γ = 1+p and identifying Γ with Gal(M_{∞}/M), Gal(M_{m}/M_{n}) is generated by γ^{p^{n}}; so, the derived group Gal(H_{m}/H_{n}M_{m}) is generated by the commutators

[γ^{p^{n}}, τ] = γ^{p^{n}}τγ^{−p^{n}}τ^{−1} = (γ^{p^{n}} − 1)τ

(writing additively on the right-hand side). This shows

Gal(H_{m}/H_{n}M_{m}) ≅ (γ^{p^{n}} − 1)C_{m}.

Since M_{m}/M_{n} fully ramifies at p, we have Gal(H_{n}M_{m}/M_{m}) ≅ Gal(H_{n}/M_{n}). Thus we have the following control theorem (cf. [CPI] [52] Sections 3–4):

Theorem 1.1 (Control). We have C_{∞} ⊗_{Λ} Λ/(γ^{p^{n}} − 1)Λ ≅ C_{n} for all n > 0.

Note that Λ ≅ ℤ_{p}[[x]] by γ ↦ 1 + x and (1 + x)^{p^{n}} − 1 is in the unique maximal ideal of Λ, by Nakayama’s lemma, the finiteness of C_{n} tells us the following

Corollary 1.2. The Iwasawa module C_{∞} is a torsion Λ-module of finite type.

Let F be a totally real field of finite degree. We now assume that M = F [μ_{p}] (note that p is tamely ramified in M/ℚ). Then Gal(M/F) ↪ μ_{p−1} by the Teichmüller character ω. Again by conjugation, Gal(M/F) acts on C_{∞}. Let C_{∞}[ω^{i}] = {x ∈ C_{∞}|σ(x) = ω^{i}(σ)x}.

Over a principal ideal domain A, any torsion module X of finite type is isomorphic to ⨁_{P} A/P^{e(P)} for finitely many prime elements P. For a regular local ring A of dimension two (including Λ), if X is a torsion A-module of finite type, we can find a homomorphism ι : X → ⨁_{P} A/P^{e(P)}A for finitely many prime elements P with finite kernel and cokernel. A generator of the ideal ∏_{P} P^{e(P)} is called the characteristic element in A and the characteristic power series of X when A = Λ, which is uniquely determined modulo Λ^{×}.

Let ω = lim_{n→∞} N^{p^{n}} be the Teichmüller character of Gal(M[μ_{p^{∞}}]/M) and regard it as an ideal character by class field theory so that ω(𝔩) = ω(F rob_{𝔩}) for the Frobenius element F rob_{𝔩} for prime ideal 𝔩 outside p. We put ω(𝔭) = 0 if 𝔭|p (but ω^{n}(𝔭) = 1 for all 𝔭 including 𝔭|p if ω^{n} is the trivial character). When F is totally real, in the late 1970’s, Deligne and Ribet (and Barsky and Cassou-Nogués; see [LFE] Chapter 3)) constructed a power series Φ_{F} (x; ω^{i}) ∈ Λ such that for even integer i with 0 ≤ i < p − 1

Φ_{F} (γ^{1−n} − 1; ω^{i}) =

\begin{cases}(γ^{1−n} − 1) ∏_{𝔭|p}(1 − ω^{−n}(𝔭)N(𝔭)^{n−1})L(1 − n, ω^{−n}) ∈ ℤ[μ_{p−1}]_{(p)} & if i = 0 \\ ∏_{𝔭|p}(1 − ω^{i−n}(𝔭)N(𝔭)^{n−1})L(1 − n, ω^{i−n}) ∈ ℤ[μ_{p−1}]_{(p)} & if i > 0. \end{cases}

for all n > 0. Here is a theorem of A. Wiles ([W]) which was originally called the main conjecture of the Iwasawa theory for F:

Theorem 1.3 (A. Wiles). Let F be a totally real number field of finite degree. Then we have char(C_{∞}[ω^{1−i}])(x) = Φ_{F} (x; ω^{i}) for even i up to a unit multiple in Λ.

It looks like to have a mismatch because the ω^{1−i}-part C_{∞}[ω^{1−i}] is described by the L-function of ω^{i}. However, we may regard L(1 − n, ω^{i}) as a function of characters N^{n−1}ω^{i−n} ≡ ω^{i−1} mod p; so, the ω^{1−i}-part C_{∞}[ω^{1−i}] is described by the L-function whose domain is made up of characters congruent to ω^{1−i} modulo p.

We can generalize the construction of the characteristic power series to a more general ring R. We consider a formal power series ring W [[x_{1}, …, x_{g}]] and assume that R is a normal integral domain finite and torsion-free over W [[x_{1}, …, x_{g}]]. Then a prime ideal P of R is called a prime divisor if dim R/P = dim R − 1 for the Krull dimension dim R of R. In this situation, the localization R_{P} of R at a prime divisor P is a discrete valuation ring. Then for a torsion R-module X of finite type, we have X_{P} = X ⊗_{R} R_{P} has finite length over R_{P}. We write ℓ_{P} (X) for the length. Then char(X) = ∏_{P} P^{ℓ_{P} (X)} is an ideal of R, which is called the characteristic ideal of X. When R = W [[x_{1}, …, x_{g}]], the characteristic ideal of X is generated by the characteristic power series char(X)(x) ∈ W [[x_{1}, …, x_{g}]] (see [BCM] Chapter 7). Basically by definition, we have a morphism of R-modules i : X → ⨁_{P} R/P^{e(P)} whose cokernel and kernel vanishes after localization at every prime divisor.

1.2. Galois theoretic interpretation of the class group. Let 𝕋_{p} = ℚ_{p}/ℤ_{p}. For any abelian p-profinite compact or p-torsion discrete module X, we define the Pontryagin dual module X^{∗} by X^{∗} = Hom_{cont}(X, 𝕋_{p}) and give X^{∗} the topology of uniform convergence on every compact subgroup of X. Then by Pontryagin duality theory (cf. [FAN]), we have (X^{∗})^{∗} ≅ X canonically. Moreover, if X is finite, X^{∗} ≅ X non-canonically.

Exercise 1.1. Prove that X^{∗} is a discrete module if X is p-profinite and X^{∗} is compact if X is discrete.

By this fact, if X is the dual of a profinite module X = \underleftarrow{lim}_{n}X_{n} for finite modules X_{n} with surjections X_{m} ↠ X_{n} for m > n, X^{∗} = ⋃_{n} X_{n}^{∗} is a discrete module which is a union of finite modules X_{n}^{∗}.

For the (profinite) Galois group G over a field inside \overline{ℚ} or a p-adic field inside \overline{ℚ}_{p} and a continuous G-module X, we denote by H^{q}(G, X) the continuous group cohomology with coefficients in X. If X is finite, H^{q}(G, X) is as defined in [MFG] 4.3.3. By a result of Tate, Galois cohomology has cohomological dimension 2; so, only H^{0}, H^{1} and H^{2} are important. Thus we have

H^{0}(G, X) = X^{G} = {x ∈ X|gx = x for all g ∈ G},

and if X is finite,

H^{1}(G, X) = \frac{\{G ⟶^{c} X : continuous|c(στ) = σc(τ) + c(σ) for all σ, τ ∈ G\}}{\{G ⟶^{b} X|b(σ) = (σ − 1)x for x ∈ X independent of σ\}}.

If G = Gal(\overline{ℚ}_{p}/K) for a finite extension K/ℚ_{p}, by Tate duality (see [MFG] 4.42),

H^{2−i}(G, X) ≅ Hom(H^{i}(G, X^{∗}(1)), ℚ/ℤ)

for finite X. If X = \underleftarrow{lim}_{n}X_{n} for finite G-modules X_{n}, we have H^{q}(G, X) = \underleftarrow{lim}_{n}H^{q}(G, X_{n}), and if X = ⋃_{n} X_{n} is discrete with finite X_{n}, we have H^{q}(G, X) = \underrightarrow{lim}_{n}H^{q}(G, X_{n}). For a general K-vector space V with a continuous action of G and a G-stable W -lattice L of V, we define H^{q}(G, V) = H^{q}(G, L) ⊗_{W} K.

For each Galois character ψ : Gal(\overline{ℚ}/F) → W^{×}, we write \overline{ψ} for the 𝔽^{×}-valued character ψ mod 𝔪_{W} for the maximal ideal 𝔪_{W} of W. For any W -module X, we write X(ψ) for the Galois module whose underlying W -module is X and Galois action is given by ψ. We simply write X(i) for X(N^{i}). In particular ℤ_{p}(1) ≅ \underleftarrow{lim}_{n}μ_{p^{n}} as Galois modules. Write 𝔊_{K} = Gal(F^{(p)}/K) for any intermediate field K of F^{(p)}/F, where F^{(p)}/F is the maximal extension unramified outside p and ∞. Fix an integer i as above, and put ψ = ω^{1−i}. We consider the (continuous) Galois cohomology group H^{1}(𝔊_{F} , 𝕋_{p}(ψ)) and define the Selmer group of 𝕋_{p}[ψ] over K by

(1.1) Sel_{K}(𝕋_{p}[ψ]) = Ker(H^{1}(𝔊_{K}, 𝕋_{p}(ψ)) ⟶^{Res} ∏_{𝔩} H^{1}(I_{𝔩}, 𝕋_{p}(ψ))),

where 𝔩 runs over all prime ideals of K and I_{𝔩} is a chosen inertia subgroup at 𝔩 of 𝔊_{K}. By the inflation-restriction sequence,

0 → H^{1}(Gal(M/F), H^{0}(𝔊_{M}, 𝕋_{p}(ψ)))

→ H^{1}(𝔊_{F} , 𝕋_{p}(ψ)) → H^{0}(Gal(M/F), H^{1}(𝔊_{M}, 𝕋_{p}(ψ)))
→ H^{2}(Gal(M/F), H^{0}(𝔊_{M}, 𝕋_{p}(ψ)))

is exact. Since Gal(M/F) has order prime to p,

H^{1}(Gal(M/F), H^{0}(𝔊_{M}, 𝕋_{p}(ψ))) = H^{2}(Gal(M/F), H^{0}(𝔊_{M}, 𝕋_{p}(ψ))) = 0.

Thus we have

H^{1}(𝔊_{F} , 𝕋_{p}(ψ)) ≅ H^{0}(Gal(M/F), H^{1}(𝔊_{M}, 𝕋_{p}(ψ))) ⟶_{ι}^{∼} Hom_{𝔊_{F}} (𝔊_{M}, 𝕋_{p}(ψ)).

Since H_{m}/M_{m} is maximal p-abelian extension unramified everywhere and elements in Sel_{F} (𝕋_{p}(ψ)) is made up of unramified cocycle, we find that

Sel_{F} (𝕋_{p}(ψ)) ≅ Hom(C_{0}[ψ], 𝕋_{p}).

Exercise 1.2. Give a detailed proof of the above identity.

This shows that C_{0}[ψ] ≅ Sel_{F} (𝕋_{p}(ψ))^{∗}, which we write as Sel_{F}^{∗} (𝕋_{p}(ψ)) hereafter. As a consequence of Theorem 1.3, we get the following very precise p-class number formula:

Corollary 1.4. Let the notation be as above. Then we have C_{0}[ψ] ≅ Sel_{F}^{∗} (𝕋_{p}(ψ)) canonically, and

|C_{0}[ψ]| = |Sel_{F} (𝕋_{p}(ψ))| = \begin{cases}|L(0, ψ)|_{p}^{−1} & if ψ ≠ ω \\ |(γ − 1)L(0, ψ)|_{p}^{−1} & if ψ = ω, \end{cases}

where |x|_{p} = |i_{p}(x)|_{p} for the p-adic absolute value | · |_{p} of \overline{ℚ}_{p} with |p|_{p} = p^{−1}.

The direction p||C_{0}[ψ]| ⇒ p|L(0, ψ) is due to Herbrand (1932), and the converse p|L(0, ψ) ⇒ p||C_{0}[ψ]| is later proven by Ribet (1976). The above precise formula follows from Theorem 1.3 because

• if X is a torsion Λ-module of finite type and has no nontrivial finite Λsubmodule, char_{Λ}(X)(α) = char_{W} (X/(x − α)X) by ring theory,

• C_{∞} does not have a nontrivial finite Λ-submodule as proven by R. Greenberg (cf., Inventiones Math. 47, 1978).

By a similar argument, taking the Pontryagin dual, we find Corollary 1.5. Let ψ_{∞} be the restriction of ψ to 𝔊_{M_{∞}}. Then we have a canonical isomorphism: C_{∞}[ψ_{∞}] ≅ Sel_{F_{∞}}(𝕋_{p}(ψ))^{∗}.

If G is a group with subgroup H and a given W [H]-module X, we define Ind_{H}^{G} X = W [G]⊗_{W [H]} X on which we let g ∈ G act by g(g′ ⊗x) = (gg′ ⊗x). Identifying X with 1 ⊗ X ⊂ Ind_{H}^{G} X, the action of H on 1 ⊗ X and X are identical. Thus Ind_{H}^{G} X is an extension of H-module X to the bigger group G. Choosing a complete representative set Q of G/H in G, we find Ind_{H}^{G} X = ⨁_{q∈Q} q ⊗ X ≅ ⨁_{q∈Q} X with q ⊗ X ≅ X as W -modules.

We can embed Ind_{H}^{G} X with Hom_{W [H]}(W [G], X) by sending ξ = ∑_{q} q ⊗ a_{q} to ϕ_{ξ}(q) = a_{q}. If we let g ∈ G act on ϕ ∈ Hom_{W [H]}(W [G], X) by (gϕ)(x) = ϕ(g^{−1}x), this injection is W [G]-equivariant. If (G : H) < ∞, this is a surjective isomorphism. Often Hom_{W [H]}(W [G], X) is called co-induced module and written as Coind_{H}^{G} X; so, Coind_{H}^{G} X ≅ Ind_{H}^{G} X if [G : H] < ∞ and Ind_{H}^{G} X ↪ Coind_{H}^{G} X in general.

When G = 𝔊_{F} and H = 𝔊_{M}, we write simply Ind_{M}^{F} X for Ind_{𝔊_{M}}^{𝔊_{F}} X. Since Γ = Gal(F_{∞}/F), we can define a character ψ : Gal(M_{∞}/F) → Λ^{×} by ψ(σ) = (σ|_{F_{∞}})ψ(σ|_{M}). In this way, we get 𝔊_{F} -module Λ^{∗}(ψ).

Exercise 1.3. Prove Λ^{∗}(ψ) ≅ Ind_{F_{∞}}^{F} 𝕋_{p}(ψ).

Here is Shapiro’s lemma:

Lemma 1.6. Let X be a discrete p-torsion 𝔊_{K}-module for an intermediate field K of F^{(p)}/F. Then we have a canonical isomorphism H^{q}(G, Ind_{K}^{F} X) ≅ H^{1}(G, X) for G = 𝔊_{K} and I_{𝔩}. If X = 𝕋_{p}(ψ) and K = F_{∞}, the above isomorphism is a homomorphism of Λ-modules, where Λ acts on the left-hand side through the coefficients Ind_{K}^{F} 𝕋_{p}(ψ) ≅ Λ^{∗}(ψ).

Proofs of this lemma can be found in any book on Galois cohomology (e.g., [CGP] III.6.2 or [MFG] Lemma 4.20). In these books, the lemma is stated using Coind_{H}^{G}, which is isomorphic to Ind_{H}^{G} X if G/H is finite. Since Galois cohomology is defined by continuous cocycles, the lemma is valid actually for Ind_{F}^{K} even if K/F is an infinite extension if the coefficient module has discrete topology. As an immediate corollary of this lemma, we have

Corollary 1.7. We have Sel_{F_{∞}}(𝕋_{p}(ψ)) ≅ Sel_{F} (Λ^{∗}(ψ)).

1.3. The Iwasawa algebra as a deformation ring. We can interpret the group algebra Λ as a universal deformation ring. In this subsection, we assume F = ℚ. We write CL_{/W} for the category of p-profinite local W -algebras A with A/𝔪_{A} = 𝔽. A character ρ : 𝔊_{ℚ} → A^{×} for A ∈ CL_{/W} is called a W -deformation (or just simply a deformation) of \overline{ψ} if (ρ mod 𝔪_{A}) = \overline{ψ}. A couple (R, ρ) made of an object R of CL_{/W} and a character ρ : 𝔊_{F} → R^{×} is called a universal couple for ψ if for any deformation ρ : 𝔊_{F} → A of \overline{ψ}, we have a unique morphism ϕ_{ρ} : R → A in CL_{/W} (so it is a local W -algebra homomorphism) such that ϕ_{ρ} ∘ ρ = ρ. By the universality, if exists, the couple (R, ρ) is determined uniquely up to isomorphisms. The ring R is called the universal deformation ring and ρ is called the universal deformation of \overline{ψ}.

Lemma 1.8. The couple (Λ_{W} , ψ) is the universal couple for \overline{ψ}.

