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Non-vanishing modulo p of Hecke L–values

Haruzo Hida ^{∗}

Contents

1 Introduction 1

2 Density of CM points 5 2.1 CM abelian varieties . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Hilbert modular Shimura varieties . . . . . . . . . . . . . . . . . . 8 2.3 Density modulo p . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.4 Linear independence . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3 Measure associated to a Hecke eigenform 24 3.1 Hecke relation among CM points . . . . . . . . . . . . . . . . . . . 24 3.2 Geometric modular forms . . . . . . . . . . . . . . . . . . . . . . . 25 3.3 Hecke operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.4 Anti-cyclotomic measure . . . . . . . . . . . . . . . . . . . . . . . . 29

4 Hecke L–values 36 4.1 Anti-cyclotomic L–functions . . . . . . . . . . . . . . . . . . . . . . 36 4.2 A generalization of a result of Hurwitz . . . . . . . . . . . . . . . . 38 4.3 Proof of Theorem 1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 42

1. Introduction

It is a classical problem to study the divisibility of the Hecke L–values by a given prime p when one varies Hecke characters of ℓ–power conductor for another prime ℓ. Washington studied such a problem for Dirichlet L–values, and as its application, he proved that the exponent of a prime p in the class number is bounded independently of ℓ–abelian extensions of ℚ unramified outside ℓ if ℓ ≠ p ([W]). When ℓ = p, this problem can be formulated as the determination of the Iwasawa μ–invariant of a given ℤ_{p}^{d}–extension. In the cyclotomic case, a solution has been given as vanishing of the μ–invariant by Ferrero and Washington ([FW]). We studied in [H03b] the case ℓ = p for p–adic Hecke L–functions of p–ordinary CM fields, and the subject of the author’s lecture at the Dwork memorial conference is on the μ–invariant of the p–adic Hecke L–functions.

∗The author is partially supported by the NSF grant: DMS 9988043 and DMS 0244401.

In this paper, we assume ℓ ≠ p and study critical values of the L–function of a Hecke character twisted by anticyclotomic finite order characters of 𝔩–power conductor. We limit our study to Hecke L–values of a p–ordinary CM field M and fix a prime factor 𝔩 of ℓ of the maximal totally real subfield F of M. Consider the torus T_{M} = Res_{M/ℚ}𝔾_{m}. Then T_{M}(𝔸) is the idele group M_{𝔸}^{×}. A continuous idele character λ : T_{M}(𝔸)/T_{M}(ℚ) → ℂ^{×} is called an arithmetic Hecke character if its restriction to T_{M}(ℝ) is induced by algebraic character λ_{∞} ∈ X^{∗}(T_{M}) = Hom_{alg-gp}(T_{M}, 𝔾_{m}). The ℤ–module ℤ[I] of formal linear combinations of elements in I = Hom_{field}(M, \overline{ℚ}) can be identified with X^{∗}(T_{M}) regarding κ = ∑_{σ} κ_{σ}σ ∈ ℤ[I] as a character x ↦ x^{κ} = ∏_{σ}(σ(x))^{κ_{σ}}. Thus λ_{∞} = λ|_{T_{M} (ℝ)} is induced by κ ∈ ℤ[I], and the element κ ∈ ℤ[I] is called the infinity type of λ. Let R be the integer ring of M. Since λ is continuous, there exists an R–ideal 𝔣 ≠ 0 such that λ(x) = 1 for x ∈ R̂^{×} if x − 1 ∈ 𝔣R̂ for R̂ = R ⊗_{ℤ} ℤ̂ (ℤ̂ = \underleftarrow{lim}_{N}ℤ/Nℤ). Among such 𝔣, there is a unique maximal one, which is called the conductor of λ. We write ℭ = ℭ(λ) for the conductor of λ. For each finite idele a with trivial ℭ-component, the value λ(a) only depends on the ideal 𝔄 = aR̂ ∩ M prime to ℭ. Thus by defining λ(𝔄) := λ(a), we get Hecke’s grössen character λ of conductor ℭ. The value of ideal character λ at a principal ideal (α) is given by λ(α^{(∞)}) if α ∈ T_{M}(ℚ) = M^{×} satisfies α ≡ 1 mod ℭ, we find λ((α))α^{κ} = λ(α) = 1 and hence λ((α)) = α^{−κ} (and because of this fact, some authors call “−κ” the infinity type of λ). The character we deal with is a Hecke character of arithmetic type associated to a p–ordinary CM-type (M, Σ), which is the type of a CM abelian variety having ordinary good reduction at p, in the following sense: the infinity type of λ is given by kΣ + κ(1 − c) with κ_{σ} ≥ 0 for all σ ∈ Σ and 0 < k ∈ ℤ, where kΣ = k ∑_{σ∈Σ} σ. Take algebraic closures \overline{ℚ} and \overline{ℚ}_{p} of ℚ and ℚ_{p}, and choose two embeddings i_{∞} : \overline{ℚ} ↪ ℂ and i_{p} : \overline{ℚ} ↪ \overline{ℚ}_{p}.

We fix a Hecke character λ of conductor prime to p and look at the critical values L(0, χλ) for χ running over anticyclotomic characters of 𝔩–power conductor. The anticyclotomy means that χ(𝔄^{c}) = χ^{−1}(𝔄) for the non-trivial automorphism c of M/F. We have a pair of the periods (Ω, Ω_{p}) ∈ (ℂ^{×})^{Σ} × (W^{×})^{Σ} of a Néron differential on the abelian scheme X_{/W} of CM type (M, Σ), where W is the Witt ring (regarded as a subring of the p–adic completion ℚ̂_{p} of \overline{ℚ}_{p}) with coefficients in an algebraic closure 𝔽 of 𝔽_{p}. The abelian variety X has a polarization ideal 𝔠 ⊂ F such that X ⊗ 𝔠 is isomorphic to the dual abelian variety of X. The strict ideal class of 𝔠 is uniquely determined by X. For a multi-index κ = ∑_{σ} κ_{σ}σ for σ ∈ Σ, we write π^{−κ} = π^{− ∑_{σ} κ_{σ}} and Ω^{κ} = ∏_{σ} Ω_{σ}^{κ_{σ}}. We often identify Σ with ∑_{σ∈Σ} σ; so, π^{kΣ} = π^{k[F:ℚ]} for an integer k. Let W be the pull-back of W in \overline{ℚ}; so, it is a discrete valuation ring with maximal ideal 𝔪 = 𝔪_{W} unramified over the localization ℤ_{(p)} = ℚ ∩ ℤ_{p}. It is known by Shimura and Katz that

\frac{Γ_{Σ}(kΣ+κ)L^{(𝔩)}(0,χλ)}{π^{−κ}Ω^{kΣ+2κ}} ∈ W if 0 is critical for χλ, where Γ_{Σ}(kΣ + κ) = ∏_{σ∈Σ} Γ(k + κ_{σ}) for the gamma function Γ(s). This normalization (in particular, the gamma-factor: Γ_{Σ}(kΣ + κ)) gives optimal p-integrality since it appears as a special value of the Katz p-adic L-function (see [K1] 5.3.5 and [HT] Theorem II). We write L^{(𝔩)}(s, χλ) for the possibly imprimitive L–function obtained from L(s, χλ) removing its 𝔩– Euler factor. We shall prove

Theorem 1.1. Suppose that p > 2 is unramified in M/ℚ and that (M, Σ) is ordinary for p. Fix a character λ of conductor 1 with infinity type kΣ + κ(1 − c), where k is a positive integer and κ = ∑_{σ∈Σ} κ_{σ}σ with integers κ_{σ} ≥ 0. Then

\frac{Γ_{Σ}(kΣ+κ)L^{(𝔩)}(0,λχ)}{π^{−κ}Ω^{kΣ+2κ}} ≢ 0 mod 𝔪 for almost all anticyclotomic characters χ of 𝔩– power conductor, unless the following three conditions are satisfied simultaneously:

(M1) M/F is unramified everywhere;

(M2) The Artin symbol (\frac{M/F}{𝔠} ) has the value −1;

(M3) For all ideal 𝔞 of F prime to p, λN(𝔞) ≡ (\frac{M/F}{𝔞} ) mod 𝔪.

Here the word “almost all” means “Zariski density of anticyclotomic characters modulo 𝔩–power” if dim_{ℚ_{ℓ}} F_{𝔩} > 1 and “except for finitely many anticyclotomic characters modulo 𝔩–power” if dim_{ℚ_{ℓ}} F_{𝔩} = 1. If the three conditions (M1-3) are satisfied, the L–value as above vanishes modulo 𝔪 for all anti-cyclotomic characters χ modulo 𝔩-power.

Since Hom(ℤ_{ℓ}^{d}, μ_{ℓ^{∞}}) can be considered as a subset of 𝔾_{m/\overline{ℚ}_{ℓ}}^{d} (by sending characters

χ to (χ(e_{i}))_{i} ∈ 𝔾_{m}^{d} for the standard basis e_{i} of ℤ_{ℓ}^{d}), the Zariski density of characters has well-defined meaning that a subset of characters is dense if its closure in 𝔾_{m}^{d} is 𝔾_{m}^{d} itself. In particular, except for the case where dim_{ℚ_{ℓ}} F_{𝔩} = 1, our notion of “almost all” is weaker than “outside a proper Zariski closed set”, and this theorem when dim F_{𝔩} > 1 has still wide room for improvement.

Our idea could be applied to Hecke characters λ with non-trivial conductor. However the non-vanishing of the L–values is even more subtle for λ with nontrivial conductor because exceptional cases governed by the conditions similar to (M1-3) occur more frequently. To present our idea in a simpler form, we have chosen to assume λ to have conductor 1 leaving its generalization to a future work [H03c]. The CM-fields satisfying (M1-3) are rare: (M1) implies that the strict class number of F and [F : ℚ] are both even, but they exist (an example is given in [H03b] Section 5.4). This example shows that the polarization ideal 𝔠 depends on the choice of Σ, and 𝔠 satisfies (M2) for one such choice and not for the other choices.

Until the above theorem, all known results of this kind dealt with either cyclotomic or elliptic cases and ℤ_{ℓ}–extensions. The imaginary quadratic case has been studied by T. Finis [F] Corollary 4.7, reducing the assertion to the non-vanishing of the theta lift from GL(2) to U(2, 1) and a rather deep Diophantine statement ([B] Théorème 1 and [F] Lemma 4.5) about a product of elliptic curves with complex multiplication by M. Actually Finis covers the case where Σ is non-ordinary at p although he needs to suppose the ordinarity for ℓ. A similar assertion for Coates-Wiles ℤ_{ℓ}–extensions had been studied much earlier by Gillard [G] who reduced the problem to a similar (but slightly stronger) Diophantine statement. Recently Vatsal has given another result of similar kind for elliptic modular L–functions over imaginary quadratic fields ([V] and [V1]).

When M is an imaginary quadratic field, we can interpret the above theorem in terms of the boundedness of the order of certain Selmer groups over the anticyclotomic ℤ_{ℓ}–tower. We hope to come back to this question for general CM fields in future.

Our proof is based on two ideas. One is the philosophy that the knowledge of reciprocity laws (studied by Shimura, e.g., [ACM] and [AAF]) at each place of an arithmetic automorphic function field is almost equivalent to the knowledge of the value of automorphic forms at the point, and the other is Sinnott’s idea ([Si]) of relating the non-vanishing modulo p of such L–values to Zariski-density (modulo p) of special points of the algebraic variety underlying the L–values. In the cyclotomic case, the variety is just 𝔾_{m} and in the elliptic case treated by Gillard and Finis, it is a product of elliptic curves with complex multiplication by M, but in our case, they are more sophisticated Hilbert modular Shimura varieties. Nondensity of characters with non-vanishing L–values modulo p leads to the vanishing of an Eisenstein series at densely many CM points with complex multiplication by M. The Zariski density of these CM points then yields the vanishing of the Eisenstein series itself. This vanishing is against the non-vanishing modulo p of the q–expansion of the Eisenstein series, and hence we obtain the desired assertion. The density of CM points of Siegel and Hilbert modular varieties modulo p has been studied by C.-L. Chai (for example [C1]). We quote from [H03b] in Section 2 a version of his result (in the Hilbert modular case) in a fashion suitable to our use. Over the complex field, not just Zariski density but stronger equidistribution of such CM points is known for many Shimura varieties by the works of a handful of mathematicians (e.g. [COU] and references therein).

A key to relating vanishing of L–values mod p to the vanishing of the Eisenstein series is the existence of a measure interpolating the special values of a Hecke eigenform at Hecke and Heegner points. The construction of the measure is a reminiscence of the modular symbol method of Mazur, and we propose to apply the method to global sections of automorphic vector bundles.

In earlier versions of this paper, the proof of Zariski density of CM points (modulo p) relied on a lifting lemma of the Zariski closure to a characteristic 0 formal scheme. Although lifting works well over the ordinary locus, C.-L. Chai pointed out a flaw in the proof that it may not fit well with a characteristic 0 formal scheme at super-singular points and suggested the author the use of his techniques in his four papers from [C1] to [C4] to recover the result. In particular, the proof of a crucial lemma (Lemma 2.4) is due to him, and also the use of Zarhin’s theorem in the proof of Proposition 2.8 is suggested by him. The author would like to thank Ching-Li Chai for his remarks and assistance. The author wishes to thank the organizers of the Dwork memorial conferences for their invitation. Finally but not least, the author would like to thank the referee of this paper for his careful reading and comments.

2. Density of CM points

Our principal tool for proving Theorem 1.1 is Proposition 2.8 which asserts the Zariski density of a small infinite set of CM points in the product of copies of the Hilbert modular variety. We study this density problem in this section.

We shall keep the notation in the introduction. In particular, M/F is a totally imaginary quadratic extension of a totally real field F with a CM type Σ. Thus Σ ⊔ Σc gives the set of all embeddings of M into \overline{ℚ}. We write R (resp. O) for the integer ring of M (resp. F). We write | |_{p} for the p–adic absolute value of \overline{ℚ}_{p} and define ℚ̂_{p} by the p–adic completion of \overline{ℚ}_{p} under | |_{p}. For an idele x and an ideal 𝔣, we write x_{𝔣} the components at places dividing 𝔣 and define x^{(𝔣)} = xx_{𝔣}^{−1}.

An abelian variety of CM type (M, Σ) over ℂ is a quotient of ℂ^{Σ} by an O-lattice of M. As we will see below, an O–lattice A is a proper ideal of an order (possibly non-maximal) of M. Although an ideal of an order is traditionally (from the time of Kronecker) denoted by a lower case Gothic letter, we will use script letters A and B for them in order to use lower case Gothic letters exclusively for ideals of rings other than orders of M (particularly, ideals of F are denoted by lower case Gothic letters). We often use capital Gothic letters for ideals of R (with some exceptions).

2.1. CM abelian varieties.

For each σ ∈ (Σ ∪ Σc), i_{p}σ induces a p–adic place 𝔭_{σ} whose p-adic absolute value is |x|_{𝔭_{σ}} = |i_{p}(σ(x))|_{p}. Set Σ_{p} = {𝔭_{σ}|σ ∈ Σ} and Σ_{p}c = {𝔭_{σc}|σ ∈ Σ}.

(ord) We assume that Σ is p–ordinary: Σ_{p} ∩ Σ_{p}c = ∅.

Such a CM type Σ is called a p–ordinary CM type. The existence of a p–ordinary CM type is equivalent to the fact that all prime factors of p in F split into a product of two distinct primes in M. We suppose

(unr) p is unramified in F/ℚ.

This condition combined with (ord) implies that p is unramified in M/ℚ.

We first recall the construction of CM abelian varieties from [ACM]. Recall the complex field embedding i_{∞} : \overline{ℚ} ↪ ℂ. For each O–lattice A ⊂ M with A_{p} = A ⊗_{ℤ} ℤ_{p} ≅ R ⊗_{ℤ} ℤ_{p} = R_{p}, we consider the complex torus X(A)(ℂ) = ℂ^{Σ}/Σ(A), where Σ(A) = {(i_{∞}(σ(a)))_{σ∈Σ}|a ∈ A}. By a theorem of Shimura-Taniyama-Weil (cf. [ACM] 12.4), this complex torus is algebraizable to an abelian variety X(A) of CM type (M, Σ) defined over a number field.

The main theorem of complex multiplication ([ACM] 18.6) combined with the criterion of good reduction over W in [ST] tells us that X(A) is actually defined over the field of fractions K of W and extends to an abelian scheme (still written as X(A)_{/W}) over W. The special fiber of the abelian scheme X(A)_{/W} is ordinary under (ord). All endomorphisms of X(A) are defined over W. In particular, we have an embedding M ↪ End(X(A)_{/W}) ⊗_{ℤ} ℚ taking α ∈ M to the complex multiplication by Σ(α) on X(A)(ℂ) = ℂ^{Σ}/Σ(A).

Let R(A) = {α ∈ R|αA ⊂ A}. Then R(A) is an order of M over O. An O–order R of M is determined by its conductor ideal 𝔣 ⊂ O (which satisfies 𝔣R = {α ∈ R|αR ⊂ R}). We have R = O + 𝔣R. The conductor 𝔣(A) of R(A) will be called the conductor of A. We thus have R(A) = O + 𝔣(A)R.

For any order R of M over O and any fractional R-ideal A, the following three conditions are equivalent (cf. [IAT] Proposition 4.11 and (5.4.2) and [CRT] Theorem 11.3):

(I1) A is R–projective;

(I2) A is locally principal (that is, localization at each prime ideal is principal);

(I3) A is a proper R–ideal (that is, R = R(A)).

Define the class group Cl(R) by the group of R–projective fractional ideals modulo globally principal ideals; so, Cl(R) = Pic(R). The group Cl(R) is finite and called the ring class group of conductor 𝔣 if R has conductor 𝔣. Since A_{p} ≅ R_{p}, the order R(A) has conductor prime to p.

For an abelian scheme X over a subring of an algebraically closed field k, we define the Tate module T (X) = \underleftarrow{lim}_{N}X[N](k) for the kernel X[N] of the multiplication by a positive integer N. In this subsection, we take k = \overline{ℚ}. We choose a base w = (w_{1}, w_{2}) of R̂ = R ⊗_{ℤ} ℤ̂ over Ô so that the p–component w_{p} is the standard basis ((1, 0), (0, 1)) of R_{p} = R_{Σ^{c}} ⊕ R_{Σ}, where R_{Σ} = ∏_{𝔭∈Σ_{p}} R_{𝔭} ≅ O_{p} ≅ ∏_{𝔭∈Σ_{p}^{c}} R_{𝔭} = R_{Σ^{c}}. The base w gives rise to a level N–structure η_{N} :

(O/NO)^{2} ≅ X(R)[N] given by η_{N}(a, b) = \frac{aw_{1}+bw_{2}}{N} ∈ X(R)[N]. Taking their limit and tensoring with 𝔸^{(∞)}, we get

η = η(R) = \underleftarrow{lim}_{N}η_{N} : F^{2} ⊗_{ℚ} 𝔸^{(∞)} ≅ V (X(R)) = T (X(R)) ⊗ℤ̂ 𝔸^{(∞)}.

