ON THE NON-COMMUTATIVE MAIN CONJECTURE FOR ELLIPTIC CURVES WITH COMPLEX MULTIPLICATION

THANASIS BOUGANIS AND OTMAR VENJAKOB

In [3] a non-commutative Iwasawa Main Conjecture for elliptic curves over ℚ has been formulated. In this note we show that it holds for all CM-elliptic curves E defined over ℚ. This was claimed in (loc. cit.) without proof, which we want to provide now assuming that the torsion conjecture holds in this case. Based on this we show firstly the existence of the (non-commutative) p-adic L-function of E and secondly that the (non-commutative) Main Conjecture follows from the existence of the Katz-measure, the work of Yager and Rubin’s proof of the 2-variable main conjecture. The main issues are the comparison of the involved periods and to show that the (non-commutative) p-adic L-function is defined over the conjectured in (loc. cit.) coefficient ring. Moreover we generalize our considerations to the case of CM-elliptic cusp forms.

Acknowledgements: We are grateful to John Coates and Sujatha for their interest and various discussions.

1. The non-commutative Main Conjecture

Let E be an elliptic curve defined over ℚ and p ≥ 5 a prime at which E has good ordinary reduction. Then the p-adic Lie extension F_{∞} := ℚ(E(p)) with Galois group G = G(F_{∞}/ℚ) contains the cyclotomic ℤ_{p}-extension ℚ_{cyc} of ℚ and the quotient of G by its closed normal subgroup H := G(F_{∞}/ℚ_{cyc}) equals Γ := G(ℚ_{cyc}/ℚ) ≅ ℤ_{p}.

We write D for the ring of integers O_{L}, where L is either a finite extension of ℚ_{p} or of the completion ℚ̂_{p}^{nr} of the maximal unramified extension ℚ_{p}^{nr} of ℚ_{p}. As usual we write

Λ := Λ_{D}(G) := D[[G]]

for the Iwasawa algebra of G with coefficients in D. Note that is a Noetherian pseudocompact semi-local ring (which is compact if L is finite over ℚ_{p}). We denote by

𝔐_{H}(G) := 𝔐_{D,H}(G)

the category of all finitely Λ(G)-modules M such that its quotient M/M(p) by its pprimary subgroup M(p) is finitely generated over the subalgebra Λ(H) of Λ(G). Also we recall from [3] the definition of the multiplicatively closed subsets

S := {λ ∈ Λ| Λ/Λλ is finitely generated over Λ(H)} and

Date: June 8, 2010.

We acknowledge support by the ERC and DFG.

S^{∗} = ⋃_{n≥0} p^{n}S.

The following theorem compromises the technical heart of [3].

Theorem 1.1. The sets S and S^{∗} are (left and right) Ore sets, i.e. the localisations Λ_{S} and Λ_{S^{∗}} of Λ exist and the following holds:

(i) The category of all finitely generated S^{∗}-torsion Λ(G)-modules coincides with 𝔐_{H}(G).

(ii) There is an long exact localisation sequence of K-groups

\xymatrix{ & Λ(G)_{S^{∗}} \ar@{->>}[d] \\ K_{1}(Λ(G)) \ar[r] & K_{1}(Λ(G)_{S^{∗}}) \ar[r]^{∂} & K_{0}(𝔐_{H}(G) \ar[r] & 0}

and analogously for Λ(G)_{S} and the category of finitely generated S-torsion modules.

(iii) There is a canonical way of evaluation an element f ∈ K_{1}(Λ(G)_{S^{∗}}) at any continuous representation ρ : G → GL_{n}(O) with n ≥ 1 and O the ring of integers of a finite extension (depending of ρ) of ℚ_{p} :

f(ρ) ∈ ℂ_{p} ∪ {∞},

i.e. f can be considered as a map on the set of such representations.

Proof. See [17, 3]. Another proof that S is an Ore-set was given by Ardakov and Brown in [?] (and Schneider). The pseudo-compact case (generalised to certain skew power series rings) is covered by [12] or can be proven in the same way as in [3]. □

By S(E/F_{∞}) and X = X(E/F_{∞}) = S(E/F_{∞})^{∨} we denote the (classical) p-primary Selmer group of E over F_{∞} and its Pontryagin dual, respectively. It is easy to see that X is a finitely generated Λ_{ℤ_{p}}(G)-module with the natural Galois-action on S(E/F_{∞}), but the following torsion-property can be interpreted as a generalization of a deep conjecture of Mazur.

Conjecture 1.2 (Torsion-conjecture). The dual of the Selmer group is S^{∗}-torsion:

X(E/F_{∞}) ∈ 𝔐_{H}(G).

Now let K(F_{∞}) be the maximal abelian extension of ℚ inside F_{∞} in which p does not ramify and L = K(F_{∞})_{𝔓} its completion at some 𝔓 lying over p. Note that L is a finite extension of ℚ_{p}. Finally we put D = O_{L}.

If ω denotes the Neron differential of E, we obtain the usual real and complex periods Ω_{±} = ∫_{γ^{±}} ω by integrating along pathes γ^{±} which generate H_{1}(E(ℂ), ℤ)^{±}. We set R =

{q prime, |j(E)|_{q} > 1} ∪ {p} and let u, w be the roots of the characteristic polynomial of the action of Frobenius on the Tate-module T_{p}E of E

1 − a_{p}T + pT^{2} = (1 − uT)(1 − wT), u ∈ ℤ_{p}^{×}.

Further let p^{𝔣_{p}(ρ)} be the p-part of the conductor of an Artin representation ρ, while P_{p}(ρ, T) = det(1 − F rob_{p}^{−1} T |V_{ρ}^{I_{p}}) describes the Euler-factor of ρ at p. We also set

d^{±}(ρ) = dim_{ℂ} V_{ρ}^{±} and denote by ρ^{∨} the contragredient representation of ρ. By e_{p}(ρ) we denote the local ϵ-factor of ρ at p. In the notation of [?] this is e_{p}(ρ, ψ(−x), dx_{1}) where ψ is the additive character of ℚ_{p} defined by x → exp(2πix) and dx_{1} is the Haar measure that gives volume 1 to ℤ_{p}. Finally, in order to express special values of complex L-functions in the p-adic world, we fix embeddings of ℚ̄ into ℂ and ℂ_{p}, the completion of an algebraic closure of ℚ_{p}.

Conjecture 1.3 (Existence of p-adic L-function). There is a (unique) L_{E} ∈ K_{1}(Λ(G)_{S^{∗}}) such that

L_{E}(ρ^{∨}) = \frac{L_{R}(E, ρ, 1)}{Ω_{+}^{d^{+}(ρ)}Ω_{−}^{d^{−}(ρ)}} e_{p}(ρ̌)\frac{P_{p}(ρ^{∨}, u^{−1})}{P_{p}(ρ, w^{−1})}u^{−𝔣_{p}(ρ̌)}

for all Artin representations ρ of G.

For a class [M] ∈ K_{0}(𝔐_{H}(G)) we denote by [M]_{D} the base change to K_{0}(𝔐_{D,H}(G)).

Conjecture 1.4 (Main Conjecture). The p-adic L-function L_{E} is a characteristic element of X(E/F_{∞}) :

∂L_{E} = [X(E/F_{∞})]_{D}.

We refer the reader to [?, ?] where a refined version involving leading terms is discussed and where some implications of the Main Conjecture are explained.

2. The CM-case

In this section we assume that E is still defined over ℚ, has conductor N := N_{E} and moreover admits complex multiplication by the ring of integers O_{K} of a quadratic imaginary field K = ℚ(\sqrt{−d_{K}}), where cf. Appendix A §3 of [14] the discriminant d_{K} > 0 lies in the finite set

{3, 4, 7, 8, 11, 19, 43, 67, 163}.

Note that since j(E) ∈ ℚ the class number of K is trivial: h_{K} = 1. As we assume that E has good ordinary reduction at our fixed prime p, the latter decomposes into two primes 𝔭 = (π) and 𝔭̄ = (π̄), such that π and π̄ are conjugate under c ∈ Δ := G(K/ℚ) which is induced by complex conjugation. We fix an embedding of K into ℚ̄ such that 𝔭 corresponds to the induced embedding of K into ℂ_{p} (using the convention of the previous section).

Setting H := G(K_{∞}/K_{cyc}), G := (K_{∞}/K) we obtain isomorphisms G/H ≅ Γ and H/H ≅ Δ ≅ G/G. The situation can be illustrated by the following diagram of field extensions:

[figure: F_{∞} = K_{∞}; K_{cyc}; Γ; H; ℚ_{cyc}; Γ; Δ; H; K; G; ℚ; Δ]

In particular the non-commutative group G is an extension of the cyclic group of order 2 by the abelian group G, i.e. very close to being abelian.

One strategy to verify the Main Conjecture for E in the previous section would be a close analysis how the groups K_{1}(Λ(G)_{S^{∗}}) and K_{0}(𝔐_{H}(G)) look like, in order to construct the (non-commutative) p-adic L-function and to know how to show that two classes in K_{0}(𝔐_{H}(G)) in the absence of a suitable structure theory. This certainly can be done, but we will follow a much simpler way.

In this work we are going to make the following assumption:

Assumption: The Conjecture 1.2 in the CM case is true.

Remark 2.1. We remark that this assumption implies that X(E/F_{∞}) is actually Storsion. For this one has simply to show that the μ-invariant of X(E/F_{∞}) is trivial since it is known by [?, théorème 2.4] that X(E/F_{∞}) contains no non-trivial pseudonull Λ(G) submodules. But the triviality of the μ-invariant follows from the results of Schneps [?] after employing a descent argument similar to the one done by Hatchimori and Venjakob in [?, theorem 5.3].

The key ingredient for our proof of the Main Conjecture is the fact that to E there is attached a A_{0}- Größencharacter, in the sense of Weil, ψ of weight (1, 0) and conductor 𝔣 = 𝔣_{ψ} with N = d_{K}N_{K/ℚ}(𝔣) such that

(2.1)  ”E = Ind_{ℚ}^{K}ψ”,

This can either be understood (i) in the sense of the attached compatible system of l-adic Galois representations. Indeed to the character ψ there is by Weil attached a compatible system of ℓ-adic representations and then the corresponding system of ℓ-adic representations of E are obtained by induction from the absolute Galois group of K to the absolute Galois group of ℚ.

(ii) or in the automorphic induction sense. Indeed the automorphic counterpart of ψ is an adelic character ψ : 𝔸_{K}^{×}/K^{×} → ℂ^{×} and the corresponding cuspidal automorphic representation π_{E} of GL_{2}(𝔸_{ℚ}) associated to E is obtained by automorphic induction from GL_{1}(𝔸_{K}) to GL_{2}(𝔸_{ℚ}).

There is also a Größencharacter ψ̄ of weight (0, 1) (attached to E). As it was first shown by Weil these characters although being adelic in nature can also be interpreted as Galois characters of G(K(𝔣p^{∞})/K) or G(K_{∞}/K) with values in ℤ_{p}^{∗}, see also [4, page 38]. Note that, since E is defined over ℚ, it holds by [8, page 559] that

(2.2)  ψ̄ = ψ^{c}

where ψ^{c}(g) = ψ(cgc^{−1}) for all g ∈ G(K_{∞}/K).

By the functorial properties of K- and representation theory and (2.1) the desired result can be reduced to the two-variable Main Conjecture as proved by Rubin. In order to state it we introduce X (K_{∞}) to be the Galois group of the maximal abelian p-extension of K_{∞} which is unramified outside 𝔭. This is a finitely generated torsion Λ(G)-module, [10, page 37]. Furthermore, let

T_{π} := T_{π}E := lim ; ⟵_{n}E[π^{n}]

be the π-primary Tate-module of E, similarly for π̄, and

T_{π}^{∗} = Hom(T_{π}, ℤ_{p})

its ℤ_{p}-dual representation of G_{K}. Note that the action of G_{K} on T_{π} and Tπ̄ is given by the characters ψ and ψ̄, respectively. Our above philosophy is confirmed by the following

Proposition 2.2. There is a natural isomorphism of Λ(G)-modules

X(E/K_{∞}) ≅ Ind_{G}^{G} (X (K_{∞}) ⊗_{ℤ_{p}} T_{π}^{∗}) ,

where G acts diagonally on the tensor product X (K_{∞}) ⊗_{ℤ_{p}} T_{π}^{∗} ≅ X (K_{∞})(ψ^{−1}).

The proof is a modification of the proof of the old observation by John Coates that the π-Selmer group S_{π}(E/K_{∞}) of E over K_{∞} is canonically isomorphic to

S_{π}(E/K_{∞}) ≅ Hom(X (K_{∞}), E(π)).