Proof. Since ℚ[μ_{p^{∞}}] is the maximal abelian extension of ℚ unramified outside p and ∞ by class field theory (or else, the theorem of Kronecker-Weber), we have 𝔊_{ℚ}/\overline{[𝔊_{ℚ}, 𝔊_{ℚ}]} = Gal(ℚ^{(p)}/ℚ[μ_{p^{∞}}]). Since each deformation ρ : 𝔊_{ℚ} → A^{×} factors through Gal(Q[μ_{p^{∞}}]/ℚ) = Γ×Gal(ℚ[μ_{p}]/ℚ), the character ρ is determined by ρ(γ), because ρ|_{ℚ[μ_{p}]} is given by ψ. Then we have ϕ_{ρ} : Λ_{W} = W [[x]] → A by sending x to ρ(γ) − 1, and we have ϕ_{ρ} ∘ ψ = ρ.

For a given non-abelian n-dimensional representation \overline{ρ} : 𝔊_{F} → GL_{n}(𝔽), a deformation ρ : 𝔊_{F} → GL_{n}(R) is a continuous representation with ρ mod 𝔪_{R} ≅ \overline{ρ}. Suppose that R is a normal domain finite type over W [[x_{1}, …, x_{g}]]. This lemma suggests that we could define Sel_{F} (V (ρ)^{∗}) as a suitable subgroup of H^{1}(𝔊_{F} , V (ρ)^{∗}), where V (ρ) = R^{n} on which 𝔊_{F} acts by ρ and V (ρ)^{∗} = V (ρ) ⊗_{R} R^{∗}. If further Sel_{F} (V (ρ)^{∗}) is a torsion R-module of finite type, we consider the characteristic ideal char(Sel_{F} (V (ρ)^{∗})) and try to compare it with the p-adic L-function L_{p}(ρ) of ρ, which has to be an element of R. Then the main conjecture would be that char(Sel_{F} (V (ρ)^{∗})) is generated by L_{p}(ρ). We describe in this lecture a non-abelian generalization of Iwasawa’s theory relating it to automorphic forms.

2. Selmer Groups

In this section, we recall the definition due to R. Greenberg of (possibly nonabelian) Selmer groups in terms of Galois cohomology groups. There are other definitions due to Bloch-Kato (which include the nonordinary cases) in terms of different cohomology theory.

2.1. Ordinary Galois representations. We recall a definition of ordinary padic Galois representations which gives a good foundation of defining Greenberg’s Selmer groups. Let K be a p-adic local field (a finite extension of ℚ_{p}). Let S be a finite set of rational primes containing p and primes ramified in F/ℚ. We consider the maximal extension ℚ^{(S)}/ℚ unramified outside S and ∞, and write 𝔊_{F}^{S} = Gal(ℚ^{(S)}/F) (so 𝔊_{F} = 𝔊_{F}^{(p)} if only p ramifies in F/ℚ). Let V be a finite dimensional vector space with a continuous action of 𝔊_{F}^{S}.

We write D_{𝔭} for the decomposition group of a prime ideal 𝔭|p in 𝔊_{F}^{S}. By local class field theory, we have the Artin reciprocity map F_{𝔭}^{×} ∋ u ↦ [u, F_{𝔭}] ∈ D_{𝔭}^{ab} such that N ([u, F_{𝔭}]) = N_{F_{𝔭}/ℚ_{p}}(u)^{−1} if u is a 𝔭-adic unit.

For each prime factor 𝔭|p of F, we assume to have a filtration of the following type for integers a = a(𝔭) and b = b(𝔭) with b ≤ 0 < a:

V = F_{𝔭}^{b}V ⊃ ⋯ ⊃ F_{𝔭}^{0}V ⊃ F_{𝔭}^{1}V ⊃ ⋯ ⊃ F_{𝔭}^{a+1}V = {0} (2.1)

stable under the decomposition group D_{𝔭} and an open subgroup of the inertia group I_{𝔭} ⊂ D_{𝔭} acts on each sub-quotient F_{𝔭}^{i}V/F_{𝔭}^{i+1}V by N^{i}. We call V p-ordinary if V satisfies (2.1) for all 𝔭|p. Once V satisfies (2.1), its dual V^{∗}(1) = Hom_{K}(V, K) ⊗ N again satisfies (2.1) for F_{𝔭}^{−i}V^{∗}(1) = (F_{𝔭}^{i}V)^{⊥}(1). If V is associated with a pure motive M, the existence of the filtration as in (2.1) is almost equivalent to the criticality of M at 0 and M^{∗}(1) at 1, though one may allow a = b = 0 for criticallity at 0 of M if complex conjugation acts by the scalar −1 on V (resp. a = b = 1 for criticality of M^{∗}(1) at 1 if complex conjugation acts trivially on V).

Example 2.1. If E is an elliptic curve defined over F having non-super-singular semi-stable reduction at all prime factors 𝔭 of p in F, then its Galois representation V (E) = T (E) ⊗_{ℤ} ℚ given by the Tate module T (E) = \underleftarrow{lim}_{r}E[p^{r}](\overline{ℚ}) satisfies the above condition for b = 0 and a = 1 with dim_{ℚ_{p}} F_{𝔭}^{i}V/F_{𝔭}^{i+1}V = 1 for 0 ≤ i ≤ 1.

Similarly, if f is an elliptic Hecke eigenform on Γ_{0}(N) with Neben character ψ, for each prime 𝔓|p of its field ℚ(f) generated by Hecke eigenvalues for f (the Hecke field of f), we have a representation V = V_{𝔓}(f) of 𝔊_{ℚ}^{S} for K = ℚ(f)_{𝔓}. Here S is the set of prime factors of Np. If furthermore, the Hecke eigenvalue a(p, f) for the Hecke operator T (p) or U(p) is a 𝔓-adic unit (we call such a form 𝔓-ordinary), V_{𝔓}(f) has a filtration as above for (a, b) = (k − 1, 0) for the weight k of f, as long as ψ is unramified at p. In this example, dim_{K} F_{𝔭}^{0}V/F_{𝔭}^{1}V = dim_{K} F_{𝔭}^{k−1}V = 1 and dim_{K} V = 2 (e.g., [GME] Theorem 4.2.7).

Example 2.2. Suppose that F is totally real. Pick a holomorphic Hilbert modular Hecke eigenform f : GL_{2}(F)\GL_{2}(F_{𝔸}) → ℂ of weight k (as defined in [MFG] 5.6), where f is called of weight (n, v) for n = (k − 2)I for I = ∑_{σ:F ↪\overline{ℚ}} σ and v = 0). Let ℚ(f) be the Hecke field of f. Take a prime factor 𝔓 of p in ℚ(f), and set K = ℚ(f)_{𝔓}. Assume that f is 𝔓-ordinary and that the central character of f is unramified at p. Then we have again a two dimensional representation V_{𝔓}(f) of 𝔊_{F}^{S} associated with f for a suitable finite set S such that the trace of Frobenius element at a prim 𝔩 ∉ S is given by the Hecke eigenvalue of T (𝔩) for f. This representation V_{𝔓}(f) has a filtration as above for (a, b) = (k − 1, 0) and dim_{K} F_{𝔭}^{0}V/F_{𝔭}^{1}V = dim_{K} F_{𝔭}^{k−1}V = 1. We refer the reader to [MFG] 5.6 and [H96] 2.8 for more details of this Galois representation.

2.2. Greenberg’s Selmer groups. Suppose V be an ordinary Galois representation of 𝔊_{F}^{S}. Let W be the p-adic integer ring of K, and take a W -lattice L in V stable under 𝔊_{F}^{S}. Here is how to find L. Take a base of V and identify End_{K}(V) = M_{2}(V). The action of 𝔊_{F}^{S} induces a continuous homomorphism ρ : 𝔊_{F} → GL_{2}(K). Since 𝔊_{F}^{S} is compact, Im(ρ) sits in a maximal compact subgroup. Since all maximal compact subgroup of GL_{2}(K) is a conjugate g · GL_{2}(W)g^{−1} of GL_{2}(W), if Im(ρ) ⊂ g · GL_{2}(W)g^{−1}, replacing ρ by the isomorphic g^{−1}ρg, we may take L = W^{2} ⊂ K^{2} = V. The isomorphism class of L is unique when L/𝔪_{W} L is absolutely irreducible 𝔊_{F}^{S} -module (a result of Carayol and Serre: see [MFG] Proposition 2.13), where 𝔪 = 𝔪_{W} ⊂ W is the maximal ideal.

We write H^{q}(F,?) for H^{q}(𝔊_{F}^{S} ,?) and H^{q}(F_{𝔩},?) for H^{q}(D_{𝔩},?). We put F_{𝔭}^{+}V = F_{𝔭}^{1}V. For prime ideals 𝔮 ∤ p and 𝔭|p of F, we put

L_{𝔮}(V) = \begin{cases}Ker(Res : H^{1}(F_{𝔮}, V) → H^{1}(I_{𝔮}, V)) & if 𝔮 ∤ p, \\ Ker(Res : H^{1}(F_{𝔭}, V) → H^{1}(I_{𝔭}, \frac{V}{F_{𝔭}^{+}(V)})) & if 𝔭|p, \end{cases}

and define L_{𝔮}(V/L) by the image of L_{𝔮}(V) in H^{1}(F_{𝔮}, V/L); so, L_{𝔮}(V/L) is a divisible submodule of H^{1}(F_{𝔮}, V/L). Then we define, for A = K and K/W

Sel_{F} (L ⊗_{W} A) = Ker(H^{1}(F, L ⊗_{W} A) → ∏_{𝔮} \frac{H^{1}(F_{𝔮}, L ⊗_{W} A)}{L_{𝔮}(L ⊗ A)}). (2.2)

The classical Selmer group of V is given by Sel_{F} (V/L), equipped with discrete topology. The strict Selmer group Sel_{F}^{str}(L ⊗ A) is defined replacing I_{𝔮} by D_{𝔮} for 𝔮|p in the above definition. The “−” Selmer group Sel_{F}^{−}(L ⊗ A) is defined by the same formula replacing F^{+}(V) by F^{−}(V) := F^{0}(V) in the definition of L_{𝔭}(V).

2.3. Selmer groups with general coefficients. In the above definition of the Selmer group, important point is to have a two step filtration V ⊃ F_{𝔭}^{+}V ⊃ {0} stable under the local Galois group D_{𝔭}. Thus if we have such a filtration, we can at least generalize the definition to any continuous p-profinite Galois module L̃. We describe it when L̃ is an R[𝔊_{F}]-module, where we assume that R is a profinite local integral domain over the power series ring W [[x]]. Let κ : 𝔊_{F} → W [[x]]^{×} be the universal character deforming the identity character of 𝔊_{F}.

Exercise 2.1. Show that there is a unique square root \sqrt{κ} : 𝔊_{ℚ} → Λ_{W}^{×} of κ (Hint: Use the assumption p > 2).

We suppose that L̃ is R-free of finite rank. For each prime factor 𝔭|p of F, we assume to have a filtration:

L̃ = F_{𝔭}^{b}L̃ ⊃ ⋯ ⊃ F_{𝔭}^{0}L̃ ⊃ F_{𝔭}^{1}L̃ ⊃ ⋯ ⊃ F_{𝔭}^{a+1}L̃ = {0} (2.3)

stable under the decomposition group D_{𝔭} and an open subgroup of the inertia group I_{𝔭} ⊂ D_{𝔭} acts on each sub-quotient F_{𝔭}^{i}L̃/F_{𝔭}^{i+1}L̃ by κ^{i}. We suppose that F_{𝔭}^{i}L̃/F_{𝔭}^{i+1}L̃ is R-free of finite rank.

Let R^{∗} be the Pontryagin dual of R. We put F^{+}L̃ = F^{1}L̃. For prime ideals 𝔮 ∤ p and 𝔭|p of F, we put

L_{𝔮}(L̃ ⊗_{R} R^{∗}) = \begin{cases}Ker(Res : H^{1}(F_{𝔮}, L̃ ⊗_{R} R^{∗}) → H^{1}(I_{𝔮}, L̃ ⊗_{R} R^{∗})) & if 𝔮 ∤ p, \\ Ker(Res : H^{1}(F_{𝔮}, L̃ ⊗_{R} R^{∗}) → H^{1}(I_{𝔭}, \frac{L̃⊗_{R}R^{∗}}{F_{𝔭}^{+}(L̃)⊗_{R}R^{∗}})) & if 𝔭|p. \end{cases}

Then we define,

Sel_{F} (L̃ ⊗_{R} R^{∗}) = Ker(H^{1}(F, L̃ ⊗_{R} R^{∗}) → ∏_{𝔮} \frac{H^{1}(F_{𝔮}, L̃ ⊗_{R} R^{∗})}{L_{𝔮}(L̃ ⊗_{R} R^{∗})}). (2.4)

Note that Sel_{ℚ}(Λ^{∗}(ψ)) we already discussed is isomorphic to Sel_{ℚ}(Λ(ψ) ⊗_{Λ} Λ^{∗}), because F_{p}^{+}Λ(ψ) = {0} in this case.

Suppose that R = Λ or its normal finite extension. To distinguish this case from the other, we write 𝕃 for L̃. Naive questions are (Q1) When Sel_{F} (𝕃⊗_{R} R^{∗}) has R-torsion Pontryagin dual module Sel_{F}^{∗} (𝕃⊗_{R} R^{∗}) of

finite type? (See Greenberg’s paper in “Motive”: Proc. Symp. Pure. Math. 55 No. 2, 1994).

It is well known that Sel_{F}^{∗} (𝕃 ⊗_{R} R^{∗}) is of finite type over R. Thus the main point of this question is the torsion property. We expect that this should be true if L̃_{k} := 𝕃 ⊗_{W [[Γ]],k} W is algebro-geometric for infinitely many positive integer k (that is, the Galois representation can be realized in the étale cohomology group of a smooth projective variety over F). Here k : W [[Γ]] → W is the W -algebra homomorphism induced by the character Γ ∋ γ ↦ γ^{k} ∈ W;

(Q2) Suppose further that R = Λ. If Sel_{F}^{∗} (𝕃 ⊗_{Λ} Λ^{∗}) is Λ-torsion, what is its characteristic power series f_{𝕃}(x) of Sel_{F}^{∗} (𝕃 ⊗_{Λ} Λ^{∗}). More generally, if R is just a normal finite extension of Λ and if Sel_{F}^{∗} (𝕃 ⊗_{R} R^{∗}) is R-torsion, is char(Sel_{F}^{∗} (𝕃 ⊗_{R} R^{∗})) principal generated by f_{𝕃} ∈ R? If that is the case, what is the function f_{𝕃} in the structure sheaf of Spec(R)? Is it related to a p-adic L-function? (As for analytic p-adic L-functions, see [LFE] Chapters 7 and 10 and [H96]).

If 𝕃 is algebro-geometric in the sense of (Q1), we expect to have a p-adic analytic method of making p-adic L-functions L_{p}(𝕃, s) such that L_{p}(𝕃, s) = ϕ_{𝕃}(γ^{s} − 1) (γ = 1 + p) for ϕ_{𝕃}(x) ∈ Ŵ[[x]] and ϕ_{𝕃}(x) = f_{𝕃}(x) up to units in Ŵ[[x]], where Ŵ is a discrete valuation ring containing W.

3. Deformation Rings and Adjoint Square Selmer Groups

We relate Galois deformation rings and adjoint square Selmer groups. Let V be a two dimensional irreducible representation of 𝔊_{F} = 𝔊_{F}^{(p)} over K and suppose L satisfies (2.1):

(ord) V ⊋ F_{𝔭}^{+}V ⊋ {0} with dim_{K} F_{𝔭}^{+}V = 1

stable under the decomposition group D_{𝔭} such that the inertia group I_{𝔭} ⊂ D_{𝔭} acts on the quotient V/F_{𝔭}^{+}V trivially. We write ε_{𝔭} (resp. δ_{𝔭}) for the character giving the action of D_{𝔭} on F_{𝔭}^{+}V and V/F_{𝔭}^{+}V. We also write \overline{ε}_{𝔭} = (ε_{𝔭} mod 𝔪_{W}) and \overline{δ}_{𝔭} = (δ_{𝔭} mod 𝔪_{W}). Let F_{𝔭}^{ur} be the maximal unramified extension of F_{𝔭} inside \overline{ℚ}_{p}. We suppose the following two conditions: (cyc) The character det(V)|_{I_{𝔭}} factors through Gal(F_{𝔭}^{ur}[μ_{p^{∞}}]/F_{𝔭}^{ur}) for each 𝔭|p; (ds) \overline{δ}_{𝔭} ≠ \overline{ε}_{𝔭} for all 𝔭|p. We fix a W -lattice L ⊂ V stable under 𝔊_{F}. Choose a base of L over W and write ρ : 𝔊_{F} → GL_{2}(W) the resulting matrix representation.