We remove the p–part of η and define

η^{(p)} = η^{(p)}(R) : F^{2} ⊗_{ℚ} 𝔸^{(p∞)} ≅ V^{(p)}(X(R)) = T (X(R)) ⊗ℤ̂ 𝔸^{(p∞)}.  (2.1)

The level structure η^{(p)}(R) is defined over W because X[N] for p ∤ N is étale hence constant over W. Since we have

T (X(A)) ⊗ℤ̂ 𝔸^{(∞)} = M ⊗_{ℚ} 𝔸^{(∞)} = T (X(R)) ⊗ℤ̂ 𝔸^{(∞)},

the level structure η(R) induces the level structure η_{1}(A) for all X(A). As we will see later, this level structure η_{1}(A) is not good for our purpose, since they do not distinguish two abelian varieties X(A) and X(R) as distinct points of the Shimura variety because of the commutativity of

V^{(p)}(X(A)) ⟶ V^{(p)}(X(R))
η_{1}(A)↑  ↑^{η(R)}
M_{𝔸}^{(∞)}  M_{𝔸}^{(∞)},

where the top arrow is the isogeny induced by the bottom equality.

We choose a “good” level structure η(A) of X(A) so that η(A)(Ô^{2}) = Â in the following way. First we choose a representative set {𝔄_{j}} of ideal classes of M (prime to p𝔣). Then we can write 𝔄̂_{j} = a_{j}R̂ for an idele a_{j} with a_{j} = a_{j}^{(𝔣p∞)} and choose α ∈ M so that AR = α𝔄_{j}. If 𝔣(A) = O (so, A is an R–ideal), we define the level structure η(A) by (F_{𝔸}^{(∞)})^{2} ∋ (a, b) ↦ aαa_{j}w_{1} +bαa_{j}w_{2} ∈ M_{𝔸}^{(∞)} = V (X(A)). When 𝔣(A) ≠ O, we first suppose that 𝔣 = (φφ^{c}) for φ ∈ M. Take α ∈ M such that AR = α𝔄_{j}, and choose a base w(A) of Â so that w(A)^{(𝔣)} = (φαa_{j}w)^{(𝔣)} and w(A)_{𝔣} = αw_{𝔣} · g for g ∈ GL_{2}(F_{𝔣}) with det(g_{𝔣}) = φφ^{c}. Then we define η(A)(a, b) = a · w_{1}(A) + b · w_{2}(A) ∈ M_{𝔸}^{(∞)}. There is an ambiguity of the choice of α and φ up to units in R, but this does not cause any trouble later.

Suppose that 𝔣(A) is not generated by a norm from M. Let G = Res_{F/ℚ}GL(2) (so, G(A) = GL_{2}(A ⊗_{ℚ} F)). We choose g ∈ G(𝔸^{(∞)}) with g^{(𝔣)} = 1 so that w(A) = αa_{j}w · g gives a base over Ô of Â, and define η(A) by using w(A). In the above two cases, we choose g independent of the ideals in the proper ideal class of A; in other words, we choose w(βA) = βαa_{j}w · g. We then define g(A) ∈ G(𝔸^{(∞)}) by η(A) = η(𝔄_{j}) · g(A). We will later specify the choice of g precisely.

We introduce a representation ρ_{A} : M_{𝔸}^{×} → G(𝔸^{(∞)}) by αη(A) = η(A) · ρ_{A}(α). By our choice, we have ρ_{A} = ρ_{R} on M_{𝔸}^{(𝔣(A))×}, and

det(g(A)) ∈ F_{+}^{×} if 𝔣(A) is generated by a norm from M.  (2.2)

We choose a totally imaginary δ ∈ M with Im(σ(δ)) > 0 for all σ ∈ Σ. Then the alternating form (a, b) ↦ (c(a)b − ac(b))/2δ gives an identity R ∧_{O} R = 𝔠^{∗} for a fractional ideal 𝔠 of F. Here 𝔠^{∗} = {x ∈ F |Tr_{F/ℚ}(x𝔠) ⊂ ℤ} = 𝔡^{−1}𝔠^{−1} for the different 𝔡 of F/ℚ. Identifying M ⊗_{ℚ} ℝ with ℂ^{Σ} by m ⊗ r ↦ (σ(m)r)_{σ∈Σ}, we find that (a, ia) = \frac{\sqrt{−1}}{δ}a\overline{a} ≫ 0 for a ∈ M^{×}. Thus Tr_{F/ℚ} ∘ (·, ·) gives a Riemann form for the lattice Σ(R), and therefore, a projective embedding of ℂ^{Σ}/Σ(R) onto a projective abelian variety X(R)_{/ℂ} (cf. [ABV] Chapter I). As we already remarked, X(R) extends to an abelian scheme over W (unique up to isomorphisms). In this way, we get a 𝔠–polarization Λ(R) : X(R)(ℂ) ⊗ 𝔠 ≅ ^{t}X(R)(ℂ) for the dual abelian scheme ^{t}X(R) = Pic_{X(R)/W}^{0} ([ABV] Section 13). The same δ induces

R ∧ R = 𝔣(O ∧ R) + 𝔣^{2}(R ∧ R) = (𝔣^{−1}𝔠)^{∗} and A ∧ A = (N_{M/F} (A)^{−1}𝔣(A)^{−1}𝔠)^{∗}, where the exterior product is taken over O. Hereafter we fix δ so that 𝔠 is prime to p𝔣(A)𝔡, and write 𝔠(A) for N_{M/F} (A)^{−1}𝔣(A)^{−1}𝔠 (so, 𝔠 = 𝔠(R)). We can always choose such a δ, since in this paper we only treat A with 𝔩–power conductor.

Since a generically defined isogeny between abelian schemes over W extends to the entire abelian scheme (e.g. [GME] Lemma 4.1.16), we have a well defined 𝔠(A)– polarization Λ(A) : X(A) ⊗ 𝔠(A) ≅ ^{t}X(A). Replacing X(A) by an isomorphic X(αA) for α ∈ M, we may assume that A_{p} = R_{p}. Then

X(A)[𝔭_{F}] = X(A)[𝔭] ⊕ X(A)[𝔭^{c}]

for 𝔭_{F} = 𝔭 ∩ F is isomorphic by Λ(A) to its Cartier dual. Since the Rosatiinvolution a ↦ a^{∗} = Λ(A) ∘^{t}a ∘ Λ(A)^{−1} is the complex conjugation c, X(A)[𝔭]_{/W} is multiplicative (étale locally) if and only if X(A)[𝔭^{c}] is étale over W.

Since X(A) has ordinary reduction over W, the connected component X(A)[p]^{∘} is isomorphic to μ_{p}^{[F:ℚ]} and shares with X(A)_{/𝔽} the tangent space Lie(X(A)) at the origin. We confirm, from Lie(X(A))⊗_{W} ℂ = ℂ^{Σ} as R–modules, that R acts on Lie(X(A)) by a representation isomorphic to Σ = ⨁_{σ∈Σ} σ. By (unr), X(A)[𝔭]_{/𝔽} is multiplicative if and only if 𝔭 ∈ Σ_{p}. Let A_{Σ} = ∏_{𝔭∈Σ_{p}} A_{𝔭}, A_{Σ^{c}} = ∏_{𝔭∈Σ_{p}^{c}} A_{𝔭}, M_{Σ^{c}} = ∏_{𝔭∈Σ_{p}^{c}} M_{𝔭} and p = ∏_{𝔭∈Σ_{p}} 𝔭. Then we may define an étale level p– structure η_{p}^{et} over W by η_{p}^{et} : M_{Σ^{c}}/A_{Σ^{c}} ≅ X(A)[(p^{c})^{∞}]. By the duality under Λ(A), this is equivalent to having η_{p}^{ord}(A) : μ_{p^{∞}} ⊗_{ℤ_{p}} A_{Σ}𝔡^{−1} ≅ X(A)[p^{∞}], which in turn induces an isomorphism η_{p}^{ord}(A) : 𝔾̂_{m} ⊗_{ℤ_{p}} A_{Σ}𝔡^{−1} ≅ X̂(A) of formal groups, where X̂(A) is the formal completion of X(A)_{/W} at the origin of its mod p fiber. By A_{p} ≅ R_{p}, we have an identity (induced by our choice of η(A)): X(R)[𝔭^{∞}](\overline{ℚ}) = M_{𝔭}/R_{𝔭} ≅ X(A)[𝔭^{∞}](\overline{ℚ}).

We choose and fix a generator ω = ω(R) of H^{0}(X(R), Ω_{X(R)/W}) over W ⊗_{ℤ} O. If A_{Σ} = R_{Σ}, X(R ∩ A) is an étale covering of the both X(A) and X(R); so, ω(R) induces a differential ω(A) so that the pull back of ω(A) and ω(R) to X(R ∩ A) coincide. We then have

H^{0}(X(A), Ω_{X(R)/W}) = (W ⊗_{ℤ} O)ω(A).

In this way, we get many quadruples: (X(A), Λ(A), η^{(p)}(A) × η_{p}^{ord}(A), ω(A))_{/W} as long as 𝔣(A) is prime to p. We call this quadruple x(A).

2.2. Hilbert modular Shimura varieties.

We shall create a lot of CM points on the Hilbert modular Shimura variety. Define an affine group scheme G_{/ℤ} by G = Res_{O/ℤ}GL(2). We identify the symmetric space ℨ for G(ℝ) with the collection of embeddings h : 𝕊 = Res_{ℂ/ℝ}𝔾_{m} ↪ G_{/ℝ} which are conjugates of h_{i}(a + b\sqrt{−1}) = ( \begin{matrix}a & −b \\ b & a \end{matrix} ) (𝕊(ℝ) = ℂ^{×}). Then the pair (G_{/ℚ}, ℨ) satisfies Deligne’s axioms for having its Shimura variety ([D2] and [D3] 2.1.1). We write ℨ^{+} for the identity connected component of ℨ containing h_{i}.

An abelian scheme X over a scheme S is called an abelian variety with real multiplication (AVRM) if O acts on X as S–endomorphisms with an O–linear isomorphism Lie(X) ≅ O ⊗_{ℤ} O_{S} Zariski locally (an AVRM is called a Hilbert-Blumenthal abelian variety in [K1]).

A polarization Λ : X → ^{t}X = Pic_{X/S}^{0} is an O–linear isogeny induced étale locally by an ample line bundle over X (see [DAV] I.1.6). Then Λ induces X ⊗ 𝔠 ≅ tX for a fractional ideal 𝔠 of F. We always assume that the polarization ideal 𝔠 is prime to p𝔡. Let O_{(p)} = O ⊗_{ℤ} ℤ_{(p)} ⊂ F (which is a semi-local Dedekind domain). A polarization class \overline{Λ} is the set \{Λ ∘ ξ\}_{ξ∈O_{(p)+}^{×}} for the multiplicative group O_{(p)+}^{×} of totally positive units in O_{(p)}.

Suppose now that the base S is a ℤ_{(p)}–scheme. Thus Λ ∈ \overline{Λ} is an étale isogeny; so, for each geometric point s ∈ S, π_{1}(S, s) acts on \overline{Λ}. We say that \overline{Λ} is defined over S if it is stable under the action of π_{1}(S, s) for all geometric points s ∈ S. In this definition, by a standard argument, we only need to require the stability taking one geometric point on each connected component of S.

Suppose that \overline{Λ} is defined over S. Then we find an irreducible étale Galois covering V/S so that we have a member Λ : X ×_{S} V → ^{t}X ×_{S} V in \overline{Λ}. The map Gal(V/S) ∋ σ ↦ Λ^{1−σ} ∈ O_{(p)+}^{×} is a homomorphism of the finite group Gal(V/S) into the torsion-free module O_{(p)+}^{×} (on which Gal(V/S) acts trivially). Thus Λ^{1−σ} = 1, and Λ is defined over S. By this descent argument, we can always find a member Λ ∈ \overline{Λ} which is defined globally over S if \overline{Λ} is defined over S. Thus our definition of S-integrality is equivalent to having a member Λ defined over S in the class \overline{Λ} (which is the definition of integrality Kottwitz used in [Ko] Section 5). As shown in [GIT] Proposition 6.10, for the universal Poincaré bundle P over X ×_{S} ^{t}X, the isogeny 2Λ is induced globally by L = (1×Λ)^{∗}P; so, if p ≠ 2, we have Λ in \overline{Λ} globally associated to an ample line bundle L.

An F –linear isomorphism η^{(p)} : (F_{𝔸}^{(p∞)})^{2} ≅ V^{(p)}(X) = T (X)⊗_{ℤ}𝔸^{(p∞)} is called a level structure of X. An adele g ∈ G(𝔸) acts on η^{(p)} by η^{(p)} ↦ η^{(p)} ∘ g^{(p∞)}. A coset \overline{η}^{(p)} = η^{(p)}K for a closed subgroup K ⊂ G(𝔸^{(∞)}) is called a level K– structure. A level K–structure \overline{η}^{(p)} is defined over S if for each geometric point s and each σ ∈ π_{1}(S, s), σ ∘ η^{(p)} = η^{(p)} ∘ k for k ∈ K.

Two triples \underline{X} = (X, \overline{Λ}, \overline{η}^{(p)})_{/S} and \underline{X}′ = (X′, \overline{Λ}′, \overline{η}′^{(p)})_{/S} are isomorphic up to prime-to–p isogeny if we have an isogeny ϕ : X_{/S} → X′_{/S} of degree prime to p such

that ϕ^{−1} ∘ \overline{Λ}′ ∘ ϕ = \overline{Λ} and ϕ ∘ \overline{η}^{(p)} = \overline{η}′^{(p)}. We write \underline{X} ≈ \underline{X}′ if this is the case. If we insist ϕ above is an isomorphism of abelian scheme (not just an isogeny), we write \underline{X} ≅ \underline{X}′. Let Sh^{(p)} = Sh^{(p)}(G, X)_{/W} be the p–integral model over W of the Shimura variety for (G, X) of prime-to–p level (e.g. [H03a] Lectures 6 and 9 and [PAF] Chapter 4), and the construction of such a model for general Shimura varieties of PEL type has been given by Kottwitz [Ko] (see [PAF] Chapter 7 for an exposition of the construction). By its construction, Sh^{(p)} represents the functor F^{(p)} : ℤ_{(p)}–SCH → SET S given by

F^{(p)}(S) = \{(X, \overline{Λ}, η^{(p)})_{/S}\} / ≈ .

The scheme Sh_{/ℤ_{(p)}}^{(p)} is smooth over ℤ_{(p)}. Each adele g ∈ G(𝔸) acts on Sh^{(p)} by

η^{(p)} ↦ η^{(p)} ∘ g^{(p∞)}. For each open compact subgroup K = K_{p} × K^{(p)} of G(𝔸^{(∞)}) with K_{p} = G(ℤ_{p}) (we call such an open compact subgroup “maximal at p”), the quotient Sh_{K}^{(p)} = Sh^{(p)}/K represents the following quotient functor

F_{K}^{(p)}(S) = \{(X, \overline{Λ}, η^{(p)} ∘ K)_{/S}\} / ≈ .

The scheme Sh_{K}^{(p)} is quasi-projective of finite type over ℤ_{(p)}, and we have Sh^{(p)} = \underleftarrow{lim}_{K}Sh_{K}^{(p)}, where K = G(ℤ_{p})×K^{(p)} and K^{(p)} runs over open-compact subgroups

of G(𝔸^{(p∞)}). If K^{(p)} is sufficiently small, Sh_{K/ℤ_{(p)}}^{(p)} is smooth.

A triple (X(A), \overline{Λ}, η^{(p)})_{/W} for an O–lattice A with A_{p} = R_{p} gives rise to a W–point of the p–integral model of the Hilbert modular Shimura variety Sh_{/ℤ_{(p)}}^{(p)}.

Here Λ may not be proportional to Λ(A) and η^{(p)} may not be equal to η^{(p)}(A) we have chosen. A point of Sh^{(p)} obtained in this way from an O–lattice A of conductor prime to p is called a CM point of Sh^{(p)}.

We fix an O–lattice A ⊂ M with 𝔣(A) generated by a norm from M and A_{p} = R_{p}. Let V_{K} be the geometrically irreducible component of Sh_{K/ℚ}^{(p)} containing

the geometric point x = x(A)_{/\overline{ℚ}}. In other words, V_{K/\overline{ℚ}} is a subscheme of Sh_{K}^{(p)}⊗_{ℚ}\overline{ℚ} which is irreducible over \overline{ℚ} and contains x = x(A) ⊗_{W} \overline{ℚ}. Since the Galois action permutes geometrically irreducible components, we can think of the stabilizer 𝔊_{K} of V_{K} in Gal(\overline{ℚ}/ℚ). The fixed field k_{K} of 𝔊_{K} in \overline{ℚ} is the field of definition of V_{K} in Weil’s sense that V_{K/\overline{ℚ}} descends to an irreducible closed subscheme V_{K/k_{K}} of Sh^{(p)} ⊗_{ℤ_{(p)}} k_{K} and that the algebraic closure of ℚ in the function field k_{K}(V_{K}) coincides with k_{K}. As shown in [Sh1] II and [MS] Theorem 4.13, k_{K} ⊂ ℚ^{(p)} = ⋃_{p∤N} ℚ[μ_{N}], and it is unramified over ℚ at p. Thus k_{K} is in the field of fractions

K of W, and the schematic closure of V_{K} ⊗_{k_{K}} K in Sh_{K/W}^{(p)} is smooth over W if K^{(p)} is sufficiently small. We write again V_{K/W} for the schematic closure. By a result of Rapoport (see [R], [C] and also [DT] in this volume), we can compactify V over W so that the projective compactification f : V^{∗} → Spec(W) (either smooth toroidal or minimal) is projective normal over W with f_{∗}(O_{V^{∗}}) = W. Then by Zariski’s connectedness theorem, V_{K}^{∗} ⊗_{W} 𝔽 and hence V_{K} ⊗_{W} 𝔽 remain irreducible. We take V^{(p)} = \underleftarrow{lim}_{K}V_{K/W}, where K runs over all open compact subgroups of G(𝔸^{(∞)}) maximal at p. The scheme V^{(p)} is therefore smooth over W, and all its geometric fibers are irreducible. In this sense, we call V_{/W}^{(p)} a geometrically connected component of Sh^{(p)} over W containing the W–point x = x(A) (so its generic and special fibers are both geometrically irreducible).

We can think of the following functor F^{ℚ} : ℚ–SCH → SET S in place of F^{(p)}:

F^{ℚ}(S) = \{(X, \overline{Λ}_{ℚ}, η)_{/S}\} / ∼,

where η : (F_{𝔸}^{(∞)})^{2} ≅ V (X) = T (X) ⊗_{ℤ} 𝔸^{(∞)} is an O–linear isomorphism and \overline{Λ}_{ℚ} = \{Λ ∘ ξ|ξ ∈ F_{+}^{×}\}. The equivalence relation (X, \overline{Λ}_{ℚ}, η) ∼ (X′, \overline{Λ}′_{ℚ}, η′) is given by an isogeny ϕ : X → X′ similarly to the case of F^{(p)} but we do not require that deg(ϕ) be prime to p. This functor is represented by a quasi-projective smooth ℚ–scheme Sh = Sh(G, X) ([Sh] and [D2] 4.16-21), and Sh^{(p)} ⊗_{ℤ_{(p)}} ℚ = Sh/G(ℤ_{p}). We consider x_{ℚ}(A) = (X(A), \overline{Λ}(A)_{ℚ}, η(A))_{/K} as a closed point x_{ℚ}(A) ∈ Sh(K). We then take the geometrically irreducible component V ⊂ Sh ⊗_{ℚ} \overline{ℚ} containing x_{ℚ}(A). Then V is actually defined over ℚ^{ab} = ⋃_{N} ℚ[μ_{N}] in the sense of Weil, and we have V^{(p)} ⊗_{ℚ^{(p)}∩W} ℚ^{(ab)} = V/G_{1}(ℤ_{p}), where G_{1} is the derived group of G. The complex point of the Shimura variety Sh has the following expression ([D3] Proposition 2.1.10 and [M] page 324 and Lemma 10.1)

Sh(ℂ) = G(ℚ)\ (ℨ × G(𝔸^{(∞)})) /\overline{Z(ℚ)},

where \overline{Z(ℚ)} is the topological closure in G(𝔸^{(∞)}) for the center Z of G and the action is given by γ(z, g)u = (γ(z), γgu) for γ ∈ G(ℚ) and u ∈ \overline{Z(ℚ)}. Thus we have another geometrically irreducible component V_{i} ⊂ Sh defined over ℚ^{ab} which contains the image of ℨ^{+} × 1. If we choose the base w = (w_{1}, w_{2}) of R̂ rationally so that R = Ow_{1} +𝔠^{∗}w_{2} with Im(w_{2}/w_{1}) ≫ 0, we actually have V = V_{i}. The right action (z, h) ↦ (z, gh) is induced by the action of G(𝔸^{(∞)}) on the level structure η. Thus G(𝔸^{(∞)}) acts on the Shimura variety by η ↦ η ∘ g as scheme automorphisms over ℚ, and G(𝔸^{(∞)}) acts transitively on the set π_{0}(Sh_{/\overline{ℚ}}) of all geometrically irreducible components of Sh_{/\overline{ℚ}}. Each ξ ∈ F^{×} gives an endomorphism End^{ℚ}(X) = End(X) ⊗_{ℤ} ℚ taking η to η ∘ ξ. Thus the center Z(ℚ) acts trivially on Sh. By the above expression of Sh(ℂ), the action of G(𝔸^{(∞)})/Z(ℚ) on Sh factors through G(𝔸^{(∞)})/\overline{Z(ℚ)} for the topological closure \overline{Z(ℚ)} of Z(ℚ).