Proof. Conferring to [4, page 124] the π-primary Selmer group is given by the following exact sequence

\xymatrix{0 \ar[r] & S_{π}(E/K_{∞}) \ar[r] & H^{1}(G_{S_{p}}(K_{∞}), E(π)) \ar[r] & Coind_{G}^{G𝔭̄}H^{1}(K_{∞,ν}, E)(π)}

where the coinduction is dual to the induction functor, ν denotes a prime of K_{∞} above 𝔭̄. Applying the exact functor Coind_{G}^{G}, using the transitivity of coinduction, the isomorphisms

Coind_{G}^{G}H^{1}(G_{S_{p}}(K_{∞}), E(π)) ≅ Hom(G_{S_{p}}(K_{∞}), Coind_{G}^{G}E(π)) ≅ H^{1}(G_{S_{p}}(K_{∞}), E(p))

and

H^{1}(K_{∞,ν}, E)(p) ≅ H^{1}(K_{∞,ν}, E)(π) ⊕ H^{1}(K_{∞,ν}, E)(π̄),

as well as the vanishing (see loc. cit.) of H^{1}(K_{∞,ν}, E)(π̄), one just obtains the defining sequence of the full p-primary Selmer group

\xymatrix{0 \ar[r] & S(E/K_{∞}) \ar[r] & H^{1}(G_{S_{p}}(K_{∞}), E(p)) \ar[r] & Coind_{G}^{G𝔭̄}H^{1}(K_{∞,ν}, E)(p).}

The result now follows by taking duals. □

In order to introduce the (commutative) two-variable p-adic L-function we choose a complex period Ω_{∞} ∈ ℂ^{×} such that Λ_{E} = Ω_{∞}𝔣_{ψ} for the period lattice Λ_{E} attached to the pair (E, ω). Here we view 𝔣_{ψ} as a lattice in ℂ with respect the chosen embedding K ↪ ℂ that correspond to the type of ψ. This determines the p-adic period Ω_{p} ∈ ℤ̂_{p}^{nr}^{×} by the procedure described in [4, page 66] such that the ratio \frac{Ω_{∞}}{Ω_{p}} is independent of any choices. Actually we can pin down the p-adic Ω_{p} up to elements in ℤ_{p}^{×} by the rule

\frac{Ω_{p}^{Φ}}{Ω_{p}} = u where Φ is the extension of Frobenious to ℤ̂_{p}^{nr}. Then we have,

Theorem 2.3 (Manin-Višik, Katz, Yager, de Shalit). Let 𝔣 be an integral ideal of K relative prime to p. For D = ℤ̂_{p}^{nr} there is a unique measure μ(𝔣𝔭̄^{∞}) ∈ Λ_{D}(G(K(𝔣p^{∞})/K)) such that for any grössencharacter ϵ of conductor dividing 𝔣p^{∞} and of type (k, j) with 0 ≤ −j and k > 0 we have,

Ω_{p}^{j−k} ∫_{G(K(𝔣p^{∞})/K)} ϵ^{−1}(σ) μ(𝔣𝔭̄^{∞}) = Ω_{∞}^{j−k}Γ(k)i^{k}(\frac{\sqrt{d_{K}}}{2π})^{j}G(ϵ)(1 − \frac{ϵ(𝔭)}{p})L_{𝔭̄𝔣}(ϵ^{−1}, 0)

where

G(ϵ) := \frac{ϕ^{k}ϕ̄^{j}(𝔭^{n})}{p^{n}}G(χ) := \frac{ϕ^{k}ϕ̄^{j}(𝔭^{n})}{p^{n}} ∑_{γ∈M} χ(γ)(ζ_{n}^{γ})^{−1}

Here we write ϵ = ϕ^{k}ϕ̄^{j}χ with ϕ a Grössencharacter of conductor prime to 𝔭 and of type (1, 0) and χ a character of finite order of conductor at 𝔭 equal to 𝔭^{n} and

M := {γ ∈ Gal(K(𝔣𝔭̄^{∞}𝔭^{n})/K)|γ_{|F′} = (𝔭^{n}, F′/K)}, F′ := K(𝔣𝔭̄^{∞})

Proof. This theorem is from [4, thm. II 4.14] (where the restriction on the type of the character is that 0 ≤ −j < k) in combination with corollary 6.7 of the same book. We note here that the relation between the Galois counterpart of ϵ that appear in the left hand side of the above formula and its automorphic counterpart that shows up in the right side are related by ϵ^{Galois}(F rob_{𝔮}^{−1}) = ϵ^{aut}(𝔮), different from deShalit who sets ϵ^{Galois}(F rob_{𝔮}) = ϵ^{aut}(𝔮). This explains the difference of the formula above with the one of deShalit in page 80 (the argument of the integral is inverted). We also need to remark that in the theorem of de Shalit also the Gamma factor of the L function appears which is equal to (2π)^{−1}. However in de Shalit the archimedean period used is equal to (2π)^{−1}Ω_{∞} (compare theorem 4.11 and 4.12 in [4]). Finally the factor i^{k} is coming from the normalization of the p-adic periods (see [4, page 70]). □

For our purposes it is important to understand the relation of the ”Gauss sum” like factor with the epsilon factor at 𝔭. To this end we have the following lemma,

Lemma 2.4. Let δ ∈ ℤ_{p}^{×} be the image of \sqrt{−d_{K}} under K ↪ K_{𝔭} = ℚ_{p} and σ_{δ} ∈ Gal(K(𝔣p^{∞})/K(𝔣𝔭̄^{∞})) such that σ_{δ}(ζ) = ζ^{δ} for all p-power roots of unity. Then with notation as in the theorem above we have

G(χ) = χ(σ_{δ})e_{𝔭}(χ)

Proof. We base the following proof on the article of Tate [?] and [4] section 6.3. By [4] (page 92) we have

e_{𝔭}(ϵ, ψ, dx_{1})^{−1} = p^{\frac{n}{2} (k+j+1)}δ^{−k}ϵ(σ_{δ})G(ϵ^{−1})

From the definition of δ ∈ ℤ_{p}^{×} we have ϕ̄(σ_{δ}) = 1 and ϕ(σ_{δ}) = δ (see also [4] (page 92). In the above equation we set ϵ = χ̄ϕ^{−1} where ϕ as always a Grössencharacter of type (1, 0). In particular we have that ϵ is a character of type (−1, 0). The above equation reads (with k = −1 and j = 0)

e_{𝔭}(χ̄ϕ^{−1}, ψ(x), dx_{1})^{−1} = δχ̄(σ_{δ})ϕ^{−1}(σ_{δ})G(χϕ)

Now we note that ϕ^{−1}(σ_{δ}) = δ^{−1} and since ϕ is unramified at 𝔭 we have that (see [?]) e_{𝔭}(ϕ^{−1}χ̄, ψ(x), dx_{1}) = ϕ(𝔭)^{−n}e_{𝔭}(χ̄, ψ(x), dx_{1}). With these remarks the above equation simplifies to

e_{𝔭}(χ̄, ψ(x), dx_{1})^{−1}ϕ(𝔭)^{n} = χ̄(σ_{δ})G(χϕ)

Using now the duality property (see [?])

e_{𝔭}(χ̄, ψ(x), dx_{1})e_{𝔭}(χ, ψ(−x), dx_{1}) = p^{n}

we obtain that

e_{𝔭}(χ, ψ(−x), dx_{1})p^{−n}ϕ(𝔭)^{n} = χ̄(σ_{δ})G(χϕ)

But G(χϕ) = \frac{ϕ(𝔭)^{n}}{p^{n}} G(χ) from which we conclude that

G(χ) = χ(σ_{δ})e_{𝔭}(χ, ψ(−x), dx_{1}) = χ(σ_{δ})e_{𝔭}(χ)

following our convention to write e_{𝔭}(·) for e_{𝔭}(·, ψ(−x), dx_{1}). □

Corollary 2.5. There is a unique L′ ; ψ̄ := μ ∈ Λ_{D}(G) such that

L′ ; ψ̄(χ̄) = ∫_{G} χdμ = \frac{L(ψ̄χ, 1)}{Ω_{∞}} e_{𝔭}(χ̄)P_{𝔭}(χ̄, u^{−1})P𝔭̄(χ, u^{−1}) u^{−𝔣_{𝔭}(χ̄)}

for all Artin-character χ of G.

Proof. This is in principle the measure mentioned in the above theorem 2.3 twisting it by the fixed character ψ. However in the theorem above it is the imprimitive L function that appears (as we remove the Euler factors at 𝔣 even if the conductor of the character is not 𝔣). For our purposes we need the primitive L functions. We now explain how it can be constructed. We consider the decomposition Gal(K_{∞}/K) = G ≅ Δ × G_{1} where Δ ≅ (ℤ/pℤ)^{×} × (ℤ/pℤ)^{×} and G_{1} ≅ ℤ_{p}^{2}. This decomposition induces a decomposition of Iwasawa algebras

Λ_{D}[[G]] = D[[G]] ≅ D[Δ][[G_{1}]] ≅ ⊕_{θ∈Δ̂}D[[G_{1}]]

where Δ̂ = Hom(Δ, D^{×}) the dual group of Δ. For a character χ of G we consider its decomposition to χ = χ_{Δ}χ_{1} according to the above decomposition of G. Similarly we have ψ = ψ_{Δ}ψ_{1}. We are going to define the element μ ∈ Λ_{D}[[G]] claimed in the proposition by defining the elements in D[[G_{1}]] in the above decomposition of the Iwasawa algebras. We define the θ^{th} component μ_{θ} as follows. We write 𝔤 for the prime to p part of the conductor of the character θψ_{Δ}. We note that 𝔤 can be different from 𝔣_{ψ}. We consider the measure δ σ_{δ}^{−1} ⋆ Ω_{p}^{−1}μ(𝔤𝔭̄^{∞}) ∈ Λ_{D}(G(K(𝔤p^{∞})/K)) (here ⋆ denotes convolution of measures) and define the measure μ_{θ} on G_{1} as

∫_{G_{1}} fdμ_{θ} = \frac{1}{Ω_{p}} ∫_{G(K(𝔤p^{∞})/K)} ψ^{−1}θ · (f ∘ pr) d(δ σ_{δ}^{−1} ⋆ μ(𝔤𝔭̄^{∞}))

for all integrable functions f on G_{1}. Here pr : G(K(𝔤p^{∞})/K) ↠ G_{1} denotes the natural projection map and ψθ is seen as a character of G(K(𝔤p^{∞})/K). We define the measure μ by putting together the components μ_{θ} according to the isomorphism Λ_{D}[[G]] ≅ ⊕_{θ∈Δ̂}D[[G_{1}]]. Now the result follows from the above theorem by taking k = 1, j = 0, ϵ = ψχ̄ and noting that 𝔣_{𝔭}(χ̄ψ) = 𝔣_{𝔭}(χ̄) = 𝔣_{𝔭}(χ) because ψ is unramified at 𝔭. Also note that from the equation

(1 − uX)(1 − wX) = 1 − a_{p}X + pX^{2} = (1 − ψ(𝔭)X)(1 − ψ(𝔭̄)X)

and the condition that u is a unit in ℤ_{p}, it follows that

(2.3)  u = ψ(𝔭̄) = π̄

and

(2.4)  u^{−1} = \frac{w}{p} = \frac{ψ(𝔭)}{p}.
□

Corollary 2.6. There is a unique Lψ̄ := μ ∈ Λ_{D}(G) such that

Lψ̄(χ̄) = ∫_{G} χ̄dμ = \frac{L(ψ̄χ, 1)}{Ω_{∞}} e𝔭̄(χ̄)P_{𝔭}(χ̄, u^{−1})P𝔭̄(χ, u^{−1}) u^{−𝔣𝔭̄(χ̄)}

for all Artin-character χ of G.

Proof. This measure is just a twist of the measure L′ ; ψ̄ by a unit in Λ_{D}(G). This follows from the functional equation of the L function involved. We defer the proof of this corollary for the next section (compare with corollary 3.6 and corollary 3.7). □

Now we are ready to state the two-variable Main Conjecture [10, page 37, Theorem 4.1] and [?, page 413, Theorem 1]:

Theorem 2.7 (Rubin,Yager). As ideals in Λ_{D}(G) there is the following equality

char (X (K_{∞}) ⊗_{ℤ_{p}} T_{π}^{∗}) = (Lψ̄),

where the left hand side denotes the characteristic ideal associated with the Λ(G)-torsion module X (K_{∞}) ⊗_{ℤ_{p}} T_{π}^{∗} by the structure theory from commutative algebra.

Lemma 2.8. The module X (K_{∞}) ⊗_{ℤ_{p}} T_{π}^{∗} is S-torsion, in particular X (K_{∞}) ⊗_{ℤ_{p}} T_{π}^{∗} ∈ 𝔐_{H}(G), and thus Lψ̄ ∈ (Λ(G)_{S})^{×} ⊆ (Λ(G)_{S^{∗}})^{×}.

Proof. By Proposition 2.2 X (K_{∞}) ⊗_{ℤ_{p}} T_{π}^{∗} being S-torsion is equivalent to X(E/K_{∞}) being S-torsion, which is just remark 2.1. The rest of the claim follows from the twovariable Main Conjecture above and the commutative diagram

[figure: 0; K_{1}(Λ(G)); K_{1}(Λ(G)_{S}); K_{0}(S−tor); 0; 0; K_{1}(Λ(G)); K_{1}(Q(G)); K_{0}(tors); 0] ,

where Q(G) denotes the maximal ring of quotients of Λ(G), while tors denotes the category of all finitely generated Λ(G)-torsion modules. □

Note that, although the Ore-sets S_{G}^{∗} and S_{G}^{∗} are different in general, we have a natural isomorphism Λ(G)_{S_{G}^{∗}} ≅ Λ(G)_{S_{G}^{∗}} by [12, Lemma 4.6] where in the second case the localization is formed with respect to the G-module structure of Λ(G). Hence we have a commutative base change square

\xymatrix{Λ(G) \ar@{^{(}->}[d]^{ι} \ar@{^{(}->}[r] & Λ(G)_{S} \ar@{^{(}->}[d]^{ι_{S}} \\ Λ(G) \ar@{^{(}->}[r] & Λ(G)_{S},}

which induces the following diagram with exact rows by functoriality of K-theory

\xymatrix{0 \ar[r] & K_{1}(Λ(G)) \ar[d]_{ι_{∗}} \ar[r] & K_{1}(Λ(G)_{S^{∗}}) \ar[d]_{(ι_{S})_{∗}} \ar[r]^{∂} & K_{0}(𝔐_{H}(G)) \ar[d]_{Λ(G)⊗_{Λ(G)}−} \ar[r] & 0 \\  & K_{1}(Λ(G)) \ar[r] & K_{1}(Λ(G)_{S^{∗}}) \ar[r] & K_{0}(𝔐_{H}(G)) \ar[r] & 0.}

It follows that L := (ι_{S})_{∗}(Lψ̄) is automatically a characteristic element for [X(E/K_{∞})] by Proposition 2.2. A naive hope would be that this is the desired p-adic L-function for E. In order to check this we need the next lemma which describes the evaluation of L at representations.

Lemma 2.9. For all Artin representations ρ of G one has

L(ρ) = Lψ̄(Res_{G}^{G}ρ),

where Res_{G}^{G}ρ denotes the restriction to the subgroup G. With other words L is induced by the measure μ_{Lψ̄} trivially extended

∫_{G} fdμ_{L} = ∫_{G} f_{|G}dμ_{Lψ̄},

i.e. from the image of μ_{Lψ̄} under the natural map Λ(G) ↪ Λ(G).