3.1. Nearly ordinary deformation rings. In this subsection, A denotes an object in CL_{/W}.

We consider universality among the deformations of L of the following type. They are rank two free A-modules L̃ with continuous 𝔊_{F} -action satisfying the following four properties: (D1) L̃/𝔪_{A}L̃ ≅ \overline{L} as 𝔊_{F} -modules for \overline{L} = L ⊗_{W} 𝔽; (D2) Writing ι : W → A for the structure homomorphism of W -algebras, we have

the identity of the determinant characters:

ι ∘ det(L) = det L̃;

(D3) Fix a decomposition group D_{𝔭} of each prime factor 𝔭|p in 𝔊_{F}. Then we have an exact sequence 0 → F_{𝔭}^{+}L̃ → L̃ → L̃/F_{𝔭}^{+}L̃ → 0 stable under D_{𝔭} for each prime factor 𝔭|p, where L̃/F_{𝔭}^{+}L̃ is free of rank one over A and the action of D_{𝔭} on F_{𝔭}^{+}L̃ ⊗_{A} 𝔽 is given by \overline{ε}_{𝔭};

(D4) The action of D_{𝔭} on L̃/F_{𝔭}^{+}L̃ factors through Gal(F^{ur}[μ_{p^{∞}}]/F^{ur}).

The condition (D3) is the near ordinarity, and L̃ is ordinary if L̃/F_{𝔭}^{+}L̃ is unramified for all 𝔭|p. The condition (D2) on the automorphic side corresponds to fixing the central character of automorphic representations. The condition (D4) is not necessary (so, it is not made in [MFG] in 5.6.3) but makes our argument far simpler as we will see later.

When we consider the matrix form of the representation L̃ (resp. \overline{L} and L) fixing a base of L̃ over A, we write it as ρ̃: 𝔊_{F} → GL_{2}(A) (resp. \overline{ρ} : 𝔊_{F} → GL_{2}(𝔽) and ρ : 𝔊_{F} → GL_{2}(W)). If the centralizer Z(\overline{ρ}) = \{x ∈ GL_{2}(𝔽)|x\overline{ρ}x^{−1} = \overline{ρ}\} of \overline{ρ} is made up of scalar matrices, we have a universal couple (R_{F} , ρ_{F} : 𝔊_{F} → GL_{2}(R_{F})) such that for any deformation ρ̃ as above, we have a unique W -algebra homomorphism φ : R_{F} → A such that φ ∘ ρ_{F} ≅ ρ̃ in GL_{2}(A) (see [MFG] 2.3 and 3.2.4). Here R_{F} is a Noetherian p-profinite local W -algebra with residue field 𝔽.

Starting from a Galois representation \overline{L} of 𝔊_{F_{𝔭}} := Gal(\overline{ℚ}_{p}/F_{𝔭}), we can think of universality among deformations L̃ of \overline{L} replacing 𝔊_{F} in (D1–4) by 𝔊_{F_{𝔭}}. Again we have a universal couple (R_{F_{𝔭}}, ρ_{F_{𝔭}}) similar to (R_{F} , ρ_{F}) if Z(\overline{ρ}) is made up of scalars.

We write Φ(A) for the set of isomorphism classes of deformations ρ̃ : 𝔊_{F} → GL_{2}(A) of \overline{ρ}. Similarly we write Φ_{𝔭}(A) for the set of isomorphism classes of deformations ρ̃: 𝔊_{F_{𝔭}} → GL_{2}(A) of \overline{ρ} if ρ and \overline{ρ} are representations of 𝔊_{F_{𝔭}}. Then Φ and Φ_{𝔭} is a covariant functor from the category CL_{/W} of profinite W -local algebras with residue field 𝔽 into the category SET S of sets.

If the universal couple (R_{F} , ρ_{F} : 𝔊_{F} → GL_{2}(R_{F})) exists, we say that the functor Φ is representable by (R_{F} , ρ_{F} : 𝔊_{F} → GL_{2}(R_{F})), and we have Hom_{W -alg}(R_{F} , A) ≅ Φ(A) by Hom_{W -alg}(R_{F} , A) ∋ φ ↦ φ ∘ ρ_{F} ∈ Φ(A).

The condition Z(\overline{ρ}) = 𝔽^{×} ⊂ GL_{2}(𝔽) follows from one of the following two conditions: (ai_{M}) \overline{ρ} restricted to Gal(\overline{ℚ}/M) is absolutely irreducible; (DS) \overline{ρ} is reducible, and Im(\overline{ρ}) contains at least one nontrivial unipotent element

and at least one semi-simple element with two distinct eigenvalues.

Exercise 3.1. Let \overline{ρ} : 𝔊_{ℚ_{p}} → GL_{2}(𝔽_{p}) be a Galois representation of the form \overline{ρ}(σ) =

( \begin{matrix}\overline{ψ} & ∗ \\ 0 & 1 \end{matrix} ) for a nontrivial character \overline{ψ} : 𝔊_{ℚ_{p}} → 𝔽_{p}^{×}. Suppose u = ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) ∈ Im(\overline{ρ}). Show that, for each element in Φ_{𝔭}(A), we can find its member ρ̃ such that ρ̃(σ) is upper triangular for all σ ∈ 𝔊_{ℚ_{p}} and Im(ρ̃) contains u = ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) ∈ GL_{2}(A).

By (D3), the space L(ρ_{F}) of ρ_{F} has a natural filtration, stable under D_{𝔭}, L(ρ_{F}) ⊃ F_{𝔭}^{+}L(ρ_{F}) ⊃ {0} with L(ρ_{F})/F^{+}L(ρ_{F}) ≅ R_{F}. We write δ_{𝔭} (resp. ε_{𝔭}) for the character giving the action of D_{𝔭} on L(ρ_{F})/F^{+}L(ρ_{F}) (resp. F^{+}L(ρ_{F})). Thus ρ_{𝔭}|_{D_{𝔭}} ≅ ( \begin{matrix}ε_{𝔭} & ∗ \\ 0 & δ_{𝔭} \end{matrix} ). In particular, δ_{𝔭} : Gal(F_{𝔭}^{ur}[μ_{p^{∞}}]/F_{𝔭}^{ur}) → R_{F}^{×} is a continuous character. Since φ_{ρ} ∘ δ_{𝔭} is trivial for φ_{ρ} : R_{F} → W inducing ρ = φ_{ρ} ∘ ρ_{F}, the image of δ_{𝔭} in R_{F}^{×} is p-profinite. Thus δ_{𝔭} factors through p-profinite part of Gal(F^{ur}[μ_{p^{∞}}]/F^{ur}). Note that N : Gal(F_{𝔭}^{ur}[μ_{p^{∞}}]/F_{𝔭}^{ur}) → ℤ_{p}^{×} identifies the p-profinite part of Gal(F_{𝔭}^{ur}[μ_{p^{∞}}]/F_{𝔭}^{ur}) with a p-profinite subgroup Γ_{𝔭} of 1 + pℤ_{p}. Thus Γ_{𝔭} is generated by γ_{𝔭} := γ^{p^{m}} for a suitable m for the generator γ = 1 + p of 1 + pℤ_{p} = γ^{ℤ_{p}}, and δ_{𝔭} induces a W [[Γ_{𝔭}]]-algebra structure on R_{F}. Let Γ_{F} = ∏_{𝔭|p} Γ_{𝔭}, and hereafter we regard R_{F} as W [[Γ_{F}]]-algebra by this algebra homomorphism. Write S = {𝔭|p}, we have an isomorphism W [[Γ_{F}]] ≅ W [[x_{𝔭}]]_{𝔭∈S}.

3.2. Adjoint square Selmer groups and differentials. Let L̃ ∈ Φ(A). Define

Ad(L̃) = {ϕ ∈ End_{A}(L̃)|Tr(ϕ) = 0} .

Then Ad(L̃) has the following three step filtration stable under D_{𝔭}:

Ad(L̃) ⊃ F^{−}Ad(L̃) ⊃ F^{+}Ad(L̃) ⊃ {0}, (3.1)

where

\begin{aligned}F^{−}Ad(L̃) & = \{ϕ ∈ Ad(L̃)|ϕ(F^{+}L̃) ⊂ F^{+}L̃\}, \\ F^{+}Ad(L̃) & = \{ϕ ∈ Ad(L̃)|ϕ(F^{+}L̃) = 0\}. \end{aligned}

If we take a base of L̃ containing a generator of F^{+}L̃ and we identify End_{K}(L̃) with M_{2}(A) by this base, F^{−}Ad(L̃) (resp. F^{+}Ad(L̃)) is made up of upper triangular matrices with trace zero (resp. upper nilpotent matrices). We thus have Sel_{F} (Ad(V/L)) for Ad(V/L) := Ad(L) ⊗_{ℤ_{p}} 𝕋_{p} = Ad(V)/Ad(L), and by a general version of the definition of Selmer groups, we have Sel_{F} (Ad(L̃) ⊗_{A} A^{∗}).

Note that D_{p} acts trivially on F^{−}Ad(V)/F^{+}Ad(V); so, F^{−}Ad(V)/F^{+}Ad(V) ≅ K as D_{p}-modules.

We want to prove

Proposition 3.1. Suppose that Φ is represented by the universal couple (R_{F} , ρ_{F}). Then the Pontryagin dual Sel_{F}^{∗} (Ad(V/L)) is canonically isomorphic to the module of 1-differentials Ω_{R_{F} /W [[Γ_{F}]]} ⊗_{R_{F} ,φ} W, where φ : R_{F} → W is the W -algebra homomorphism such that ρ ≅ φ ∘ ρ_{F}. More generally, for any L̃ ∈ Φ(A), we have Sel_{F}^{∗} (Ad(L̃) ⊗_{A} A^{∗}) = Hom(Sel_{F} (Ad(L̃) ⊗_{A} A^{∗}), 𝕋_{p}) ≅ Ω_{R_{F} /W [[Γ_{F}]]} ⊗_{R_{F} ,ϕ} A, where φ : R_{F} → A is the W -algebra homomorphism such that ρ̃ ≅ ϕ ∘ ρ_{F}.

To prove this proposition, we recall here the definition of Kähler 1-differentials and some of their properties. Let A be a B-algebra, and suppose that A and B are objects in CL_{/W}.

We consider the module of continuous 1-differentials Ω_{A/B} with the universal B-derivation d : A → Ω_{A/B} for a B-algebra A (A, B ∈ CL_{/W}). Here the continuity is with respect to the profinite topology on A.

For a module M with continuous A-action (in short, a continuous A-module), let us define the module of B-derivations by

Der_{B}(A, M) = \{δ : A → M ∈ Hom_{B}(A, M)|\begin{matrix}δ: continuous \\ δ(ab) = aδ(b) + bδ(a) \\ for all a, b ∈ A \end{matrix}\} .

Here the B-linearity of a derivation δ is equivalent to δ(B) = 0, because

δ(1) = δ(1 · 1) = 2δ(1) ⇒ δ(1) = 0.

Then the module Ω_{A/B} has the following universal property: For any given δ ∈ Der_{B}(A, M), there exists a unique continuous A-linear map ϕ : Ω_{A/B} → M such that δ = ϕ ∘ d. Thus Hom_{A}(Ω_{A/B}, M) ≅ Der_{B}(A, M) by ϕ ↦ ϕ ∘ d. In other words, Ω_{A/B} represents the covariant functor M ↦ Der_{B}(A, M) from the category of continuous A-modules into ℤ-MOD. Since Der_{B}(A, M) only depends on the image of B in A under the algebra homomorphism ι : B → A giving the B-algebra structure of A, we have

Ω_{A/B} ≅ Ω_{A/ι(B)}. (3.2)

The construction of Ω_{A/B} is easy. The multiplication a ⊗ b ↦ ab induces a B-algebra homomorphism m : Aˆ⊗_{B}A → A taking a ⊗ b to ab. We put I = Ker(m), which is an ideal of Aˆ⊗_{B}A. Then we define Ω_{A/B} = I/I^{2}. It is easy to check that the map d : A → Ω_{A/B} given by d(a) = a ⊗ 1 − 1 ⊗ a mod I^{2} is a continuous B-derivation (Exercise 3.2). Thus we have a morphism of functors: Hom_{A}(Ω_{A/B},?) → Der_{B}(A,?) given by ϕ ↦ ϕ ∘ d. Since Ω_{A/B} is generated by d(A) as A-modules (Exercise 3.2), the above map is injective. To show that Ω_{A/B} represents the functor, we need to show the surjectivity of the above map.

Proposition 3.2. The above morphism of two functors M ↦ Hom_{A}(Ω_{A/B}, M) and M ↦ Der_{B}(A, M) is an isomorphism, where M runs over the category of continuous A-modules. Thus the functor: M ↦ Der_{B}(A, M) is represented by Ω_{A/B} in the category of continuous A-modules.

Proof. Define ϕ : A × A → M by (a, b) ↦ aδ(b) for δ ∈ Der_{B}(A, M). Then ϕ(ab, c) = abδ(c) = ϕ(b, c) and ϕ(a, bc) = aδ(bc) = abδ(c) = bϕ(a, c) for a, c ∈ A and b ∈ B. Thus ϕ gives a continuous B-bilinear map. By the universality of the tensor product, ϕ : A × A → M extends to a B-linear map ϕ : Aˆ⊗_{B}A → M. Now we see that ϕ(a ⊗ 1 − 1 ⊗ a) = aδ(1) − δ(a) = −δ(a) and

ϕ((a ⊗ 1 − 1 ⊗ a)(b ⊗ 1 − 1 ⊗ b)) = ϕ(ab ⊗ 1 − a ⊗ b − b ⊗ a + 1 ⊗ ab)

= −aδ(b) − bδ(a) + δ(ab) = 0.

This shows that ϕ|_{I}-factors through I/I^{2} = Ω_{A/B} and δ = ϕ ∘ d, as desired.

Corollary 3.3. Let the notation be as in the proposition.

(i) Suppose that B is a C-algebra for an object C ∈ CL_{/W}. Then we have the following natural exact sequence:

Ω_{B/C} ˆ⊗_{B}A ⟶ Ω_{A/C} ⟶ Ω_{A/B} → 0.

Moreover, if Ω_{A/B} = 0, we have an isomorphism: Ω_{B/C} ˆ⊗_{B}A ≅ Ω_{A/C}. (ii) Let π : A ↠ C be a surjective morphism in CL_{/W}, and write J = Ker(π).

Then we have the following natural exact sequence:

J/J^{2} ⟶ Ω_{A/B} ˆ⊗_{A}C ⟶ Ω_{C/B} → 0.

Moreover, if B = C, then J/J^{2} ≅ Ω_{A/B} ˆ⊗_{A}C.

Proof. By assumption, we have algebra morphisms C → B → A in Case (i) and B → A ↠ C = A/J in Case (ii). By Yoneda’s lemma, we only need to prove that

0 → Der_{B}(A, M) ⟶^{α} Der_{C}(A, M) ⟶^{β} Der_{C}(B, M)

is exact in Case (i) for all continuous A-modules M and that

0 → Der_{B}(C, M) ⟶^{α} Der_{B}(A, M) ⟶^{β} Hom_{C}(J/J^{2}, M)

is exact in Case (ii) for all continuous C-modules M. The first α is just the inclusion and the second α is the pull back map. Thus the injectivity of α is obvious in two cases. Let us prove the exactness at the mid-term of the first sequence. The map β is the restriction of derivation D on A to B. If β(D) = D|_{B} = 0, then D kills B and hence D is actually a B-derivation, i.e. in the image of α. The map β in the second sequence is defined as follows: For a given B-derivation D : A → M, we regard D as a B-linear map of J into M. Since J kills M, D(jj′) = jD(j′) + j′D(j) = 0 for j, j′ ∈ J. Thus D induces C-linear map: J/J^{2} → M. Then for b ∈ B and x ∈ J, D(bx) = bD(x) + xD(b) = bD(x). Thus D is B-linear, and β(D) = D|_{J}. Now prove the exactness at the mid-term of the second exact sequence. The fact β ∘α = 0 is obvious. If β(D) = 0, then D kills J and hence a derivation well defined on C = A/J. This shows that D is in the image of α.

Now suppose that Ω_{A/B} = 0 in the assertion (i). We only prove the isomorphism when all the rings involved are p-profinite local and Noetherian. Then writing I for the kernel of the multiplication Aˆ⊗_{B}A ↠ A, we find I/I^{2} = Ω_{A/B} = 0. Thus by Nakayama’s lemma, I = 0, which implies the B-algebra structure B → A is surjective. From this, the isomorphism follows.

Now suppose that B = C in the assertion (ii). We need to create a surjective C-linear map: γ : Ω_{A/B} ⊗ C ↠ J/J^{2} such that γ ∘ α = id. Let π : A → C be the projection and ι : B = C ↪ A be the structure homomorphism giving the B-algebra structure on A. To do that, we first look at the map δ : A → J/J^{2} given by δ(a) = a − P (a) mod J^{2} for P = ι ∘ π. Then

aδ(b) + bδ(a) − δ(ab) = a(b − P (b)) + b(a − P (a)) − ab − P (ab)

= (a − P (a))(b − P (b)) ≡ 0 mod J^{2}.