Let

E = \{x ∈ G(𝔸)| det(x) ∈ \overline{𝔸^{×}F^{×}F_{ℝ+}^{×}} ⊂ F_{𝔸}^{×}\} /\overline{Z(ℚ)G(ℝ)^{+}} ↪ \frac{G(𝔸^{(∞)})}{\overline{Z(ℚ)}},

where G(ℝ)^{+} is the identity connected component of G(ℝ), and F_{ℝ+}^{×} (for F_{ℝ} = F ⊗_{ℚ} ℝ) is the identity connected component of the multiplicative group F_{ℝ}^{×}.

The stabilizer of the geometrically connected component V_{i} is given by E (cf.

[Sh1] II, [MS] and [PAF] 4.2.2). Actually Aut(V_{i/ℚ}) = E ⋊ Aut(F) (cf. [IAT] Theorem 6.23, [Sh1] II 6.5, [D3] 2.4-6, [MS] 4.13 and [PAF] Theorem 4.14), where σ ∈ Aut(F) acts on Sh through its Galois action on G(𝔸). Since V is a conjugate of V_{i} under an element of G(𝔸^{(∞)}) and E is normal in G(𝔸^{(∞)})/\overline{Z(ℚ)}, we also have Aut(V_{/ℚ}) = E ⋊ Aut(F) in the same way. We write τ(g) (g ∈ E) for the automorphism of V induced by the action of g (if it is necessary to indicate that we regard g ∈ G(𝔸) as an automorphism of the Shimura variety V). We define

E^{(p)} = \frac{\{x ∈ G(𝔸^{(p)} × ℤ_{p})| det(x) ∈ \overline{(𝔸^{(p)})^{×}O_{(p)}^{×}F_{ℝ+}^{×}}\}}{\overline{G(ℤ_{p})G(ℝ)^{+}Z(ℤ_{(p)})}}.  (2.3)

Then g ∈ E^{(p)} acts on V_{/ℚ}^{(p)} by η^{(p)} ↦ η^{(p)} ∘ g^{(p)} as scheme automorphisms, and if

x ∈ E^{(p)} has det(x) ∈ \overline{O_{(p)}^{×}F_{ℝ+}^{×}}, the action of x is an automorphism of the scheme V^{(p)} over W. Thus putting

D^{(p)} = \frac{\{x ∈ G(𝔸^{(p)} × ℤ_{p})| det(x) ∈ \overline{O_{(p)}^{×}F_{ℝ+}^{×}}\}}{\overline{G(ℤ_{p})G(ℝ)^{+}Z(ℤ_{(p)})}},  (2.4)

one can show that Aut(V_{/\overline{𝔽}_{p}}^{(p)}) = D^{(p)} ⋊ Aut(F) (see the proof of Proposition 2.8 in the text, [PAF] Theorem 4.17 and [H03d]).

If B · R is in the ideal class of A · R and 𝔣(B) is generated by a norm from M, x(A) and x(B) are on the same geometrically irreducible component V, because η(B) = η(A) ∘ g for g ∈ D^{(p)} by (2.2). We have infinitely many CM points in the component V. Recall that 𝔩 is a prime factor in F of the prime ℓ ≠ p fixed in the introduction. Let L be the closed subgroup of D^{(p)} made up of elements represented by g ∈ G(𝔸) whose prime-to–𝔩 component g^{(𝔩)} is 1. We would like to state

Conjecture 2.1. If X(R)_{/𝔽} is an ordinary abelian variety with End(X(R)_{/𝔽}) = R, then any infinite subset of {g(x(R))|g ∈ L} is Zariski dense in V^{(p)}(𝔽).

This type of questions has been studied by C.-L. Chai (e.g. [C1] and [C4]) in the Hilbert and Siegel modular cases. This conjecture is related to the following folkloric version of a conjecture of André and Oort (cf. [COU]):

Conjecture 2.2. For a geometrically irreducible component V_{/ℂ} of Sh(G, X)_{/ℂ}, if a subvariety V_{/ℂ} ⊂ V contains infinitely many special points, then V contains an irreducible component of a Shimura subvariety of Sh(G, X)_{/ℂ}.

When we drop the assumption: End(X(R))_{/𝔽} = R, the assertion of Conjecture 2.1 is not true for an arbitrary infinite subset of {g(x(R))|g ∈ L}. One of the ways to remedy this is to take an infinite sequence of orders R_{j} with x(B) ∈ V^{(p)} for a proper ideal B of R_{j} and to assert the density in V^{(p)}(𝔽) of {x(B) ∈ V^{(p)}(𝔽)}_{j,B}. We will study this version in Proposition 2.7 via the method of Chai. In Subsection 2.4, we generalize Proposition 2.7 to the case of a product of copies of the Hilbert modular variety.

2.3. Density modulo p.

Let K be an open compact subgroup of G(𝔸^{(∞)}) maximal at p. We suppose that K is sufficiently small so that V_{K/W}^{(p)} is smooth. In this setting, fixing a lattice L ⊂ F^{2}

such that KL̂ = L̂ and the alternating form ⟨·, ·⟩ induces Hom(L_{p}, O_{p}) ≅ L_{p}, we have an isomorphism of F_{K}^{(p)} onto the following functor F_{K}:

F_{K}(S) = \{\underline{X}_{/S}|\underline{X} ≈ ∃\underline{X}′_{/S} ∈ F^{(p)}(S) and η^{(p)}(L̂^{(p)}) = T^{(p)}(X)\} / ≅

where T^{(p)}(X) = T (X) ⊗ ℤ̂^{(p)}, \underline{X} = (X, Λ, η^{(p)}) and “≅” is induced by the isomorphisms of abelian schemes (not just an isogeny). The point here is that we imposed extra condition: η^{(p)}(L̂^{(p)}) = T^{(p)}(X) but also tighten the equivalence from “prime-to–p isogenies” to “isomorphisms”. Thus F_{K}^{(p)} ≅ F_{K} can be shown by finding a unique triple (X, Λ, η^{(p)}) (up to isomorphisms) in a given prime-to–p Sisogeny class of a given \underline{X}, which can be done in a standard way of the construction of Shimura varieties (see [PAF] 4.2.1 for the argument).

If we choose K suitably (see [PAF] 4.2.1), V_{K} is isomorphic to the (fine or coarse) moduli 𝔐(Γ(N))_{/ℤ_{(p)}[μ_{N}]} representing the following functor F_{N} : ℤ_{(p)}– SCH → SET S (cf. [K1] and [PAF] 4.1.2):

F_{N}(S) = {(X, Λ, ϕ_{N})_{/S}} / ≅,

where Λ is a O–polarization and ϕ_{N} is an isomorphism: (NO)^{∗}/O^{∗} × (O/NO) ≅ A[N] of locally free group schemes with e_{N}(ϕ(a, b), ϕ(c, d)) = exp(2πiTr(ad − bc)) (see [ABV] Section 20). Thus V_{/W}^{(p)} is isomorphic to V_{i}^{(p)} = \underleftarrow{lim}_{p∤N}𝔐(Γ(N))_{/W}.

Recall that V_{K}(W) has the CM point x = x(A). By the Serre-Tate deformation theory (cf. [H03a] Lecture 8, [H03b] Section 2 and [PAF] 8.2), we have an identification depending on η_{p}^{ord}(A):

Ô_{V,x/𝔽} ≅ Ô_{𝔾_{m}⊗_{ℤ}O,1} = \underleftarrow{lim}_{n}𝔽[t^{ξ}]_{ξ∈O}/(t^{ξ_{1}} − 1, ⋯, t^{ξ_{d}} − 1)^{n}

for a ℤ–base ξ_{1}, ξ_{2}, ⋯, ξ_{d} of O. We write the ring at the right-hand-side as 𝔽[[t^{ξ}]]_{ξ∈O} symbolically (though it is not exactly a power series ring). The ring O_{p} acts on Ĝ_{m} ⊗_{ℤ} O through the right factor which is induced by the variable change t ↦ t^{a} in the affine ring 𝔽[[t^{ξ}]]_{ξ}. The coordinate t therefore depends on η_{p}^{ord}(A), since by the Serre-Tate theory, the deformation space is identified via η_{p}^{ord}(A) with

Hom(T_{p}(X(A)_{/𝔽}) ⊗_{O} T_{p}(X(A)_{/𝔽}), 𝔾̂_{m}) for T_{p}(X(A)_{/𝔽}) = \underleftarrow{lim}_{n}X(A)[p^{n}](𝔽),

and the identification of T_{p}(X(A)) with O_{p} depends on the level structure η_{p}^{ord}(A). We consider a torus T_{/ℤ} defined by the following exact sequence:

1 → Res_{O/ℤ}𝔾_{m} → Res_{R/ℤ}𝔾_{m} → T → 1.

By Hilbert’s theorem 90, the homomorphism α ↦ α^{1−c} induces an isomorphism of group schemes from T_{/ℤ_{(p)}} to the kernel of the norm map: Res_{R_{(p)}/ℤ_{(p)}}𝔾_{m} → Res_{O_{(p)}/ℤ_{(p)}}𝔾_{m}. We identify the two tori by this isomorphism. The embedding ρ_{A} :

(M_{𝔸}^{(∞)})^{×} → G(𝔸^{(∞)}) induces an embedding of T (ℤ_{(p)}) into D^{(p)} ⊂ Aut(V_{/ℤ_{(p)}}^{(p)}) by composing T (ℤ_{(p)}) ↪ G(ℤ_{(p)})/Z(ℤ_{(p)}) with a diagonal inclusion as principal ideles. Then T (ℤ_{(p)}) fixes the point x = x(A) ∈ V (𝔽) because α ∈ R_{(p)} induces a prime-to–p isogeny

α : (X(A), \overline{Λ}(A), η^{(p)}(A)) → (X(A), \overline{Λ}(A), αη^{(p)} = η^{(p)}(A) ∘ ρ_{A}(α)).

Let \underline{𝕏} = (𝕏, \overline{Λ}, η^{(p)}) be the universal triple over Sh^{(p)}. Then the Igusa tower over Sh_{/𝔽}^{(p)} is defined by \underleftarrow{lim}_{n}Isom_{gp-sch}(μ_{p^{n}} ⊗𝔡_{/𝔽}^{−1}, 𝕏[p^{n}]_{/𝔽}^{∘}) for the connected component 𝕏[p^{n}]_{/𝔽}^{∘} of the locally free group scheme 𝕏[p^{n}]_{/𝔽}. We write Ig_{/V^{(p)}} for the pull back of this Igusa tower to V_{/𝔽}^{(p)}. The scheme Ig is étale faithfully flat over

the ordinary locus of V_{/𝔽}^{(p)}. The p–ordinary level structure η_{p}^{ord}(A) defines a point, again denoted by x = x(A), of the Igusa tower Ig/V^{(p)} (see [H03a] Lectures 7 and 10 for the Igusa tower). Since R_{Σ} ≅ O_{p} canonically, the multiplicative group R_{(p)}^{×} acts on ordinary level structures and hence on the stalk O_{Ig,x/𝔽} through the quotient T (ℤ_{(p)}). Thus T (ℤ_{(p)}) is identified with a p–adically dense subgroup of Aut_{O}(Ŝ) = O_{p}^{×} for Ŝ = 𝔾̂_{m} ⊗_{ℤ} O which is isomorphic via η_{p}^{ord}(A) to the formal completion of Ig at x (see [H03a] Lecture 8 and [H03b] Section 2). By the Serre-Tate theory, this identification is equivariant under the action of α ∈ T (ℤ_{(p)}) through the automorphism ρ_{A}(α) of Ig and multiplication by α^{1−c} on the right factor 𝔾̂_{m} ⊗_{ℤ} O (that is, sending the Serre-Tate coordinate t to t^{α^{1−c}}). We can guess this fact heuristically from the identity: R_{(p)}^{×}/O_{(p)}^{×} ⊂ R_{p}^{×}/O_{p}^{×} and α ≡ α^{1−c} mod O_{p}^{×}, although the real proof is more complicated (via a full use of the Serre-Tate theory), because a non-trivial element in O_{p}^{×}/O_{(p)}^{×} (which is the center of G(ℤ_{p})/Z(ℤ_{(p)})) really moves x to another point on Ig.

We state here formally the fact as a lemma and give a brief outline of the proof (see [H03b] Proposition 3.3 for more details):

Lemma 2.3. We have a canonical identification:

Îg_{/𝔽} ≅ Ŝ = 𝔾̂_{m} ⊗_{ℤ} O = ∏_{𝔭∈Σ_{p}} 𝔾̂_{m} ⊗_{ℤ_{p}} O_{𝔭},

where Îg is the formal completion of Ig along x = x(A). Under this identification, the action on Ig of ρ_{A}(α) ∈ D^{(p)} for an element α ∈ R_{(p)}^{×} induces on Ŝ an action

given by t ↦ t^{α^{1−c}} for the Serre-Tate coordinate t of Ŝ, where c is the generator of Gal(M/F) (that is, “complex conjugation”).

Here is an outline of the proof: In the proof, we use Katz’s notation on the Serre-Tate moduli in [K2] without much explanation to make our exposition short. In particular, we write q for the Serre-Tate coordinate (keeping t exclusively for the corresponding coordinate on Ig centered at x). The first assertion follows from the universality of the Serre-Tate local moduli and the global universality of Ig. To show the second assertion, let X_{0} = X(A) ⊗_{W} 𝔽. By the Serre-Tate theory, the abelian variety (X_{0}, \overline{Λ}(A), η^{ord}(A)) is sitting on the origin 1 of 𝔾̂_{m} ⊗ O.

We now compute the effect of the isogeny α : X_{0} → X_{0} (for α ∈ R) on the deformation space Ŝ. Pick a deformation X_{/A} of X_{0} = A_{x/𝔽} for an artinian W - algebra A with residue field 𝔽, and we look into the following diagram with exact rows:

\begin{matrix}Hom(T X_{0}[p^{∞}]^{et}, 𝔾̂_{m}(A)) & ↪ & X[p^{n}](A) & ↠ & X_{0}[p^{n}]^{et}(A) \\ α^{−c}↑ & & & & ↓α \\ Hom(T X_{0}[p^{∞}]^{et}, 𝔾̂_{m}(A)) & ↪ & X[p^{n}](A) & ↠ & X_{0}[p^{n}]^{et}(A). \end{matrix}  (2.5)

Take u = \underleftarrow{lim}_{n}u_{n} ∈ T X_{0}[p^{∞}]^{et}, and lift it to v = \underleftarrow{lim}_{n}v_{n} for v_{n} ∈ X[p^{n}]. Then q(u) = \underleftarrow{lim}_{n}q_{n}(u_{n}) ∈ Hom(T X_{0}[p^{∞}]^{et}, 𝔾̂_{m}) for q_{n}(u_{n}) =“p^{n}”v_{n}, where “p^{n}” is the Drinfeld lift of multiplication by p^{n} in [K2] Lemma 1.1.2. Note that the identification of Hom(T X_{0}[p^{∞}]^{et}, 𝔾̂_{m}) with the formal group X̂ of X is given by the Cartier duality composed with the polarization; so, given by α^{−c} because the complex conjugation is the Rosati involution. Thus the effect of α on q is given by q ↦ q^{α^{1−c}}. Once the identification of Ŝ with 𝔾̂_{m} ⊗ 𝔡^{−1} is given (via η_{p}^{ord}(A)), α ∈ R prime to p acts on the coordinate t by t ↦ t^{α^{1−c}}. ⊓ ⊔

Let L be a ℤ_{p}–free module of finite rank on which T (ℤ_{p}) acts by a ℚ_{p}–rational linear representation. Then Ŝ_{L/𝔽} = 𝔾̂_{m} ⊗_{ℤ_{p}} L inherits the action of T from L. Here we quote a lemma whose proof has been supplied to me by Ching-Li Chai:

Lemma 2.4. Suppose that the trivial representation of T (ℤ_{p}) is not a subquotient of L ⊗_{ℤ_{p}} ℚ_{p}. If Z_{/𝔽} is a reduced irreducible formal subscheme of Ŝ_{L} stable under the action of an open subgroup of T (ℤ_{p}), then there exists a ℤ_{p}–direct summand L_{Z} ⊂ L stable under T (ℤ_{p}) such that Z = 𝔾̂_{m} ⊗_{ℤ_{p}} L_{Z}, and Z is a smooth formal subtorus of T̂.

This is proven in [C1] Proposition 4 assuming further the smoothness of Z at the origin of Ŝ_{L}. Recently Chai has removed the smoothness assumption from [C1] Proposition 4 and has actually generalized his result in much more general situations (see [C3]). His proof in the above setting can be found in [H03b] Lemma 3.5. We actually need the lemma only under the smoothness assumption.

Let 𝔟_{K} be a prime ideal of O_{V_{K},x/𝔽} for x = x(A). We consider the Zariski closure X_{K} of Spec(O_{V_{K},x}/𝔟_{K}) in V_{K/𝔽}. Since O_{Ig,x} is étale over O_{V_{K},x}, we have a unique prime ideal 𝔟 ⊂ O_{Ig,x} with 𝔟 ∩ O_{V_{K},x} = 𝔟_{K}. We write X for the Zariski closure of Spec(O_{Ig,x}/𝔟) in Ig. Then X_{K} is the Zariski closure of the image of X in V_{K} (supplemented with the non-ordinary loci). Since T (ℤ_{(p)}) (strictly speaking τ(ρ_{A}(T (ℤ_{(p)})))) fixes x, it acts on the stalk O_{Ig,x} as endomorphisms of the ring O_{Ig,x}, and in particular, T (ℤ_{(p)}) acts by automorphisms.

Proposition 2.5. Let T be a subgroup of T (ℤ_{(p)}) whose p–adic closure contains an open subgroup of T (ℤ_{p}). Suppose that dim X_{/𝔽} > 0 and X is stable under T . Then we have X_{/𝔽} = Ig_{/𝔽} and hence X_{K/𝔽} = V_{K/𝔽}.

We shall repeat the proof given in [H03b] 3.4 because this is crucial in the sequel.