Proof. Upon comparing with the definition of evaluation in [3] the statement follows from the following diagram, which is obviously commutative:

[figure: Λ(G)_{S^{∗}}; (Res_{G}^{G}ρ)⊗pr_{|G}; M_{n}(O) ⊗_{ℤ_{p}} Λ(Γ)_{S^{∗}}; Λ(G)_{S^{∗}}; ρ⊗pr; M_{n}(O) ⊗_{ℤ_{p}} Λ(Γ)_{S^{∗}},]

where pr : Λ(G)_{S^{∗}} ↠ Λ(Γ)_{S^{∗}} denotes the canonical projection. □

The next task is to understand the (irreducible) Artin representations of G. By [13, page 62] and the fact that G is a semi-direct product

G ≅ G ⋊ Δ

one immediately obtains that an irreducible such representation is either of

Typ A : ρ is one dimensional,

or

Typ B: ρ = Ind_{G}^{G}χ is of dimension two where χ is a one-dimensional character of G with χ ≠ χ^{c}.

Indeed to see that the only irreducible representations of dimension bigger than one are those of dimension two and they are of the form described above one has simply to use Frobenious reciprocity. For if an Artin representation ρ of G has dimension n with n ≥ 2 and we consider its restriction to G then Res_{G}^{G}ρ = ⊕_{i=1}^{n}χ_{i} for n one dimensional representations of G. But then for every χ_{j}, with 1 ≤ j ≤ n, by applying Frobenious Reciprocity we have

0 ≠ (Res_{G}^{G}ρ, χ_{j})_{G} = (ρ, Ind_{G}^{G}χ_{j})_{G}

and the representation Ind_{G}^{G}χ_{j} is of dimension two. That is, every representation of dimension greater (or equal) than two has a subrepresentation of dimension two. The irreducibility assumption allows us to conclude our claim.

From Lemma 2.9 it is now clear that

L(ϵ) = Lψ̄(Res_{G}^{G}ϵ) = L(1)

where 1 is the trivial representation. But this is not compatible with the interpolation property in Conjecture 1.3 because the periods Ω_{+} and Ω_{−} are interchanged. This is clear from an philosophical point of view as the periods Ω_{±} arise from paths γ_{±} ∈ H_{1}(E(ℂ), ℤ)^{±} ⊆ H_{1}(E(ℂ), ℤ_{p})^{±} ≅ T_{p}E, while the period Ω_{∞} corresponds to a choice of γ ∈ T_{π}E (via the same identification). Thus we define the following correction term which describes the change of complex periods

L_{Ω} := \frac{Ω_{+}}{Ω_{∞}} \frac{1 + c}{2} + \frac{Ω_{−}}{Ω_{∞}} \frac{1 − c}{2}

Lemma 2.10. L_{Ω} ∈ Λ_{ℤ_{p}}(G)^{×}.

Proof. In order to prove this lemma we need to understand the relation between the Neron periods Ω_{+}, Ω_{−} and the period Ω_{∞}, and in particular show that the ratios \frac{Ω_{+}}{Ω_{∞}} and \frac{Ω_{−}}{Ω_{∞}} belong to ℤ_{p}^{×}. Then we have that L_{Ω} ∈ Λ_{ℤ_{p}}(G) and it is easily seen then that the element \frac{Ω_{∞}}{Ω_{+}} \frac{1+c}{2} + \frac{Ω_{∞}}{Ω_{−}} \frac{1−c}{2} ∈ Λ_{ℤ_{p}}(G) is its inverse. Recall that we write Λ for the lattice associated to E and we have defined Ω_{∞} by Λ = Ω_{∞}𝔣 : ψ. For our aims we may actually assume that Λ = Ω_{∞}O_{K} as 𝔣 = (f) for some f ∈ K ↪ K_{𝔭} ≅ ℤ_{p} which is a p-adic unit since (𝔣, 𝔭) = 1. Recall that we view O_{K} = ℤ + ℤ\sqrt{−d_{K}} as a lattice in ℂ by our fixed embedding K ↪ ℂ.

It is a well-known fact that the Neron periods have the property that τ := \frac{Ω_{−}}{Ω_{+}} is a totally imaginary number and moreover the lattice Λ′ := ℤΩ_{+} + ℤΩ_{−} is isomorphic to Λ as ℤ[\frac{1}{2}]-lattices. As p ≠ 2 we may work for our purposes with the lattice Λ′ which we keep denoting by Λ. As the lattice Λ has CM by O_{K} we have that \frac{1}{Ω_{+}} Λ ⊆ K is a fractional ideal of K [?, page 164]. In particular we have that τ ∈ K and that ℤ + ℤτ = O_{K}α for some α ∈ K. We write τ = s\sqrt{−d_{K}} and α = α_{1} + α_{2}\sqrt{−d_{K}} with s, α_{1}, α_{2} ∈ ℚ. We will show that s ∈ ℤ_{(p)}^{×} and α ∈ O_{𝔭}^{×}. Note that this is enough for our purposes because we have

ℤ + ℤτ = ℤ + ℤ\frac{Ω_{−}}{Ω_{+}} = \frac{Λ}{Ω_{+}} = \frac{Ω_{∞}}{Ω_{+}} O_{K}

The lattice ℤ + ℤτ has CM by O_{K}. That means that we have that

O_{K} = ℤ + ℤ\sqrt{−d_{K}} = \{a + bτ|a, b ∈ ℤ, 2br ∈ ℤ and b(r^{2} + d_{K}s^{2}) ∈ ℤ\}

As \sqrt{−d_{K}} should belong to the set on the right we have that bs = 1 and d_{K}s ∈ ℤ. That is, s = \frac{s′}{d_{K}} for some s′ ∈ ℤ. As b = \frac{1}{s} = \frac{d_{K}}{s′} ∈ ℤ we conclude that s′|d_{K}. Hence s = (\frac{d_{K}}{s′})^{−1} ∈ ℤ_{p}^{×} as D ∈ ℤ_{p}^{×}. That is, τ ∈ O_{𝔭}^{×}.

Now we show that α ∈ O_{𝔭}^{×}. We have ℤ + ℤτ = O_{K}α. We take the completion at 𝔭. It is enough to show that (ℤ + ℤτ) ⊗_{O_{K}} O_{𝔭} = O_{𝔭}. This follows trivially from the fact that τ ∈ O_{𝔭}^{×}. Hence we obtain that also α is a unit. □

Finally we set

L′_{E} := LL_{Ω}^{−1} ∈ K_{1}(Λ_{D}(G)_{S^{∗}}).

Proposition 2.11. L′_{E} satisfies the interpolation property in Conjecture 1.3.

Proof. We calculate the fraction \frac{L(ρ̌)}{L′_{E}(ρ̌)}, where the denominator here is the right hand side of the desired interpolation formula in Conjecture 1.3, in a purely formal way in order to compare all terms showing up, in particular the possible presence of zeroes does not matter at all.

We start with representations ρ = Ind_{G}^{G}χ of type B. Then ρ̌_{|_{G}} = χ̄ ⊕ χ̄^{c} where χ^{c}(g) := χ(cgc^{−1}) with complex conjugation c ∈ G.

\frac{L(ρ̌)}{L′_{E}(ρ̌)} = \frac{Lψ̄(χ̄) · Lψ̄(χ̄^{c})}{L′_{E}(ρ̌)} = \frac{\frac{1}{Ω_{∞}^{2}}}{Ω_{+}^{−1}Ω_{−}^{−1}} \frac{u^{−(𝔣𝔭̄(χ̄)+𝔣𝔭̄(χ̄^{c}))}e𝔭̄(χ̄)e𝔭̄(χ̄^{c})}{u^{−𝔣_{p}(ρ̌)} e_{p}(ρ̌)}

· \frac{P_{𝔭}(χ̄, u^{−1})P𝔭̄(χ, u^{−1})P_{𝔭}(χ̄^{c}, u^{−1})P𝔭̄(χ^{c}, u^{−1})}{P_{p}(ρ̌, u^{−1})P_{p}(ρ, w^{−1})^{−1}P_{p}(E, ρ, \frac{1}{p})} \frac{L(ψ̄χ, 1) L(ψ̄χ^{c}, 1)}{L(E, ρ, 1)}

= \frac{Ω_{+}Ω_{−}}{Ω_{∞} Ω_{∞}} \frac{u^{−(𝔣_{𝔭}(χ̄)+𝔣𝔭̄(χ̄))}e_{𝔭}(χ̄)e𝔭̄(χ̄)}{u^{−𝔣_{p}(ρ̌)} e_{p}(ρ̌)}

· \frac{P_{𝔭}(χ̄, u^{−1})P𝔭̄(χ̄, u^{−1})P_{𝔭}(χ^{c}, u^{−1})P𝔭̄(χ^{c}, u^{−1}) P_{p}(ρ, w^{−1})}{P_{p}(ρ̌, u^{−1}) P𝔭̄(ψ̄χ, \frac{1}{p}) P𝔭̄(ψ̄χ^{c}, \frac{1}{p}) P_{𝔭}(ψ̄χ^{c}, \frac{1}{p})P_{𝔭}(ψ̄χ, \frac{1}{p})}

· \frac{L(ψ̄χ, 1) · L(ψ̄χ^{c}, 1)}{L((Indψ̄) ⊗ Indχ, 1)} = α^{2}τ

with τ := \frac{Ω_{−}}{Ω_{+}} , α :=\frac{Ω_{+}}{Ω_{∞}} ∈ O_{𝔭}^{×} ∩ K.

Let us comment on the calculations above. We start with our considerations on the ϵ-factors. Note that as the representation ρ̌ is induced from χ̄ we have for the conductor of ρ̌, Nρ̌ = DN_{K/ℚ}(Nχ̄) where Nχ̄ the conductor of χ̄. Locally at p that means 𝔣_{p}(ρ̌) = 𝔣_{𝔭}(χ̄) + 𝔣𝔭̄(χ̄) as p splits in K. Concerning the epsilon factors note that we know that they are inductive in degree zero. In particular we have

e_{p}(Ind_{G}^{G}(χ̄ ⊖ 1)) = e_{𝔭}(χ̄ ⊖ 1)e𝔭̄(χ̄ ⊖ 1) = \frac{e_{𝔭}(χ̄)e𝔭̄(χ̄)}{e_{𝔭}(1)e𝔭̄(1)}

But also

e_{p}(Ind_{G}^{G}(χ̄ ⊖ 1) = \frac{e_{p}(Ind_{G}^{G}(χ̄))}{e_{p}(1 ⊕ ϵ)} = \frac{e_{p}(ρ̌)}{e_{p}(1)e_{p}(ϵ)}

where as always ϵ is the non-trivial character of Gal(K/ℚ). Hence we obtain

e_{p}(ρ̌) = \frac{e_{𝔭}(χ̄)e𝔭̄(χ̄)}{e_{𝔭}(1)e𝔭̄(1)}e_{p}(1)e_{p}(ϵ)

But note that p split in K we have e_{p}(1) = e_{p}(ϵ) = e_{𝔭}(1) = e𝔭̄(1) = 1. Hence e_{p}(ρ̌) = e_{𝔭}(χ̄)e𝔭̄(χ̄). Now we explain the ratio of the Euler factors. The starting point is the inductive properties of the Euler factors. In particular as p splits in K we have that P_{p}(ρ, X) = P_{𝔭}(χ, X)P𝔭̄(χ, X). Using this and the relations

u^{−1} = \frac{ψ(𝔭)}{p} = \frac{ψ̄(𝔭̄)}{p}, w^{−1} = \frac{ψ(𝔭̄)}{p} = \frac{ψ̄(𝔭)}{p}.

we see that the Euler factors cancel (the Euler factors which are opposite to each other with respect to the fraction line cancel each other). Finally for the last equation we use the inductiveness of L-functions and the following well-known fact from representation theory implied by Frobenius reciprocity

(Indψ̄) ⊗ Indχ = Ind(ψ̄ ⊗ Res Indχ) = Ind(ψ̄ ⊗ (χ ⊕ χ^{c})).

Note that, since ψ̄ = ψ^{c} by [8, page 559] also E = Indψ̄ holds.

Now we turn to those ρ of type A, i.e ρ is now one dimensional. We have

\frac{L(ρ̌)}{L′_{E}(ρ̌)} = \frac{Lψ̄(ρ̄_{|G})}{L′_{E}(ρ̌)} =

= \frac{\frac{1}{Ω_{∞}}}{Ω_{ρ(c)}^{−1}} \frac{u^{−𝔣𝔭̄(ρ̄_{|G})}e𝔭̄(ρ̄_{|G})}{u^{−𝔣_{p}(ρ̄)} e_{p}(ρ̄)} \frac{P_{𝔭}(ρ̄_{|G}, u^{−1})P𝔭̄(ρ̄_{|G}, u^{−1})P_{p}(ρ, w^{−1})}{P_{p}(ρ̄, u^{−1})P_{p}(E, ρ, p^{−1})} \frac{L(ψ̄ρ_{|G}, 1)}{L(E, ρ, 1)} =

= \frac{Ω_{ρ(c)}}{Ω_{∞}} \frac{u^{−𝔣𝔭̄(ρ̄_{|G})}e𝔭̄(ρ̄_{|G})}{u^{−𝔣_{p}(ρ̄)} e_{p}(ρ̄)} \frac{P_{𝔭}(ρ̄_{|G}, u^{−1})P𝔭̄(ρ̄_{|G}, u^{−1})P_{𝔭}(ρ_{|G}, w^{−1})}{P_{𝔭}(ρ̄_{|G}, u^{−1})P_{𝔭}(ψ̄ρ_{|G}, p^{−1})P𝔭̄(ψ̄ρ_{|G}, p^{−1})} \frac{L(ψ̄ρ_{|G}, 1)}{L(E, ρ, 1)} =

\frac{Ω_{ρ(c)}}{Ω_{∞}} \frac{L(ψ̄ρ_{|G}, 1)}{L(Ind(ψ̄ ⊗ ρ_{|G}), 1)}

= \{ \begin{matrix}α & & ρ(c) = +1 \\ & if \\ ατ & & ρ(c) = −1 \end{matrix}

Recall that τ = \frac{Ω_{−}}{Ω_{+}} , α = \frac{Ω_{+}}{Ω_{∞}} , ατ = \frac{Ω_{−}}{Ω_{∞}} all belong to O_{𝔭}^{×} ∩ K. Let us comment

on the above computations. We start with the epsilon factors. First note that as the extension K is unramified at p we have that 𝔣_{p}(ρ̄) = 𝔣𝔭̄(ρ̄_{|G}) since as adelic characters ρ̄_{|G} = ρ̄∘ N_{K/ℚ} and p splits in K, that is locally when we identify K𝔭̄ with ℚ_{p} the two characters are equal. This explains also the equality e_{p}(ρ̄) = e𝔭̄(ρ̄_{|G}). Note that also this remark explains the equality of Euler factors P_{p}(ρ, X) = P_{𝔭}(ρ_{|G}, X) and similarly for ρ̄. The rest is just trivial inspection of the formula. Finally for the L-functions we use as before Frobenius reciprocity and inductive properties of L-functions.