Thus δ is a B-derivation. By the universality of Ω_{A/B}, we have an A-linear map ϕ : Ω_{A/B} → J/J^{2} such that ϕ ∘ d = δ. By definition, δ(J) generates J/J^{2} over A, and hence ϕ is surjective. Since J kills J/J^{2}, the surjection ϕ factors through Ω_{A/B} ⊗_{A} C and induces γ as desired.

For any continuous A-module X, we write A[X] for the A-algebra with square zero ideal X. Thus A[X] = A ⊕ X with the multiplication given by

(r ⊕ x)(r′ ⊕ x′) = rr′ ⊕ (rx′ + r′x).

It is easy to see that A[X] ∈ CL_{/W}, if X is of finite type, and A[X] ∈ CL_{/W} if X is a p-profinite A-module. By definition,

Der_{B}(A, X) ≅ {ϕ ∈ Hom_{B−alg}(A, A[X])|ϕ mod X = id} ,

where the map is given by δ ↦ (a ↦ (a ⊕ δ(a)). Note that i : A → Aˆ⊗_{B}A given by i(a) = a⊗1 is a section of m : Aˆ⊗_{B}A → A. We see easily that Aˆ⊗_{B}A/I^{2} ≅ A[Ω_{A/B}] by x ↦ m(x) ⊕ (x − i(x)). Note that d(x) = x − i(x).

Exercise 3.2. 1. Show that the map d : A → Ω_{A/B} given by d(a) = a ⊗ 1 − 1 ⊗ a

mod I^{2} is a continuous B-derivation;

2. Prove that Ω_{A/B} is generated by d(A) as A-modules and that it is an Amodule of finite type if A is Noetherian.

Proof of Proposition 3.1. Write simply (R, ρ) for (R_{F} , ρ_{F}). Let X be a profinite R-module. Then R[X] is an object in CL_{/W}. We consider the W -algebra homomorphism ξ : R → R[X] with ξ mod X = id. Then we can write ξ(r) = r ⊕ d_{ξ}(r) with d_{ξ}(r) ∈ X. By the above definition of the product, we get d_{ξ}(rr′) = rd_{ξ}(r′)+r′d_{ξ}(r) and d_{ξ}(W) = 0. Thus d_{ξ} is a W -derivation, i.e., d_{ξ} ∈ Der_{W} (R, X). For any derivation d : R → X over W, r ↦ r ⊕ d(r) is obviously an W -algebra homomorphism, and we get

(3.3) {ρ̃ ∈ Φ(R[X])|ρ̃ mod X = ρ} / ≈_{X}

≅ {ρ̃ ∈ Φ(R[X])|ρ̃ mod X ≈ ρ} / ≈
≅ {ξ ∈ Hom_{W -alg}(R, R[X])|ξ mod X = id}

≅ Der_{W} (R, X) ≅ Hom_{R}(Ω_{R/W} , X),

where “≈_{X}” is conjugation under (1⊕M_{n}(X))∩GL_{2}(R[X]), and “≈” is conjugation by elements in GL_{2}(R[X]).

Let ρ̃ be the deformation in the left-hand side of (3.3). Then we may write ρ̃(σ) = ρ(σ) ⊕ u′(σ). We see

ρ(στ) ⊕ u′(στ) = (ρ(σ) ⊕ u′(σ))(ρ(τ) ⊕ u′(τ)) = ρ(στ) ⊕ (ρ(σ)u′(τ) + u′(σ)ρ(τ)),

and we have

u′(στ) = ρ(σ)u′(τ) + u′(σ)ρ(τ).

Define u(σ) = u′(σ)ρ(σ)^{−1}. Since det ρ̃ = det ρ = det ρ, x(σ) = ρ̃(σ)ρ(σ)^{−1} has values in SL_{2}(R[X]), and x = 1 ⊕ u ↦ u = x − 1 is an isomorphism from the multiplicative group of the kernel of the reduction map SL_{2}(R[X]) ↠ SL_{2}(R): {x ∈ SL_{2}(R[X])|x ≡ 1 mod X} onto the additive group

Ad(X) = {x ∈ M_{2}(X)|Tr(x) = 0} = L(Ad(ρ)) ⊗_{R} X.

Thus we may regard that u has values in Ad(X) = L(Ad(ρ)) ⊗_{R} X.

We also have

(3.4) u(στ) = u′(στ)ρ(στ)^{−1}

= ρ(σ)u′(τ)ρ(στ)^{−1} + u′(σ)ρ(τ)ρ(στ)^{−1} = Ad(ρ)(σ)u(τ) + u(σ).

Hence u : 𝔊_{F} → Ad(X) is a 1-cocycle. It is a straightforward computation to see the injectivity of the map:

{ρ̃ ∈ F(R[X])|ρ̃ mod X ≈ ρ} / ≈_{X}↪ H_{ct}^{1}(𝔊_{F} , Ad(X))

given by ρ̃ ↦ [u] (Exercise 2). We put F_{𝔭}^{±}(Ad(X)) = F_{𝔭}^{±}L(Ad(ρ)) ⊗_{R} X. Then we see from the fact that Tr(u) = 0 that

u(I_{𝔭}) ⊂ F_{𝔭}^{+}(Ad(X)) ⟺ u′(I_{𝔭}) ⊂ F_{𝔭}^{+}(Ad(X)) ⟺ d_{ξ}(W [[Γ_{F}]]) = 0 (3.5)

if ξ ∈ Hom_{W -alg}(R, R[X]) induces ρ̃.

Note that R[W^{∗}] = ⋃_{n} R[p^{−n}W/W], since W^{∗} = Hom(W, 𝕋_{p}) ≅ K/W. We put the profinite topology on the individual R[p^{−n}W/W]. On R[W^{∗}], we give a injective-limit topology. A map ϕ :? → R[W^{∗}] is continuous if ϕ^{−1}(R[p^{−n}W/W]) → R[p^{−n}W/W] is continuous for all n with respect to the induced topology from ? on the source and the profinite topology on the target. From this, any deformation (continuous with respect to having values in GL_{2}(R[W^{∗}]) gives rise to a continuous 1-cocycle with values in the discrete 𝔊-module Ad(V/L). In this way, we get

(Ω_{R/W [[Γ_{F}]]} ⊗_{R} W)^{∗} ≅ Hom_{R}(Ω_{R/W [[Γ_{F}]]}, W^{∗}) ↪ H_{ct}^{1}(𝔊_{F} , Ad(L)^{∗}). (3.6)

By definition, ρ is nearly p-ordinary if and only if u restricted to I_{𝔭} has values in F_{𝔭}^{−}(Ad(L)^{∗}) = F_{𝔭}^{−}(Ad(V/L)). This proves the result for Sel_{F}^{∗} (Ad(V/L)). The proof is the same for the general case, replacing W^{∗} as above by A^{∗} and taking X = A^{∗}.

In the above proof, we only used the fact that (R_{F} , ρ_{F}) is universal among the deformations satisfying (D1-4). If we add more conditions on deformations, we get a smaller universal couple (R′_{F} , ρ′_{F}). By the same argument, we often get the formula Sel_{F}^{∗} (Ad(L̃)⊗_{A}A^{∗} = Ω_{R′_{F} /W [[Γ_{F}]]}⊗_{R′_{F} ,ϕ}A. If R′_{F} is small enough, we could conclude that the Selmer group Sel_{F}^{∗} (Ad(V/T)) is finite and expect that |Sel_{F}^{∗} (Ad(V/T))| = |L_{p}(1, Ad(V))|_{p}^{−[K:ℚ_{p}]} (a generalized class number formula). This type of formula is verified by Wiles and others adding the following condition to the deformations L̃ satisfying (D1-4). Fix an integer k_{𝔭} for each 𝔭 ∈ S, and write k for the tuple (k_{𝔭})_{𝔭}. As an extra condition, we now consider

(D5) On L̃/F_{𝔭}^{+}L̃, Gal(F_{𝔭}^{ur}[μ_{p^{∞}}]/F_{𝔭}^{ur}) acts by the character N^{−k_{𝔭}} for all 𝔭|p. We write Φ_{k}(A) for the set of isomorphism classes of deformations ρ̃ : 𝔊_{F} → GL_{2}(A) of \overline{ρ} satisfying (D1-5). Under (ai_{F}) or (DS), we have the universal couple (R_{k,F} , ρ_{k,F}) among the deformations satisfying (D1-5).

Exercise 3.3. Show R_{k,F} = R_{F} ⊗_{W [[Γ_{F}]],k} W, where k : Γ_{F} → W^{×} is given by (γ_{𝔭}) ↦ ∏_{𝔭} N (γ_{𝔭})^{k_{𝔭}}.

We call c ∈ 𝔊_{F} a complex conjugation, if c is in the conjugacy class of a complex conjugation in Gal(\overline{ℚ}/ℚ).

Conjecture 3.4. Suppose (ds) and (ai_{F}) for \overline{ρ} and that F is totally real. If det(ρ)(c) = −1 for any complex conjugation c, the universal ring R_{k,F} is free of finite rank over W. If det(ρ) = N^{k_{0}} for an integer k_{0} on an open subgroup of Γ_{𝔭} for all 𝔭, R_{k,F} is a reduced local complete intersection if k_{𝔭} ≥ 1 − \frac{k_{0}}{2} for all 𝔭 ∈ S.

Here a reduced algebra A free of finite rank over W [[x_{1}, …, x_{t}]] is a local complete intersection over W [[x_{1}, …, x_{t}]] if A ≅ A[[T_{1}, …, T_{r}]]/(f_{1}(T), …, f_{r}(T)) for r power series f_{i}(T), where r is the number of variables in A[[T_{1}, …, T_{r}]].

Theorem 3.5 (Wiles, Taylor, Fujiwara). Suppose that the starting representation ρ is associated to a Hilbert modular form of p-power level (in this case, we call ρ is modular). If (ai_{M}) holds for M = F [μ_{p}], Conjecture 3.4 holds.

See [MFG] Theorem 3.31 for a proof of this fact for F = ℚ.

Proposition 3.6. Assume Conjecture 3.4. Then 1. Sel_{F} (Ad(ρ̃) ⊗_{W} W^{∗}) is finite for any ρ̃ ∈ Φ_{k}(W) if k_{𝔭} ≥ 1 − \frac{k_{0}}{2} for all 𝔭 ∈ S,

2. R_{F} is a reduced local complete intersection free of finite rank over W [[Γ_{F}]],

3. Sel_{F}^{∗} (Ad(ρ_{F}) ⊗_{R_{F}} R_{F}^{∗}) is a torsion R_{F} -module,

4. For an irreducible component Spec(𝕀) of Spec(R_{F}), write ρ_{𝕀} = π ∘ ρ_{F} for the projection π : R_{F} ↠ 𝕀. Then Sel_{F}^{∗} (Ad(ρ_{𝕀}) ⊗_{𝕀} 𝕀^{∗}) is a torsion 𝕀-module.

Proof. If R is reduced and free of finite rank over W, Ω_{R/W} is a finite module. Thus the first assertion follows. Note that P_{k} = Ker(k : W [[Γ_{F}]] → W) is generated by ((1 + x_{𝔭}) − N (γ_{𝔭})^{k_{𝔭}}) for 𝔭 ∈ S. Thus ∩_{k}P_{k} = {0}. Since R_{F} /P_{k}R_{F} ≅ R_{k,F} which is free of finite rank s over W, by Nakayama’s lemma, R_{F} is generated by s elements r_{1}, …, r_{s} over W [[Γ_{F}]] which give a base of R_{k,F} over W. Thus we have a surjective W [[Γ_{F}]]-linear map ι : W [[Γ_{F}]]^{s} → R_{F} sending (a_{1}, …, a_{s}) to ∑_{j} a_{j}r_{j}. Taking other k′, we find that R_{F} /P_{k′}R_{F} ≅ R_{k′,F} which is free over W; so, it has to be free of rank s over W. Thus Ker(ι) ⊂ P_{k′}^{s} for all k′; so, ι has to be an isomorphism. This shows the freeness of the second assertion.

Let C be the set of all k = (k_{𝔭})_{𝔭} such that k_{𝔭} ≥ 1− \frac{k_{0}}{2} for all 𝔭. Then we still have ⋂_{k∈C} P_{k} = {0}. Thus the natural W -algebra homomorphism R_{F} → ∏_{k∈C} R_{k,F} is an injection. The right-hand side is reduced (i.e., no nilpotent radical), and R_{F} is reduced.

We write R_{k} = R/P_{k} = W [[T_{1}, …, T_{r}]]/(\overline{f}_{1}(T), …, \overline{f}_{r}(T)). Write \overline{t}_{j} ∈ 𝔪_{R_{k}} for the image of T_{j} in R_{k}. Take a lift t_{j} in 𝔪_{R} of \overline{t}_{j} so that \overline{t}_{j} = (t_{j} mod P_{k}R). Then we define φ : W [[Γ_{F}]][[T_{1}, …, T_{r}]] ↠ R_{F} by φ(f(T_{1}, …, T_{r})) = f(t_{1}, …, t_{r}). Since R_{F} is W [[Γ_{F}]]-free, we have Ker(φ) ⊗_{W [[Γ_{F}]],k} W = (\overline{f}_{1}, …, \overline{f}_{r}); so, taking a lift f_{j} ∈ Ker(φ) of \overline{f}_{j}, we find Ker(φ) = (f_{1}, …, f_{r}) by Nakayama’s lemma, and hence R_{F} is a local complete intersection over W [[Γ_{F}]].

Since R_{F} is reduced, Ω_{R_{F} /W [[Γ_{F}]]} is a torsion R_{F} -module. From this, the last two assertions follow.

Since R_{F} is reduced and free of finite rank over W [[Γ_{F}]], its total quotient ring Q is a product of fields of finite dimension over the field K of fractions of W [[Γ_{F}]]. In particular, writing 𝕂 for the field of fractions of 𝕀, we have Q = 𝕂 ⊕ X for a complementary ring direct summand X. Let 𝕀′ be the projection of R_{F} to X. Then Spec(R_{F}) = Spec(𝕀)∪Spec(𝕀′) (and Spec(𝕀′) is the union of irreducible components other than Spec(𝕀)). We take the intersection Spec(C_{0}) = Spec(𝕀) ∩ Spec(𝕀′); so, C_{0} = 𝕀 ⊗_{R_{F}} 𝕀′, which is a torsion 𝕀-module.

Exercise 3.4. Show that 𝕀 ⊗_{R_{F}} 𝕀′ ≅ 𝕀/ ((𝕂 ⊕ 0) ∩ R_{F}) (cf. [H88] Lemma 6.3).

By the above expression, the W [[Γ_{F}]]-freeness tells us that the 𝕀-ideal char(C_{0}) = (𝕂 ⊕ 0) ∩ R_{F} is an intersection of a power of prime divisors (cf. [BCM] 7.4.2). Suppose that 𝕀 is a unique factorization domain, which follows if 𝕀 ≅ W [[Γ_{F}]] = W [[x_{𝔭}]]_{𝔭} (this is usually the case) and, hence, char(C_{0}) is principal generated by f ∈ 𝕀. For this conclusion, we do not need the isomorphism 𝕀 ≅ W [[Γ_{F}]] = W [[x_{𝔭}]]_{𝔭} but a milder condition that 𝕀 is a Gorenstein ring over W [[Γ_{F}]] is enough (that is, Hom_{W [[Γ_{F}]]}(𝕀, W [[Γ_{F}]]) ≅ 𝕀 as 𝕀-modules; see [H88] Theorem 6.8). Note that a local complete intersection over W [[Γ_{F}]] is a Gorenstein ring (e.g., [CRT] Theorem 21.3). Now by a theorem of Tate (e.g., [MFG] 5.3.4),

char(Ω_{R_{F} /W [[Γ_{F}]]} ⊗_{R_{F}} 𝕀) = char(C_{0}).

Thus we find

Corollary 3.7. Suppose either that 𝕀 is a unique factorization domain or that 𝕀 is a Gorenstein ring. Then char(Sel_{F}^{∗} (Ad(ρ_{𝕀}) ⊗_{𝕀} 𝕀^{∗})) is a principal ideal in 𝕀 generated by f ∈ 𝕀.

By Corollary 3.3, we have for any prime ideal P ∈ Spec(𝕀) with ι : 𝕀/P ≅ W, writing ρ_{P} = ι ∘ ρ_{𝕀} : 𝔊_{F} → GL_{2}(W)

Sel_{F}^{∗} (Ad(ρ_{P}) ⊗_{W} W^{∗}) ≅ Ω_{R_{F} /W [[Γ_{F}]]} ⊗_{R_{F} ,P} W ≅ Sel_{F}^{∗} (Ad(ρ_{𝕀}) ⊗_{𝕀} 𝕀^{∗}) ⊗_{𝕀} 𝕀/P.