Proof. We follow the argument in [C1] Section 4 and 5. The proof is separated into the following three steps:

1. To show the existence of a subset Σ′ ⊂ Σ_{p} such that the formal completion X̂_{y} of X along any ordinary point y ∈ X (𝔽) is given by ∏_{𝔭∈Σ′} 𝔾̂_{m} ⊗_{ℤ_{p}} O_{𝔭} in the Serre-Tate deformation space giving an infinitesimal neighborhood of y. In particular, for the pull-back A to X_{K} of the universal abelian scheme, we show that A[𝔭^{∞}] for 𝔭 ∉ Σ_{p} − Σ′ gives an étale constant Barsotti-Tate group over the ordinary locus of X_{K};

2. To show that X_{K} contains a super-singular point z;

3. Draw a curve C from an ordinary closed point to the super-singular z inside X_{K}. Then by Step 1, A[𝔭^{∞}] for 𝔭 ∉ Σ_{p} − Σ′ is an étale constant deformation of A_{z}[𝔭^{∞}] (which is impossible because A_{z} is super-singular); so, we conclude Σ′ = Σ_{p} and hence X = Ig.

Step 1: By the stability of X under T , the formal completion X̂ along the point x is stable under its closure \overline{T} ⊂ T (ℤ_{p}). Then, applying Lemma 2.4 to Ŝ (L = O_{p}), we find that

X̂ = ∏_{𝔭∈Σ′} 𝔾̂_{m} ⊗_{ℤ_{p}} O_{𝔭}  (2.6)

for a subset Σ′ ⊂ Σ_{p}. In particular, X̂ = X̂_{K} is a smooth formal subgroup of Ŝ, and X is stable under T (ℤ_{(p)}).

On Ig, by the argument in [C1] page 473, the tangent bundle T_{Ig} is decomposed into the direct sum of eigenspaces under the O–action:

T_{Ig} ≅ O_{Ig} ⊗_{ℤ} O locally, and T_{Ig} = ⨁_{𝔭∈Σ_{p}} T_{𝔭},

where T_{𝔭} is a locally free O_{Ig} ⊗_{ℤ_{p}} O_{𝔭}-module of rank 1. By the above expression, the tangent space TX̂ of X̂ coincides with ⊕_{𝔭∈Σ′}T_{𝔭}⊗_{O_{Ig,x}} Ô_{Ig,x}. Through faithfully flat descent, we have T_{X} ⊗_{O_{Ig}} O_{Ig,x} = ⊕_{𝔭∈Σ′}T_{𝔭} ⊗_{O_{Ig}} O_{Ig,x}. Thus on an open dense subscheme X^{∘} ⊂ X inside the smooth locus X^{sm} ⊂ X, we have

T_{X^{∘}} = ⊕_{𝔭∈Σ′}T_{𝔭} ⊗_{O_{Ig}} O_{X^{∘}}.

Since the tangent space is locally defined, if T_{X^{∘},y} = ⊕_{𝔭∈Σ′}T_{𝔭}⊗_{O_{Ig}} O_{X^{∘},y} at y ∈ X, the same identity hold at the image of y under the toric action of T (ℤ_{(p)}). Thus we may suppose that X^{∘} is stable under T (ℤ_{(p)}).

Suppose that Σ′ ≠ Σ_{p} and let 𝔭 ∈ Σ_{p}−Σ′. We only need to prove that V_{K} = X_{K} for a choice of an open subgroup K maximal at p. Choosing K sufficiently small, we may assume that V_{K} is smooth over 𝔽 = \overline{𝔽}_{p}. Let 𝕏_{/V_{K}} be the universal abelian scheme, and define A = A_{K} = 𝕏 ×_{V_{K}} X_{K}. Write 𝔭_{F} = F ∩ 𝔭, and consider the 𝔭_{F} –divisible group A[𝔭_{F}^{∞}]. We choose the connected component Y_{η,r} = Y_{K,η,r} of the generic fiber A[𝔭_{F}^{r}]_{η} (for the generic point η of X_{K}) whose projection to the maximal étale quotient A[𝔭_{F}^{r}]_{η}^{et} gives the generator over O/𝔭_{F}^{r}. We then take the schematic closure Y_{r} = Y_{K,r} of Y_{η,r} in A[𝔭_{F}^{r}]. The subscheme Y_{K,r} ⊂ A[𝔭_{F}^{r}] projects down to the generator of A[𝔭_{F}^{r}]^{et} in the sense of [AME] Sections 1.9-10. We arrange {Y_{K,r}}_{r} coherently so that it gives a tower of irreducible varieties over X_{K}. The reduced scheme Y_{K,η,r}^{red} is étale over X_{K,η}; they give rise to the tower of function fields 𝔽(X_{K}^{red}) ⊂ 𝔽(Y_{K,1}^{red}) ⊂ ⋯ ⊂ 𝔽(Y_{K,r}^{red}) ⊂ · · ·. Since Y_{K,r}^{red} is the generator of A[𝔭_{F}^{r}]^{et}, the fiber Y_{K,r} of A[𝔭_{F}^{r}] over Y_{K,r}^{red} ⊂ A[𝔭_{F}^{r}]^{et} can be identified with the connected component A[𝔭_{F}^{r}]^{∘}. Thus the scheme A[𝔭_{F}^{r}]^{∘} is a connected finite flat group scheme over Y_{K,r}^{red}.

Pick any function ϕ ∈ 𝔽(Y_{K,r}^{red}) finite at x, and expand ϕ into a power series ϕ(t_{𝔭′}) around x using the variables t_{𝔭′} of 𝔾̂_{m} ⊗_{ℤ_{p}} O_{𝔭′} for 𝔭′ ∈ Σ_{p}. Since 𝔭 ∉ Σ′, ϕ(t_{𝔭′}) is constant with respect to t_{𝔭}. Since t_{𝔭} is the variable of the universal deformation space over 𝔽 of X(A)[𝔭_{F}^{∞}] = A[𝔭_{F}^{∞}]×_{X_{K},x} 𝔽, A[𝔭_{F}^{r}] as a deformation of X(A)[𝔭_{F}^{r}] has to be constant on X̂_{K}. Since X(A) is ordinary, we have X(A)[𝔭^{∞}] ≅ (μ_{p^{∞}} ⊗ O_{𝔭}) × F_{𝔭}/O_{𝔭}. Thus over X̂, A[𝔭_{F}^{∞}]_{/X̂} is again constant, and A[𝔭_{F}^{∞}]_{/X̂} ≅

(μ_{p^{∞}} ⊗ O_{𝔭}) × F_{𝔭}/O_{𝔭}. In other words, the morphism of X̂_{K} into the universal deformation space Ŝ_{𝔭} = 𝔾̂_{m} ⊗_{ℤ_{p}} O_{𝔭} of X(A)[𝔭_{F}^{∞}]_{/𝔽} induced by A[𝔭_{F}^{∞}]_{/X̂} is the

constant morphism factoring through Spec(𝔽) ↪ Ŝ_{𝔭}. In particular, A[𝔭_{F}^{r}]_{/X̂} is defined by equations with coefficients in 𝔽; so, it is defined over a finite field 𝔽_{q} (q = p^{f}). Therefore the q–th power relative Frobenius map Φ̂ of A[𝔭_{F}^{r}]_{/X̂}^{et} is

constant. Since the relative Frobenius map Φ̂ is the base-change (from O_{X_{K},x} to X̂_{K}) of the relative Frobenius map Φ of A[𝔭_{F}^{r}]_{/O_{X ,x}}^{et}, Φ has to be constant, since

Ô_{X ,x} is faithfully flat over O_{X ,x}. Thus we conclude that A[𝔭_{F}^{r}]_{/O_{X ,x}}^{et} is a constant étale group (before completion) defined over 𝔽_{q}. Hence it is constant over an open dense subscheme X_{K}^{∘} of X_{K} (see Remark in [C1] page 473). We hereafter rewrite X^{∘} for \underleftarrow{lim}_{K}X_{K}^{∘} (for K maximal at p).

By the constancy of A[𝔭_{F}^{r}]^{et} over X_{K}^{∘}, we find that 𝔽(Y_{K,r}^{red}) = 𝔽(X_{K}) for all r > 0, because 𝔽(Y_{K,r}^{red}) is the function field of an irreducible component of A[𝔭_{F}^{r}]^{et}. Since Y_{K,r}^{red} is étale over the ordinary locus of X_{K}, the two ordinary loci coincide, and X_{K}^{∘} is the ordinary locus of X_{K}, which is stable under T (ℤ_{(p)}) and is open dense in X_{K}. The étale group scheme A[𝔭_{F}^{r}]^{et} is constant over X^{∘} (independent of r). Thus for any geometric point y ∈ X^{∘}, the formal completion X̂_{y} along y is isomorphic to ∏_{𝔭′∈Σ′} 𝔾̂_{m} ⊗_{ℤ_{p}} O_{𝔭′} as formal O–modules.

Step 2: We proceed in a manner similar to the proof of Proposition 7 and Theorem 1 of [C1], though in our case, we take a slightly different approach, because T (ℤ_{(p)}) is much smaller than the full group of prime-to–p Hecke operators treated in [C1].

For simplicity, hereafter we write V for V^{(p)} = \underleftarrow{lim}_{K:maximal at p}V_{K} if no confusion is likely. Let V^{∗} be the Satake compactification of V. The automorphism group D^{(p)} still acts on the compactification V^{∗} since V_{K}^{∗} is given by the “Proj” of the graded algebra of parallel weight modular forms (see [C]). The Hasse-Witt matrix extends to the compactification V^{∗} and is invertible at the cusp. To see this, we first make the smooth toroidal compactification \overline{V}_{K} which carries a universal semi-abelian scheme (depending on the toroidal compactification data: see [C]). Thus the Hasse-Witt matrix extends to the toroidal compactification \overline{V}_{K}. It is constant on the boundary divisors, since in the Hilbert modular case, the abelian variety fully degenerates towards the cusp. Thus the matrix actually factors through the Satake compactification. Then V^{∗} can be stratified so that at each stratum the rank of the Hasse-Witt matrix is constant 0 ≤ r ≤ [F : ℚ], that is, one of r × r minors of the matrix does not vanish on the stratum but (r + 1) × (r + 1) minors all vanish. Each stratum is known to be quasi affine (see [O] and the discussion in [C1] Section 5). Each stratum is stable under the action of D^{(p)} because r gives the p–rank of the abelian varieties over the stratum. Let X^{∗} be the closure of the image of X in V^{∗}. Since X^{∗} is proper and the cusp is contained in the generic stratum, X^{∗} cannot be contained in the generic stratum (because of affineness of the ordinary locus). Let S be one of the non-generic stratum of V^{∗} such that X^{∗}∩S is non-empty. Note that S = \underleftarrow{lim}_{K}S_{K} for S_{K} ⊂ V_{K}. Then X^{∗}∩S is stable under T (ℤ_{(p)}), and S does not contain any cusp. For each y ∈ S, the abelian variety A_{y} sitting over y is non-ordinary. If \overline{α} = τ(ρ_{A}(α)) for α ∈ T (ℤ_{(p)}) fixes the point y, it is induced by a prime-to–p isogeny: 𝕏 → \overline{α}^{∗}𝕏 = 𝕏 ×_{V,\overline{α}} V by the universality of the Hilbert modular Shimura variety, which induces an isogeny α : A_{y} → A_{y}. This implies M = F [α] ↪ End(A_{y}) ⊗_{ℤ} ℚ. By [C1] Lemma 6, if the p–rank of A_{y} is positive, the commutant of F in End(A_{y}) ⊗ ℚ is equal to M, and the relative Frobenius q–th power endomorphism ϕ ∈ End(A_{y}) (for a sufficient large p–power q) generates M over F. Thus, the multiplication by M determines (through the level structure η_{y}^{(p)} of A_{y}) a unique subtorus T_{y} ≅ Res_{M/ℚ}𝔾_{m} of G, which is the centralizer of ϕ and is different from T_{x} since the p–rank of A_{y} is less than [F : ℚ]. The quotient torus T_{y}/Z for the center Z of G gives the stabilizer of y; so, T_{x} = T_{y}. This is a contradiction, since the p–rank is less than [F : ℚ]. So, the stabilizer in T (ℤ_{(p)}) of A_{y} is trivial. Thus S_{K} ∩ X_{K} is an infinite set, and X^{∗} meets some of the lower strata. This shows that X^{∗} contains a point z which carries a super-singular AVRM (a p–rank 0 abelian scheme).

Write (A_{z}, i_{z} : O ↪ End(A_{z}), ϕ) for the abelian variety sitting over the supersingular point z ∈ V_{K}, where ϕ is a level K–structure. Now we denote by X̂_{K} (resp. V̂_{K}) the formal completion of X_{K} (resp. V_{K}) along z ∈ V_{K}. We have a splitting V̂_{K} = ∏_{𝔭∈Σ_{p}} V̂_{𝔭}, where V̂_{𝔭} is the deformation space of the p–divisible group of A_{z}[𝔭_{F}^{∞}] for 𝔭_{F} = 𝔭∩F. Choose z ∈ X^{∗} projecting down to z ∈ X_{K}. Then z ∈ V carries the triple (A_{z}, \overline{λ}, η_{z}^{(p)}).

Step 3: Suppose (contrary to the desired conclusion) that X is a proper subscheme of Ig; so, X_{K} is a proper subscheme of V_{K}. Since z is in the Zariski closure of X_{K}^{∘} in V_{K}, we have a formal curve ξ : Spf(𝔽[[t]]) → X̂_{K} such that

1. The abelian scheme A_{ξ} = A ×_{X̂_{K},ξ} Spf(𝔽[[t]]) over ξ is generically ordinary. In particular, the projection ξ_{𝔭} : Spf(𝔽[[t]]) → X̂ → V̂_{𝔭} is determined by A_{ξ}[𝔭_{F}^{∞}] which is a generically ordinary p–divisible group;

2. Over 𝔽((t)), the image of ξ is contained in ∏_{𝔭∈Σ′} 𝔾̂_{m}⊗_{ℤ_{p}} O_{𝔭}, and the tangent space T_{ξ}^{∘} over the generic point is given by ⊕_{𝔭∈Σ′}T_{𝔭} ⊗_{O_{V}} 𝔽((t)) for Σ′ ⊊ Σ_{p};

3. ξ is the formal completion around z of an irreducible smooth algebraic curve C → X_{K} with open dense subset C^{∘} = C ×_{X_{K}} X_{K}^{∘}.

Choose 𝔭 ∉ Σ′. Since A_{ξ}[𝔭_{F}^{∞}] is generically ordinary but deforms into a supersingular p–divisible group at special fiber, ξ_{𝔭} is non-constant. The formal group A_{C}[𝔭_{F}^{∞}] = A[𝔭_{F}^{∞}] ×_{X_{K}} C is constant on the dense open subscheme C^{∘} in C. We choose a minimal non-trivial p–divisible subgroup Γ ⊂ A_{C}[𝔭_{F}^{∞}] and consider its generator C_{r} ∈ Γ[p^{r}] (similarly defined as Y_{K,r} in the sense of [AME] Sections 1.9-10) as a covering of C. Then the relative dimension of the formal group Γ̂ of Γ is equal to 1, and we have a tower of algebraic curves {C_{r}^{red}}_{r} which carries the connected finite flat group scheme Γ[p^{r}]^{∘}. We consider the function field 𝔎 = 𝔽(C_{∞}^{red}) = ∪_{r}𝔽(C_{r}^{red}), its Galois closure 𝔎^{gal} over 𝔽(C) and the Galois group 𝔊 = Gal(𝔎^{gal}/𝔽(C)). We may assume that the formal group Γ̂ is of height 2 over z and is of height one at the generic point, because it is constant and ordinary generically. By a classical result of Igusa, the formal deformation space of a height 2 formal group over 𝔽 is a smooth formal curve Spf(𝔽[[T]]), and the Galois representation on the p–adic Tate module T_{p}(G) ≅ ℤ_{p} of the universal p–divisible group G has full image ℤ_{p}^{×} (cf. [I] and [K] Theorem 4.3). There is a generalization of this fact to height h > 0 formal groups over Spf(𝔽[[t]]) in [C2]. This Galois action of course factors through the inertia group I ⊂ 𝔊 of z (by the very definition of C_{r}^{red}/C). By the universality, we have a morphism Spf(𝔽[[t]]) → Spf(𝔽[[T]]), which is non-constant since the special fiber of Γ is non-ordinary. Thus the Galois representation of I on T_{p}(Γ) has open image in ℤ_{p}^{×}, and hence 𝔊 is non-trivial. Since A_{C}[𝔭_{F}^{∞}]^{et} is constant over C^{∘}, the image of the representation of 𝔊 on T_{p}(Γ) has to be trivial, hence a contradiction. Thus we conclude that Σ′ = Σ_{p}, which finishes the proof. ⊓ ⊔

Recall that O_{x/𝔽} = O_{x/W} ⊗_{W} 𝔽 is the stalk of x = x(A) of O_{Ig/𝔽}. The following result is a key to prove the linear independence.

Proposition 2.6. Let 𝔽 = \overline{𝔽}_{p} and the notation be as in Proposition 2.5 and its proof. We consider O_{x/𝔽} = O_{Ig,x/𝔽} and write S = Spec(O_{x}). For a positive integer m, let X be a reduced irreducible closed subscheme of S^{m} containing x^{m} = (x, x, …, x) for the closed point x ∈ S. Suppose that X is stable under the diagonal action of T . If the projection to the first (m − 1)–factor S^{m−1} induces a surjective morphism π_{X} : X → S_{/𝔽}^{m−1}, then either X = S^{m} or X is integral over S^{m−1} under the projection to the first (m − 1)–factor. If further π_{X} induces a surjection of the tangent space at x^{m} onto that of S^{m−1} at x^{m−1} and X is a proper subscheme of S^{m}, π_{X} : X → S^{m−1} is étale, where the 0 fold product S^{0} is supposed to be equal to Spec(𝔽).

Proof. Let Ŝ = Spf(Ô_{x}) for the formal completion of S along x. We have Ŝ ≅ 𝔾̂_{m} ⊗_{ℤ} O. Since the case m = 1 has already been taken care of by Proposition 2.5, we may assume that m ≥ 2. If dim X = 0, we see that Supp(X) is made up of the point induced by x, which is impossible since X covers Spec(O_{x})^{m−1}. Thus we have dim X > 0. Since X is stable under the diagonal action of T ⊂ Aut(O_{x}) (which is p–adically dense in Aut_{O}(Ŝ) = O_{p}^{×}) and X is irreducible reduced, by Chai’s lemma (Lemma 2.4), X̂ is a formal O_{p}–submodule of Ŝ^{m} of the form 𝔾̂_{m} ⊗ L for an O_{p}–direct summand L of the cocharacter group X_{∗}(Ŝ^{m}) ≅ O_{p}^{m}. In particular, X is stable under T (ℤ_{(p)}).

Since π_{X} : X̂ → Ŝ^{m−1} is surjective, the projected image L^{X} = π_{X} (L) of L in the first (m−1)–factor O_{p}^{m−1} ≅ X_{∗}(Ŝ^{m−1}) is an O_{p}–submodule of finite index, and L_{X} = Ker(π_{X} : L → O_{p}^{m−1}) is an O_{p}–direct summand of O_{p} = Ker(π : X_{∗}(Ŝ^{m}) → X_{∗}(Ŝ^{m−1})) given by ∏_{𝔭∈Σ′} O_{𝔭} for a subset Σ′ of Σ_{p}.