Since it is easily checked that

L_{Ω}(ρ̌) = \{ \begin{matrix}α^{2}τ & & ρ & = & Indχ \\ α & ∈ D^{×}, if & ρ(c) & = & +1 \\ ατ & & ρ(c) & = & −1 \end{matrix}

for irreducible ρ, we obtain that

L′_{E}(ρ̌)L_{Ω}(ρ̌) = L(ρ̌)

for all Artin representations ρ as desired. □

Now let K(F_{∞}) be the maximal abelian extension of ℚ inside F_{∞} in which p does not ramify and L = K(F_{∞})_{𝔓} its completion at some 𝔓 lying over p. Then L′_{E} can be replaced by an element L_{E} defined already over O, the ring of integers of L.

Theorem 2.12. Assuming Conjecture 1.2 there exists L_{E} ∈ K_{1}(Λ_{O}(G)_{S}) satisfying Conjecture 1.3 and 1.4.

Proof. This follows from the explanation before Lemma 2.9 and Theorem 4.3 in the appendix. Indeed assumption (ii) holds because O(ρ) is contained in O(μ(p)) as the values of χ (and thus ρ = Indχ or χ)are in μ_{(p−1)p^{∞}}; (iii) is clear from the construction, (iv) follows from the Lemma below while (v) again follows from the explanation before Lemma 2.9. □

Lemma 2.13 (Deligne-Conjecture, Blasius). In the CM setting the following algebraicity result holds:

\frac{L_{\{p\}}(E, ρ, 1)}{Ω_{+}^{d^{+}(ρ)}Ω_{−}^{d^{−}(ρ)}} e_{p}(ρ̌)\frac{P_{p}(ρ^{∨}, u^{−1})}{P_{p}(ρ, w^{−1})}u^{−𝔣_{p}(ρ̌)} ∈ L(ρ).

Moreover these values are p-adically integral.

Proof. That the values are p-integral follows from the fact that are obtained as values of a p-integral valued measure. So we need to prove that actually the values are in L(ρ) and of course it is enough to prove it for ρ an irreducible Artin representation. Note that as p splits we can identify ℚ_{p} = K_{𝔭}. Moreover we have that u, w ∈ ℚ_{p} hence we need simply to prove that

\frac{L(E, ρ, 1)}{Ω_{+}^{d^{+}(ρ)}Ω_{−}^{d^{−}(ρ)}} e_{p}(ρ̌) ∈ L(ρ)

If ρ is one dimensional then this is well known. Hence we consider the case where ρ = Ind_{G}^{G}χ. Then the above statement is equivalent to

\frac{L(E/K, χ, 1)}{Ω_{∞}^{2}} (e_{𝔭}(χ)e𝔭̄(χ))^{−1} ∈ L(χ)

using the fact the Ω_{+} = Ω_{−} = Ω_{∞} up to elements in K^{×}, e_{𝔭}(χ̄)e𝔭̄(χ̄) = e_{p}(ρ̌) and the duality of the epsilon factors. Moreover we have L(E/K, χ, 1) = L(ψ, χ, 1)L(ψ̄, χ, 1). From Blasius proof of Deligne’s conjecture for Hecke characters (of CM fields) [?] we know that

\frac{L(ψ, χ, 1)}{Ω_{∞}}c(ψ, χ)^{−1} ∈ K(χ), \frac{L(ψ̄, χ, 1)}{Ω_{∞}}c(ψ̄, χ)^{−1} ∈ K(χ)

where c(ψ, χ) and c(ψ̄, χ) are defined as in [?] (see page 65 for the definition and page 67 for the factorization of Deligne’s periods for the Hecke characters ψχ and ψ̄χ). Moreover we have that c(ψ, χ)c(ψ̄, χ) = e_{𝔭}(χ)e𝔭̄(χ) up to elements in L(χ)^{×}. This can be seen from the definition of the periods c(ψ, χ) and c(ψ̄, χ). From [?, page 65] we have that the element c(ψ, χ) is an element in K(χ) ⊗_{ℚ} ℚ̄ characterized from the reciprocity law

(1 ⊗ τ)c(ψ, χ) = (χ ∘ V er_{ψ})(τ)c(ψ, χ)

where V er_{ψ} : G_{ℚ} → G_{K}^{ab} is the half-transfer map of Tate associated to the CM-type of the character ψ. As it is explained by Blasius this reciprocity law characterizes the element c(ψ, χ) up to elements in K(χ)^{×}. From the properties of the half-transfer map of Tate one has that

V er_{ψ}(τ) · V erψ̄(τ) = V er(τ), τ ∈ G_{ℚ}

where V er : G_{ℚ} → G_{K}^{ab} is the classical transfer map. But then as is shown in [?, page 70], the product of the all epsilon factors ∏_{𝔮} e_{𝔮}(χ) has exactly the same reciprocity law as the product c(ψ, χ)c(ψ̄, χ) with respect the operation of G_{ℚ} and hence they are equal up to elements in K(χ)^{×}. The characters χ that we consider are only ramified at p or at the primes 𝔮 that divide 𝔣_{ψ}. But for the epsilon factors e_{𝔮}(χ) for 𝔮|𝔣_{ψ} we have that e_{𝔮}(χ) = e_{q}(ρ) up to elements in K(χ)^{×} where q := 𝔮∩ℚ. But moreover it is well known [?, page 330] that e_{q}(ρ) = e_{q}(det(ρ)) up to element in ℚ(ρ)^{×}. But e_{q}(det(ρ)) is a Gauss sum hence an element in L(ρ)^{×} see [?, pages 103 and 104]. Hence we conclude that also e_{𝔮}(χ) ∈ L(χ)^{×}. Hence putting all the above considerations together we conclude that e_{𝔭}(χ)e𝔭̄(χ) is equal up to elements in L(χ)^{×} to c(ψ, χ)c(ψ̄, χ). □

We finish this section with a last comment. It seems natural to ask if the measure Lψ̄ has already values in ℤ_{p} or could be suitably modified by a unit in the Iwasawa algebra Λ_{D}(G) to still interpolate the critical values and take values in ℤ_{p}. Let us first try to explain why we believe that this measure does not take values in ℤ_{p}. We recall its interpolation property:

Lψ̄(χ̄) = ∫_{G} χ̄dμ = \frac{L(ψ̄χ, 1)}{Ω_{∞}} e_{𝔭}(χ̄)P_{𝔭}(χ̄, u^{−1})P𝔭̄(χ, u^{−1}) u^{−𝔣_{𝔭}(χ̄)}

for all Artin-character χ of G. Now we note that in order to prove that this measure lies in Λ_{ℤ_{p}}(G) is equivalent to show the following rationality result

\frac{L(ψ̄χ, 1)}{Ω_{∞}} e_{𝔭}(χ)^{−1} ∈ K(χ)

since we are in the ordinary case i.e. K_{𝔭} = ℚ_{p}. As we explained in the proof above we know from Blasius work that

\frac{L(ψ̄χ, 1)}{Ω_{∞}}c(ψ̄, χ)^{−1} ∈ K(χ)

hence one needs to understand if e_{𝔭}(χ) = c(ψ̄, χ) up to elements in K(χ)^{×} for finite characters χ of G. However this cannot be the case for all finite characters of Gal(K^{ab}/K). (However we remark that it is the case when χ is cyclotomic i.e. χ^{c} = χ.) Indeed from Blasius [?] (page 66) we have that the extension of K defined by adjoining to K the values c(ψ̄, χ)) for all finite order characters of Gal(K^{ab}/K) is an abelian extension of K not included in ℚ^{ab}. However the epsilon factors are just Gauss sums hence they can generate only extensions in ℚ^{ab}. In particular we have that the two “periods” cannot be equal up to elements in K(χ)^{×} ⊆ ℚ^{ab} (see also the comment in [?] page 109). Concerning the other question one may speculate that there is a measure L? ; ψ̄ of G such that

L? ; ψ̄(χ̄) = ∫_{G} χ̄dμ = \frac{L(ψ̄χ, 1)}{c_{p}(ψ̄, χ)Ω_{∞}} P_{𝔭}(χ̄, u^{−1})P𝔭̄(χ, u^{−1}) u^{−𝔣_{𝔭}(χ̄)}

for some canonically normalized element c_{p}(ψ̄, χ) equal to c(ψ̄, χ) up to elements in K(χ)^{×}. This measure would have values in ℤ_{p}.

3. CM-modular forms

In this section we would like to indicate how most of our results can be extended to the case of CM-modular forms. Our reference for the theory of CM-modular forms is Ribet’s aricle [?] as well as the last section of Schappacher’s book [?].

We start by fixing our setting. Let f be cuspidal newform of weight k ≥ 2, Nebentypus ϵ_{f} and level N. We pick a number field F that contains all a_{n} of f = ∑_{n≥1} a_{n}q^{n}. We want now to formulate a GL_{2}-Main Conjecture for f. Our starting point is the following theorem.

Theorem 3.1 (Eichler-Shimura-Deligne-Scholl-Jannsen). There exists a motive M(f) defined over ℚ with coefficients in F of rank two over F such that

L_{N}^{∗}(M(f), s) := ( ∏_{(p,N)=1} det_{F} (1 − F r_{p} p^{−s}|H_{ℓ}(M(f))^{−1})_{τ} = ( ∑_{(n,N)=1} a_{n}^{τ}n^{−s})_{τ}

where τ ∈ Hom(F, ℂ), ℓ ≠ p prime, F r_{p} is a geometric Frobenius element at p and Re(s) ≫ 0

Proof. We refer to the book of Schappacher [?] in page 139 and the references there. □

It is known [7, page 240, 14.10] that the Kummer dual M(f)^{∗}(1) of M(f) is isomorphic to M(f^{∗})(k), where f^{∗} = ∑_{n≥1} ā_{n}q^{n} is the dual cusp form.

We fix an embedding τ : F ↪ ℚ̄, we let p ≥ 5 be a rational prime and let λ be the prime of F above p corresponding to our fixed embedding F ↪ ℚ̄ ↪ ℚ̄_{p}. We write ρ_{λ} := ρ_{f,λ} : G_{ℚ} → GL_{2}(F_{λ}) for the associated G_{ℚ}- representation given by the λadic realisation V := V (f) := V_{F_{λ}}(f) of M(f). Moreover we assume that p is a good ordinary prime for M(f), this is equivalent to p being relative prime to N and a_{p} a λ-adic unit. Note that with f also f^{∗} is good ordinary at p, see [7, prop. 17.1].

We consider the p-adic Lie extension F_{∞} of ℚ determined by the image of ρ_{λ}, i.e. G := Gal(F_{∞}/ℚ) ≅ Im(ρ_{λ}). We note that the determinant of ρ_{λ} is of the form

det(ρ_{λ}) ≅ χ_{cycl}^{1−k}ϵ_{f}^{−1}

and hence G contains a closed normal subgroup H such that G/H ≅ ℤ_{p}. In particular the setting of Theorem 1.1 of the introduction apply to the above defined group G.

Next we define the λ-primary Selmer group attached to f as follows: By V′(f) we denote the (unique) unramified one dimensional G_{ℚ_{p}}-subrepresentation of V (f) (restricted to G_{ℚ_{p}}), which exists due to [7, prop. 17.1] f being good ordinary at λ. We fix an O := O_{F_{λ}}- lattice T (f) ⊆ V (f) and set T′(f) := T (f) ∩ V′(f) and T′′(f) := T (f)/T′(f) ⊆ V′′(f) := V (f)/V′(f). Then, for any integer r, we define

Sel^{ord}(T (f)(r)/F_{∞}) := ker (H^{1}(G_{S_{p}}(F_{∞}), T (f)(r)⊗ℚ/ℤ) → Coind_{G}^{G_{p}}H^{1}(F_{∞,ν}, T′′(f)(r)⊗ℚ/ℤ))

where G_{S_{p}}(F_{∞}) denotes the Galois group of the maximal outside p unramified extension of F_{∞} and ν is a fixed place of F_{∞} over p. Using the same arguments as in the proof of [7, prop. 17.2] one easily shows that this Selmer group coincides with the Bloch-Kato Selmer group Sel_{(1)}(T (f)(r) ⊗ ℚ/ℤ, F_{∞}) in [5, 4.2.28]. Finally, we write

X := X(T (f)(r)/F_{∞}) := Sel^{ord}(T (f)(r)/F_{∞})^{∨}

for its Pontryagin dual. Then the Torsion-Conjecture reads as follows

Conjecture 3.2 (Torsion-Conjecture). For one (and hence any) r, the dual of the Selmer group is S^{∗}-torsion:

X(T (f^{∗})(r)/F_{∞}) ∈ 𝔐_{H}(G).