This shows that if char(Sel_{F}^{∗} (Ad(ρ_{𝕀}) ⊗_{𝕀} 𝕀^{∗})) = (f) for f ∈ 𝕀, we have

char(Sel_{F}^{∗} (Ad(ρ_{P}) ⊗_{W} W^{∗})) = (f(P)),

where f(P) = (f mod P) ∈ W. Thus we get

Corollary 3.8. Let the notation and the assumption be as in Corollary 3.7. Then we have |Sel_{F}^{∗} (Ad(ρ_{P}) ⊗_{W} W^{∗})| = |f(P)|_{p}^{−[K:ℚ_{p}]} for all P ∈ Spec(𝕀)(W).

If we can show that for densely populated points P ∈ Spec(𝕀)(W), the size of the Selmer group Sel_{F}^{∗} (Ad(ρ_{P}) ⊗_{W} W^{∗}) is given by the p-adic L-value L_{p}(1, Ad(ρ_{P})), f(P) = L_{p}(1, Ad(ρ_{P})) up to units in W; so, f gives a p-adic L-function of the adjoint square representation Ad(ρ_{𝕀}). This holds if F = ℚ and the starting ρ is modular (associated to an elliptic modular form; see [MFG] Theorem 5.20). A generalization of this type of formulas, see papers by E. Ghate posted in his web page (http://www.math.tifr.res.in/˜ eghate) when [F : ℚ] = 2 and papers by M. Dimitrov in his web page (http://www-math.univ-paris13.fr/˜ dimitrov) for general F under some assumptions.

In any case, we expect to have a p-adic L-function L_{p}(Ad(ρ_{𝕀})) such that the characteristic ideal char(Sel_{F}^{∗} (Ad(ρ_{𝕀}) ⊗_{𝕀} 𝕀^{∗})) ⊂ 𝕀 is generated by L_{p}(Ad(ρ_{𝕀})) := f. This L-function should interpolate the L-values \frac{L(1,Ad(ρ_{k}))}{period} for ρ_{k} = k ∘ ρ_{𝕀} : 𝔊_{F} → GL_{2}(W). Note that Ad(ρ⊗χ) = Ad(ρ) for any Galois character χ : Gal(F [μ_{p^{∞}}]/F), and thus, this interpolation does not involve the character twists Ad(ρ)⊗χ. In other words, we should have the Selmer group of |S|+1 variables (S = {𝔭|p}): |S| variables coming from 𝕀 (or R_{F}) and another coming from the (cyclotomic) character twists. Accordingly we should have two variable p-adic L-function L_{p} ∈ 𝕀ˆ⊗_{W} W [[Γ]] = 𝕀[[x]] such that L_{p}(P_{k}, P_{χ}) = L_{p} mod (P_{k} ⊗ Ker(χ)) giving rise to L_{p}(1, Ad(ρ_{k}) ⊗ χ). In order to include the character twists, we need to study R_{F_{n}} for each layer of the cyclotomic ℤ_{p}-extension F_{∞}.

In this section, we have related the adjoint square Selmer group with the differentials of the deformation rings. It is an interesting problem to find such a link for the Selmer group of Galois representations functorially obtained from V. More plainly, one could ask: Is there any good ring invariant of the deformation ring which describe the standard Selmer group of ρ or ρ_{F}?

4. Iwasawa Theory for Deformation Rings

In this section, we study the relation among R_{n} = R_{ℚ_{n}} for the cyclotomic ℤ_{p}- extension ℚ_{∞}/ℚ. We start from a two dimensional representation V of 𝔊_{ℚ} over K satisfying (ord), (ds) and (ai_{ℚ}). We assume Conjecture 3.4 for ρ over F = ℚ_{n} for all n; so, it holds for modular Galois representation satisfying (ai_{ℚ[μ_{p}]}) and (ds) by Theorem 3.5, because (ai_{F [μ_{p}]}) and (ds) over F = ℚ are equivalent to these conditions over ℚ_{n} ([MFG] (AI) in page 205), and if the modularity for ρ is valid over ℚ, it is valid over ℚ_{n} by the existence of base-change of a given elliptic modular form to a Hilbert modular form over ℚ_{n} (cf. [BCG]).

4.1. Galois action on deformation rings. We write ρ_{n} and \overline{ρ}_{n} for the restriction of ρ and \overline{ρ} to 𝔊_{ℚ_{n}}. Then we consider the universal couple (R_{n} = R_{ℚ_{n}}, ρ_{n} = ρ_{ℚ_{n}}) staring from ρ_{n}. It is easy to see that (ai_{ℚ}) implies (ai_{ℚ_{n}}) and also once (ds) holds for \overline{ρ}, it is still valid for \overline{ρ}_{n}. Thus R_{n} is a reduced algebra free of finite rank over Λ_{n} := W [[Γ^{p^{n}}]] ⊂ W [[Γ]] = Λ, because Γ_{ℚ_{n}} = Γ^{p^{n}} ⊂ Γ_{ℚ} = Γ.

We write the functor Φ for ℚ_{n} as Φ_{n}. Let m > n. Starting from ρ̃ ∈ Φ_{n}(A), the restriction ρ̃_{m} of ρ̃ to 𝔊_{m} = 𝔊_{ℚ_{m}} is a deformation of \overline{ρ}_{m}; so, we have a natural transformation Φ_{n} → Φ_{m}. In particular, we have a unique W -algebra homomorphism π_{m,n} : R_{m} → R_{n} such that ρ_{ℚ_{n}}|_{𝔊_{m}} = π_{m,n} ∘ ρ_{ℚ_{m}}. Thus we get a tower of rings:

R_{m} ⟶^{π_{m,m−1}} R_{m−1} → ⋯ → R_{n} → ⋯ → R_{0}.

We define R_{∞} = \underleftarrow{lim}_{n}R_{n}, which represents the functor Φ_{∞} defined for 𝔊_{∞}. We write π_{n} : R_{∞} → R_{n} for the projection.

For σ ∈ Gal(ℚ_{n}/ℚ), write σ̃ for the lifting σ to 𝔊_{0}; so, σ̃|_{ℚ_{n}} = σ. Then the isomorphism class of ρ^{σ}(g) = ρ̃(σ̃gσ^{−1}) is uniquely determined by σ for ρ̃ ∈ Φ_{n}(A). Thus Gal(ℚ_{n}/ℚ) acts on Φ_{n}. In particular, we have a unique W -algebra homomorphism ⟨σ⟩ : R_{n} → R_{n} such that ρ_{ℚ_{n}}^{σ} ≅ ⟨σ⟩ ∘ ρ_{n}. Since ⟨σ⟩ ∘ ⟨σ^{−1}⟩ is the identity map, ⟨σ⟩ is an automorphism of the W -algebra R_{n}. Hereafter if confusion is unlikely, we simply write σ for ⟨σ⟩, and in this way, Gal(ℚ_{n}/ℚ) and hence 𝔊_{0} acts on R_{n}.

Theorem 4.1. Let R_{n}(σ − 1)R_{n} be the ideal of R_{n} generated by σ(r) − r for all r ∈ R_{n}. Write γ = N^{−1}(1+p) for the generator of Gal(ℚ_{∞}/ℚ). Then π_{m,n} induces an isomorphism R_{n} ≅ R_{m}/R_{m}(γ^{p^{n}} − 1)R_{m} for all m ≥ n including m = ∞.

To prove this theorem, we admit the following well-known fact:

Lemma 4.2. Suppose p > 2. Then if ρ̃ ∈ Φ_{m}(A) is fixed by Gal(ℚ_{m}/ℚ_{n}), then ρ̃ extends to a unique ρ′ ∈ Φ_{n}(A).

See [MFG] Lemma 5.32 for a proof.

Proof of Theorem 4.1. Let σ = γ^{p^{n}}. Since

σ^{n}(r) − r = (σ − 1)(1 + σ + ⋯ + σ^{n−1})(r),

we find that R_{m}(σ − 1)R_{m} is stable under the action of Gal(ℚ_{m}/ℚ_{n}), the group Gal(ℚ_{m}/ℚ_{n}) acts trivially on R′ = R_{m}/R_{m}(σ − 1)R_{m}. For the projection π′ : R_{m} ↠ R′, we write ρ′_{m} = π′ ∘ ρ_{m}. Then ρ′_{m}^{σ} ≅ ρ′_{m} and hence ρ′_{m} extends uniquely to a Galois representation ρ′ ∈ Φ_{n}(R′). We only need to prove that the couple (R′, ρ′) is universal for Φ_{n}. If ρ̃ ∈ Φ_{n}(A), we restrict ρ̃ to 𝔊_{m}, getting ρ̃_{m} : 𝔊_{m} → GL_{2}(A). By the universality of ρ_{m}, we find a unique morphism φ : R_{m} → A such that φ∘ρ_{m} ≅ ρ_{m}. Since φ∘ρ_{m}^{σ} = (φ∘ρ_{m})^{σ} ≅ ρ_{m}^{σ} ≅ ρ_{m} ≅ φ∘ρ_{m}, we find that φ ∘ σ = φ. Thus φ factors through R′. By the uniqueness of the extension to 𝔊_{n}, we find that φ ∘ ρ′ ≅ ρ̃; so, (R′, ρ′) is universal for Φ_{n}.

4.2. Control of adjoint square Selmer groups. Out of Theorem 4.1, we can create a lot of well controlled Iwasawa modules. We suppose that we have an algebra A in CL_{/W} so that

(i) R_{∞} is an A-algebra ; (ii) Γ acts trivially on A.

We can obviously take A to be W. There seems no other canonical choice of A, but we anyway state our result in this generality. Take an A-algebra B in CL_{/W}, and suppose we have an A-algebra homomorphism π : R_{0} → B. Then we have a unique Galois representation φ = π ∘ ρ_{ℚ}. We then consider the module of continuous 1-differentials Ω_{R_{j}/A} and its B-part: Ω_{R_{j}/A} ⊗_{R_{j}} B. Then from the above control theorem, we have

Proposition 4.3. Let A be a closed W -subalgebra of R_{∞} (in CL_{/W}) on which Γ acts trivially. Let B be an A-algebra in CL_{/W} and π : R_{0} → B be an A-algebra homomorphism. Then we have for 0 ≤ j ≤ k ≤ ∞ and γ_{j} = γ^{p^{j}},

\frac{Ω_{R_{k}/A} ˆ⊗_{R_{k}}B}{(γ_{j} − 1)Ω_{R_{k}/A} ˆ⊗_{R_{k}}B} ≅ Ω_{R_{j}/A} ˆ⊗_{R_{j}}B. (Ct)

Proof. We write R for R_{k} ˆ⊗_{A}B and R′ for R_{j} ˆ⊗_{A}B. Then R/R(γ_{j} − 1)R ≅ R′. Write α for the projection: R → R′ and π′ for

m ∘ ((π ∘ α_{j}) ⊗ id) : R′ = R_{k} ˆ⊗_{A}B → B ˆ⊗_{A}B ⟶^{m} B

for multiplication m : B ˆ⊗_{A}B → B. Let λ = π′ ∘ α. We have from Corollary 3.3 that

Ker(λ) ⊗_{R} B = Ker(λ)/ Ker(λ)^{2} = Ω_{R/B} ⊗_{R} B = Ω_{R_{k} ˆ⊗_{A}B/A⊗_{A}B} ⊗_{R} B

≅ (Ω_{R_{k}/A} ˆ⊗_{A}B) ⊗_{R} B ≅ Ω_{R_{k}/A} ˆ⊗_{R_{k}}B.

Similarly, we have Ker(π′) ⊗_{R} B ≅ Ω_{R_{j}/A} ˆ⊗_{R_{j}}B. We have the following exact sequence:

0 ⟶ R(γ_{j} − 1)R ⟶ Ker(λ) ⟶^{α} Ker(π′) ⟶ 0. (4.1)

Tensoring B over R to (4.1) and writing J = R(γ_{j} − 1)R, we get another exact sequence:

(J/J^{2}) ⊗_{R} B = J ⊗_{R} B ⟶^{i} Ω_{R_{k}/A} ˆ⊗_{R_{k}}B ⟶ Ω_{R_{j}/A} ˆ⊗_{R_{j}}B ⟶ 0.

We look into the B-linear map γ_{j} −1 : Ker(λ) → R. Write B′ for the image of B in R. Then B′ ⊂ Ker(γ_{j} −1) and R = Ker(λ)+B′. Thus (γ_{j} −1)R = (γ_{j} −1) Ker(λ). Since γ_{j} is a B′-algebra automorphism of R and J/J^{2} is a B′-module, we have

r(γ_{j} − 1)r′ ≡ (γ_{j} − 1)rr′ mod J^{2} (r, r′ ∈ J).

This shows that γ_{j} − 1 : Ker(λ) → R induces a surjective morphism of B′-modules: Ker(λ)/ Ker(λ)^{2} → J/J^{2}; thus, Im(i) = (γ_{j} − 1)(Ω_{R_{k}/A} ˆ⊗_{R_{k}}B), which shows the result.

In particular, taking A = W and B = 𝔽, we see that Ω_{R_{∞}/W} ⊗_{R_{∞}} 𝔽 is an 𝔽[[Γ]]-module of finite type. It is of torsion if and only if R_{∞} is Noetherian. We might ask (Q3) In addition to the assumptions (ai_{ℚ}) and (ds), suppose Conjecture 3.4 for ρ

over ℚ_{n} for all n. Then, is R_{∞} Noetherian? Conjecturing R_{∞} to be Noetherian would be too wild, because Ω_{R_{∞}/W} ⊗_{R_{∞}} 𝔽 having a quotient isomorphic to 𝔽[[Γ]] is just the positivity of the μ-invariant of the characteristic power series of Sel_{ℚ_{∞}}^{∗}(Ad(ρ_{∞}) ⊗_{R_{∞}} R_{∞}^{∗}) which could possibly occur (although for good p-adic L-functions, such as, p-adic Dirichlet L-functions, their μ-invariants do vanish; see [ICF]).

We write Λ_{n} for the subalgebra of R_{n} topologically generated by the image of δ_{∞} over W (writing ρ_{n} = ( \begin{matrix}ε_{n} & ∗ \\ 0 & δ_{n} \end{matrix} )). By (D4), δ_{n} restricted to the p-wild inertia subgroup factors through Γ_{n} = Γ^{p^{n}} and the tame part has values in W. Thus Λ_{n} = Λ_{n}[δ_{n}(F rob_{p})] for any lift of the Frobenius element in the decomposition group at p in 𝔊_{∞}. We can specifically take F rob_{p} = [p, ℚ_{p}] for the local Artin symbol [p, ℚ_{p}] = [p_{p}, ℚ], where p_{p} is the idele having p at the place p and 1 at all other places. We write δ_{0}(F rob_{p}) = a(p) ∈ R_{0}.

Proposition 4.4. Suppose that R_{0} ≅ Λ = W [[x]] and \frac{da(p)}{dx} ∈ Λ^{×}. Then we have R_{n} ≅ R_{0} ≅ Λ for all n.

Proof. For a B-algebra A with A, B ∈ CL_{/W}, we have Ω_{A/B} = 0 ⇔ A is a surjective image of B. Note that Ω_{W [[x]]/W} = W [[x]]dx and by our assumption, d(a(p)) = \frac{da(p)}{dx}dx generates Ω_{W [[x]]/W}. Thus by Corollary 3.3 (i), we find

Ω_{Λ_{0}/Λ_{∞}} = W [[x]]dx/W [[x]]d(a(p)) = 0,

and Λ_{∞} = Λ_{0}. Since σ(δ_{n}(F rob_{p})) = δ_{n}(σ̃F rob_{p}σ^{−1}) = δ_{n}(F rob_{p}), we find that Γ acts trivially on Λ_{n}. Thus applying the above proposition to A = Λ_{∞} and B = R_{0} = Λ_{0}, we find that

\frac{Ω_{R_{n}/Λ_{∞}} ˆ⊗_{R_{n}}R_{0}}{(γ − 1)Ω_{R_{n}/Λ_{∞}} ˆ⊗_{R_{n}}R_{0}} ≅ Ω_{R_{0}/Λ_{∞}} = 0.

Then by Nakayama’s lemma applied to the W [[Γ]]-module Ω_{R_{n}/Λ_{∞}} ˆ⊗_{R_{n}}R_{0}, we find that Ω_{R_{n}/Λ_{∞}} ˆ⊗_{R_{n}}R_{0} = 0. Then by applying the Nakayama’s lemma applied to R_{n}- module Ω_{R_{n}/Λ_{∞}}, we find that Ω_{R_{n}/Λ_{∞}} = 0; so, R_{n} = Λ_{n} = Λ_{∞} ≅ W [[x]].