We consider the diagonal formal torus Δ̂ ⊂ Ŝ^{m} (Δ̂ ≅ Ŝ) and its image α(Δ̂) under the component-wise action of α ∈ T (ℤ_{(p)})^{m}. Since a formal subtorus Ŝ′ ⊂ Ŝ^{m} is determined by its cocharacter group X_{∗}(Ŝ′) = Hom(𝔾̂_{m}, Ŝ′), the dimension of Ŝ′ is given by rank_{ℤ_{p}} X_{∗}(Ŝ′). The dimension of the intersection of two tori is determined by the rank of the intersection of the cocharacter groups of the two tori in X_{∗}(Ŝ^{m}). As we have already seen, the cocharacter group of X̂ at x^{m} contains the subspace ∏_{𝔭∈Σ′} O_{𝔭}^{m} in X_{∗}(Ŝ^{m}) = O_{p}^{m}. Thus rank(X_{∗}(α(Δ̂)) ∩ X_{∗}(X̂)) ≥ |Σ′| = ∑_{𝔭∈Σ′}[F_{𝔭}, ℚ_{p}], and by the p–adic density of T (ℤ_{(p)}) in Aut_{O}(Ŝ), the equality is attained for some α ∈ T (ℤ_{(p)})^{m}. Thus the intersection α(Δ̂) ∩ X̂ gives rise to an irreducible subscheme in Spec(O_{x}) of dimension |Σ′| stable under the action of T (ℤ_{(p)}). Then by Proposition 2.5, we conclude that |Σ′| = 0 or Σ′ = Σ_{p}. If Σ′ = Σ_{p}, we conclude X = S^{m}.

Now suppose |Σ′| = 0. Thus X = Spec(R) is integral over S^{m−1}. If further π_{X} : L → X_{∗}(S^{m−1}) is surjective (⇔ π_{X} is surjective at the level of tangent spaces), π_{X} : L → X_{∗}(S^{m−1}) is an isomorphism; so, X̂ ≅ Ŝ^{m−1} by π_{X}. Thus we conclude that X is étale over S^{m−1}. ⊓ ⊔

Remark 2.1. Suppose that X ⊂ V^{m} is the Zariski closure of an infinite set of distinct points of the form x = (x(A_{1}), …, x(A_{m})) for O–lattices A_{j} ⊂ M of conductor prime to p and is stable under a subgroup U_{0} of T (ℤ_{(p)}) whose p–adic closure is an open subgroup of T (ℤ_{p}). Here T acts diagonally on V^{m}. Then we can apply Proposition 2.6, because we can use any point x as above on an irreducible component of X which is fixed by a subgroup of finite index in U_{0}. In particular, by changing x, we may assume that the formal completion X̂ of X along x is smooth; so, we conclude X = V^{m} (if dim X > (m − 1) dim V and X covers any (m − 1)– factor V^{m−1}) from Proposition 4 of [C1] (by the same proof of Proposition 2.5 and Proposition 2.6) without using the refined version: Lemma 2.4.

Recall the prime factor 𝔩 in O of ℓ ≠ p fixed in the introduction. Let R_{n} = O + 𝔩^{n}R, and write Cl_{n} = Pic(R_{n}). Let A be a proper ideal of R_{n}. Each α ∈ R_{(p)}^{×} induces an isomorphism:

(X(A), \overline{Λ}(A), η^{(p)}(A)) ≅ (X(αA), \overline{Λ}(αA), η^{(p)}(αA)).

Thus the isomorphism class of the triple x(A)_{/𝔽} = x(A)×_{W} 𝔽 for a proper R_{n}–ideal A ⊂ M only depends on the class [A] ∈ Cl_{n}. Let C^{alg} ⊂ Cl_{∞} be the subgroup made up of the image [x] = \underleftarrow{lim}_{n}[xR̂_{n}∩M] of x ∈ M_{𝔸}^{×} with x_{𝔩} = 1. Since the image of F^{×} ⊂ M_{𝔸}^{×} is trivial in Cl_{∞}, we may embed the image T_{𝔩} of M ∩R_{𝔩}^{×} in T (ℚ) into Cl_{∞} by α ↦ [α^{(𝔩)}]. Then we have an exact sequence: 1 → T_{𝔩} → C^{alg} → Cl_{M} → 1 for the class group Cl_{M} of M.

Remark 2.2. As remarked by the referee of this paper, this group C^{alg} is obviously close to the rational points S_{1}^{M}(ℚ) of the Serre group S_{1}^{M} relative to the CM field M (of the empty modulus of support; see [ALR] II.2.2) and is isomorphic to S_{1}^{M}(ℚ)/T_{1}^{F} (ℚ) when 𝔩 = (ℓ) and 𝔩 remains prime in M, where T_{1}^{F} is the quotient of Res_{F/ℚ}𝔾_{m} by the Zariski closure of O^{×} (regarded as a subgroup of S_{1}^{M}). In any case, C^{alg} is a countable subgroup in the compact profinite group Cl_{∞}, in a way similar to the inclusion of the Serre groups: S_{𝔪}(ℚ) ⊂ S_{𝔪}(ℚ_{ℓ}).

Proposition 2.7. Suppose (unr) and (ord) for p. Let \underline{n} = 0 < n_{1} < n_{2} < ⋯ be an infinite sequence of integers with 𝔩^{n_{j}} generated by an element in N_{M/F} (R_{(p)}^{×}), and let Ξ = {x(A) ∈ V (𝔽)|A ∈ Cl_{n_{j}} with [A] ∈ Ker(Cl_{n_{j}} ↠ Cl_{r}) (j = 1, 2, …)} for an integer r with 0 ≤ r ≤ n_{1}. Then Ξ is Zariski dense in V (𝔽).

Proof. Let K = G_{1}(ℤ̂). Then V_{1/𝔽} = V/K is the coarse moduli scheme of abelian variety with real multiplication by O with a given polarization ideal (cf. [H03a] Lecture 9 or [PAF] Chapter 4). The image Ξ_{1} of Ξ in V_{1}(𝔽) has infinitely many points (see the remark after the proposition); so, the dimension of an irreducible component of the Zariski closure X of Ξ is positive. We pick such a component X_{2}, and suppose that x(A) ∈ Ξ is in X_{2}. For a sufficiently small open compact subgroup S of K (maximal at p), V_{/𝔽} is étale over V_{S/𝔽}. Thus if X has a singularity at y ∈ X, it has singularity at the image of y in V_{S}(𝔽), which is a smooth irreducible variety over 𝔽 of finite type. Thus the singular locus in the image X_{S} of X in V_{S/𝔽} is a proper closed subscheme, and we may assume that x = x(A)_{/𝔽} is a smooth point of X (by changing A if necessary). We suppose that A is a proper R_{n}–ideal (for n = n_{j} with an index j).

Since τ(ρ_{B}(α))(x(B)) = x(B) for α ∈ R_{(p)}^{×}, we find that

τ(ρ_{B}(α_{𝔩})^{−1})(x(B)) = τ(ρ_{B}(α^{(𝔩)}))(x(B)).

By our construction, we have ρ_{B}(α^{(𝔩)}) = ρ_{A}(α^{(𝔩)}) = ρ_{R}(α^{(𝔩)}). Since we have a commutative diagram for a ∈ M_{𝔸}^{×} with a_{p} = a_{𝔩} = 1:

(F_{𝔸}^{(p∞)})^{2} ⟶

multiplication by ρ_{R}(a)

(F_{𝔸}^{(p∞)})^{2}
η^{(p)}(B)↓  ↓_{η^{(p)}(aB)}
V^{(p)}(X(B))  ⟶^{ϕ_{a}}  V^{(p)}(X(aB))

for a prime-to–p isogeny ϕ_{a} : X(B) → X(aB), the action of τ(ρ_{R}(α^{(𝔩)})) brings the point x(B) to x([α^{(𝔩)}]B), and hence it induces a permutation of Ξ as long as (α^{(𝔩)}R_{r}) is in the identity class in Cl_{r} (⇔ α ≡ 1 mod 𝔩^{r}). If α ≡ 1 mod 𝔩^{n}, the class [αR_{n}] in Cl_{n} is trivial and [αA] = [α^{(𝔩)}][A] = [A], which implies τ(ρ_{A}(α^{(𝔩)}))(x(A)) = x(A). The image T of {α ∈ R_{(p)}^{×} ∩ R_{𝔩}|α ≡ 1 mod 𝔩^{max(r,n)}} in T (ℤ_{p}) is p–adically dense and preserves X and fixes x(A). Since t ∈ T fixes x = x(A), t ∈ T permutes irreducible components of the Zariski closure X of Ξ passing through x. In particular, if t really moves the irreducible component X_{2} of X, x ∈ X_{2} ∩ t(X_{2}) is not a smooth point of X. Thus we conclude that X_{2} is stable under T . By our choice, we have dim X_{2} > 0. Thus we can apply Proposition 2.5 to the ideal 𝔟_{K} of X_{K} in O_{V_{K},x/𝔽}, which is stable under T . By Proposition 2.5, we have X_{2} = X = V. ⊓ ⊔

Remark 2.3. By a result of Deligne and Tate ([D1] and [T]), each isomorphism class of an abelian variety over 𝔽 is determined by the action of the relative Frobenius map on its 𝔩–adic Tate module (over the field of definition), all abelian varieties sitting over the points in Ξ are non-isomorphic.

2.4. Linear independence.

Keeping the notation and the assumption of Proposition 2.7, let C = C_{Ξ} denote the space of functions defined over Ξ with values in P^{1}(𝔽) = 𝔽 ⊔ {∞}. The class group Cl_{∞} acts on C by translation: f(x) ↦ f(xy) (y ∈ Cl_{∞}). By Zariski density of Ξ in V = V_{/𝔽}^{(p)}, we can embed into C the function field 𝔽(V) of V.

Proposition 2.8. For a finite set Δ = {γ_{1}, …, γ_{m}} ⊂ Cl_{∞} independent modulo C^{alg}, the fields γ_{1}(𝔽(V)), …, γ_{m}(𝔽(V)) are linearly disjoint over 𝔽 in C_{Ξ}. In other words, the subset {(x(δ(A))_{δ∈Δ}|x(A) ∈ Ξ} is Zariski dense in the product V_{/𝔽}^{Δ} of Δ copies of V_{/𝔽}.

The following proof is modeled after an argument of C.-L. Chai ([C4] Section 8) proving a special subvariety (called a Tate linear subvariety) of a product of copies of Hilbert modular varieties is actually a Shimura subvariety. Particularly the use of Zarhin’s theorem is his idea, which shortened substantially the original argument in the older versions of this paper.

Proof. We write x_{m}(A) for (x(γ_{j}A)) ∈ V^{m}. Let X be the Zariski closure of

{x_{m}(A) ∈ V_{/𝔽}^{m}|x(A) ∈ Ξ}

in V^{m}. The assertion follows from the density of the above set: X = V^{m}. We shall prove this by induction on m. The case m = 1 is already treated (by Proposition 2.7); so, we assume that m ≥ 2.

The projection of X to any of the factor V^{m−1} is surjective by the induction hypothesis. Thus we can choose an irreducible component X_{1} of X so that the projection to each factor V^{m−1} is onto. By the same argument as in the proof of Proposition 2.7, we may assume that x_{m}(A) is a smooth point on X. As we have seen, the action of τ(ρ_{R}(α^{(𝔩)})) sends x(B) to x([α^{(𝔩)}]B). Recall the image T ⊂ C^{alg} of α^{(𝔩)} for α ∈ M^{×} prime to 𝔩 with (α) trivial in Cl_{r} and Cl_{n}, where n is the index such that A is a proper R_{n}–ideal. Then the action of α^{(𝔩)} ∈ T as described above coincides with the action of τ(ρ_{R}(α^{(𝔩)})). Thus (X, x_{m}(A)) is stable under the diagonal action T . Since there are only finitely many irreducible components of X crossing at x_{m}(A), the smoothness of X at x_{m}(A) tells us that X_{1} is stable under T . The p–adic closure of T coincides with T (ℤ_{p}). Let O_{Ig^{m},x^{m}/𝔽} for x = x(A) be the stalk at x^{m}. We apply Proposition 2.6 to X_{1}, which claims that either X_{1} = Ig^{m} or X_{1} is algebraic over each factor Ig_{/𝔽}^{m−1}.

By changing γ_{j} in the coset γ_{j}C^{alg}, we may assume that γ_{j}(x(A)) = x(A). We first suppose that m = 2. If X_{1} = Ig^{2}, there is nothing to prove. Suppose X_{1} is a proper subscheme. Let i : X_{1} ↪ Ig^{2} and 𝕐 = i^{∗}𝕏^{2} = 𝕏^{2} ×_{Ig^{2}} X_{1}. Since the two projections π_{j} : X_{1} → Ig are dominant, End(π_{j}^{∗}𝕏) ⊗ ℚ = F for π_{j}^{∗}𝕏 = 𝕏 ×_{Ig} X_{1}. Thus there are only two possibilities of End^{ℚ}(𝕐) = End(𝕐) ⊗ ℚ: Either End^{ℚ}(𝕐) = F × F or End^{ℚ}(𝕐) = M_{2}(F).

We suppose End^{ℚ}(𝕐) = F × F and try to get a contradiction (in order to prove that End^{ℚ}(𝕐) = M_{2}(F)). We pick a sufficiently small K^{(p)} so that V_{K} is smooth, and write 𝕐_{K} for the abelian scheme i^{∗}𝕏_{K}^{2} over the image X_{1,K} of X_{1} in Ig_{K}^{2} for the Igusa tower Ig_{K} over V_{K}. Then X_{1,K} and 𝕐_{K} are varieties over a finite field 𝔽_{q} for a p–power q. Let η be the generic point of X_{1,K/𝔽_{q}}, and write \overline{η} for the geometric point over η so that 𝔽_{q}(\overline{η}) is a separable algebraic closure of 𝔽_{q}(η). Take an odd prime ℓ, and consider the ℓ–adic Tate module T_{ℓ}𝕐\overline{η} for the generic fiber 𝕐\overline{η} of 𝕐. We consider the image of the Galois action Im(Gal(𝔽_{q}(\overline{η})/𝔽_{q}(η))) in GL_{O_{ℓ}×O_{ℓ}}(T_{ℓ}𝕐\overline{η}). Then by a result of Zarhin ([Z] and [DAV] Theorem V.4.7) combined with a standard argument, Im(Gal(𝔽_{q}(\overline{η})/𝔽_{q}(η))) is an ℓ–adically open subgroup of ℚ_{ℓ}–points of a reductive group G defined over ℚ. By Zarhin’s theorem, the centralizer of G is End(𝕐) ⊗ ℚ_{ℓ}, and hence the derived group G_{1}(ℚ_{ℓ}) of G(ℚ_{ℓ}) has to be SL_{2}(F_{ℓ} × F_{ℓ}). Then by Chebotarev’s density theorem, we can find a set of closed points x ∈ X_{1,K}(𝔽) with positive density such that the Zariski closure in G of the subgroup generated by the Frobenius element F rob_{x} ∈ Im(Gal(𝔽_{q}(\overline{η})/𝔽_{q}(η))) at x is a torus containing a maximal torus T_{x} of the derived group G_{1} of G. In particular the centralizer of T_{x} in G_{1} is itself. Thus 𝕐_{x} is isogenous to a product of two non-isogenous absolutely simple abelian varieties Y_{1} and Y_{2} with multiplication by F defined over a finite field. The endomorphism algebra M_{j} = End^{ℚ}(Y_{j}) is a CM quadratic extension of F generated over ℚ by the relative Frobenius map ϕ_{j} induced by F rob_{x}. The relative Frobenius map F rob_{x} acting on X_{∗}(Îg_{x_{1}}) ≅ O_{p} has [F : ℚ] distinct eigenvalues {ϕ_{1}^{(1−c)σ}|σ ∈ Σ_{1}} for the CM type Σ_{1} of Y_{1}, which differ from the eigenvalues of F rob_{x} ∈ End(Y_{2}) on X_{∗}(Îg_{x_{2}}) ≅ O_{p}. Since it has been proven in [H03b] Proposition 3.11 (using many results of Chai in [C4]) that over an open dense subscheme U of X_{1}, the formal completion of U at (x_{1}, x_{2}) ∈ X_{1} ⊂ Ig^{2} is canonically isomorphic to a formal subtorus Ẑ ⊂ Îg_{x_{1}} × Îg_{x_{2}} with co-character group X_{∗}(Ẑ) ≅ O_{p}, we may assume that our point x = (x_{1}, x_{2}) as above is in U (because such x has positive density). Projecting X_{∗}(Ẑ) down to the left and the right factor Ig, the projection map X_{∗}(Ẑ) → X_{∗}(Îg_{x_{j}}) is actually an injection commuting with the action of F rob_{x}. Thus F rob_{x} has more than [F : ℚ] distinct eigenvalues on X_{∗}(Ẑ), which is a contradiction. Thus we conclude that End^{ℚ}(𝕐) = M_{2}(F). This implies that we have a prime-to–p isogeny α : π_{1}^{∗}𝕏_{K} → π_{2}^{∗}𝕏_{K}. Writing η_{j} for the level structure (including the ordinary level structure at p) of π_{j}^{∗}𝕏, we find that α ∘ η_{1} = η_{2} ∘ g_{K}^{−1} for g_{K} ∈ O_{p}^{×} × G(𝔸^{(p∞)}). Thus X_{1,K} is a graph of the action of g_{K} in Ig_{K} × Ig_{K}.

Shrinking K and taking the limit, we have g = lim_{K→{1}} g_{K} in O_{p}^{×} × G(𝔸^{(p∞)}), and X_{1} is the graph of the action of g. Thus gγ_{1} = γ_{2}. Since g fixes x(A), we find that it is induced by α ∈ End^{ℚ}(X(A)) = M prime to p. In other words, γ_{1}^{−1}γ_{2} ∈ C^{alg}, which is a contradiction. Thus X_{1} = Ig^{2}, and we get the desired assertion when m = 2.

We now deal with the case where m > 2. By the induction hypothesis (and Proposition 2.6), we know that dim_{𝔽} X_{1} = d(m − 1) or dm for d = [F : ℚ]. We need to prove dim_{𝔽} X_{1} = dm. Suppose now that dim_{𝔽} X_{1} = d(m − 1) to get a contradiction. We consider the partial product for S = Spec(O_{x}):

T_{i,j} = ({x_{i−1}(A)}× S^{i} ×{x_{j−i−1}(A)}× S^{j} ×{x_{m−j}(A)}) ⊂ S^{m}

for 1 ≤ i < j ≤ m. We see that dim(X̂_{1} ∩ T̂_{i,j}) ≥ d by Chai’s lemma (Lemma 2.4), taking intersection of the formal cocharacter groups of T̂_{i,j} and X̂_{1}. If the strict inequality holds for all (i, j), by Proposition 2.6 applied to (X_{1} ∩T_{i,j}) ⊂ T_{i,j} ≅ S^{2}, we find that dim(X̂_{1} ∩ T̂_{i,j}) = 2d and X_{1} ⊃ T_{i,j}. Again by Chai’s lemma, this implies X_{1} = S^{m}, because the cocharacter group of X̂_{1} contains the cocharacter group of T̂_{i,j} for all (i, j). This contradicts our assumption: dim_{𝔽} X_{1} = d(m − 1). Thus we find that dim(T_{i,j} ∩ X_{1}) = d for at least one pair (i, j). The argument in the case of m = 2 tells us that γ_{i}^{−1}γ_{j} ∈ C^{alg}, which is a contradiction. ⊓ ⊔

The linear independence applied to the global sections of a line bundle (regarded as sitting inside the function field) yields the following result:

Corollary 2.9. Let the notation and the assumption be as in Proposition 2.8. Let L be a line bundle over Ig_{/𝔽}. Then for a finite set Δ ⊂ Cl_{∞} independent modulo C^{alg} and a set {s_{δ} ∈ L}_{δ∈Δ} of non-constant global sections s_{δ} of L finite at Ξ, the functions s_{δ} ∘ δ (δ ∈ Δ) are linearly independent in C_{Ξ}.