We let L denote the same field as in the introduction in our setting and we write Λ(G) for the Iwasawa algebra of G with coefficients in D := O_{L}. We denote by Ω_{±} the periods of Deligne associated to M(f) with respect to our fixed τ ∈ Hom(F, ℂ), that is determined up to elements in O_{F}^{×}, see below for more details. We set

R = {p} ∪ {l ≠ p| the ramification index of l in F_{∞}/ℚ is infinite}

and we define u ∈ ℤ_{p}^{×} by

1 − a_{p}T + p^{k−1}ϵ_{f}(p)T^{2} = (1 − uT)(1 − wT)

Conjecture 3.3 (Existence of p-adic L-function). There is a L_{f} ∈ K_{1}(Λ(G)_{S^{∗}}) such that

L_{f}(ρ) = \frac{L_{R}(M(f), ρ^{∨}, 1)}{Ω_{+}^{d^{+}(ρ)}Ω_{−}^{d^{−}(ρ)}} e_{p}(ρ)\frac{P_{p}(ρ, u^{−1})}{P_{p}(ρ^{∨}, w^{−1})}u^{−𝔣_{p}(ρ)}

for all Artin representations ρ of G.

And similarly,

Conjecture 3.4 (Main Conjecture). The p-adic L-function L_{f} is a characteristic element of X(T (f^{∗})(k − 1)/F_{∞}) :

∂L_{f} = [X(M(f)/F_{∞})]_{D}.

Actually, the last two conjectures are a consequence of the much more general conjectures in [5]. Indeed, theorem 4.2.22 or 4.2.26 in (loc. cit.) applied to the motive M = M(f)(1) with coefficients in F predicts the existence of L_{f} such that, for 0 ≤ i ≤ k − 2, the following more general interpolation property holds (at least for i + 1 ≠ \frac{k−1}{2})^{1}

L_{f}(ρκ^{−i}) = \frac{L_{R}(M(f), ρ^{∨}, i + 1)}{Ω_{+}^{d^{+}(ρ,i)}Ω_{−}^{d^{−}(ρ,i)}(2πι)^{di}} (i!)^{d} e_{p}(ρ)\frac{P_{p}(ρ, u^{−1}p^{i})}{P_{p}(ρ^{∨}, up^{−i−1})}(u^{−1}p^{i})^{𝔣_{p}(ρ)}

where d = dim ρ while d^{+}(ρ, i) and d^{−}(ρ, i) denote the dimension of the part of ρ n which complex conjugation acts as (−1)^{i} and (−1)^{i−1}, respectively. To this end note that the eigenvalue u of the geometric Frobenius automorphism acting on V′(f) equals pν for the ν in (loc. cit.) due to the compatibility conjecture C_{W D} in [?, 2.4.3], which is known for modular forms (loc.cit., rem 2.4.6(ii)) and for Artin motives, and that

h(r) := dim gr^{r}((M ⊗ ρ)_{dR}) = \{ \begin{matrix}d, & if r = −1 or r = k − 2; \\ 0, & otherwise. \end{matrix}

because the de Rham realisation M(f)(j)_{dR} has the following decreasing filtration

M(f)(j)_{dR}^{i} = \{ \begin{matrix}M(f)_{dR}, & if i ≤ −j; \\ M(f)_{dR}^{1}, & if 1 − j ≤ i ≤ k − 1 − j; \\ 0, & if k − j ≤ i. \end{matrix}

see [7, §11.3]. In particular, all the twists M(f)(j), 1 ≤ j ≤ k −1 are critical. Moreover M(f) is pure of weight k − 1 with Hodge decomposition of type (k − 1, 0) + (0, k − 1).

empty set and replacing \frac{P_{p}(ρ,u^{−1}p^{i})}{P_{p}(ρ^{∨},up^{−i−1})} by \{P_{L,p}(W, u)P_{L,p}(Ŵ , u)^{−1}\}_{u=1}·P_{L,p}(Ŵ^{∗}(1), 1)·∏_{l∈B} P_{L,l}(W, 1),

1Otherwise some Euler factors might be zero and the formula can be rewritten by replacing R by the where W := M(ρ^{∗})_{λ} = [ρ^{∗}]_{λ} ⊗ M_{λ} and Ŵ := [ρ^{∗}]_{λ} ⊗ V̂′(f)(1).

The existence of good basis γ^{+}, γ^{−} and δ of M_{B}^{+}, M_{B}^{−} and the tangent space t_{M} := M_{dR}/M_{dR}^{0}, respectively, in the sense of [5, 4.2.24] follows from [7, 17.5] where the dual situation is discussed, in particular we have chosen Ω_{±} = Ω(γ^{±}, δ) in the notation of [5].

Assuming the conjectures in [5], Conjecture 3.4 is a direct consequence of theorem 4.2.22, proposition 4.3.15/16 in (loc. cit.) (observing that V′(f) grants an infinite residue extension of p in F_{∞}/ℚ) once we have seen that T induces the zero class in K_{0}(𝔐_{H}(G)). If f is not CM, this follows from proposition 4.3.17 in (loc. cit.) while in the CM-case we give an argument in the proof of Proposition 3.5.

Now we focus on the case where f is a CM-modular form. We will say that f has CM by a non-trivial (quadratic) Dirichlet character ϵ if

ϵ(q)a_{q} = a_{q}

for a set of primes q of density 1. If we write K for the quadratic extension that corresponds to ϵ we say that f has CM by K. For example in our previous setting if we write f_{E} for the newform that corresponds to E we have that f_{E} has CM by the non-trivial quadratic character ϵ of Gal(K/ℚ) as in that case a_{q} = 0 for the primes that inert in K. From now on our fixed modular form f will have CM by some quadratic field K and we will write ϵ for the associated character. From [?] (proposition 4.4 and theorem 4.5) we know:

(i) The field K is imaginary.

(ii) Let G := Gal(F_{∞}/K). Then ρ_{λ}_{|_{G}} is abelian.

(iii) f is (automorphic)-induced by a Grössencharacter ψ over K of type (k − 1, 0) and after fixing an embedding of E in ℚ̄ we have L(f, s) = L(ψ, s). Henceforth we assume that F contains K(ψ(K̂^{×}), where K̂ denotes the finite adeles of K, e.g. we can just take F = K(ψ(K̂^{×}) by [7, 15.10]. (iv) For the λ-adic representation attached to ψ we have

ρ_{λ}_{|_{G}} = ψ_{λ} ⊕ ψ_{λ}^{c}

where ψ_{λ} is the F̄_{λ}^{×}-valued λ-adic counterpart of the Grössencharacter ψ. In particular ρ_{λ} = Ind_{G}^{G}ψ_{λ} = Ind_{G}^{G}ψ_{λ}^{c}, see also [7, (15.11.2)]. Here our convention is that ψ_{λ}((𝔞, K_{∞}/K)) = ψ(𝔞)^{−1} and we write V (ψ) = V_{F_{λ}}(ψ) for the corresponding representation space.

(v) ψψ^{c} = N_{K}^{k−1}(ϵ_{f} ∘ N_{K}).

Assumption: We are going to assume all along that the size of the torsion part of G is relative prime to p.

Now we pick a prime p that splits in K and write p = 𝔭𝔭̄ for a prime 𝔭 of K. We make the standard assumption that 𝔭 is the prime that corresponds to the p-adic embedding

K ↪ ℚ̄ ↪ ℚ̄_{p}

with respect to our fixed embedding ℚ̄ ↪ ℚ̄_{p}, in particular λ|𝔭. We now note that

1 − a_{p}T + p^{k−1}ϵ_{f}(p)T^{2} = (1 − ψ(𝔭̄)T)(1 − ψ(𝔭)T)

and claim that ψ(𝔭̄) is a λ-adic unit. Indeed, as the character ψ is of type (k − 1, 0) we have that its λ-adic counterpart ψ_{λ} factors through Gal(K(𝔣_{ψ}𝔭^{∞})/K), i.e. is it is a character of the form

ψ_{λ} : Gal(K(𝔣_{ψ}𝔭^{∞})/K) → Ō_{λ}^{×}

where Ō_{λ} the ring of integers of F̄_{λ}^{×}. But then ψ(𝔭̄) = ψ_{λ}(F rob𝔭̄) ∈ Ō_{λ}^{×} and we have u = ψ(𝔭̄).

For a G_{K}-representation ρ : G_{K} → Aut(V) on a finite dimensional F_{λ}-vector space V and any Galois stable O := O_{F_{λ}}-lattice T ⊆ V we define

S(T/K_{∞}) := ker (H^{1}(G_{S_{p}}(K_{∞}), T ⊗ ℚ/ℤ) → Coind_{G}^{G𝔭̄}H^{1}(K_{∞,ν}, T ⊗ ℚ/ℤ))

where ν as before denotes any fixed place of K_{∞} lying over 𝔭̄. Its Pontryagin-dual

X (T/K_{∞}) = S(T/K_{∞})^{∨} ≅ X (K_{∞}) ⊗_{ℤ_{p}} T^{∗}

is a finitely generated Λ_{O}(G)-module. We fix an O-lattice T (ψ)^{∗} ⊆ V (ψ)^{∗} and assume that

T := T (f^{∗})(k − 1)

coincides with

Ind_{ℚ}^{K}T (ψ)^{∗} ⊆ V (f)^{∗} = V (f^{∗})(k − 1).

As noted above T (ψ)^{∗} is unramified at 𝔭̄, whence the free rank one O-module T (ψ^{c})^{∗} with Galois action given by the complex conjugate character (ψ^{c})^{−1} is unramified at 𝔭. It follows immediately that

T (ψ^{c})^{∗} = T′ := T′(f^{∗})(k − 1) and T (ψ)^{∗} = T′′ := T′′(f^{∗})(k − 1)

as G_{K}-modules.

Proposition 3.5. There is a natural isomorphism of Λ(G)-modules

X(T/K_{∞}) ≅ Ind_{G}^{G} (X (K_{∞}) ⊗_{ℤ_{p}} T (ψ)) ,

where G acts diagonally on the tensor product.

Proof. Applying the exact functor Coind_{G}^{G} to the defining sequence

\xymatrix{0 \ar[r] & S(T (ψ)^{∗}/K_{∞}) \ar[r] & H^{1}(G_{S_{p}}(K_{∞}), T (ψ)^{∗} ⊗ ℚ/ℤ) \ar[r] & Coind_{G}^{G𝔭̄}H^{1}(K_{∞,ν}, T (ψ)^{∗} ⊗ ℚ/ℤ)}

using the transitivity of coinduction and the isomorphisms

Coind_{G}^{G}H^{1}(G_{S_{p}}(K_{∞}), T (ψ)^{∗} ⊗ ℚ/ℤ) ≅ Hom(G_{S_{p}}(K_{∞}), Coind_{G}^{G}(T (ψ)^{∗} ⊗ ℚ/ℤ))
≅ H^{1}(G_{S_{p}}(K_{∞}), T ⊗ ℚ/ℤ)

as well as

H^{1}(K_{∞,ν}, T (ψ)^{∗} ⊗ ℚ/ℤ) ≅ H^{1}(K_{∞,ν}, T′′ ⊗ ℚ/ℤ),

one just obtains the defining sequence of the full Selmer group

\xymatrix{0 \ar[r] & Sel^{ord}(T/K_{∞}) \ar[r] & H^{1}(G_{S_{p}}(K_{∞}), T ⊗ ℚ/ℤ) \ar[r] & Coind_{G}^{G𝔭̄}H^{1}(K_{∞,ν}, T′′ ⊗ ℚ/ℤ).}

The result now follows by taking duals. Finally, we give the promised proof that the class of X(T/K_{∞}) coincides with that of the Selmer complex used in [5]: Let H′ be the (open) maximal torsionfree pro-p (abelian) subgroup of H, i.e. H = H′ × H′′ for some finite abelian group H′′. Thus we have a natural functor from the category Λ(H′)- mod of finitely generated Λ(H′)-modules to 𝔐_{H}(G) by extending the H′-action to a G-action letting G/H′ ≅ H′′ × Γ act trivially, which induces the first homomorphism in the following composition

K_{0}(Λ(H′)-mod) → K_{0}(𝔐_{H}(G)) → K_{0}(𝔐_{H}(G)) → K_{0}(𝔐_{H}(G)),

where the second map is induced by twisting with T (ψ)^{∗} while the last one is induced by tensoring with Λ(G) ⊗_{Λ(G)} −. As, Λ(H′) being a regular local ring, the Λ(H′)-rank induces an isomorphism K_{0}(Λ(H′)-mod) ≅ ℤ, one sees that the class of the trivial H′-module ℤ_{p}, which is sent to the class of T = Ind_{G}^{G}(T (ψ)^{∗} under the above map, is zero. □

Now we explain the construction of the non-abelian p-adic L-function. We start by fixing archimedean and p-adic periods that correspond canonical to the Grössencharacter ψ that we have associated to our CM modular form f. We pick a prime ideal 𝔣 of K that is relative prime to 𝔭 and with the property that the integer w_{𝔣} defined as the number of roots of unity in K congruent to 1 modulo 𝔣 is equal to 1. Then from [4] (Lemma in page 41) we know that there is a Grössencharacter ϕ of K of conductor 𝔣 and type (1, 0). Moreover from the same lemma in [4] we know that if ϕ is a Grössencharacter of K of type (1, 0) then there exists an elliptic curve defined over K(𝔣_{ϕ}) where 𝔣_{ϕ} the conductor of ϕ such that E has CM by O_{K} and its associated Grössencharacter is given by ψ_{E} = ϕ∘N_{K(𝔣_{ϕ})/K}. Now we are ready to define our periods for our Grössencharacter ψ. We distinguish two cases

(i) ψ is of type (1, 0), i.e. f is of weight 2. Then we write E for the elliptic curve defined over K(𝔣_{ψ}) and Λ for its corresponding lattice in ℂ. Then we define Ω_{∞} by

Ω_{∞}Λ = 𝔣

where we implicity assuming that we see 𝔣 as a lattice in ℂ with respect the embedding K ↪ ℂ imposed by the CM type of the character ψ (see also [4] page 66). The p-adic periods Ω_{p} are defined using the elliptic curve E as is done in [4] page 66)

(ii) If the character ψ is of type (k − 1, 0) for k > 2 then we pick some other Grössencharacter ϕ of type (1, 0) and of conductor 𝔣_{ϕ} prime to 𝔭 and write

ψ = θϕ^{k−1}

where θ is some finite order character of conductor relative prime to 𝔭. We define then the periods Ω_{∞} and Ω_{p} as in (i) using the character ϕ.