The condition R_{0} = Λ follows if and only if ρ is associated to a Hecke eigenform f in S_{k}(SL_{2}(ℤ)) and for any other Hecke eigenform g ∈ S_{k}(SL_{2}(ℤ)) f ≢ g mod 𝔪_{W}. If we start with ρ associated to a Hecke eigenform f in S_{2}(Γ_{0}(p)), we can verify by computation the assumption of the corollary holds for primes p = 11, 17, 19, 23, 29, 31, 37, 41, 43, 47, 59, 61 and 67. Note that S_{2}(Γ_{0}(13)) = 0. As for p = 53, there are two Hecke eigenforms modulo Galois action, and one of them with f|T (p) = −f has non-unit \frac{da(p)}{dx}. In this case, for the cusp form g ∈ S_{54}(SL_{2}(ℤ)) of weight 54 giving a deformation of ρ, λ+1 for g|T (p) = λg is divisible by a square of a prime factor of 53 in the Hecke field of g; so, a(p)(γ^{p−1}−1) = λ and a(p)(0) = −1, which imply that \frac{da(p)}{dx} ∈ 𝔪_{Λ}. For a given Hecke eigenform f ∈ S_{k}(SL_{2}(ℤ)), write f|T (p) = λ(p)f. Then the associated p-adic Galois representation is ordinary if λ(p) is a p-adic unit in W. We call such a prime p ordinary for f. Serre has predicted

Conjecture 4.5. Ordinary primes for f ∈ S_{k}(SL_{2}(ℤ)) have Kronecker density 1.

This fact is known for weight 2 Hecke eigenform associated with a rational elliptic curve without complex multiplication by the Ramanujan bound combined with a work of Serre ([S]). Now we may ask

(Q4) For a given Hecke eigenform f ∈ S_{k}(SL_{2}(ℤ)) with Galois representation ρ, assuming Conjecture 4.5, does \frac{da(p)}{dx} ∈ Λ^{×} hold for density one ordinary primes?

Let h_{k}(ℤ) ⊂ End(S_{k}(SL_{2}(ℤ))) be the algebra generated over ℤ by Hecke operators T (n). It is well known that h_{k} is free of finite rank over ℤ (see [IAT] Chapter 3). We put h_{k}(W) = h_{k} ⊗_{ℤ} W. We fix a weight k_{0} and pick a Hecke eigenform f of weight k_{0} in S_{k_{0}}(SL_{2}(ℤ)). Defining λ : h_{k_{0}} → W (for sufficiently large W) by f|T = λ(T)f for T ∈ h_{k_{0}}, we have a W -algebra homomorphism λ : h_{k_{0}}(W) → W. We pick a unique local ring 𝕋_{k_{0}} ⊂ h_{k_{0}}(W) through which λ factors. Suppose the p-adic Galois representation ρ associated to f is ordinary and \overline{ρ} satisfies (ai_{ℚ[μ_{p}]}) and (ds) (which hold for almost all primes p by a result of Serre and Ribet). Since we have an Eisenstein series E of weight p − 1 with E ≡ 1 mod p (if p ≥ 5), multiplying by E^{m} sends f to g = f · E^{m} ∈ S_{k}(SL_{2}(ℤ)) for k = k_{0} + m(p − 1), and g|T ≡ λ(T)g mod 𝔪_{W}. Thus we have a W -algebra homomorphism \overline{λ} : h_{k}(W) → 𝔽 by sending T to \overline{λ}(T) = (λ(T) mod 𝔪_{W}) ∈ 𝔽. Thus there exists a unique local ring 𝕋_{k} ⊂ h_{k}(W) through which \overline{λ} factors. The ring 𝕋_{k} is free of finite rank over W and carries a Galois representation ρ_{k}^{ord} : 𝔊_{ℚ} → GL_{2}(𝕋_{k}) with det(X − ρ_{k}^{ord}(F rob_{ℓ})) = X^{2} − t(ℓ)X + ℓ^{k−1} for all primes ℓ ∤ p, where t(ℓ) is the image of the Hecke operator T (ℓ) in 𝕋_{k}. Thus det(ρ_{k}^{ord}) = N^{k}. Since Tr(ρ_{k}^{ord}(F rob_{ℓ})) ≡ \overline{λ}(T (ℓ)) ≡ Tr(ρ_{k}^{ord}(F rob_{ℓ})) mod 𝔪_{W}, by Chebotarev density, ρ_{k}^{ord} ≡ \overline{ρ} mod 𝔪_{W}; so, ρ_{k}^{ord} satisfies (D1–2) and (D4). Since det(ρ_{k}^{ord}) ≡ det(ρ_{k_{0}}^{ord}) mod 𝔪_{W}, we see ψ = det(ρ_{k}^{ord})^{−1} det(ρ_{k_{0}}^{ord}) ≡ 1 mod 𝔪_{W}; so, we have a unique square root \sqrt{ψ} : 𝔊_{ℚ} → W^{×}. Then ρ_{k} = \sqrt{ψ} ⊗ ρ_{k}^{ord} satisfies (D1–4), and hence ρ_{k} ∈ Φ(W). Thus we have a unique algebra homomorphism τ_{k} : R_{0} → 𝕋_{k} with τ_{k} ∘ ρ_{0} ≅ ρ_{k}. Indeed, Wiles, Taylor and Fujiwara proved that τ_{k} induces an isomorphism R_{k,ℚ} ≅ 𝕋_{k} for all k as above (this is the proof of Theorem 3.5). Thus R_{0} ↪ ∏_{k} 𝕋_{k}, and writing 𝕋 for its image, we have R_{0} ≅ 𝕋. Without using the existence of R_{0}, we may define 𝕋 to be the p-profinite W -subalgebra of ∏_{k} 𝕋_{k} topologically generated by t(n) for all n prime to p (then, 𝕋 is Λ_{W} -free of finite rank; see [LFE] Chapter 7). The proof over general F due to Fujiwara is done basically by the same techniques. Anyway, τ_{k}(a(p)) are all distinct (because a(p) is the unit root of X^{2} − t(p)X + p^{k−1}). Thus shows

Proposition 4.6. Let a(p) = δ_{0}(F rob_{p}) ∈ R_{0}. Then \frac{da(p)}{dx} is a non-zero divisor of R_{0} = 𝕋.

Thus our question (Q4) is about when \frac{da(p)}{dx} is a unit in 𝕋.

Theorem 4.7. Suppose that ρ is modular as above associated to a Hecke eigenform f ∈ S_{k_{0}}(SL_{2}(ℤ)). Let 𝕀 be a residue ring of R_{0} modulo a minimal prime ideal; so, Spec(𝕀) is an irreducible component of Spec(R_{0}). Then we have

1. For 𝕁 = R_{0} or 𝕀, Ω_{R_{∞}/Λ_{∞}} ˆ⊗_{R_{∞}}𝕁 is a torsion 𝕁[[Γ]]-module;

2. We have a pseudo isomorphism of 𝕀[[Γ]]-modules:

Sel_{ℚ_{∞}}^{∗}(Ad(ρ_{𝕀}) ⊗_{𝕀} 𝕀^{∗}) ∼ 𝕀 × (Ω_{R_{∞}/Λ_{∞}} ˆ⊗_{R_{∞}}𝕀);

in particular, Sel_{ℚ_{∞}}^{∗}(Ad(ρ_{𝕀}) ⊗_{𝕀} 𝕀^{∗}) is a torsion 𝕀[[Γ]]-module.

3. If λ(τ_{k_{0}}(\frac{da(p)}{dx})) ≠ 0 in W, we have a W [[Γ]]-homomorphism

Sel_{ℚ_{∞}}^{∗}(Ad(ρ) ⊗_{W} W^{∗}) ∼ W × (Ω_{R_{∞}/Λ_{∞}} ˆ⊗_{R_{∞},λ∘τ_{k_{0}}} W )

with finite kernel and cokernel. Moreover Sel_{ℚ_{∞}}^{∗}(Ad(ρ) ⊗_{W} W^{∗}) is a torsion W [[Γ]]-module whose characteristic power series is of the form xφ(x) with φ(0) ≠ 0.

Here is a sketch of a proof.

Proof. Since \frac{da(p)}{dx} ≠ 0, Ω_{R_{0}/Λ_{0}} is a torsion R_{0}-module. Then by Proposition 4.3, Ω_{R_{∞}/Λ_{∞}} ⊗_{R_{∞}} R_{0} is a torsion R_{0}[[Γ]]-module. This shows (1). As long as we have λ(τ_{k_{0}}(\frac{da(p)}{dx})) ≠ 0, the same argument works, and we get the torsion property of Ω_{R_{∞}/Λ_{∞}} ˆ⊗_{R_{∞},λ∘τ_{k_{0}}} W over W [[Γ]] and its characteristic power series φ does not vanish at 0, because φ(0) is the size of the finite module Ω_{R_{0}/Λ_{0}} ˆ⊗_{R_{0},λ∘τ_{k_{0}}} W = Ω_{𝕋_{k_{0}} /W} ˆ⊗_{𝕋_{k_{0}}} W.

As for the product expression, we look into the following commutative diagram with exact rows and columns for 𝕁 = 𝕀 or W:

0  ⟶ Ω_{Λ_{∞}/W} ⊗_{Λ_{∞}} 𝕁  Ω_{Λ_{∞}/W} ⊗_{Λ_{∞}} 𝕁 ⟶ 0
↓  ↓^{e}  ↓^{f}
(J_{∞}/J_{∞}^{2}) ⊗_{𝕀} 𝕁 ⟶ Ω_{R_{∞}/W} ⊗_{R_{∞}} 𝕁 ⟶^{b} Ω_{𝕀/W} ⊗_{𝕀} 𝕁 ⟶ 0
≀∥↓  ↓^{g}  ↓^{h}
(J_{∞}/J_{∞}^{2}) ⊗_{𝕀} 𝕁 ⟶ Ω_{R_{∞}/Λ_{∞}} ⊗_{R_{∞}} 𝕁 ⟶^{d} Ω_{𝕀/Λ_{∞}} ⊗_{𝕀} 𝕁 ⟶ 0
↓  ↓  ↓
0  ⟶  0  ⟶  0.

(4.2)

Since \frac{da(p)}{dx} ≠ 0 implies injectivity of f and e, we have the following exact sequence for M = Ω_{R_{∞}/Λ_{∞}} ⊗_{R_{∞}} 𝕁:

0 → Ω_{R_{∞}/O[[I_{∞}]]} ⊗ 𝕁 ⟶^{β} M × (Ω_{𝕀/W} ⊗_{𝕀} 𝕁) ⟶^{α} Ω_{𝕀/Λ_{0}} ⊗_{𝕀} 𝕁 → 0,

where α(m, a) = d(m) − h(a) and β(a) = (g(a), b(a)). Since Ω_{𝕀/Λ_{0}} ⊗_{𝕀} 𝕁 is 𝕁-torsion by \frac{da(p)}{dx} ≠ 0, it is 𝕁[[x]]-pseudo-null. Again by \frac{da(p)}{dx} ≠ 0, 𝕁d(a(p))\(Ω_{𝕀/W} ⊗_{𝕀} 𝕁) is 𝕁-torsion and hence is 𝕁[[x]]-pseudo-null.

There is a two variable p-adic L-function L_{p}(Ad(ρ_{𝕀})) ∈ 𝕀[[Γ]] constructed in [H90], and recently E. Urban has shown (cf., [U]):

Theorem 4.8 (Urban). Let the assumption be as in Theorem 4.7. Then we have 1. (L_{p}(Ad(ρ_{𝕀}))) ⊃ char(Sel_{ℚ_{∞}}^{∗}(Ad(ρ_{𝕀}) ⊗_{𝕀} 𝕀^{∗})) in 𝕀[[Γ]];

2. If ρ is associated to an elliptic curve which has split multiplicative reduction at p and is indecomposable over the inertia subgroup I_{q} of a prime q ≠ p, (L_{p}(Ad(ρ_{𝕀}))) = char(Sel_{ℚ_{∞}}^{∗}(Ad(ρ_{𝕀}) ⊗_{𝕀} 𝕀^{∗})).

The proof of the stronger assertion (2) relies on the fact \frac{da(p)}{dx} |_{x=0} ≠ 0 for ρ associated to the elliptic curve with multiplicative reduction (see Theorem 5.5). If this non-vanishing holds for any ρ associated to an abelian variety with multiplicative reduction at p, the assertion (2) remains true for such ρ.

4.3. Λ-Adic forms. We constructed a big Hecke algebra 𝕋 isomorphic R_{0} = R_{ℚ}. Since Gal(ℚ_{p}^{ur}[μ_{p^{∞}}]/ℚ_{p}^{ur}) ≅ Gal(ℚ[μ_{p^{∞}}]/ℚ) ≅ ℤ_{p}^{×}, we may regard the local character δ_{0} : Gal(ℚ_{p}^{ur}[μ_{p^{∞}}]/ℚ_{p}^{ur}) → R_{0}^{×} as a global character δ_{cyc} : Gal(ℚ[μ_{p^{∞}}]/ℚ) →

R_{0}^{×}. Define ρ^{ord} = δ_{cyc}^{−1} ⊗ ρ_{0}. Then we have ρ^{ord}|_{D_{p}} ≅ ( \begin{matrix}ε^{ord} & ∗ \\ 0 & δ^{ord} \end{matrix} ), and δ^{ord}

is unramified (thus ρ^{ord} is p-ordinary). By our construction, for the projection π_{k} : 𝕋 ↠ 𝕋_{k}, π_{k} ∘ ρ^{ord} ≅ ρ_{k}^{ord}. By this fact, we find from the properties satisfied by ρ_{k}^{ord} (cf. [GME] Theorem 4.2.7):

Theorem 4.9. The Galois representation ρ^{ord} : 𝔊_{ℚ} → GL_{2}(𝕋) is irreducible pordinary and satisfies:

• Tr(ρ^{ord}(F rob_{q})) = t(q) for all primes q ∤ p;

• δ^{ord}(F rob_{p})) = u(p), where the image of u(p) in 𝕋_{k} is the unit root of the equation X^{2} − T (p)X + p^{k−1}.

Let Spec(𝕀) be the normalization of an irreducible component of Spec(𝕋) for 𝕋 ≅ R_{ℚ}. Then 𝕀 is a normal integral domain finite and torsion-free over Λ. In this case, it is known that 𝕀 is free of finite rank over Λ. Writing a(n) for the image of the Hecke operators t(n) in 𝕀, we consider a formal series F_{𝕀}(q) = ∑_{n=1}^{∞} a(n)q^{n}, which is called an 𝕀-adic form associated to the irreducible component Spec(𝕀). By a duality found by Hecke, for any given algebra homomorphism λ : h_{k} → W, the value λ(T (n)) ∈ (\overline{ℚ} ∩ W) ⊂ ℂ gives the eigenvalue of a modular form f ∈ S_{k}(SL_{2}(ℤ)) and f = ∑_{n=1}^{∞} λ(T (n))q^{n} for q = exp(2πiz). If λ ∈ Spec(𝕀)(W) factors through R_{k,ℚ} = 𝕋_{k}, λ induces an algebra homomorphism λ_{k} : h_{k} → 𝕋_{k} ⟶^{λ} W, which gives rise to f_{λ} = ∑_{n=1}^{∞} λ(a(n))q^{n} ∈ S_{k}(SL_{2}(ℤ)). Thus the formal expansion F_{𝕀} gives rise to a family of modular forms {f_{λ}}_{k≡k_{0} mod p−1}.

Now we renormalize W [[x]]-algebra structure of 𝕋 so that 𝕋/((1+x)−γ^{k}))𝕋 ≅ 𝕋_{k} (it was originally normalized so that 𝕋/x𝕋 = 𝕋_{k_{0}}. Then as long as Ker(λ) ∋ ((1 + x) − ε(γ)γ^{k}) for k ≥ 2 and a character ε : (ℤ/p^{r}ℤ)^{×} → W^{×}, we can find a Hecke eigenform f_{λ} ∈ S_{k}(Γ_{1}(p^{r})) such that f_{λ} = ∑_{n=1}^{∞} λ(a(n))q^{n}. The point λ ∈ Spec(𝕋)(\overline{ℚ}_{p}) satisfying the condition: Ker(λ) ∋ ((1 + x) − ε(γ)γ^{k}) for k ≥ 2 and a character ε : (ℤ/p^{r}ℤ)^{×} is called arithmetic. We write Spec^{arith}(𝕀) for the set of all arithmetic points in Spec(𝕀)(\overline{ℚ}_{p}). Thus the 𝕀-adic form F_{𝕀} gives rise to a p-adic family of Hecke eigenform {f_{λ}}_{λ∈Spec^{arith}(𝕀)}. If 𝕀 ≅ Λ, an arithmetic point λ is given by the evaluation W [[x]] ∋ ϕ(x) ↦ ϕ(ε(γ)γ^{k} − 1) for an integer k ≥ 2; so, the Fourier coefficients of f_{λ} depends p-adic analytically on λ ∈ Spec(𝕀); so, the family {f_{λ}}_{λ∈Spec^{arith}(𝕀)} is called a p-adic analytic family of p-ordinary Hecke eigenforms. Here is a result you can find in [LFE] Chapter 7:

Theorem 4.10. If k ≥ 1, for any given Hecke eigenform f ∈ S_{k}(Γ_{1}(p^{r})) with f|U(p) = λ(U(p))f for λ(U(p)) ∈ W^{×}, we have an irreducible component of Spec(𝕋) giving rise to a finite flat Λ-algebra 𝕀 such that f = f_{λ} for a unique arithmetic point λ in Spec(𝕀). If k ≥ 2, 𝕀 is uniquely determined by f, and the localization-completion of 𝕀 at P = Ker(λ) for λ ∈ Spec^{arith}(𝕀) is isomorphic to the localization-completion of Λ at P (which is in turn isomorphic to K[[x]]).