3. Measure associated to a Hecke eigenform

To each Hecke eigenform f, we associate a measure supported on Cl_{∞}.

3.1. Hecke relation among CM points.

We write the left action: G(𝔸^{(∞)}) × Sh^{(p)} → Sh^{(p)} simply as (g, x) ↦ g(x) := τ(g)^{−1}(x). Here the action of τ(g) is a right action induced by η ↦ η ∘ g. For each point x = (X, Λ, η) ∈ Sh, we can associate a lattice L̂ = η^{−1}(T (X)) ⊂ (F_{𝔸}^{(∞)})^{2}. Then the level structure η is determined by the choice of a base w = (w_{1}, w_{2}) of L̂ over Ô. In view of the base w, the inverted action x ↦ g(x) is matrix multiplication: ^{t}w ↦ g^{t}w, because (η ∘ g^{−1})^{−1}(T (X)) = gη^{−1}(T (X)) = gL̂.

Recall the order R_{n} = O + 𝔩^{n}R ⊂ M and the class groups Cl_{n} = Pic(R_{n}) and Cl_{∞} = \underleftarrow{lim}_{n}Cl_{n}. By class field theory, Cl_{n} gives the Galois group Gal(H_{n}/M) of the ring class field H_{n} of conductor 𝔩^{n}. The ideal 𝔩_{n} = 𝔩 + 𝔩^{n}R = 𝔩R_{n−1} is a prime ideal of R_{n} but is not proper (it is a proper ideal of R_{n−1}). Since X(R_{n})[𝔩_{n}] ≅ R_{n}/𝔩_{n} = O/𝔩 and 𝔩_{n}R_{n−1} ⊂ R_{n}, we find that X(R_{n})[𝔩_{n}] = R_{n−1}/R_{n}. In other words, we have X(R_{n})/X(R_{n})[𝔩_{n}] ≅ X(R_{n−1}). We pick a subgroup C ⊂ X(R_{n})[𝔩] isomorphic to O/𝔩 but different from X(R_{n})[𝔩_{n}]. We look into X(R_{n})/C. Take a lattice 𝔄 so that X(R_{n})/C = X(𝔄) ⇔ 𝔄/R_{n} = C. Since C is an O–submodule, 𝔄 is an O–lattice of M. Since 𝔩C = 0, we find 𝔩R_{n}𝔄 ⊂ 𝔄. Thus 𝔄 is R_{n+1}–ideal, because R_{n+1} = O + 𝔩R_{n}. Since C is not an R_{n}–submodule, the ideal 𝔄 is not R_{n}–ideal; so, it is a proper R_{n+1}–ideal. Since C generates over R_{n} all 𝔩–torsion points of X(R_{n}), we find R_{n}𝔄 = 𝔩^{−1}R_{n}. In this way, we have created ℓ proper R_{n+1}–ideals 𝔄 with 𝔄R_{n} = 𝔩^{−1}R_{n}.

We have chosen a base w = (w_{1}, w_{2}) of R̂ over Ô in 2.1. We also specified the base of R̂_{n}^{(𝔩)} to be w^{(𝔩)} there, because R̂_{n}^{(𝔩)} = R̂^{(𝔩)}. To specify the base w_{𝔩} of R_{𝔩}, we take d ∈ O_{𝔩} so that R_{𝔩} = O_{𝔩}[\sqrt{d}] ⊂ M_{𝔩}. We assume that d is a 𝔩–adic unit if 𝔩 is unramified in M/F and d generates 𝔩 if 𝔩 ramifies in M/F. Then we choose w_{𝔩} = (1, \sqrt{d}).

Since the base of R_{n,𝔩} is given by α_{n}^{t}(1, \sqrt{d}) for α_{n} = ( \begin{matrix}1 & 0 \\ 0 & ϖ_{𝔩}^{n} \end{matrix} ) with a prime element ϖ_{𝔩} of O_{𝔩}, we find that α_{n}(x(R)) = x(R_{n}) and α_{1}(x(R_{n−1})) = x(R_{n}). Moreover, for a suitable u ∈ O

ϖ_{𝔩}(x(A)) = ( \begin{matrix}1 & \frac{u}{ϖ_{𝔩}} \\ 0 & 1 \end{matrix} ) (x(R_{n+1})) if x(A) = x(R_{n})/C for C as above,  (3.1)

because the base of ϖ_{𝔩}A_{𝔩} is given by ( \begin{matrix}1+ϖ_{𝔩}^{n}u\sqrt{d} \\ ϖ_{𝔩}^{n+1}\sqrt{d} \end{matrix} ) = ( \begin{matrix}1 & \frac{u}{ϖ_{𝔩}} \\ 0 & 1 \end{matrix} ) α_{n+1} ( \begin{matrix}1 \\ √ & \overline{d} \end{matrix} ) . Here the

action of ϖ_{𝔩} : x(A) ↦ ϖ_{𝔩}(x(A)) may bring x(A) on a geometrically irreducible component of Sh^{(p)} to a different one.

Now we consider x(A) in V_{K} for an open subgroup K ⊂ G(𝔸^{(∞)}) containing Z(ℤ̂). By repeating (3.1), if x(A) = x(R_{n})/C for C ≅ O/𝔩^{m} with C ∩X(R_{n})[𝔩_{n}] = {0}, then A is a proper R_{n+m}–ideal. If further 𝔩^{m} is generated by an element ϖ ∈ F, we get x(A) = x(ϖA) = ϖ_{𝔩}^{m}(x(A)) in V_{K} (because ϖ/ϖ_{𝔩}^{m} ∈ K) and

x(A) = ( \begin{matrix}1 & \frac{u}{ϖ^{m}_{𝔩}} \\ 0 & 1 \end{matrix} ) (x(R_{n+m})) = ( \begin{matrix}1 & \frac{u}{ϖ} \\ 0 & 1 \end{matrix} ) (x(R_{n+m})) for a suitable u ∈ O. (3.2)

The set {x(A)|[AR_{n}] = [𝔄]}/ ≅ for A ∈ Cl_{n+m} running through ideal classes A projecting down to a given ideal class [𝔄] ∈ Cl_{n} is in bijection with O/𝔩^{m} by associating u to A in (3.2) (see Proposition 4.2).

3.2. Geometric modular forms.

Let k be a weight of T = Res_{O/ℤ}𝔾_{m}, that is, k : T (A) = (A ⊗_{ℤ} O)^{×} → A^{×} is a homomorphism given by (a ⊗ ξ)^{k} = ∏(aξ^{σ})^{k_{σ}} for integers k_{σ} indexed by field embeddings σ : F ↪ \overline{ℚ}. Let B be a base ring, which is a W–algebra. We consider quadruples (X, \overline{Λ}, η^{(p)}, ω)_{/A} for a B–algebra A with a differential ω generating H^{0}(X, Ω_{X/A}) over A ⊗_{ℤ} O. We impose the following condition:

η^{(p)}(L̂_{𝔠}^{(p)}) = T (X) ⊗_{ℤ} ℤ̂^{(p)} for L_{𝔠} = O ⊕ 𝔠^{∗} with a fixed 𝔠.  (3.3)

Under this condition, as seen in 2.3, the classification up to prime-to–p isogenies of the quadruples is equivalent to the classification up to isomorphisms. A modular form f (integral over B) of weight k is a functorial rule of assigning a value f(X, \overline{Λ}, η^{(p)}, ω) ∈ A to (the A–isomorphism class of) each quadruple (X, \overline{Λ}, η^{(p)}, ω)_{/A} (called a test object) defined over a B–algebra A. Here Λ is a 𝔠–polarization which (combined with η^{(p)}) induces L_{𝔠} ∧ L_{𝔠} ≅ 𝔠^{∗} given by ((a ⊕ b), (a′ ⊕ b′)) ↦ ab′ − a′b. The Tate test object at the cusp (𝔞, 𝔟) for two factional ideals with 𝔞^{∗}𝔟 = 𝔠^{∗} is an example of such test objects. The Tate semi-AVRM Tate_{𝔞,𝔟}(q) is defined over ℤ[[q^{ξ}]]_{ξ∈(𝔞𝔟)_{+}} and is given by the algebraization of the formal quotient (𝔾̂_{m} ⊗ 𝔞^{∗})/q^{𝔟} (see [K1] Chapter I and [HT] 1.7 for details of this construction). The rule f is supposed to satisfy the following three axioms:

(G1) For a B–algebra homomorphism ϕ : A → C, we have

f((X, \overline{Λ}, η^{(p)}, ω) ×_{A,ϕ} C) = ϕ(f(X, \overline{Λ}, η^{(p)}, ω)).

(G2) f is finite at all cusps, that is, the q–expansion of f at every Tate test object does not have a pole at q = 0 (see [K1] Chapter I and [HT] 1.7).

(G3) f(X, \overline{Λ}, η^{(p)}, αω) = α^{−k}f(X, \overline{Λ}, η^{(p)}, ω) for α ∈ T (A).

We write G_{k}(𝔠; B) for the space of all modular forms f satisfying (G1-3) for B– algebras A. We put

G_{k}(B) = ⨁_{𝔠} G_{k}(𝔠; B),  (3.4)

where 𝔠 prime to ℓp runs over a representative set of strict ideal classes of F. An element g ∈ G(𝔸^{(∞)}) fixing L̂_{𝔠} acts on f ∈ G_{k}(𝔠; B) by

f|g(X, \overline{Λ}, η^{(p)}, ω) = f(X, \overline{Λ}, η^{(p)} ∘ g, ω).

For a closed subgroup K ⊂ K_{𝔠} = GL(L̂_{𝔠}) ∩ G_{1}(𝔸^{(∞)}), we write G_{k}(𝔠; K; B) for the space of all K–invariant modular forms; thus,

G_{k}(𝔠; K; B) = H^{0}(K, G_{k}(𝔠; B)).

Take an O–ideal 𝔑 prime to p𝔠. Then the 𝔑–component of K_{𝔠} is SL_{2}(O_{𝔑}). Let

Γ_{0}(𝔑) = \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL_{2}(O_{𝔩})|c ∈ 𝔑O_{𝔑}\} ,

and define for an open subgroup Γ ⊂ SL_{2}(O_{𝔑})

G_{k}(Γ; B) = ⨁_{𝔠} G_{k}(𝔠; Γ × K_{𝔠}^{(𝔑)}; B).

A W –algebra B is called a p–adic algebra if B = \underleftarrow{lim}_{n}B/p^{n}B. We write η_{ord} for the pair of level structures (η_{p}^{ord} : μ_{p^{∞}} ⊗𝔡^{−1} ↪ X[p^{∞}], η^{(p)}). A p–adic modular form f over a p–adic W –algebra B is a functorial rule of assigning a value in A to triples (X, \overline{Λ}, η_{ord})_{/A} with 𝔠–polarization class \overline{Λ} satisfying an obvious version of (G1-2) for p–adic B–algebras A (not just B–algebras). In general, we do not impose (G3) on p–adic modular forms. We write V (𝔠; B) for the space of p–adic modular forms defined over B. We again define

V (B) = ⨁_{𝔠} V (𝔠; B) and V (Γ; B) = ⨁_{𝔠} V (𝔠; Γ × K_{𝔠}^{(p𝔑)}; B),  (3.5)

where V (𝔠; K; B) = H^{0}(K, V (𝔠; B)). For f ∈ V (B), we write f_{𝔠} ∈ V (𝔠; B) for the 𝔠–component of f, and we say that f is of level 𝔑 if f in either in G_{k}(Γ; B) or in V (Γ; B) for Γ ⊂ SL_{2}(O_{𝔑}).

Since η_{p}^{ord} induces the identification η_{p}^{ord} : 𝔾̂_{m} ⊗O^{∗} ≅ X̂ for the formal completion of X along the origin, by pushing forward the differential \frac{dt}{t}, we can associate (X, \overline{Λ}, η^{(p)}, η_{p,∗}^{ord} \frac{dt}{t}) to a quadruple (X, \overline{Λ}, η_{p}^{ord}, η^{(p)}). In this way, any modular form f satisfying (G1-3) can be regarded as a p–adic modular form by

f(X, \overline{Λ}, η_{ord}) = f(X, \overline{Λ}, η^{(p)}, η_{p,∗}^{ord} \frac{dt}{t}).

By the q–expansion principle (cf. [K1] (1.2.15-16) and [PAF] Corollary 4.23), we thus have a canonical embedding of G_{k}(B) into V (B) which keeps the q– expansion. A p–adic modular form associated to a modular form in G_{k}(B) satisfies the following replacement of (G3):

(g3) f(X, \overline{Λ}, α · η_{p}^{ord}, η^{(p)}) = α^{−k}f(X, \overline{Λ}, η_{p}^{ord}, η^{(p)}) for α ∈ O_{p}^{×}.

Although we do not impose the condition (G3) on p–adic modular forms f, we limit ourselves to the study of forms satisfying the following condition (G3′) in order to define the modified value f([A]) later at CM points x(A) truly independent of the choice of A in its proper ideal class. Here abusing our notation, x(A) is the quadruple (X(A), Λ(A), η_{ord}(A), ω(A))_{/W} introduced in 2.1. We consider the torus T_{M} = Res_{R/ℤ}𝔾_{m} and identify its character group X^{∗}(T_{M}) with the module ℤ[Σ ⊔ Σc] of formal linear combinations of embeddings of M into \overline{ℚ}. By the identity: (X(αA), Λ(αA), αη_{ord}(αA))_{/W} ≅ (X(A), αα^{c}Λ(A), η_{ord}(A) ∘ ρ_{A}(α))_{/W}, we may assume that for k, κ ∈ ℤ[Σ],

(G3′) f(x(αA)) = f(ρ_{R}(α^{(𝔩)})(x(A))) = α^{−k−κ(1−c)}f(x(A)) for α ∈ T_{M}(ℤ_{(ℓ)}).

It is known that for the p–adic differential operator d_{σ} of Dwork-Katz ([K1] 2.5-6) corresponding to \frac{1}{2πi} \frac{∂}{∂z_{σ}} for σ ∈ Σ, d^{κ}f satisfies (G3′) if f ∈ G_{k}(B).

3.3. Hecke operators.

Suppose that the 𝔩–component Γ_{𝔩} of the level subgroup Γ is equal to Γ_{0}(𝔩^{r}) (r ≥ 0). Let e_{1} = ^{t}(1, 0), e_{2} = ^{t}(0, 1) be the standard basis of F^{2} ⊗ 𝔸^{(p∞)}. Then, under (3.3), for each triple (X, \overline{Λ}, η_{ord})_{/A} with η_{ord} = η_{p}^{ord} × η^{(p)},

C = η_{𝔩}(𝔩^{−r}O_{𝔩}e_{1} + O_{𝔩}e_{2})/η_{𝔩}(O_{𝔩}^{2})

gives rise to an A–rational cyclic subgroup of X of order 𝔩^{r}, that is, a finite group subscheme defined over A of X_{/A} isomorphic to O/𝔩^{r} étale locally. Since Γ_{0}(𝔩^{r}) fixes (𝔩^{−r}O_{𝔩}e_{1} + O_{𝔩}e_{2})/O_{𝔩}^{2}, the level Γ_{0}(𝔩^{r}) moduli problem is equivalent to the classification of quadruples (X, \overline{Λ}, C, \overline{η}_{ord}^{(𝔩)})_{/A} for a subgroup C of order 𝔩^{r} in X, where η_{ord}^{(𝔩)} is the (p–ordinary) level structure outside 𝔩. Thus we may define for f ∈ G_{k}(Γ; B) with Γ as above the value of f at (X, \overline{Λ}, C, η^{(p𝔩)}, ω) by f(X, \overline{Λ}, C, η^{(p𝔩)}, ω) := f(X, \overline{Λ}, η^{(p)}, ω). When f is a p–adic modular form of level Γ, we replace the ingredient ω by the ordinary level structure η_{p}^{ord} in order to define the value f(X, \overline{Λ}, C, η^{(p𝔩)}, η_{p}^{ord}).

We shall define Hecke operators T (1, 𝔩^{n}) and U(𝔩^{n}) over (p–adic) modular forms of level Γ (with Γ_{𝔩} = Γ_{0}(𝔩^{r})). The operator U(𝔩^{n}) is defined when r > 0, and T (1, 𝔩^{n}) is defined when r = 0. Since 𝔩 is prime to p (and B is a W–algebra), any cyclic subgroup C′ of X of order 𝔩^{n} is isomorphic to O/𝔩^{n} étale locally. We make the quotient π : X ↠ X/C′, and Λ, η_{p}^{ord} and ω induce canonically a polarization π_{∗}Λ a canonical level structure π_{∗}η_{p}^{ord} = π ∘ η_{p}^{ord}, π_{∗}η^{(p𝔩)} = π ∘ η^{(p𝔩)} and a differential (π^{∗})^{−1}ω on X/C′. If C′ ∩ C = {0} for the Γ_{0}(𝔩^{r})–structure C (in this case, we call that C′ and C are disjoint), π(C) = C + C′/C′ gives rise to the level Γ_{0}(𝔩^{r})– structure on X/C′. We write \underline{X/C′} for the new test object of the same level as the

test object \underline{X} = (X, \overline{Λ}, C, η_{ord}^{(𝔩)}, ω) we started with. When f is p–adic, we suppose not to have ω in \underline{X}, and when f is classical, we ignore the ingredient η_{p}^{ord} in \underline{X}. Then we define (for r > 0)

f|U(𝔩^{n})(\underline{X}) = \frac{1}{N(𝔩^{n})} ∑_{C′} f(\underline{X/C′}),  (3.6)

where C′ runs over all étale cyclic subgroups of order 𝔩^{n} disjoint from C. We also define (for r = 0)

f|T (1, 𝔩^{n})(\underline{X}) = \frac{1}{N(𝔩^{n})} ∑_{C} f(\underline{X/C}),  (3.7)

where C runs over all étale cyclic subgroups of order 𝔩^{n} and \underline{X} and \underline{X/C} do not contain any datum of cyclic 𝔩–subgroups. The newly defined f|U(𝔩^{n}) and f|T (1, 𝔩^{n}) are modular forms of the same level as f and U(𝔩^{n}) = U(𝔩)^{n}. Since the polarization ideal class of X/C′ is given by 𝔠𝔩^{n} for the polarization ideal class 𝔠 of X, the operators U(𝔩^{n}) and T (1, 𝔩^{n}) permute the components f_{𝔠}.

We recall some other isogeny actions on modular forms. For ideals 𝔄 in F, we can think of the association X ↦ X ⊗_{O} 𝔄 for each AVRM X. This operation will be made explicit in terms of the lattice L = π_{1}(X) in Lie(X) in (4.12). There are a natural polarization and a level structure on X ⊗ 𝔄 induced by those of X (as specified later below (4.12)). Writing (X, Λ, η) ⊗ 𝔄 for the triple made out of (X, Λ, η) after tensoring 𝔄, we define f|⟨𝔄⟩(X, Λ, η) = f((X, Λ, η) ⊗ 𝔄) (see [PAF] 4.1.9 for more details of this definition). For X(A), we have ⟨𝔄⟩(\underline{X}(A)) = \underline{X}(𝔄A). The effect of the operator ⟨𝔄⟩ on the Fourier expansion at (𝔞, 𝔟) is given by that at (𝔄^{−1}𝔞, 𝔄𝔟) (by [DR] 5.8; see also [PAF] 4.2.9).