Let K(𝔭^{∞}) be the maximal ℤ_{p}-extension of K inside K(𝔣_{ψ}𝔭^{∞}) and set G′ := Gal(K(𝔣_{ψ}𝔭^{∞})/K(𝔭^{∞})). Let m := |G′|, then we define D to be the ring generated over ℤ̂_{p}^{nr} by the m^{th} roots of unity. Using exactly the same construction as in section 2 we conclude the following corollary

Corollary 3.6. There exists a unique Lψ̄ := μ ∈ Λ_{D}(G) such that for 0 ≤ −j and 0 < k − 1 + j

Lψ̄(χκ^{j}) = ∫_{G} χκ^{j}dμ = Γ(k − 1 + j)i^{k−1+j} \frac{L(ψ̄χ, k − 1 + j)}{(2π)^{j}Ω_{∞}^{k−1}}×

e_{𝔭}(χ̄)P_{𝔭}(χ̄, \frac{w}{p^{−j+1}} )P𝔭̄(χ, u^{−1}p^{−j}) (\frac{ψ(𝔭)}{p})^{𝔣_{𝔭}(χ̄)} p^{j𝔣_{𝔭}(χ̄)}

for all Artin-character χ of G and the cyclotomic character κ : F rob_{𝔮}^{−1} ↦ N_{K}(𝔮).

Proof. We write 𝔤 for the maximal ideal that is contained in 𝔣 and 𝔣̄. From 2.3 we know that there exists a measure μ(𝔤𝔭̄^{∞}) of G(K(𝔤p^{∞})/K) so that

Ω_{p}^{j−k} ∫_{G(K(𝔤p^{∞})/K)} ϵ^{−1}(σ) μ(𝔤𝔭̄^{∞}) = Γ(k)i^{k}Ω_{∞}^{j−k}(\frac{\sqrt{d_{K}}}{2π})^{j}G(ϵ)(1 − \frac{ϵ(𝔭)}{p})L_{𝔭̄𝔤}(ϵ^{−1}, 0)

for ϵ of type (k, j) with 0 ≤ −j and k > 0. As explained above the character ψ is of the form ϕ^{k−1}θ for ϕ of type (1, 0) and θ a finite character, both unramified at p. Moreover we have the relation ϕϕ̄ = N_{K}. In particular for a finite order character χ of Gal(K(p^{∞})/K) the character

ϵ := ψN_{K}^{j} χ̄ = ϕ^{k−1+j}ϕ̄^{j}θχ̄

is a valid choice for ϵ above provided that 0 ≤ −j and 0 < k − 1 + j. But then for the L function we have the equalities

L(ϵ^{−1}, 0) = L(ψ^{−1}N_{K}^{−j}χ, 0) = L(ψ̄χN_{K}^{−(k−1+j)}) = L(ψ̄χ, k − 1 + j)

Then the proof is the same as in the case of elliptic curves with CM that we did in corollary 2.6. □

We note that in the more general case that we consider now we do not have ψ̄ = ψ^{c} but only that ψ^{c} = ψ̄(ϵ_{f} ∘ N_{K}) which means that the motive M(f) is not self-dual. We now use the functional equation in order to get the critical values at the point s = 1. For the character ψχ̄ of type (k − 1, 0) we have that its dual representation is given by the character (ψχ̄)^{−1} = ψ^{−1}χ for which we have that L(ψ^{−1}χ, s) = L(ψ̄χ, s + (k − 1)). Then the functional equation reads (see [?] page 16 or [4] page 37)

Γ_{ψ}(s)L(ψχ̄, s) = e(ψχ̄ω_{s})Γψ̄((k − 1) + 1 − s)L(ψ̄χ, (k − 1) + 1 − s)

where for a character ϕ of type (k, j) we write Γ_{ϕ}(s) := \frac{Γ(s−min(k,j))}{(2π)^{s−min(k,j)}} and e(ψχ̄ω_{s}) = ∏_{𝔮} e_{𝔮}(ψχ̄ω_{s}, ψ_{ad}, dx_{ψ_{ad}}) where ω_{s} as in Tate’s [?]. In particular for s := 1 − j with j ≤ 0 we have

L(ψ̄χ, k − 1 + j) = \frac{Γ_{ψ}(1 − j)}{Γψ̄(k − 1 + j)}e(ψχ̄ω_{1−j})^{−1}L(ψχ̄, 1 − j)

Hence

L(ψ̄χ, k−1+j) e_{𝔭}(χ̄) (\frac{ψ(𝔭)}{p})^{𝔣_{𝔭}(χ̄)} p^{j𝔣_{𝔭}(χ̄)} = \frac{Γ_{ψ}(1 − j)}{Γψ̄(k − 1 + j)}e(ψχ̄ω_{1−j})^{−1}e_{𝔭}(χ̄) (\frac{ψ(𝔭)}{p})^{𝔣_{𝔭}(χ̄)} p^{j𝔣_{𝔭}(χ̄)}L(ψχ̄, 1)

But we have

e(ψχ̄ω_{1−j})^{−1}e_{𝔭}(χ̄) (\frac{ψ(𝔭)}{p})^{𝔣_{𝔭}(χ̄)} p^{j𝔣_{𝔭}(χ̄)} = ∏_{𝔮}e_{𝔮}(ψχ̄ω_{1−j}, ψ_{ad}, dx_{ψ_{ad}})^{−1}e_{𝔭}(ψχ̄ω_{1−j}) = ∏_{𝔮≠𝔭}e_{𝔮}(ψχ̄ω_{1−j}, ψ_{ad}, dx_{ψ_{ad}})^{−1}

where we have used the fact that dx_{ψ_{ad}} = dx_{1} at 𝔭 as this prime is unramified in K. Now we observe that

e𝔭̄(ψχ̄ω_{1−j}, ψ) = e𝔭̄(χ̄ω_{1−j}, ψ_{ad}, dx_{1})ψ(𝔭̄)^{𝔣𝔭̄(χ̄)} = e𝔭̄(χω_{j}, ψ_{ad}^{−1})^{−1}ψ(𝔭̄)^{𝔣𝔭̄(χ̄)} = e𝔭̄(χω_{j}, ψ_{ad}^{−1})^{−1}u^{𝔣𝔭̄(χ̄)} where we have used the duality e𝔭̄(χω_{j}, ψ_{ad}^{−1}, dx_{1})e𝔭̄(χ̄ω_{1−j}, ψ_{ad}, dx_{1}) = 1. Hence we obtain

∏_{𝔮≠𝔭}e_{𝔮}(ψχ̄ω_{1−j}, ψ_{ad}, dx_{ψ_{ad}})^{−1} = ∏_{𝔮≠𝔭,𝔭̄}e_{𝔮}(ψχ̄ω_{1−j}, ψ_{ad}, dx_{ψ_{ad}})^{−1}e𝔭̄(χ)u^{−𝔣𝔭̄(χ̄)}p^{−j𝔣_{𝔭}(χ̄)}

We can now state the following corollary

Corollary 3.7. There exists a unique L(tw) ; ψ̄ := μ ∈ Λ_{D}(G) such that for 0 ≤ −j and 0 < k − 1 + j we have

L(tw) ; ψ̄(χκ^{j}) = ∫_{G} χκ^{j}dμ = Γ(k − 1 + j)i^{j} \frac{L(ψχ̄, 1 − j)}{(2π)^{j}Ω_{∞}^{k−1}} e𝔭̄(χ)×

P_{𝔭}(χ̄, \frac{w}{p^{−j+1}} )P𝔭̄(χ, u^{−1}p^{−j}) u^{−𝔣𝔭̄(χ̄)}p^{−j𝔣_{𝔭}(χ̄)} (\frac{Γ_{ψ}(1 − j)}{Γψ̄(k − 1 + j)})

for all Artin-character χ of G.

Proof. We have already constructed a measure with the interpolation property

Lψ̄(χκ^{j}) = ∫_{G} χκ^{j}dμ = Γ(k−1+j)\frac{L(ψ̄χ, k − 1 + j)}{(2π)^{j}Ω_{∞}^{k−1}}e_{𝔭}(χ̄)P_{𝔭}(χ̄, \frac{w}{p^{−j+1}} )P𝔭̄(χ, u^{−1}p^{−j}) (\frac{ψ(𝔭)}{p})^{𝔣_{𝔭}(χ̄)} p^{j𝔣_{𝔭}(χ̄)}

and using the above computations we can rewrite it as

Lψ̄(χκ^{j}) = ∫_{G} χκ^{j}dμ = (\frac{Γ_{ψ}(1 − j)}{Γψ̄(k − 1 + j)} ∏_{𝔮≠𝔭,𝔭̄}e_{𝔮}(ψχ̄ω_{1−j}, ψ_{ad}, dx_{ψ_{ad}})^{−1}) ×

Γ(k − 1 + j)i^{j} \frac{L(ψχ̄, 1 − j)}{(2π)^{j}Ω_{∞}^{k−1}} e𝔭̄(χ)P_{𝔭}(χ̄, \frac{w}{p^{−j+1}} )P𝔭̄(χ, u^{−1}p^{−j}) u^{−𝔣𝔭̄(χ̄)}p^{−j𝔣_{𝔭}(χ̄)}

Now we claim that the mapping χκ^{j} ↦ ∏_{𝔮≠𝔭,𝔭̄} e_{𝔮}(ψχ̄ω_{1−j}, ψ_{ad}, dx_{ψ_{ad}}) is a unit in the Iwasawa algebra Λ_{D}(G) and hence we can twist our measure by this element to conclude the proposition.

We write K_{∞} ⊆ K(p^{∞}) for the ℤ_{p}^{2}-extension of K and define Γ := Gal(K_{∞}/K) ≅ ℤ_{p}^{2}. We decompose G = Δ × Γ for Δ finite of order relative prime to p. Then we have that Λ_{D}(G) = D[[G]] = D[Δ][[Γ]]. We write Δ̂ for the group of characters of Δ. Then by our assumptions on D we have that D[Δ] ≅ ⊕_{θ∈Δ̂}D given by α ↦ (…, θ(α), …) and hence Λ_{D}(G) ≅ ⊕_{θ}D[[Γ]]. Every Artin character χ of G can be decomposed as χ = θχ_{Γ} for θ a character of Δ and χ_{Γ} a character of Γ. We write 𝔡_{K} for the different of K over ℚ We note that θ can be ramified at p or at 𝔮 with 𝔮|𝔣_{ψ} for the conductor of ψ. We write 𝔣_{ψθ} for the non-p part of the conductor of ψθ̄. Then we have

∏_{𝔮≠𝔭,𝔭̄}e_{𝔮}(ψχ̄ω_{1−j}, ψ, dx_{ψ}) = i^{k−1} ∏_{𝔮|𝔣_{ψ}𝔡_{K}} e_{𝔮}(ψχ̄ω_{1−j}, ψ_{ad}, dx_{ψ_{ad}}) =

χ̄_{Γ}(𝔣_{ψθ}𝔡_{K})N_{K}(𝔣_{ψθ}𝔡_{K})^{−(1−j)}i^{k−1} ∏_{𝔮|𝔣_{ψ}𝔡_{K}} e_{𝔮}(ψθ̄, ψ_{ad}, dx_{ψ_{ad}})

since χ_{Γ} is only at p ramified and (p, 𝔣_{ψ}𝔡_{K}) = 1 under our assumptions. But ∏_{𝔮|𝔣_{ψ}𝔡_{K}} e_{𝔮}(ψθ̄, ψ_{ad}, dx_{ψ_{ad}}) is a 𝔭-adic unit (see [4] page 94) and hence we can define the measure

\frac{1}{N_{K}(𝔣_{ψθ}𝔡_{K})} ∏_{𝔮|𝔣_{ψ}𝔡_{K}} e_{𝔮}(ψθ̄, ψ_{ad}, dx_{ψ_{ad}})σ_{𝔣_{ψθ}𝔡_{K}}^{−1} ∈ Λ_{D}[[Γ]]^{×}

where σ_{𝔣_{ψθ}𝔡_{K}} ∈ Γ corresponds through the Artin reciprocity to 𝔣_{ψθ}𝔡_{K}, well defined as K_{∞} ramifies only at p. We then define the element E ∈ Λ_{D}[[G]]^{×} to be the element that corresponds to

(…, \frac{1}{N_{K}(𝔣_{ψθ}𝔡_{K})} ∏_{𝔮|𝔣_{ψ}𝔡_{K}} e_{𝔮}(ψθ̄, ψ_{ad}, dx_{ψ_{ad}})σ_{𝔣_{ψθ}𝔡_{K}}^{−1}, …) ∈ ⊕_{θ}D[[Γ]]

under the above mentioned isomorphism Λ_{D}(G) ≅ ⊕_{θ}D[[Γ]]. We then obtain for a character χ of G

E(χκ^{j}) = i^{1−k} ∏_{𝔮≠𝔭,𝔭̄}e_{𝔮}(ψχ̄ω_{1−j}, ψ, dx_{ψ})

which it allows us to conclude the proposition. □

Using now the natural map

(ı_{S})_{∗} : K_{1}(Λ(G)_{S^{∗}}) → K_{1}(Λ(G)_{S^{∗}})

we define

L := (ı_{S})_{∗}(Lψ̄) and L^{(tw)} := (ı_{S})_{∗}(L(tw) ; ψ̄).

As in the case of elliptic curves with CM we need to understand the following correction term which describes the change of complex periods

L_{Ω} := \frac{(2π)^{k−2}Ω_{+}}{Ω_{∞}^{k−1}} \frac{1 + c}{2} + \frac{(2π)^{k−2}Ω_{−}}{Ω_{∞}^{k−1}} \frac{1 − c}{2}.

Conjecture 3.8 (Period Relation). We conjecture

\frac{(2π)^{k−2}Ω_{+}}{Ω_{∞}^{k−1}} ∈ O_{E_{λ}}^{×} and \frac{(2π)^{k−2}Ω_{−}}{Ω_{∞}^{k−1}} ∈ O_{E_{λ}}^{×}

and hence L_{Ω} ∈ Λ_{O_{E_{λ}}} (G)^{×}.