The 𝕀-adic form F_{𝕀} is called the 𝕀-adic lift of f, and the family {f_{λ}}_{λ∈Spec^{arith}(𝕀)} is called the p-adic analytic family of f. A naive question is (Q5) If (1 + x) − γ ∋ Ker(λ), what is f_{λ}, a modular form of weight 1? The formal q-expansion f_{λ} for all λ ∈ Spec(𝕀) gives rise to a p-adic modular form in the sense of Serre. If its weight k is 1, it could be a classical modular form of weight 1 but rarely. A sufficient condition for this to happen can be found in [BT].

Since Question (Q4) is related to the question about how the values λ(a(p)) for λ ∈ Spec^{arith}(𝕀) are distributed at randam in \overline{ℚ}, we might ask, for a given positive integer n, (Q6) Is the field ℚ[λ(a(n))|λ ∈ Spec^{arith}(𝕀)] an infinite extension of ℚ? Related to this question, Maeda conjectured that the subalgebra

ℚ[T (n)|n = 1, 2, …] ⊂ End(S_{k}(SL_{2}(ℤ)))

is a field whose Galois closure over ℚ has Galois group isomorphic to the symmetric group 𝔖_{d} of d letters for d = dim S_{k}(SL_{2}(ℤ)).

Corollary 4.11. For a given p-adic analytic family {f_{λ}}_{λ∈Spec^{arith}(𝕀)} associated

to an 𝕀-adic form F_{𝕀}, writing F_{𝕀}|U(p) = a(p)F_{𝕀}, we have λ (\frac{da(p)}{dx}) ≠ 0 except for finitely many λ.

This tells us that char(Sel_{ℚ_{∞}}^{∗}(Ad(ρ_{λ}) ⊗_{W} W^{∗})) is divisible by x only once for almost all λ, where ρ_{λ} is the Galois representation of f_{λ}.

Proof. We already know that \frac{da(p)}{dx} ∈ 𝕀 is non-constant. Then the zero set of \frac{da(p)}{dx} in Spec(𝕀)(W) is a finite set by the Weierstrass preparation theorem.

5. L-Invariants of Adjoint Square Galois Representations

After the conjecture of Mazur-Tate-Teitelbaum, many number theorists have proposed diverse definitions of the L-invariant which are expected to give the error term (or the difference) of the conjectural arithmetic part of the leading term of the Taylor expansion of a given p-adic motivic L-function at an exceptional zero from its archimedean counter-part. For an elliptic curve E_{/ℚ} with multiplicative or ordinary good reduction modulo p, its p-adic L-function L_{p}(s, E) has the following evaluation formula at s = 1 ([LFE] 6.5):

L_{p}(1, E) = (1 − a_{p}^{−1})\frac{L_{∞}(1, E)}{period},

where L_{∞}(s, E) is the archimedean L-function of E, and a_{p} is the eigenvalue of the arithmetic Frobenius element at p on the unramified quotient of the p-adic Tate module T (E) of E. Thus if E has split multiplicative reduction, a_{p} = 1, and L_{p}(s, E) has zero at s = 1. This type of zero of a p-adic L-function resulted from the modification Euler p-factor is called an exceptional zero, and it is generally believed that if the archimedean L-values does not vanish, the order of the zero is the number e of such Euler p-factors; so, in this case, e = 1. Then L′_{p}(1, E) = \frac{dL_{p}(s,E)}{ds}|_{s=1} is conjectured to be equal to the archimedean value \frac{L_{∞}(1,E)}{period} times an error factor L(E), the so-called L-invariant:

L′_{p}(1, E) = L(E)\frac{L_{∞}(1, E)}{period}.

The problem of L-invariant is to find an explicit formula (without recourse to p-adic L-functions) for general motivic p-adic Galois representations V. In the case of E_{/ℚ} split multiplicative at p, writing E(\overline{ℚ}_{p}) = \overline{ℚ}_{p}^{×}/q^{ℤ} for the Tate period q ∈ pℤ_{p}, the solution conjectured by Mazur-Tate-Teitelbaum and proved by Greenberg-Stevens [GS] is

L(E) = \frac{log_{p}(q)}{ord_{p}(q)}.

Since E is modular, it is associated to an elliptic Hecke eigenform f_{E} of weight 2 with q-expansion ∑_{n=1}^{∞} a(n, f_{E})q^{n}. In particular, a(p, f_{E}) = a_{p} = 1 and a(1, f_{E}) = 1. We can lift f_{E} to a unique 𝕀-adic Hecke eigenform F_{𝕀} = ∑_{n=1}^{∞} a(n)q^{n} ∈ 𝕀[[q]] for a finite flat extension 𝕀 of ℤ_{p}[[x]] (étale around x = 0) so that f_{E} is a specialization of F_{𝕀} at x = 0. Then one of the key ingredient of their proof is the following formula:

L(E) = −2 log_{p}(γ)\frac{da(p)}{dx}|_{x=0},

where γ is the generator of Γ = 1 + pℤ_{p} corresponding to 1 + x under the identification: ℤ_{p}[[Γ]] = ℤ_{p}[[x]].

Greenberg has generalized in [G] the conjectural formula of his L-invariant to general V when V is p-ordinary and has a subquotient on which D_{p} acts trivially. We write L(V) for the L-invariant of Greenberg and recall his definition in the following sections. Suppose that V = V (ρ) is a modular ordinary two dimensional Galois representation associated to a p-ordinary elliptic Hecke eigenform f of weight k ≥ 2 with “Neben” character having conductor prime to p. Thus we have a(p, f) ∈ W^{×}. By definition, Ad(V) has the middle subquotient on which D_{p} acts trivially; so, L_{p}(s, Ad(V)) (see [H90]) has to have an exceptional zero at s = 1. Under this circumstance, a version of the conjecture of Greenberg is

Conjecture 5.1. L(Ad(V)) ≠ 0.

For a suitable finite flat extension 𝕀 of W [[x]], we have a unique 𝕀-adic Hecke eigenform F_{𝕀} lifting f so that its specialization at a point P_{f} of Spf(𝕀) over (x) gives f (see Theorem 4.10). Then we have an 𝕀-adic ordinary Galois representation ρ_{𝕀}^{ord} : 𝔊^{S} → GL_{2}(𝕀) associated to F_{𝕀} acting on 𝕃 = 𝕀^{2} (Theorem 4.9). Assume p > 2 and S = {p, ∞} for simplicity. The Galois character det(ρ_{𝕀}^{ord})^{−1} det(ρ) has values in the p-profinite group 1 + 𝔪_{𝕀} for the maximal ideal 𝔪_{𝕀} of 𝕀, and hence we have its unique square root ψ. Recall the representation ρ_{𝕀} : 𝔊_{ℚ} → GL_{2}(𝕀) with det(ρ_{𝕀}) = det(ρ) satisfying (ρ_{𝕀}^{ord} ⊗ ψ)(σ) = ψ(σ)ρ_{𝕀}^{ord}(σ). Note that ρ_{𝕀} ≡ ρ_{𝕀}^{ord} mod P_{f}.

We would like to prove in this section

Theorem 5.2. Let p be an odd prime, and assume that f is p-ordinary, has weight k ≥ 2 and that the “Neben” character of f has conductor prime to p. Let F_{𝕀} = ∑_{n=1}^{∞} a(n)q^{n} ∈ 𝕀[[q]] be the 𝕀-adic lift of f. Assume (ai_{ℚ}) and (ds) for the Galois representation ρ of f. Suppose that R_{k,ℚ} ≅ 𝕋_{k} (so, R ≅ 𝕋). Then we have

L(Ad(V)) = −2 log_{p}(γ)\frac{da(p)}{dx}|_{x=0}.

Corollary 4.11 tells us

Corollary 5.3. Let the notation and the assumption be as in the theorem. Then L(Ad(ρ_{λ})) ≠ 0 for almost all arithmetic points λ ∈ Spec(𝕀), where ρ_{λ} is the Galois representation of the member f_{λ} in the p-adic analytic family associated to F_{𝕀}.

There is one more corollary. Suppose that f is of weight 2 and is associated to an abelian variety A_{/ℚ} with multiplicative reduction modulo p. Then twisting A by a finite order character χ, we can bring A to another abelian variety A′_{/ℚ} with split multiplicative reduction modulo p. We may assume that dim A′ = [M : ℚ] with a field M ⊂ End(A′_{/ℚ}) ⊗_{ℤ} ℚ and that the integer ring R of M is contained in End(A′). Write A′(ℚ_{p}) = 𝔾_{m}(ℚ_{p}) ⊗ R/q^{R} for q ∈ ℚ_{p}^{×}. Greenberg in [G] proved

that L(Ad(V)) = \frac{log_{p}(q)}{ord_{p}(q)}. Thus we have

Corollary 5.4. Let the notation be as above. Then

L(Ad(V)) = −2 log_{p}(γ)\frac{da(p)}{dx}|_{x=0} = \frac{log_{p}(q)}{ord_{p}(q)}.

(Q7) log_{p}(q) ≠ 0? As for this question, only known result so far is the one in [BDGP]:

Theorem 5.5. If A′_{/ℚ} is an elliptic curve, then q is transcendental over ℚ; in particular, log_{p}(q) ≠ 0.

Theorem 5.2 can be proven under the assumptions milder than the one stated above (see [H04] for a more general version of the theorem). Actually what we need is only the two facts that the localization of R at the point P corresponding to ρ has the tangent space of dimension 1 over K and is isomorphic to the tangent space at the point P_{f} corresponding to f of the nearly p-ordinary Hecke algebra with fixed determinant character det(ρ) (of an appropriate prime-to-p level).

Since there is a categorical equivalence between deformations of \overline{L} with determinant det(ρ) and ordinary deformations of \overline{L} (if p > 2), the ring R can be also identified with the versal ordinary deformation ring of \overline{L}. If one of the deformation problem is representable, the other is also representable, and we have ρ = ρ^{ord} ⊗ ψ for the universal ordinary deformation ρ^{ord}, where ψ is the unique square root of det(ρ^{ord})^{−1} det(ρ). See [MFG] Proposition 5.43 for more details of this fact.

In the following sections, we shall start with a brief review of the definition by Greenberg of the balanced Selmer group and the L-invariant of the adjoint square of a two dimensional modular ordinary p-adic Galois representation. After the review, we shall give a proof of the theorem. Hereafter we suppose that p > 2.

5.1. Balanced Selmer groups. As already remarked, if we take a base of V containing a generator of F^{+}V and we identify End_{K}(V) with M_{2}(K) by this base, F^{−}Ad(V) (resp. F^{+}Ad(V)) is made up of upper triangular matrices with trace zero (resp. upper nilpotent matrices). Note that D_{p} acts trivially on \frac{F^{−}Ad(V)}{F^{+}Ad(V)}; so, F^{−}Ad(V)/F^{+}Ad(V) ≅ K as D_{p}-modules. In particular, the p-adic L-function of Ad(V) has an exceptional zero at s = 1.

Taking the dual Ad(V)^{∗}(1) = Hom_{K}(Ad(V), K) ⊗ N ≅ Ad(V)(1), we define subspaces F^{−}Ad(V)^{∗}(1) = F^{+}Ad(V)^{⊥}(1) = F^{−}Ad(V)(1) and F^{+}Ad(V)^{∗}(1) = F^{−}Ad(V)^{⊥}(1) = F^{+}Ad(V)(1). Thus we have

Ad(V)^{∗}(1) ⊃ F^{−}Ad(V)^{∗}(1) ⊃ F^{+}Ad(V)^{∗}(1) ⊃ {0}. (5.1)

On \frac{F^{−}Ad(V)^{∗}(1)}{F^{+}Ad(V)^{∗}(1)}, D_{p} acts by χ; so, \frac{F^{−}Ad(V)^{∗}(1)}{F^{+}Ad(V)^{∗}(1)} ≅ K(1).

We define Ad(L) = End_{W} (L) ∩ Ad(V) and Ad(L)^{∗}(1) = Hom(Ad(L), W) ⊗ N ⊂ Ad(V)^{∗}(1). Taking intersection with Ad(L) and Ad(L)^{∗}(1), we have three step filtrations of Ad(L) and Ad(L)^{∗}(1) induced from the above filtrations.

We need balanced Selmer groups \overline{Sel}_{ℚ}(Ad(V)) and \overline{Sel}_{ℚ}(Ad(V)^{∗}(1)) introduced in [G] (16) under the notation of S_{A}(ℚ). We call this Selmer group balanced since \overline{Sel}_{ℚ}(Ad(V)) and the dual \overline{Sel}_{ℚ}(Ad(V)^{∗}(1)) have equal dimension (cf., [G] Proposition 2). We define \overline{Sel}_{ℚ}(Ad(V)) by slightly shrinking L_{p}(Ad(V)) to \overline{L}_{p}(Ad(V)) ⊂ L_{p}(Ad(V)) and keeping L_{q}(Ad(V)) = \overline{L}_{q}(Ad(V)) for q ≠ p intact. Then we define \overline{L}_{q}(Ad(V)^{∗}(1)) by the orthogonal complement L_{q}(Ad(V))^{⊥} under the Tate pairing for all q including p. The new Selmer group is defined by the same formula as in (2.2): for X = Ad(V) and Ad(V)^{∗}(1),

\overline{Sel}_{ℚ}(X) = Ker(H^{1}(ℚ, X) → ∏_{q} \frac{H^{1}(ℚ_{q}, X)}{\overline{L}_{q}(X)}). (5.2)

Actually, the new Selmer group \overline{Sel}_{ℚ}(Ad(V)) often coincides with our Selmer group defined by (2.2). Indeed, unless f is multiplicative at p, we simply set \overline{L}_{q}(Ad(V)) = L_{q}(Ad(V)) for all q; thus, \overline{Sel}_{ℚ}(A) = Sel_{ℚ}(A) for A = Ad(V) and Ad(V/L) in this case. Here we call f multiplicative at p if f is of weight 2 and is associated to an abelian variety with multiplicative reduction at p (not necessarily split). Thus we only need to define \overline{L}_{p}(Ad(V)) if f is multiplicative. In this case, we simply put \overline{L}_{p}(Ad(V)) = F^{+}H^{1}(ℚ_{p}, Ad(V)) ⊂ L_{p}(Ad(V)), which is the image of H^{1}(ℚ_{p}, F^{+}Ad(V)) in H^{1}(ℚ_{p}, Ad(V)).

The above definition coincides with the one given in [G], because as we already remarked, F^{+}Ad(V)^{∗}(1) (resp. \frac{F^{−}Ad(V)}{F^{+}Ad(V)}) is the smallest (resp. the largest) sub-

space of F^{−}Ad(V)^{∗}(1) (resp. \frac{Ad(V)}{F^{+}Ad(V))}) stable under D_{p} so that D_{p} acts on it

by χ (resp. by the trivial character); so, \frac{F^{−}Ad(V)^{∗}(1)}{F^{+}Ad(V)^{∗}(1)} ≅ K(1) and \frac{F^{−}Ad(V)}{F^{+}Ad(V)} ≅ K. Therefore the space F^{+}Ad(V)^{∗}(1) (resp. F^{−}Ad(V)) is the subspace written as F^{11}Ad(V)^{∗}(1) (resp. F^{00}Ad(V)) in [G].

We now verify the following condition in [G] necessary to define L(U):

Lemma 5.6. Suppose that f satisfies the assumptions of Theorem 5.2. Then we have Sel_{ℚ}(Ad(V)) = 0 and

\overline{Sel}_{ℚ}(Ad(V)) = \overline{Sel}_{ℚ}(Ad(V)^{∗}(1)) = 0. (V)

Proof. Using the global Tate duality (e.g., [MFG] Theorem 4.50), Greenberg ([G] Proposition 2) has shown dim_{K} \overline{Sel}_{ℚ}(Ad(V)) = dim_{K} \overline{Sel}_{ℚ}(Ad(V)^{∗}(1)). Since the Selmer group Sel_{ℚ}(Ad(V/L)) contains \overline{Sel}_{ℚ}(V/L), we have

|Sel_{ℚ}(Ad(V/L))| < ∞ (⇒ \overline{Sel}_{ℚ}(Ad(V)) = 0). (V1)

Then the lemma follows from Corollary 3.8.