Let 𝔮 be a prime ideal of F outside p𝔩. For a test object (X, \overline{Λ}, C, η_{ord}^{(𝔮)}, ω) of level Γ_{0}(𝔮), we can construct canonically its image under 𝔮–isogeny:

[𝔮](X, \overline{Λ}, C, η_{ord}^{(p𝔮)}, ω) = (X′, \overline{Λ}, π_{∗}η_{ord}^{(p𝔮)}, \overline{η}_{𝔮}, (π^{∗})^{−1}ω)

for the projection π : X ↠ X′ = X/C, where \overline{η}_{𝔮} = η_{𝔮} · GL_{2}(O_{𝔮}) for any level 𝔮–structure η_{𝔮} identifying T_{𝔮}(X′) with O_{𝔮}^{2}. Then we have a linear operator [𝔮] : V (Γ; B) → V (Γ_{0}(𝔮) × Γ; B) given by f|[𝔮](\underline{X}) = f([𝔮](\underline{X})). See (4.14) for the description of this operator in terms of the lattice of X.

If 𝔮 splits into 𝔔\overline{𝔔} in M/F, choosing η_{𝔮} induced by

X(A)[𝔮^{∞}] ≅ M_{𝔔}/R_{𝔔} × M\overline{𝔔}/R\overline{𝔔} ≅ F_{𝔮}/O_{𝔮} × F_{𝔮}/O_{𝔮},

we always have a canonical level 𝔮–structure on X(A) dependent on the choice of the factor 𝔔. Then [𝔮](X(A)) = X(A[𝔔]^{−1}) for [𝔔] ∈ Cl_{∞}. When 𝔮 ramifies in M/F as 𝔮 = 𝔔^{2}, X(A) has a subgroup C = X(A)[𝔔_{n}] isomorphic to O/𝔮 for 𝔔_{n} = 𝔔 ∩ R_{n}; so, we can still define [𝔮](X(A)) = X(A𝔔_{n}^{−1}) = X(A[𝔔]^{−1}).

The effect on the q–expansion of the operator [𝔮] can be computed similarly to ⟨𝔄⟩ (e.g. [DR] 5.8; see also [PAF] 4.2.9), and the q–expansion of f|[𝔮] at the cusp (𝔞, 𝔟) is given by the q–expansion of f at the cusp (𝔮𝔞, 𝔟).

These operators [𝔮] and ⟨𝔄⟩ change polarization ideals (as we will see later in 4.2); so, they permute components f_{𝔠}. By the q–expansion principle ([K1] (1.2.15)), f ↦ f|[𝔮] and f ↦ f|⟨𝔄⟩ are injective.

3.4. Anti-cyclotomic measure.

Choose a Hecke character λ with infinity type k + κ(1 − c) of conductor 1. Then by (G3′), f([A]) = λ(A)^{−1}f(x(A)) for A prime to p depends only on the class of A in Cl_{n} if n ≥ m for the conductor 𝔩^{m} of λ. For the p–adic avatar λ̂(x) = λ(xR)x_{p}^{k+κ(1−c)}, we also have f([A]) = λ̂(A)^{−1}f(x(A)). This new definition is valid even for A with non-trivial common factor with p. Hereafter we regard f as a function of Cl^{(∞)} = ⨆_{n} Cl_{n} (embedded into Sh_{/W}^{(p)} or Ig_{/𝔽} by A ↦ x(A)).

Suppose that f is defined over a W–algebra A in which ℓ is invertible. By the result in 3.1 combined with (3.6), we have, for an integer n > m,

∑_{A∈Cl_{n}, A↦𝔩^{m−n}𝔄∈Cl_{m}} f([A]) = (λN(𝔩))^{n−m}f|U(𝔩^{n−m})([𝔄]),  (3.8)

where A runs over all elements in Cl_{n} which project down to 𝔩^{m−n}𝔄 ∈ Cl_{m}. Let h be the strict class number of F. If h|m − n, the sum on the left-hand-side of (3.8) is over A ∈ Cl_{n} projecting down to 𝔄 ∈ Cl_{m}, because [𝔩^{m−n}𝔄] = [𝔄] in Cl_{m}.

We suppose that f|U(𝔩) = (a/λN(𝔩))f with a unit a ∈ A. For each function ϕ : Cl_{∞} → A factoring through Cl_{mh}, we define

∫_{Cl_{∞}} ϕdφ_{f} = a^{−mh} ∑_{A∈Cl_{mh}} ϕ(A^{−1})f([A]).  (3.9)

Then for n > m, we find

a^{−nh} ∑_{A∈Cl_{nh}} ϕ(A^{−1})f([A]) = a^{−mh} ∑_{𝔄∈Cl_{mh}} ϕ(𝔄^{−1})a^{(m−n)h} ∑_{A∈Cl_{nh}, A↦𝔄} f([A])

=^{(3.8)} a^{−mh} ∑_{𝔄∈Cl_{mh}} ϕ(𝔄^{−1})a^{(m−n)h}f|U((π)^{(n−m)h})([𝔄]) = ∫_{Cl_{∞}} ϕ(x)dφ_{f}(x).

Thus φ_{f} gives an A–valued distribution on Cl_{∞} well defined independently of the choice of m for which ϕ factors through Cl_{m}, because U(𝔩^{α}) = U(𝔩)^{α}.

Classical modular forms are actually defined over a number field; so, we assume that f is defined over the localization V of the integer ring in a number field E. We assume that E contains M′ for the reflex (M′, Σ′) of (M, Σ). We write P|p for the prime ideal of the p–integral closure Ṽ of V in \overline{ℚ} corresponding to i_{p} : \overline{ℚ} ↪ \overline{ℚ}_{p}. More generally, if f = d^{κ}g for a classical modular form g integral over V, the value f([A]) is algebraic, abelian over M′ and P–integral over V by a result of Shimura and Katz (see [Sh2] and [K1]).

Let f = d^{κ}g for g ∈ G_{k}(Γ_{0}(𝔩); V). Suppose f|U(𝔩) = (a/N(𝔩))f for a giving a unit of Ṽ/P. For the moment, let φ be the measure associated to f with values in A = Ṽ. We have a well defined measure φ mod P. Let E_{f} be the field generated by f([A]) over E[μ_{ℓ^{∞}}]. Then E_{f}/E is an abelian extension unramified outside ℓ, and we have the Frobenius element σ_{𝔟} ∈ Gal(E_{f}/E) (that is, the image of 𝔟 under the Artin reciprocity map) for each ideal 𝔟 of E prime to ℓ. By Shimura’s reciprocity law ([ACM] 26.8), we find for σ = σ_{𝔟}, x(A)^{σ} = x(N(𝔟)^{−Σ′}A) for the norm N : E → M′. As for η_{p}^{ord}(A), we find σ ∘ η_{p}^{ord}(A) = uη_{p}^{ord} for u ∈ R_{Σ_{p}}^{×}. Since A_{p} ≅ R_{p}, we have X(R)[p^{∞}] ≅ X(A)[p^{∞}] as a Galois module. Thus we conclude u = ψ(𝔟) for the Hecke character ψ of E_{𝔸}^{×}/E^{×} giving rise to the zeta function of X(R). From this, if we extend E further if necessary, we see f([A])^{σ} = f([N(𝔟)^{−Σ′}A]) for any ideal 𝔟, since ψ(𝔟) ∈ M generates the ideal N(𝔟)^{Σ′} ⊂ M ([ACM] Sections 13 and 19) and hence ψ(𝔟)^{kΣ+κ(1−c)} = λ(N(𝔟)^{Σ′}). We then have

σ · (∫_{Cl_{∞}} ϕ(x)dφ_{f}(x)) = ∫_{Cl_{∞}} σ ∘ ϕ(N(𝔟)^{Σ′}x)dφ_{f}(x),  (3.10)

where N(𝔟) is the norm of 𝔟 over M′. Writing 𝔽_{p^{r}} for V/P ∩ V, any modular form defined over 𝔽_{p^{r}} is a reduction modulo P of a classical modular form defined over V of sufficiently high weight. Thus the above identity is valid for σ = Φ^{s} (s ∈ ℤ) for the Frobenius element Φ ∈ Gal(𝔽/𝔽_{p^{r}}). In this case, N(𝔟) is a power of a prime ideal 𝔭|p in M′.

We now assume that A = 𝔽 = Ṽ/P and regard the measure φ_{f} as having values in 𝔽. Then (3.10) shows that if ϕ is a character χ of Cl_{∞}, for σ ∈ Gal(𝔽/𝔽_{p^{r}}),

∫_{Cl_{∞}} χ(x)dφ_{f}(x) = 0 ⟺ ∫_{Cl_{∞}} σ ∘ χ(x)dφ_{f}(x) = 0.  (3.11)

Decompose Cl_{∞} into a product of the maximal torsion-free ℓ–profinite subgroup Γ_{f} and a finite group Δ. Let 𝔽[f, λ] be the finite subfield of 𝔽 generated by all ℓ|Δ|–th roots of unity over the field 𝔽_{p^{r}} of rationality of f and λ. For any finite extension κ/𝔽[f, λ], we consider the trace map: Tr_{κ/𝔽[f,λ]}(ξ) = ∑_{σ∈Gal(κ/𝔽[f,λ])} σ(ξ) for ξ ∈ κ. If χ : Cl_{n} → 𝔽^{×} is a character, we find, for d = [Im(χ) : Im(χ) ∩ 𝔽[f, λ]^{×}],

∫_{Cl_{∞}} Tr_{𝔽[f,λ](χ)/𝔽[f,λ]} ∘ χ(y^{−1}x)dφ_{f}(x) = \frac{d}{a^{n}} ∑_{A∈Cl_{n}:χ(Ay^{−1})∈𝔽[f,λ]} χ(y^{−1}A)f([A]), (3.12)

because for an ℓ–power root of unity ζ ∈ μ_{ℓ^{n}} − μ_{ℓ},

Tr_{𝔽[f,λ](ζ)/𝔽[f,λ]}(ζ^{s}) = \begin{cases}ℓ^{n−m}ζ^{s} & if ζ^{s} ∈ 𝔽[f, λ] and 𝔽[f, λ] ∩ μ_{ℓ^{∞}} = μ_{ℓ^{m}} \\ 0 & otherwise. \end{cases}

For the moment, suppose that Γ_{f} ≅ ℤ_{ℓ}. Suppose also ∫_{Cl_{∞}} χ(x)dφ_{f}(x) = 0 for an infinite set Ξ of characters χ. For sufficiently large m, we always find a character χ ∈ Ξ such that Ker(χ) ⊂ Γ_{f}^{ℓ^{m}}. Then writing Ker(χ) = Γ_{f}^{ℓ^{n}} for n ≥ m, we have the vanishing from (3.11)

∫_{Cl_{∞}} σ ∘ χdφ_{f} = 0 for all σ ∈ Gal(𝔽/𝔽[f, λ]).

This combined with (3.12), we find ∑_{y∈χ^{−1}(𝔽[f,λ]^{×})} χ(yA)f(y[A])) = 0 for all A ∈ Γ_{n}, where Γ_{n} is the image of Γ_{f} in Cl_{n}.

Since each fractional R–ideal 𝔄 prime to 𝔩 defines a class [𝔄] in Cl_{∞}, we can embed the ideal group of fractional ideals prime to 𝔩 into Cl_{∞}. We write C^{alg} for its image. Thus the projection of [𝔔] in Cl_{n} is [𝔔_{n}] as specified for the integral ideal 𝔔 above. Then Δ^{alg} = Δ ∩ C^{alg} is generated by prime ideals of M nonsplit over F. We choose a complete representative set for Δ^{alg} made of product of prime ideals in M non-split over F prime to p𝔩. We may choose this set as {𝔰ℜ^{−1}|𝔰 ∈ S, 𝔯 ∈ R}, where S is made of prime ideals of F, R is made of squarefree product of primes outside 𝔩 in F ramifying in M/F, and ℜ is a unique ideal in M with ℜ^{2} = 𝔯. The set S is a complete representative set for the image Cl_{F}^{0} of the class group of F in Cl_{0} and {ℜ|𝔯 ∈ R} is a complete representative set for 2–torsion elements in the quotient Cl_{0}/Cl_{F}^{0} . We fix a character ψ : Δ → 𝔽^{×}, and define

f_{ψ} = ∑_{𝔯∈R} λψ^{−1}(ℜ) (∑_{𝔰∈S} ψλ^{−1}(𝔰)f|⟨𝔰⟩) |[𝔯].  (3.13)

Choose a complete representative set Q for Cl_{∞}/Γ_{f}Δ^{alg} made of primes 𝔔 of M split over F outside p𝔩. We choose η_{n}^{(p)} out of the base (w_{1}, w_{2}) of R̂_{n} so that at 𝔮 = 𝔔 ∩ F, w_{1,𝔮} = (1, 0) ∈ R_{𝔔} × R_{𝔔^{c}} = R_{𝔮} and w_{2,𝔮} = (0, 1) ∈ R_{𝔔} × R_{𝔔^{c}} = R_{𝔮}. Since all operators ⟨𝔰⟩, [𝔮] and [𝔯] involved in this definition commutes with U(𝔩), f_{ψ}|[𝔮] is still an eigenform of U(𝔩) with the same eigenvalue as f. Thus in particular, we have a measure φ_{f_{ψ}|[𝔮]}. We then define another measure φ_{f}^{ψ} on Γ_{f} by

∫_{Γ_{f}} ϕdφ_{f}^{ψ} = ∑_{𝔔∈Q} λψ^{−1}(𝔔) ∫_{Γ_{f}} ϕ|𝔔dφ_{f_{ψ}|[𝔮]},  (3.14)

where ϕ|𝔔(y) = ϕ(y[𝔔]_{f}^{−1}) for the projection [𝔔]_{f} in Γ_{f} of the class [𝔔] ∈ Cl_{∞}.

Lemma 3.1. If χ : Cl_{∞} → 𝔽^{×} is a character inducing ψ on Δ, we have

∫_{Γ_{f}} χdφ_{f}^{ψ} = ∫_{Cl_{∞}} χdφ_{f}.

Proof. Write Γ_{f,n} for the image of Γ_{f} in Cl_{n}. For a proper R_{n}–ideal A, by the above definition of these operators,

f|⟨𝔰⟩|[𝔯]|[𝔮]([A]) = λ(A)^{−1}f(x(𝔔^{−1}ℜ^{−1}𝔰A)).

For sufficiently large n, χ factors through Cl_{n}. Since χ = ψ on Δ, we have

\begin{aligned}∫_{Γ_{f}} χdφ_{f}^{ψ} & = ∑_{𝔔∈Q} ∑_{𝔰∈S} ∑_{𝔯∈R} ∑_{A∈Γ_{f,n}} λχ^{−1}(𝔔ℜ𝔰^{−1}A)f|⟨𝔰⟩|[𝔯]|[𝔮]([A]) \\ & = ∑_{A,𝔔,𝔰,𝔯} χ(𝔔ℜ𝔰^{−1}A)f([𝔔^{−1}ℜ^{−1}𝔰A]) = ∫_{Cl_{∞}} χdφ_{f}, \end{aligned}

because Cl_{∞} = ⨆_{𝔔,𝔰,ℜ}[𝔔^{−1}ℜ^{−1}𝔰]Γ_{f}. ⊓ ⊔

We write 𝔽_{p}[f] for the minimal field of definition of f ∈ V (𝔽). Similarly 𝔽_{p}[λ] (resp. 𝔽_{p}[ψ]) is the subfield of 𝔽 generated by the values λ([A]) mod P (resp. ψ([A])) for all [A] ∈ C^{alg}. Define 𝔽_{p}[f, λ, ψ] by the composite of these fields. Note that 𝔽_{p}[f, λ, ψ] is a finite extension of 𝔽_{p} if f = d^{κ}g for a classical modular form g ∈ G_{k}(Γ; 𝔽) of finite level. For general h ∈ V (𝔽), we write the q–expansion at the cusp (O, 𝔠^{−1}) of the 𝔠–component h_{𝔠} as ∑_{ξ∈𝔠^{−1}} a(ξ, h_{𝔠})q^{ξ}.

Theorem 3.2. Let f ∈ V (𝔽) of level Γ_{0}(𝔩) at 𝔩 be an eigenform for U(𝔩) with non-zero eigenvalue a ∈ 𝔽. Suppose that f = d^{κ}g for a classical modular form g ∈ G_{k}(𝔽) rational over a finite field 𝔽_{p}[f]. We fix a character ψ : Δ → 𝔽^{×}. Suppose the following conditions in addition to (unr) and (ord):

(H) Write the order of the Sylow ℓ–subgroup of 𝔽_{p}[f, ψ, λ]^{×} as ℓ^{r(ψ)}. Then there exists a strict ideal class 𝔠 ∈ Cl_{F} such that the polarization ideal of x(𝔔^{−1}ℜ^{−1}𝔰) is in 𝔠 for some (𝔔, ℜ, 𝔰) ∈ Q × S × R and for every u ∈ O prime to 𝔩, we can find ξ ≡ u mod 𝔩^{r(ψ)} with a(ξ, f_{ψ,𝔠}) ≠ 0.

Suppose further that the torsion-free part Γ_{f} of Cl_{∞} has ℤ_{ℓ} rank 1. Then the integral ∫_{Cl_{∞}} χdφ_{f} vanishes for only finitely many characters χ with χ|_{Δ} = ψ.

Proof. By definition, the projection {[𝔔]_{f}}_{𝔔∈Q} of [𝔔] in Γ_{f} are all distinct in Cl_{∞}/C^{alg}. By Lemma 3.1, we need to prove that the integral ∫_{Γ_{f}} χdφ_{f}^{ψ} vanishes only for finitely many characters χ of Γ_{f}. Suppose towards contradiction that the integral vanishes for characters χ in an infinite set X.

Let Γ(n) = Γ_{f}^{ℓ^{n−r}}/Γ_{f}^{ℓ^{n}} for r = r(ψ). By applying (3.12) to a character in X with Ker(χ) = Γ_{f}^{ℓ^{n}}, we find

∑_{𝔔∈Q} ψ(𝔔)^{−1} ∑_{A∈yχ^{−1}(μ_{ℓ^{r}})} χ(A)f_{ψ}([A𝔔^{−1}][𝔔]_{f}) = 0.  (3.15)

Note here f_{ψ}([A𝔔^{−1}][𝔔]_{f}) = f_{ψ,𝔠}([A𝔔^{−1}][𝔔]_{f}) for the strict class 𝔠 of 𝔠(A𝔔^{−1}).

Fix 𝔔 ∈ Q. By (3.2), \{x(A)|[A] ∈ yχ^{−1}(μ_{ℓ^{r}})\} is given by α(\frac{u}{ϖ_{𝔩}^{r}})(x(A_{0})) for any member A_{0} ∈ yχ^{−1}(μ_{p^{r}}), where α(t) = ( \begin{matrix}1 & t \\ 0 & 1 \end{matrix} ). Actually A ↦ u mod 𝔩^{r} gives a bijection of yχ^{−1}(μ_{ℓ^{r}}) onto O/𝔩^{r}. We write the element A corresponding to u as α(\frac{u}{ϖ_{𝔩}^{r}})A_{0}. This shows, choosing a primitive ℓ^{r}–th root of unity ζ = exp(2πi/ℓ^{r}) and A_{y} ∈ yχ^{−1}(μ_{ℓ^{r}}) so that χ(α(\frac{u}{ϖ_{𝔩}^{r}})A_{y}) = ζ^{uv} for an integer 0 < v < ℓ^{r} prime to ℓ (independent of y), the inner sum of (3.15) is equal to, for 𝔠 in (H),

∑_{u mod 𝔩^{r}} ζ^{uv}(f_{ψ,𝔠}|α(\frac{u}{ϖ_{𝔩}^{r}}))([A_{y}𝔔^{−1}][𝔔]_{f}).