The difficulty in proving the above conjecture for k ≥ 3 is due to the fact that it is not clear whether the motive M(f) that it is attached to f (a direct summand of the motive associated to a Kuga-Sato variety) coincides with the motive that we have associated to the Grössencharacter ψ as k − 1-fold tensor power of the elliptic curve associated to the character ϕ of type (1, 0). For a similar discussion see also [?, page 263].

We define

L′_{f_{ψ}} := LL_{Ω}^{−1} and L^{(tw)}′_{f_{ψ}} := L^{(tw)}L_{Ω}^{−1}

Assuming the above conjecture we have

Proposition 3.9. Let M(f)^{∨} be the dual motive to M(f). Then L′_{f_{ψ}} satisfies the following interpolation property for 0 ≤ −j and 0 < k − 1 + j.

L′_{f_{ψ}}(ρκ^{j}) = Γ(k−1+j)^{d(ρ)}i^{d(ρ)j} \frac{L_{\{p\}}(M(f)^{∨}, ρ, k − 1 + j)}{(2π)^{2(k−2)+2j}Ω_{+}^{d^{+}(ρ)}Ω_{−}^{d^{−}(ρ)}} e_{p}(ρ̌)\frac{P_{p}(ρ^{∨}, \frac{w}{p^{−j+1}} )}{P_{p}(ρ, w^{−1}p^{−j})} (\frac{w}{p} )^{𝔣_{p}(ρ̌)} p^{j𝔣_{p}(ρ̌)}

for all Artin representations ρ of G.

Proof. As in the proof in the elliptic curve case we consider two cases of artin representations, those that are one-dimensional and those that are induced from a character from K to ℚ. We explain here the case where ρ = Indχ to indicate the similarities and differences with above. We compute

L(ρκ^{j}) = Lψ̄(χκ^{j})Lψ̄(χ^{c}κ^{j}) =

Γ(k − 1 + j)^{2}i^{2j} \frac{L(ψ̄χ, k − 1 + j)L(ψ̄χ^{c}, k − 1 + j)}{(2π)^{2j}Ω_{∞}^{2(k−1)}} e_{𝔭}(χ̄)e𝔭̄(χ̄) (\frac{ψ(𝔭)p^{j}}{p} )^{𝔣_{𝔭}(χ̄)+𝔣𝔭̄(χ̄)} ×

P_{𝔭}(χ̄, \frac{w}{p^{−j+1}} )P𝔭̄(χ, u^{−1}p^{−j})P𝔭̄(χ̄, \frac{w}{p^{−j+1}} )P_{𝔭}(χ, u^{−1}p^{−j})

= Γ(k−1+j)^{2}i^{2j} \frac{L_{\{𝔭,𝔭̄\}}(ψ̄χ, k − 1 + j)L_{\{𝔭,𝔭̄\}}(ψ̄χ^{c}, k − 1 + j)}{(2π)^{2j}Ω_{∞}^{2(k−1)}} e_{𝔭}(χ̄)e𝔭̄(χ̄) (\frac{ψ(𝔭)p^{j}}{p} )^{𝔣_{𝔭}(χ̄)+𝔣𝔭̄(χ̄)} ×

\frac{P_{𝔭}(χ̄, \frac{w}{p^{−j+1}} )P𝔭̄(χ, u^{−1}p^{−j})P𝔭̄(χ̄, \frac{w}{p^{−j+1}} )P_{𝔭}(χ, u^{−1}p^{−j})}{P_{𝔭}(χ, w^{−1}p^{−j})P𝔭̄(χ, u^{−1}p^{−j})P𝔭̄(χ, w^{−1}p^{−j})P_{𝔭}(χ, u^{−1}p^{−j})}

= Γ(k−1+j)^{2}i^{2j} \frac{L_{\{𝔭,𝔭̄\}}(ψ̄χ, k − 1 + j)L_{\{𝔭,𝔭̄\}}(ψ̄χ^{c}, k − 1 + j)}{(2π)^{2j}Ω_{∞}^{2(k−1)}} e_{𝔭}(χ̄)e𝔭̄(χ̄) (\frac{ψ(𝔭)p^{j}}{p} )^{𝔣_{𝔭}(χ̄)+𝔣𝔭̄(χ̄)}

×\frac{P_{𝔭}(χ̄, \frac{w}{p^{−j+1}} )P𝔭̄(χ̄, \frac{w}{p^{−j+1}} )}{P_{𝔭}(χ, w^{−1}p^{−j})P𝔭̄(χ, w^{−1}p^{−j})}

= e_{p}(ρ̌) (\frac{ψ(𝔭)p^{j}}{p} )^{𝔣_{p}(ρ)} Γ(k − 1 + j)^{2}i^{2j} \frac{L_{\{p\}}(M(f)^{∨}, ρ, k − 1 + j)}{(2π)^{2j}Ω_{∞}^{2(k−1)}} \frac{P_{p}(ρ̌, \frac{w}{p^{−j+1}} )}{P_{p}(ρ, w^{−1}p^{−j})}

The last equation follows from the fact that since ψ induces the cusp form f_{ψ} = f = ∑_{n≥1} a_{n}q^{n}, the character ψ̄ induces a cuspidal newform fψ̄ = ∑_{n≥1} \overline{a_{n}}q^{n} where the complex conjugation on the coefficients a_{n} is with respect to our fixed embedding F ↪ ℚ̄. Actually one has that \overline{a_{n}} = ϵ_{f}^{−1}(n)a_{n}. But then fψ̄ corresponds to the dual motive M(f)^{∨}. Then as above we conclude that L′_{f_{ψ}} = LL_{Ω}^{−1} has the claimed interpolation property. □

Similarly we have

Proposition 3.10. The measure L^{(tw)}′_{f_{ψ}} satisfies the following interpolation property for 0 ≤ −j and 0 < k − 1 + j,

L^{(tw)}′_{f_{ψ}}(ρκ^{j}) = Γ(1 − j)^{d(ρ)}i^{d(ρ)j} \frac{L_{\{p\}}(M(f), ρ̌, 1 − j)}{(2π)^{−2j}Ω_{+}^{d^{+}(ρ)}Ω_{−}^{d^{−}(ρ)}} e_{p}(ρ)\frac{P_{p}(ρ, u^{−1}p^{−j})}{P_{p}(ρ̌, \frac{u}{p^{−j+1}} )}(up^{j})^{−𝔣_{p}(ρ̌)}

for all Artin representations ρ of G.

Proof. As above we have

L^{(tw)}(ρκ^{j}) = L(tw) ; ψ̄(χκ^{j})L(tw) ; ψ̄(χ^{c}κ^{j}) =

Γ(k − 1 + j)^{2}i^{2j} \frac{L(ψχ̄, 1 − j)L(ψχ̄^{c}, 1 − j)}{(2π)^{2j}Ω_{∞}^{2(k−1)}} e_{𝔭}(χ)e𝔭̄(χ) (up^{j})^{−𝔣_{𝔭}(χ̄)−𝔣𝔭̄(χ̄)}

×P_{𝔭}(χ̄, \frac{w}{p^{−j+1}} )P𝔭̄(χ, u^{−1}p^{−j})P𝔭̄(χ̄, \frac{w}{p^{−j+1}} )P_{𝔭}(χ, u^{−1}p^{−j}) (\frac{Γ_{ψ}(1 − j)}{Γ_{ψ}(k − 1 + j)})^{2}

= Γ(k − 1 + j)^{2}i^{2j} \frac{L_{\{𝔭,𝔭̄\}}(ψχ̄, 1 − j)L_{\{𝔭,𝔭̄\}}(ψχ̄^{c}, 1 − j)}{(2π)^{2j}Ω_{∞}^{2(k−1)}} e_{𝔭}(χ)e𝔭̄(χ) (up^{j})^{−𝔣_{𝔭}(χ̄)−𝔣𝔭̄(χ̄)}×

\frac{P_{𝔭}(χ̄, \frac{w}{p^{−j+1}} )P𝔭̄(χ, u^{−1}p^{−j})P𝔭̄(χ̄, \frac{w}{p^{−j+1}} )P_{𝔭}(χ, u^{−1}p^{−j})}{P_{𝔭}(χ̄, \frac{w}{p^{−j+1}} )P𝔭̄(χ̄, \frac{u}{p^{−j+1}} )P𝔭̄(χ̄, \frac{w}{p^{−j+1}} )P_{𝔭}(χ̄, \frac{u}{p^{−j+1}} )} (\frac{Γ_{ψ}(1 − j)}{Γψ̄(k − 1 + j)})^{2}

= Γ(k − 1 + j)^{2}i^{2j} \frac{L_{\{𝔭,𝔭̄\}}(ψχ̄, 1 − j)L_{\{𝔭,𝔭̄\}}(ψχ̄^{c}, 1 − j)}{(2π)^{2j}Ω_{∞}^{2(k−1)}} e_{𝔭}(χ)e𝔭̄(χ) (up^{j})^{−𝔣_{𝔭}(χ̄)−𝔣𝔭̄(χ̄)}

×\frac{P_{𝔭}(χ, u^{−1}p^{−j})P𝔭̄(χ, u^{−1}p^{−j})}{P_{𝔭}(χ̄, \frac{u}{p^{−j+1}} )P𝔭̄(χ̄, \frac{u}{p^{−j+1}} )} (\frac{Γ_{ψ}(1 − j)}{Γψ̄(k − 1 + j)})^{2}

= e_{p}(ρ)(up^{j})^{−𝔣_{p}(ρ̌)}Γ(k−1+j)^{2}i^{2j} \frac{L_{\{p\}}(M(f), ρ̌, 1 − j)}{(2π)^{2j}Ω_{∞}^{2(k−1)}} \frac{P_{p}(ρ, u^{−1}p^{−j})}{P_{p}(ρ̌, \frac{u}{p^{−j+1}} )} (\frac{Γ_{ψ}(1 − j)}{Γψ̄(k − 1 + j)})^{2}

We recall that Γ_{ϕ}(s) = \frac{Γ(s−min(k,j))}{(2π)^{s−min(k,j)}} for ϕ a Grössencharacter of type (k, j). In particular for the character ψ of type (k − 1, 0) (hence ψ̄ of type (0, k − 1)) we have that

\frac{Γ_{ψ}(1 − j)}{Γψ̄(k − 1 + j)} = \frac{Γ(1 − j)}{Γ(k − 1 + j)}(2π)^{k−2+2j}

This allows us to conclude the proof of the proposition. □

Altogether we obtain the following (under our assumption over the torsion part of G),

Theorem 3.11. Assuming Conjectures 3.2 and 3.8 there exists L′_{M(f)} ∈ K_{1}(Λ_{D}(G)_{S}) satisfying Conjecture 3.3 and 3.4.

4. Appendix: Galois descent for Iwasawa algebras

The aim of this appendix is to provide some results for Galois descent of (non-commutative) p-adic L-functions, by which we mean the following general question:

Assume that G is a compact p-adic Lie group which possesses an closed normal subgroup H such that G/H =: Γ ≅ ℤ_{p} and let O = O_{L} denote the ring of integers of a finite extension L over ℚ_{p}. Then we denote by D the discrete valuation ring of the completion L̃ of the maximal unramified extension L^{nr} of L. If L ∈ K_{1}(Λ_{D}(G)_{S^{∗}}) has the property

L(ρ) ∈ O(ρ) for all ρ ∈ IrrG,

does it hold that there exists an element L′ ∈ K_{1}(Λ_{O}(G)_{S^{∗}}) with

L′(ρ) = L(ρ) for all ρ ∈ IrrG?

Here IrrG denotes the set of isomorphism classes of irreducible Artin representations of G with values in \overline{ℚ_{p}} while for every ρ ∈ IrrG we write O(ρ) for the discrete valuation ring of L(ρ) = \overline{ℚ_{p}}^{G_{L,ρ}}, where G_{L,ρ} is the stabilizer subgroup of ρ in G_{L} = G(\overline{ℚ_{p}}/L) with respect to the natural action on the coefficients of ρ.

We start with a simple observation in the abelian case:

Lemma 4.1. Let G be a topological finitely generated virtual pro-p, abelian group, such that the exponent of its torsion part divides p−1. Assume that for a given f ∈ Λ_{D}(G) the values f(ρ) belongs to L(ρ) (and hence to O(ρ)), for all (irreducible) Artin characters ρ of G. Then f is already contained in Λ_{O}(G).

Proof. With out loss of generality me may assume that G ≅ ℤ_{p}^{d} for some natural number d, because otherwise me can decompose Λ_{D}(G) into a product of such rings by our assumption on the torsion part of G. Also by taking inverse limits afterwards we may and do assume that G is a finite p-group. Then it is well-known (e.g. a variant of prop. 4.5.14 in [15]) that we have a G_{L}-invariant isomorphism

ϕ : ℂ_{p}[G] ≅ Hom(R(G), ℂ_{p}), f = ∑ m_{g}g ↦ (ρ ↦ f(ρ) = ∑ m_{g}ρ(g)) ,

where on the left hand side the Galois action is just an the coefficients while on the right hand side h^{g}(ρ) = gh(g^{−1}ρ) for all g ∈ G_{ℚ_{p}}, and h any homomorphism from the group of virtual representations R(G) := R(G) of G defined over \overline{ℚ_{p}}. Recall that by the Ax-Sen-Tate theorem

ℂ_{p}^{H} = L̃ :

for H := G(\overline{ℚ_{p}})/L_{p}^{nr}). Similarly, if H_{ρ} denotes the stabiliser of ρ we get

L̃(ρ) := ℂ_{p}^{H_{ρ}} = \begin{matrix} \\ \overline{ℚ_{p}} & H_{ρ} \end{matrix} = { \atop L^{nr}(ρ)}.