We have the Poitou-Tate exact sequence (e.g., [MFG] Theorem 4.50 (5)):

0 → \overline{Sel}_{ℚ}(Ad(V)) → H^{1}(𝔊_{ℚ}, Ad(V)) → \frac{H^{1}(ℚ_{p}, Ad(V))}{\overline{L}_{p}(Ad(V))} → \overline{Sel}_{ℚ}(Ad(V)^{∗}(1))^{∗}.

Thus by (V), we have

H^{1}(𝔊_{ℚ}, Ad(V)) ≅ \frac{H^{1}(ℚ_{p}, Ad(V))}{\overline{L}_{p}(Ad(V))}. (5.3)

5.2. Greenberg’s L-invariant. Greenberg defined in [G] his invariant L(Ad(V)) in the following way. By the definition of \overline{L}_{p}(Ad(V)), the subspace \frac{F^{−}H^{1}(ℚ_{p},Ad(V))}{\overline{L}_{p}(Ad(V))} inside the right-hand side of (5.3) is isomorphic to F^{−}Ad(V)/F^{+}Ad(V) ≅ K (see just below (21) in Section 2 of [G]). By (5.3), we have a unique subspace T of H^{1}(𝔊_{ℚ}, Ad(V)) projecting down isomorphically onto

\frac{F^{−}H^{1}(ℚ_{p}, Ad(V))}{\overline{L}_{p}(Ad(V))} ↪ \frac{H^{1}(ℚ_{p}, Ad(V))}{\overline{L}_{p}(Ad(V))}.

Then by the restriction, T gives rise to a subspace L of

Hom(D_{p}^{ab}, F^{−}Ad(V)/F^{+}Ad(V))

isomorphic to F^{−}Ad(V)/F^{+}Ad(V) ≅ K. Note that

Hom(D_{p}^{ab}, \frac{F^{−}Ad(V)}{F^{+}Ad(V)}) ≅ (\frac{F^{−}Ad(V)}{F^{+}Ad(V)})^{2} ≅ K^{2}

canonically by ϕ ↦ (\frac{ϕ([u,ℚ_{p}])}{log_{p}(u)}, ϕ([p, ℚ_{p}])) for any u ∈ ℤ_{p}^{×} of infinite order. Here [x, ℚ_{p}] is the local Artin symbol (suitably normalized).

If a cocycle c representing an element in T is unramified, it gives rise to an element in \overline{Sel}_{ℚ}(Ad(V)). By the vanishing (V) of \overline{Sel}_{ℚ}(Ad(V)), this implies c = 0; so, the projection of L to the first factor F^{−}Ad(V)/F^{+}Ad(V) (via ϕ ↦ ϕ(\frac{[γ,ℚ_{p}])}{log_{p}(γ)}) is surjective. Thus this subspace L is a graph of a K-linear map

L : F^{−}Ad(V)/F^{+}Ad(V) → F^{−}Ad(V)/F^{+}Ad(V),

which is given by the multiplication by an element L(Ad(V)) ∈ K.

The cocycle c as above becomes unramified at p after restriction to 𝔊_{ℚ_{∞}} (because the p-ramification of c is consumed by ℚ_{∞}/ℚ), it gives rise to an element c_{∞} in Sel_{ℚ_{∞}}(V). The map c ↦ c_{∞} is injective (under the triviality of H^{0}(𝔊_{ℚ_{∞}}, V)) by the inflation-restriction sequence. Thus the image T_{∞} ⊂ Sel_{ℚ_{∞}}(V) gives rise to a zero of the characteristic power series of Sel_{ℚ_{∞}}(Ad(V/L)) at the augmentation ideal of W [[Γ]] (see [G] Proposition 1).

5.3. Proof of the theorem. We take a p-ordinary Hecke eigenform f of weight k ≥ 2 as in the theorem and its two dimensional Galois representation V. We take a matrix form ρ : 𝔊_{ℚ} → M_{2}(W) of the Galois representation L so that its restriction to D_{p} is given by ρ(σ) = ( \begin{matrix}ϵ(σ) & β(σ) \\ 0 & δ(σ) \end{matrix} ). We now identify Ad(L) with the following subspace of M_{2}(W):

{ξ ∈ M_{2}(W) = End_{W} (L)|Tr(ξ) = 0} .

Then F^{−}Ad(L) is the subspace of Ad(L) made up of upper triangular matrices, and F^{+}Ad(L) is made up of upper nilpotent matrices on which D_{p} acts by the character ϵδ^{−1}.

Recall the versal nearly ordinary deformation ρ : 𝔊_{ℚ} → GL_{2}(R) with det(ρ) = det(ρ) and the point P ∈ Spf(R)(W) carrying ρ. Recall the subspace T of H^{1}(𝔊_{ℚ}, Ad(V)) studied in the previous section. We know

1. dim_{K} T = 1,

2. T is made up of the classes of cocycles c which is upper triangular on D_{p} and unramified at the ℓ-inertia group for ℓ ∈ S different from p,

3. Sel_{ℚ}(Ad(V)) = {0} (Lemma 5.6). We recall the definition of the “−” Selmer group given just below (2.2): replacing L_{p}(Ad(V)) in (2.2) by the bigger

L_{p}^{−}(Ad(V)) = Ker(Res : H^{1}(ℚ_{p}, Ad(V)) → H^{1}(I_{p}, \frac{Ad(V)}{F^{−}(Ad(V))}))

(and keeping L_{q}(Ad(V)) intact for q ≠ p), we defined the bigger “−” Selmer group Sel_{ℚ}^{−}(Ad(V)) ⊃ Sel_{ℚ}(Ad(V)).

By (2), we have T ⊂ Sel_{ℚ}^{−}(Ad(V)). By (5.3), Sel_{ℚ}^{−}(Ad(V)) = T⊕Sel_{ℚ}(Ad(V)) =

T, because the image of T in \frac{H^{1}(ℚ_{p},Ad(V))}{L_{p}(Ad(V))} is given by \frac{L_{p}^{−}(Ad(V))}{L_{p}(Ad(V))}. This fact can be also seen from [H00] Corollary 5.4.

By our assumption, the deformation functor Φ of \overline{L} defined by (D1-3) is representable by R. Then by an argument similar to the proof of Proposition 3.1 (see Theorem 5.14 of [MFG]), the K-dual Sel_{ℚ}^{−}(Ad(V))^{∗} of Sel_{ℚ}^{−}(Ad(V)) is canonically isomorphic to Ω_{R/W}^{1} ⊗_{R} R_{P} /P R_{P} where P is the point of Spf(R) corresponding to ρ and R_{P} is the localization completion of R at P (so R_{P} /P R_{P} = K). Here Ω_{R/W}^{1} is the module of continuous differentials of R over W under the profinite topology of R. Thus T = Sel_{ℚ}^{−}(Ad(V)) is isomorphic to the tangent space at P of R_{P}. To describe this point more, we first remark that (R_{P} , ρ : 𝔊_{ℚ} → GL_{2}(R_{P})) is the universal couple pro-representing the following localized functor Φ_{P} (see [HT] Section 2 for a general discussion of the “localized” deformation theory). The functor Φ_{P} associates to each local Artinian K-algebra A with residue field K the set of isomorphism classes of p-adically continuous defomations ρ̃ : 𝔊_{ℚ} → GL_{2}(A) satisfying the following three conditions: (P1) ρ̃ mod 𝔪_{A} ≅ ρ for the maximal ideal 𝔪_{A} of A; (P2) Writing ι : K → A for the structure homomorphism of K-algebras, we have

the identity of the determinant characters:

ι ∘ det(ρ) = det ρ̃;

(P3) ρ̃|_{D_{p}} ≅ ( \begin{matrix}ϵ̃ & ∗ \\ 0 & δ̃ \end{matrix}) with δ̃ mod 𝔪_{A} = δ, writing ρ|_{D_{p}} ≅ ( \begin{matrix}ϵ & ∗ \\ 0 & δ \end{matrix}).

Indeed, if we have such ρ̃: 𝔊_{ℚ} → GL_{2}(A), moving ρ̃ by conjugation in GL_{2}(A) if necessary, we find by continuity a compact W -subalgebra B ⊂ A such that ρ̃ has image in GL_{2}(B) and ρ̃ mod (𝔪_{A} ∩ B) ≅ ρ. This point can be also verified by using the technique of pseudo representation of Wiles (e.g., [MFG] 2.2.1). Then by the universality of R, we have a W -algebra homomorphism φ : R → B such that φ ∘ ρ ≅ ρ̃ in GL_{2}(B). This morphism induces φ_{P} : R_{P} → A after localization. Since R_{P} is topologically generated by the traces of ρ, the localized version φ_{P} is uniquely determined by ρ̃, and hence (R_{P} , ρ) prorepresents Φ_{P}.

Here is how the isomorphism of T with the tangent space is computed, for example, in [MFG] 5.2.4: Each inhomogeneous cocycle c (representing an element of T) gives rise to an infinitesimal nearly ordinary deformation ρ̃_{c} with det(ρ̃) = det ρ:

ρ̃_{c} : 𝔊_{ℚ} → GL_{2}(K[x]/(x^{2})))

given by ρ̃_{c}(σ) = ρ(σ) + c(σ)ρ(σ)x. The tangent space at P of Spf(R)_{/W} is isomorphic to

\frac{\{ρ̃ ∈ Φ(K[x]/(x^{2}))|ρ̃ mod (x) = ρ\}}{∼} = Φ_{P} (K[x]/(x)^{2})

by a standard argument (which can be found in [MFG] 5.2.4). Here “∼” is the conjugation under 1 + xM_{2}(K) ⊂ GL_{2}(K[x]/(x^{2})) and (as one can check easily) corresponds to the cohomologus relation in T. Thus c ↦ ρ̃_{c} induces the isomorphism from T to the tangent space of Spf(R_{P}) at P.

Taking an inhomogeneous cocycle c : 𝔊_{ℚ} → Ad(V) representing a generator of T, we write c(σ) = ( \begin{matrix}−a(σ) & b(σ) \\ 0 & a(σ) \end{matrix} ) for σ ∈ D_{p}. If c restricted to D_{p} modulo upper nilpotent cocycles is unramified, it gives rise to a nontrivial element of Sel_{ℚ}(Ad(V)). By the vanishing of Sel_{ℚ}(Ad(V)) (Lemma 5.6), we find a ≠ 0 on I_{p}. Then L(Ad(V)) = a([p, ℚ_{p}])· \frac{log_{p}(γ)}{a([γ,ℚ_{p}])} for a generator γ of 1+pℤ_{p}. We therefore need to compute this value.

Since P R_{P} /P^{2}R_{P} ≅ T as already remarked, \frac{dρ}{dx} gives rise to a generator of T by the universality of (R_{P} , ρ = ρ_{𝕀}); so, we have c(σ)ρ(σ) = C · \frac{dρ}{dx}(σ) with a constant

C ∈ K^{×}. Writing ρ(σ) = ( \begin{matrix}ϵ(σ) & β(σ) \\ 0 & δ(σ) \end{matrix} ) for σ ∈ D_{p}, we have therefore

a([p, ℚ_{p}]) = C · \frac{dδ([p, ℚ_{p}])}{dx}|_{x=0} and a([γ, ℚ_{p}]) = C · \frac{dδ([γ, ℚ_{p}])}{dx}|_{x=0}. (5.4)

For the modular p-ordinary deformation ρ_{𝕀}^{ord}, its determinant det(ρ_{𝕀}^{ord}) is the universal deformation on 𝔊_{ℚ} of det(ρ) mod 𝔪 (at least locally around P_{f}). Recall the universal deformation of the trivial character κ of 𝔊_{ℚ}: κ(γ^{s}) = (1 + x)^{s}. Thus the character \sqrt{κ}^{−1} sends σ ∈ 𝔊 inducing γ^{s} on Gal(ℚ_{∞}/ℚ) to (1 + x)^{−s/2}. This shows ρ = ρ_{𝕀} = ρ_{𝕀}^{ord} ⊗ \sqrt{κ}^{−1} after localization at P. Then the character δ is given by δ([γ^{s}, ℚ_{p}]) = κ^{−1} because the character δ^{ord} at the lower right corner of ρ_{𝕀}^{ord}|_{D_{p}} is unramified. Then we have

\frac{dδ([γ^{s}, ℚ_{p}])}{dx}|_{x=0} = −\frac{s}{2} and \frac{dδ([γ^{s}, ℚ_{p}])}{dx}|_{x=0} log_{p}(γ^{s})^{−1} = −\frac{1}{2 log_{p}(γ)}.

As for the value at [p, ℚ_{p}], we have δ^{ord}([p, ℚ_{p}]) = a(p) by Theorem 4.9. Since χ([p, ℚ_{p}]) = 1 and the character (1 + x)^{s/2} interpolates χ^{m(p−1)} for all integer m, (1 + x)^{s/2} has value 1 at [p, ℚ_{p}]. Thus we get the desired result from the definition of L(Ad(V)).

References

Books

[BCM] N. Bourbaki, Algèbre Commutative, Hermann, Paris, 1961–83

[BCG] R. P. Langlands, Base Change for GL(2), Annals of Math. Studies 96, Princeton University Press, 1980

[CGP] K. S. Brown, Cohomology of Groups, Graduate texts in Math. 87, Springer, 1982 [CPI] K. Iwasawa, Collected Papers, Vol. 1-2, Springer, 2001

[CRT] H. Matsumura, Commutative Ring Theory, Cambridge studies in advanced mathematics 8, Cambridge Univ. Press, 1986 [FAN] D. Ramakrishnan and R. J. Valenza, Fourier Analysis on Number Fields, GTM 186,

1999, Springer

[GME] H. Hida, Geometric Modular Forms and Elliptic Curves, 2000, World Scientific Publishing Co., Singapore (a list of errata downloadable at www.math.ucla.edu/ hida)

[IAT] G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton University Press and Iwanami Shoten, 1971, Princeton-Tokyo

[ICF] L. C. Washington, Introduction to Cyclotomic Fields, Graduate Text in Mathematics, 83, Springer, 1980

[LFE] H. Hida, Elementary Theory of L–functions and Eisenstein Series, LMSST 26, Cambridge University Press, Cambridge, 1993

[MFG] H. Hida, Modular Forms and Galois Cohomology, Cambridge Studies in Advanced Mathematics 69, 2000, Cambridge University Press

Articles

[BDGP] K. Barré-Sirieix, G. Diaz, F. Gramain and G. Philibert, Une preuve de la conjecture de Mahler-Manin, Inventiones Math 124 (1996), 1-9

[BT] K. Buzzard and R. Taylor, Companion forms and weight one forms, Ann. of Math. 149 (1999), 905–919.

[G] R. Greenberg, Trivial zeros of p-adic L-functions, Contemporary Math. 165 (1994), 149–174

[GS] R. Greenberg and G. Stevens, p-adic L-functions and p-adic periods of modular forms, Inventiones Math. 111 (1993), 407–447

[H88] H. Hida, Modules of congruence of Hecke algebras and L-functions associated with cusp forms, Amer. J. Math. 110 (1988) 323–382

[H90] H. Hida, p-adic L-functions for base change lifts of GL_{2} to GL_{3}, in Proc. of Conference on “Automorphic forms, Shimura varieties, and L-functions”, Perspectives in Math. 11 (1990), 93–142

[H96] H. Hida, On the search of genuine p-adic modular L-functions for GL(n), Mémoire SMF 67, 1996

[H00] H. Hida, Adjoint Selmer groups as Iwasawa modules, Israel Journal of Math. 120 (2000), 361–427

[H04] H. Hida, Greenberg’s L-invariants of adjoint square Galois representations, 2004, (preprint downloadable at www.math.ucla.edu/ hida)

[HT] H. Hida and J. Tilouine, On the anticyclotomic main conjecture for CM fields, Inventiones Math. 117 (1994), 89–147

[M] B. Mazur, Deforming Galois representations, MSRI Publications 16 (1989), 385–437 [S] J.-P. Serre, Quelques applications du théorème de densité de Chebotarev, Publ. IHES.

54 (1981), 323–401

[T] R. Taylor, On Galois representations associated to Hilbert modular forms II, in Series in Number Thory 1 (1995): ”Elliptic curves, Modular forms, & Fermat’s last theorem”, pp.185–191

[TW] R. Taylor and A. Wiles, Ring theoretic properties of certain Hecke modules, Ann. of Math. 141 (1995), 553–572

[U] E. Urban, Selmer groups and the Eisenstein-Klingen ideal. Duke Math. J. 106 (2001), 485–525

[W] A. Wiles, The Iwasawa conjecture for totally real fields, Ann. of Math. 131 (1990), 493–540

[W1] A. Wiles, Modular elliptic curves and Fermat’s last theorem, Ann. of Math. 141 (1995), 443–551