Taking H- and G_{L}-invariants of ϕ thus induces the following commutative diagram

[figure: L̃[G]; ℂ_{p}[G]^{H}; Hom_{H}(R(G), ℂ_{p}); ∏_{ρ∈(IrrG)/H} L̃(ρ); L[G]; ℂ_{p}[G]^{G_{L}}; Hom_{G_{L}}(R(G), ℂ_{p}); ∏_{ρ∈(IrrG)/G_{L}} L(ρ)]

where (IrrG)/U denotes the Galois-orbits with respect to some closed subgroup U ⊆ G_{L}. Since L(ρ)/L is totally ramified, it is easy to see that

(IrrG)/H = (IrrG)/G_{L}.

Now, by assumption f ∈ D[G] ⊆ L̃[G] satisfies f(ρ) ∈ L(ρ) for all ρ ∈ IrrG, i.e. ϕ(f) ∈ ∏_{ρ∈(IrrG)/G_{L}} L(ρ). Hence f ∈ L[G] ∩ D[G] = O[G], because L ∩ D = {x ∈ L| |x|_{p} ≤ 1} = O coefficientwise. □

In the following we will use Fröhlich’s Hom-description as it has been adapted to Iwasawa theory by Ritter and Weiss [9, §3]. We have the following commutative diagram

\xymatrix{K_{1}(Λ_{D}(G)) \ar[r]^{Det} & Hom_{H}(R(G), O_{ℂ_{p}}^{×}) \\ K_{1}(Λ_{O}(G)) \ar[r]^{Det} \ar[u] & Hom_{G_{L}}(R(G), O_{ℂ_{p}}^{×}), \ar@{^{(}->}[u]}

where the homomorphism Det is defined as follows: for ρ ∈ IrrG we obtain a homomorphism of rings

ρ : Λ_{D}(G) → M_{n_{ρ}}(\overline{ℤ_{p}}),

taking K_{1}(−) of which gives a group homomorphism

ψ_{ρ} : K_{1}(Λ_{D}(G)) → K_{1}(M_{n_{ρ}}(\overline{ℤ_{p}})) ≅ K_{1}(\overline{ℤ_{p}}) ⊆ O_{ℂ_{p}}^{×}.

Now we set Det(L)(ρ) := ψ_{ρ}(L).

Setting Δ := G(L^{nr}/L) we obtain the following generalisation of a theorem of M. Taylor [16, §8, thm. 1.4]:

Theorem 4.2. For O unramified over ℤ_{p} we have

Det(K_{1}(Λ_{D}(G)))^{Δ} = Det(K_{1}(Λ_{O}(G))).

Proof. In Taylor’s theorem (loc. cit.) G is finite and D is the valuation ring of a finite unramified extension of L. His proof generalizes immediately to the case of our more general D, see [6] for details, thus we only have to show how the general case can be reduced to the case of finite groups. To this end write G = lim ; ⟵_{n}G_{n} as inverse limit of

finite groups. By Taylor’s result we have compatible continuous maps

[figure: O[G_{n}]^{×}; Det; Det(K_{1}(Λ_{D}(G_{n})))^{Δ}; Hom_{H}(R(G_{n}), O_{ℂ_{p}}^{×})^{Δ}]

where the topology on Hom_{H}(R(G_{n}), O_{ℂ_{p}}^{×}) ≅ ∏_{ρ∈IrrG_{n}/H}(O_{ℂ_{p}}^{×})^{H_{ρ}} is induced from the valuation topology on ℂ_{p}. Taking the inverse limit yields, by the compactness of Λ_{O}(G)^{×} = lim ; ⟵_{n}(O/π^{n}[G_{n}])^{×} and by letting R(G) := lim ; ⟶_{n}R(G_{n}) denote the free

abelian group on the isomorphism classes of irreducible Artin representations of G, a factorization of the homomorphism Det into

[figure: Λ_{O}(G)^{×}; Det; (lim ; ⟵_{n}Det(K_{1}(Λ_{D}(G_{n}))))^{Δ}; Hom_{G_{L}}(R(G), O_{ℂ_{p}}^{×}).]

The claim follows because denoting by res_{n} : Hom_{H}(R(G), O_{ℂ_{p}}^{×}) → Hom_{H}(R(G_{n}), O_{ℂ_{p}}^{×}) the restriction we obtain from the universal mapping property for

lim ; ⟵_{n}Hom_{H}(R(G_{n}), O_{ℂ_{p}}^{×}) ≅ Hom_{H}(R(G), O_{ℂ_{p}}^{×})

the inclusions

Det(K_{1}(Λ_{D}(G))) ⊆ lim ; ⟵_{n}im(res_{n} ∘ Det)
⊆ lim ; ⟵_{n}Det(K_{1}(Λ_{D}(G_{n}))),

whence the obvious inclusion

Det(K_{1}(Λ_{O}(G))) ⊆ Det(K_{1}(Λ_{D}(G)))^{Δ} ⊆ (lim ; ⟵_{n}Det(K_{1}(Λ_{D}(G_{n}))))^{Δ}

is surjective. □

There are (at least) two obvious questions: Firstly whether this descent result does also hold for the full K_{1}-groups ( this amounts to an analogous statement for the SK_{1}- terms) and secondly whether the analogue for the localisations holds, too. We shall at least show a weak version towards the second issue. To this end we give a variant of Ritter and Weiss’ Hom-description: Consider the following commutative diagram

\xymatrix{K_{1}(Λ_{D}(G)) \ar[d]_{π} \ar[r]^{det} & Hom_{H,R_{Γ}}(R(G), Λ_{O_{ℂp}}(Γ)^{×}) \ar@{^{(}->}[d] \ar@{^{(}->}[r]^{ev} & Hom_{H}(R(G), O_{ℂ_{p}}^{×}) \ar@{^{(}->}[d] \\ K_{1}(Λ_{D}(G)_{S^{∗}}) \ar[r]^{det} & Hom_{H,R_{Γ}}(R(G), Q_{O_{ℂp}}(Γ)^{×}) \ar[r] & Maps(IrrG, ℂ_{p} ∪ \{∞\}),}

where

• Q_{O_{ℂp}}(Γ) denotes the quotient field of Λ_{O_{ℂp}}(Γ),

• for A either Λ_{O_{ℂp}}(Γ)^{×} or Q_{O_{ℂp}}(Γ)^{×} we denote by Hom_{H,R_{Γ}}(R(G), A) the both H-invariant and R(Γ)-twist-invariant homomorphisms. The latter means that f(ρ ⊗ χ) = tw_{χ^{−1}}(f(ρ)) for all χ in R(Γ); here χ is considered as element in R(G) via the fixed surjection G ↠ Γ and tw_{χ} : A → A is induced by the action γ ↦ χ(γ^{−1})γ on Λ_{O_{ℂp}}(Γ), compare with [9, thm. 8],

• - analogously as for Det - the homomorphisms det are induced from the ring homomorphisms

Λ_{D}(G) → M_{n_{ρ}}(O_{ℂ_{p}})ˆ⊗_{D}Λ_{D}(Γ) ≅ M_{n_{ρ}}(Λ_{O_{ℂp}}(Γ)), g ↦ ρ(g) ⊗ ḡ,

and its localisation at S^{∗}

Λ_{D}(G)_{S^{∗}} → M_{n_{ρ}}(Λ_{O_{ℂp}}(Γ))_{S^{∗}} ≅ M_{n_{ρ}}(Q_{O_{ℂp}}(Γ)),

respectively.

• the homomorphism ev : Hom_{H,R_{Γ}}(R(G), Λ_{O_{ℂp}}(Γ)^{×}) → Hom_{H}(R(G), O_{ℂ_{p}}^{×}), which is induced by the augmentation map Λ_{O_{ℂp}}(Γ)^{×} → O_{ℂ_{p}}^{×}, i.e. ev(f)(ρ) = f(ρ)(χ_{triv}), where the latter means evaluation at the trivial character χ_{triv} of Γ, is injective by the Weierstrass preparation theorem and the twist invariance of f (in the kernel of ev):

f(ρ)(χ) = tw_{χ^{−1}}(f(ρ))(χ_{triv}) = f(ρ ⊗ χ)(χ_{triv}) = 1

for all χ, whence f(ρ) = 1 for all ρ,

• we do not know whether (need that) the map Hom_{H,R_{Γ}}(R(G), Q_{O_{ℂp}}(Γ)^{×}) → Maps(IrrG, ℂ_{p} ∪ {∞}) is injective.

Note that

• a similar diagram exists with O-coefficients (instead of D) and H replaced by G_{L} and that by construction the morphisms det commute with the canonical change of coefficients maps.

• the composition of the top-line of the above diagram equals Det.

• the image of L in Maps(IrrG, ℂ_{p} ∪ {∞}) is the map which attaches to ρ the value L(ρ) of L at ρ and the map into this target is multiplicative at least in the following sense: if f(ρ) ≠ 0, ∞ for all ρ, then (gf)(ρ) = g(ρ)f(ρ) for all ρ (with ∞ · a = a · ∞ = ∞ and 0 · a = a · 0 = 0 for a ≠ 0, ∞).

By ι we denote the canonical map K_{1}(Λ_{O}(G)_{S^{∗}}) → K_{1}(Λ_{D}(G)_{S^{∗}}).

Theorem 4.3. Assume that

(i) O is absolutely unramified, (ii) L(ρ)/L is totally ramified (or trivial) for all ρ ∈ IrrG, (iii) L ∈ K_{1}(Λ_{D}(G)_{S^{∗}}) is induced from an element in Λ_{D}(G) ∩ (Λ_{D}(G)_{S^{∗}})^{×} (iv) L satisfies

L(ρ) ∈ L(ρ) for all ρ ∈ IrrG,

(v) there is an F ∈ K_{1}(Λ_{O}(G)_{S^{∗}}) such that ∂(L · ι(F)^{−1}) = 0 (e.g. if L is the characteristic element of the base change from a module in 𝔐_{O,H}(G)).

Then there exists L′ ∈ K_{1}(Λ_{O}(G)_{S^{∗}}) with

L′(ρ) = L(ρ) for all ρ ∈ IrrG

and

∂(L′) = ∂(F),

in particular

ι∂(L′) = ∂(L).

Proof. Consider the following commutative diagram with exact rows

\xymatrix{K_{1}(Λ_{D}(G)) \ar[r]^{π_{D}} & K_{1}(Λ_{D}(G)_{S^{∗}}) \ar[r]^{∂_{D}} & K_{0}(𝔐_{D,H}(G)) \ar[r] & K_{0}(Λ_{D}(G)) \\ K_{1}(Λ_{O}(G)) \ar[r]^{π_{O}} \ar[u]^{ι} & K_{1}(Λ_{O}(G)_{S^{∗}}) \ar[r]^{∂_{O}} \ar[u]^{ι} & K_{0}(𝔐_{O,H}(G)) \ar[r] \ar[u]^{ι} & 0.}

By assumption there exists D ∈ K_{1}(Λ_{D}(G)) such that π_{D}(D) = L · ι(F)^{−1}. By Lemma 4.4 below Det(D) belongs to Det(K_{1}(Λ_{D}(G)))^{Δ} = Det(K_{1}(Λ_{O}(G))), i.e. there is a D′ ∈ K_{1}(Λ_{O}(G)) such that D(ρ) = D′(ρ) for all ρ in R(G). Setting L′ := π_{O}(F)D′ and recalling that D′(ρ) = D(ρ) ≠ 0, ∞ we calculate

L′(ρ) = F (ρ)D′(ρ)
= F (ρ)D(ρ)
= det(F)(ρ)(χ_{triv}) · det(D)(ρ)(χ_{triv})
= ( det(F)(ρ) det(D)(ρ))(χ_{triv})
= det (ι(F)π(D))(ρ)(χ_{triv})
= det(L)(ρ)(χ_{triv})
= L(ρ),

whence the theorem is proven. □

Lemma 4.4. With notation as in the previous proof we have (i) det(D)(ρ) ∈ O(ρ)[[Γ]] for all ρ ∈ R(G), (ii) D(ρ) = Det(D)(ρ) ∈ O(ρ) for all ρ ∈ R(G), (iii) Det(D) ∈ Det(K_{1}(Λ_{D}(G)))^{Δ}.

Proof. By construction and assumption (L being induced from ....) we have det(L)(ρ) ∈ D(ρ)[[Γ]] with D(ρ) := (O_{ℂ_{p}})^{H_{ρ}} and thus by Lemma 4.1 det(L)(ρ) belongs to O(ρ)[[Γ]]. On the other hand det(F)(ρ) belongs to Q_{O(ρ)}(Γ)^{×}, whence

det(D)(ρ) ∈ Q_{O(ρ)}(Γ)^{×} ∩ Λ_{D(ρ)}(Γ)^{×}.

We claim that this intersection equals Λ_{O(ρ)}(Γ)^{×}. One inclusion being obvious we assume that q belongs to the intersection. It follows immediately that q(χ) belongs to O(ρ, χ) := (O_{ℂ_{p}})^{G_{L,ρ}∩G_{L,χ}} for all χ ∈ R(Γ). Thus the claim follows Lemma 4.1 (with O(ρ) for the base ring O). This proves (i) and as D(ρ) = det(D)(ρ)(χ_{triv}) also (ii) follows. In order to show (iii) observe first that Δ acts trivially on IrrG/GL̃ = IrrG/G_{L}. Hence from the following commutative diagram the statement is clear:

[figure: Hom_{H}(R(G), O_{ℂ_{p}}^{×}); ∏_{ρ∈(IrrG)/G_{L}} D(ρ)^{×}; Hom_{H}(R(G), O_{ℂ_{p}}^{×})^{Δ}; Hom_{G_{L}}(R(G), O_{ℂ_{p}}^{×}); ∏_{ρ∈(IrrG)/G_{L}} O(ρ)^{×}.]

□

[1, 2, 11, 12, 18]

References

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Universität Heidelberg, Mathematisches Institut, Im Neuenheimer Feld 288, 69120 Heidelberg, Germany.

E-mail address: bouganis@mathi.uni-heidelberg.de

Universität Heidelberg, Mathematisches Institut, Im Neuenheimer Feld 288, 69120 Heidelberg, Germany.

E-mail address: venjakob@mathi.uni-heidelberg.de
URL: http://www.mathi.uni-heidelberg.de/~otmar/