The choice of v depends on χ. Since X is infinite, we can choose an infinite subset X′ of X for which v is independent of the element in X′. Then write n_{j} for the integers given by Γ_{f}^{ℓ^{n_{j}}} = Ker(χ) for χ ∈ X′ (in increasing order), and define Ξ to be the set of points x(A) for A ∈ Cl_{n_{j}} with [AR_{n_{1}}] = [R_{n_{1}}] in Cl_{n_{1}}. Define also g_{𝔔} = ∑_{u mod 𝔩^{r}} ζ^{uv}f_{ψ,𝔠}|α(\frac{u}{ϖ_{𝔩}^{r}}), because (3.15) is now the sum:

∑_{𝔔∈Q} ψ(𝔔)^{−1}g_{𝔔}|[𝔮]([A][𝔔]_{f}) = 0,

where 𝔮 = 𝔔 ∩ F. By Corollary 2.9, this implies that g_{𝔔} = 0.

The q–expansion coefficient a(ξ, g_{𝔔}) of g_{𝔔} is given by ℓ^{r}a(ξ, f_{ψ,𝔠}) if ξ ≡ −v mod 𝔩^{r} and vanishes otherwise. This contradicts to the assumption (H). ⊓ ⊔

We now treat the general case. A naive question when rank_{ℤ_{ℓ}} Γ_{f} > 1 is:

Question: Let f ≠ 0 be an eigenform of U(𝔩) which is rational over a finite extension 𝔽_{p^{r}} of 𝔽_{𝔭}). Suppose (unr) and (ord). For each ℤ_{ℓ}–rank 1 quotient π : Γ_{f} ↠ ℤ_{ℓ}, does ∫_{Γ_{f}} χ ∘ πdφ_{f} vanish only for finitely many characters of ℤ_{ℓ}?

Under the condition (H), we are tempted to believe that the above question is affirmative. However our result in the general case is weaker than this expectation.

Since Hom(Γ_{f}, 𝔽^{×}) ≅ Hom(Γ_{f}, μ_{ℓ^{∞}}), we may regard Hom(Γ_{f}, μ_{ℓ^{∞}}) as a subset of 𝔾_{m}(\overline{ℚ})^{d}. We call a subset X of characters of Γ_{f} Zariski-dense if it is Zariskidense as a subset of the algebraic group 𝔾_{m/\overline{ℚ}_{ℓ}}^{d}. What we can prove is Theorem 3.3. Suppose (unr) and (ord) for p. Let the notation be as in Theorem 3.2. In addition to the finiteness of 𝔽_{p}[f], if f_{ψ} as in (3.13) satisfies the following condition:

(h) There exists a strict ideal class 𝔠 of F such that 𝔠(𝔔^{−1}ℜ^{−1}𝔰) is in 𝔠 for some (𝔔, ℜ, 𝔰) ∈ Q × S × R and for any given integer r > 0, the N(𝔩)^{r} modular forms f_{ψ,𝔠}| ( \begin{matrix}1 & u \\ 0 & 1 \end{matrix} ) for u ∈ 𝔩^{−r}/O are linearly independent,

then the set of characters χ : Γ_{f} → 𝔽^{×} with non-vanishing ∫_{Cl_{∞}} ψχdφ_{f} ≠ 0 is Zariski dense. Here ψχ is the character of Cl_{∞} = Γ_{f} × Δ given by ψχ(γ, δ) = ψ(δ)χ(γ) for γ ∈ Γ_{f} and δ ∈ Δ.

Proof. By the same argument in the proof of Theorem 3.2, we create f_{ψ} and work with the measure φ_{f}^{ψ} on Γ_{f}. Identify Hom(Γ_{f}, μ_{ℓ^{∞}}) with μ_{ℓ^{∞}}^{d} ⊂ 𝔾_{m}^{d} by sending a character to its value at d independent generators of Γ_{f}. Write φ = φ_{f}^{ψ}. Suppose that the Zariski closure X in 𝔾_{m/\overline{ℚ}_{ℓ}}^{d} of the set of all characters χ with non-vanishing

integral ∫_{Γ_{f}} χdφ ≠ 0 for characters χ is a proper subset of 𝔾_{m}^{d}. Since the case d = 1 is already proven, we may assume that d ≥ 2.

For a sufficiently large p–power P, 𝔽_{p}[f_{ψ}, λ] = 𝔽_{P}. Since χ^{P} = σ ∘ χ ∈ X ⇔ χ ∈ X for the Frobenius automorphism σ ∈ Gal(𝔽/𝔽_{P}), we find X^{P} ⊂ X. Thus X is stable under any power of the P –th power homomorphism: t ↦ t^{P} of 𝔾_{m/\overline{ℚ}_{ℓ}}^{d}.

Let t ∈ X be a point of infinite order. We consider the ℓ–adic logarithm map log : 𝔾̂_{m/ℚ_{ℓ}}^{d} → 𝔾̂_{a/ℚ_{ℓ}}^{d}. The subvariety X contains the Zariski closure of the subset {t^{P^{n}}}, whose image under log is the one-parameter subgroup generated by log(t). Thus X is contained in a proper closed subset which is a finite union of translations by elements in μ_{ℓ^{∞}}^{d} of proper irreducible closed subschemes stable under scalar multiplication.

By the lemma following this theorem, we can find an infinite sequence of d– tuple of characters Ξ = {(χ_{1,j}, ⋯, χ_{d,j})}_{j=1,2,...} such that K_{j} = ⋂_{i=1}^{d} Ker(χ_{i,j}) gives rise to a system of open neighborhoods of the identity of Γ_{f}, and

∫_{Γ_{f}} ∏_{i=1}^{d} χ_{i,j}^{P^{k_{i}}} dφ_{f}^{ψ} = 0

for all (k_{1}, …, k_{d}) ∈ ℤ^{d}. We define for each fixed j a finite subset of Ξ by

Ξ_{j} = {(χ_{1,j}^{P^{k_{1}}} , …, χ_{d,j}^{P^{k_{d}}})|(k_{1}, …, k_{d}) ∈ ℤ^{d}}.

From this, we find

\begin{aligned}0 & = ∑_{𝔔∈Q} ψ(𝔔)^{−1} ∑_{(χ_{1},...,χ_{d})∈Ξ_{j}} ∫_{Γ_{f}} χ_{1}χ_{2} ⋯ χ_{d}dφ_{f_{ψ}} \\ & = ∑_{𝔔∈Q} ψ(𝔔)^{−1} ∑_{A} ∏_{i} ( ∑_{k_{i}=1}^{r_{i}} χ_{i,j}^{P^{k_{i}}} (A)) f_{ψ}([A𝔔^{−1}][𝔔]_{f}), \end{aligned}

where {χ_{i,j}^{P^{k_{i}}} |k_{i} = 1, ⋯, r_{i}} is the set of all distinct conjugates of χ_{i,j} under the power of the P –power Frobenius element, and A runs over all classes in Γ_{f}/Γ_{f}^{P^{N}} for N sufficiently large depending on j. For an ℓ–power root of unity ζ ∈ μ_{ℓ^{n}} − μ_{ℓ}, we write r = |{ζ^{P^{j}}|j ∈ ℤ}|. Again we use the following fact: we have ∑_{k=1}^{r} ζ^{sP^{k}} = 0 unless ζ^{s} ∈ μ_{ℓ}. From this, we have ∑_{k_{i}=1}^{r_{i}} χ_{i,j}^{P^{k_{i}}} (A) = 0 unless A ∈ χ_{i,j}^{−1}(𝔽_{P}). Writing μ_{ℓ^{r}} = μ_{ℓ^{∞}} ∩ 𝔽_{P}, we then find for any y ∈ Γ_{f}/Γ_{f}^{P^{N}} and j = 1, 2, …

∑_{𝔔} ψ(𝔔)^{−1} ∑_{A∈yχ_{j}^{−1}(μ_{ℓ^{r}}^{d})} χ_{1,j}(A)χ_{2,j}(A) ⋯ χ_{d,j}(A)f_{ψ,𝔠}([A𝔔^{−1}][𝔔]_{f}) = 0.

where χ̃_{j}(x) = (χ_{1,j}(x), …, χ_{d,j}(x)) ∈ 𝔽^{d}. From this by the same argument which proves the previous theorem, we deduce a contradiction against (h), which shows that X is equal to 𝔾_{m}^{d} as desired. ⊓ ⊔

Lemma 3.4. Let p and ℓ be distinct primes and r > 0 be an integer. Let X ⊂ 𝔾_{m/\overline{ℚ}_{ℓ}}^{d} for d ≥ 2 be a proper Zariski closed subset which is a finite union of

translations by ζ ∈ μ_{ℓ^{∞}}^{d} of closed subschemes stable under t ↦ t^{p^{rn}} for all n ∈ ℤ. Identify μ_{ℓ^{∞}}^{d} with (ℚ_{ℓ}/ℤ_{ℓ})^{d} as ℓ–divisible groups. Then we can find a sufficiently large p^{r}–power P and an infinite sequence \underline{n} = \{n_{1} < n_{2} < ⋯ \} of positive integers such that there exists a sequence of subsets Ξ_{j} outside X such that

Ξ_{j} = \{(\frac{P^{k_{1}}e_{1}}{ℓ^{n_{j}}} , …, \frac{P^{k_{d}}e_{n}}{ℓ^{n_{j}}}) mod ℤ_{ℓ}^{d}|(k_{j}) ∈ ℤ^{d}\}

if we choose a base {e_{j}} of ℤ_{ℓ}^{d} suitably.

Proof. We choose a p^{r}–power P so that P ≡ 1 mod ℓ. Let Γ_{P} = P^{ℤ_{ℓ}}, which is an open neighborhood of 1 in 1 + ℓℤ_{ℓ}.

Let V = (ℚ_{ℓ}/ℤ_{ℓ})^{d}, and write V [ℓ^{n}] for the kernel inside V of the multiplication by ℓ^{n}. By the argument using the ℓ–adic logarithm in the theorem, we may assume that X = ⋃_{ζ∈V [ℓ^{N}]}(ζ + C_{ζ}) for finitely many proper irreducible closed subschemes C_{ζ}. We first assume that N = 0. By this assumption, X is stable under t ↦ t^{s} for s ∈ ℤ; so, it is something like a projective cone centered at the origin. In other words, X is a union of one parameter subgroups of 𝔾_{m}. In particular, if we put X_{n} = V [ℓ^{n}] ∩ X, the scalar multiplication leaves stable X (because raising the power on μ_{ℓ^{∞}} is scalar multiplication on V), and hence multiplication by ℓ^{m−n} : X_{m} → X_{n} for m > n is surjective and induces a projective system {X_{n}}_{n}. We consider X̂ = \underleftarrow{lim}_{n}X_{n}, which is a projective cone in V̂ = \underleftarrow{lim}_{n}V [ℓ^{n}]. Then by definition, X_{n} is the image of X̂ in V [ℓ^{n}]. In other words, the image of the cone C(X) = {x ∈ V̂ ⊗_{ℤ_{ℓ}} ℚ_{ℓ}|tx ∈ X̂} in V̂ ⊗ (ℚ_{ℓ}/ℤ_{ℓ}) = V is the subset X. Since X̂ is a proper closed analytic subset in a compact space V̂, we can find a base e_{1}, …, e_{n} of V̂ over ℤ_{ℓ} outside X so that ℚ_{ℓ}e ∩ C(X) = {0} for e = e_{1} + e_{2} + ⋯ + e_{d}. Then the ℓ–adic distance from C(X) to the point \frac{e}{ℓ^{n}} is larger than or equal to cℓ^{n} for a positive constant c independent of n. Thus we can find sufficiently large power P of p^{r} (ℓ–adically very close to 1) so that Ξ̂_{n} = Γ_{P} \frac{e_{1}}{ℓ^{n}} + ⋯ + Γ_{P} \frac{e_{d}}{ℓ^{n}} gives rise to an open neighborhood of \frac{e}{ℓ^{n}} disjoint from C(X). Then the image Ξ_{n} of Ξ̂_{n} in V satisfies the desired property.

When N > 0, we consider the projection π : V → V/V [ℓ^{N}]. The image of X under π is stable under scalar multiplication. Using the result proven under the condition N = 0 applied to V/V [ℓ^{N}], we write Ξ′_{n} for the sets constructed for V/V [ℓ^{N}]. Then we find that for n > N any Γ_{P}^{d} –orbit of an element in the pull-back image π^{−1}(Ξ_{n}) gives a desired set Ξ_{n} ⊂ V. This finishes the proof. ⊓ ⊔

4. Hecke L–values

We compute integrals of characters for the measure associated to Eisenstein series in terms of L–values. Then Theorem 1.1 follows from Theorem 3.3.

4.1. Anti-cyclotomic L–functions.

We determine the type of Hecke L–function obtained by values of Eisenstein series at CM points. A result equivalent to the one presented here has also been obtained independently by H. Yoshida [LAP] V.3.2; so, our exposition will be brief. We leave the proof of the following two propositions to the reader.

Proposition 4.1. Recall R_{n} = O + 𝔩^{n}R. Then we have the identity:

{non-proper R_{n+1}–ideals} = {𝔩A|A is an R_{n}–ideal}.

Proposition 4.2. Let I_{n} be the group of all proper fractional R_{n}–ideals. Associating to each R_{n+1}–ideal A the R_{n}–ideal R_{n}A, we get the following homomorphism of groups π_{n} : I_{n+1} → I_{n}. The homomorphism π is surjective, and the kernel of π is isomorphic to R_{n,𝔩}^{×}/R_{n+1,𝔩}^{×}. After taking the quotient by principal ideals, we still have the following exact sequence:

1 → R_{n,𝔩}^{×}/R_{n+1,𝔩}^{×}R_{n}^{×} → Cl_{n+1} → Cl_{n} → 1.

Writing a prime element of 𝔏 and 𝔩 as ϖ and ϖ_{F}, respectively, we find that

R_{n+1,𝔩}/R_{n+1,𝔩}^{×} = {1} ⊔ ϖ_{F} R_{n,𝔩}/R_{n+1,𝔩}^{×}.  (4.1)

When 𝔩 splits into a product of two primes 𝔏\overline{𝔏} in M, we have

R_{n,𝔩}/R_{n,𝔩}^{×} ≅ (⊔_{j=0}^{n−1}(ϖ\overline{ϖ})^{j}R_{n−j,𝔩}^{×}/R_{n,𝔩}^{×}) ⨆ (⊔_{i,j=0}^{∞}ϖ^{n+i}\overline{ϖ}^{n+j}R_{𝔩}^{×}/R_{n,𝔩}^{×}) . (4.2)

When 𝔩 ramifies in M/F,

R_{n,𝔩}/R_{n,𝔩}^{×} ≅ (⊔_{j=0}^{n−1}ϖ^{2j}R_{n−j,𝔩}^{×}/R_{n,𝔩}^{×}) ⨆ (⊔_{j=0}^{∞}ϖ^{j}R_{𝔩}^{×}/R_{n,𝔩}^{×}) .  (4.3)

When 𝔩 is inert in M/F, we have

R_{n,𝔩}/R_{n,𝔩}^{×} ≅ (⊔_{j=0}^{n−1}ϖ^{j}R_{n−j,𝔩}^{×}/R_{n,𝔩}^{×}) ⨆ (⊔_{j=0}^{∞}ϖ^{j}R_{𝔩}^{×}/R_{n,𝔩}^{×}) .  (4.4)

Let χ be a character of the group of fractional proper ideals of R_{n}. By the above proposition, χ gives rise to a unique character of the full group of fractional ideals of M. In particular, taking an idele generator a ∈ M_{𝔸}^{×} with a_{∞} = 1 of a proper R_{n}–ideal A, we define N(A) = |a|_{𝔸}^{−1}. We note that N(A) = [R_{n} : A] = [R : RA]. We then define a formal L–function:

L^{n}(s, χ) = ∑_{A⊂R_{n}} χ(A)N(A)^{−s},  (4.5)

where A runs over all proper R_{n}–ideals. We write L(s, χ) for L^{0}(s, χ), which is the classical abelian L–function. This L–function depends on n, because the set of proper R_{n}–ideals depends on n. Since each proper ideal B ⊂ R_{n} equivalent to A^{−1} can be written as B = αA^{−1} with α ∈ A, we have

∑_{α∈A} χ((α))N(α)^{−s} = χ(A)N(A)^{−s} ∑_{B∼A^{−1}} χ(B)N(B)^{−s}.

We write L_{A^{−1}}^{n}(s, χ) = ∑_{B∼A^{−1}} χ(B)N(B)^{−s}. Then we have

L^{n}(s, χ) = ∑_{A∈Cl_{n}} L_{A^{−1}}^{n}(s, χ) = ∑_{A∈Cl_{n}} χ(A)^{−1}N(A)^{s} ∑_{α∈A} χ((α))N(α)^{−s}.

For a primitive character χ of Cl_{n}, we shall compute L^{n+k}(s, χ) (k > 0). The L–function has Euler product, and we only need to compute the 𝔩–factor L_{𝔩}^{n+k}(s, χ) given by ∑_{A∈I_{n+k,𝔩}} χ(A)N(A)^{−s}, where I_{n+k,𝔩} = (R_{n+k,𝔩} − {0})/R_{n+k,𝔩}^{×}. We first deal with the case where 𝔩 splits in M/F. Then by (4.2), we have

L_{𝔩}^{n+k}(s, χ) = ∑_{j=0}^{n+k−1} χ(ϖ\overline{ϖ})^{j}N(𝔩)^{−2sj} ∑_{u∈U_{n+k−j}} χ(u)

+ χ(ϖ\overline{ϖ})^{n}N(𝔩)^{−2(n+k)s} ∑_{i,j} χ(ϖ^{i}\overline{ϖ}^{j})N(𝔩)^{−(i+j)s} ∑_{u∈U_{n+k}} χ(u),

where U_{j} = R_{n+k−j,𝔩}^{×}/R_{n+k,𝔩}^{×}. We have ∑_{u∈U_{n+k−j}} χ(u) = 0 unless j = 0, 1, …, k, and we get

L_{𝔩}^{n+k}(s, χ) = ∑_{j=0}^{k} χ(ϖ\overline{ϖ})^{j}N_{F/ℚ}(𝔩)^{j−2sj} if 𝔩 splits in M/F and n > 0. (4.6)

Similarly, for ramified and inert primes, we have, for n > 0,

L_{𝔩}^{n+k}(s, χ) = ∑_{j=0}^{k} χ(ϖ)^{2j}N_{F/ℚ}(𝔩)^{j−2sj} if 𝔩 ramifies in M/F,

L_{𝔩}^{n+k}(s, χ) = ∑_{j=0}^{k} χ(ϖ)^{j}N_{F/ℚ}(𝔩)^{j−2sj} if 𝔩 remains prime in M/F.

(4.7)

When χ is of conductor 1, we have

L_{𝔩}^{k}(s, χ) − (N_{F/ℚ}(𝔩) − (\frac{M/F}{𝔩} )) χ(𝔩)^{k}N_{F/ℚ}(𝔩)^{k−1−2ks}L_{𝔩}^{0}(s, χ)

= 1 + ∑_{j=1}^{k−1} χ(𝔩)^{j}N_{F/ℚ}(𝔩)^{j−2sj} (k > 0), (4.8)

L_{𝔩}^{n+1}(s, χ) − χ(𝔩)N_{F/ℚ}(𝔩)^{1−2s}L_{𝔩}^{n}(s, χ) = 1 for n > 0.  (4.9)

and

L_{𝔩}^{1}(s, χ) − (N_{F/ℚ}(𝔩) − (\frac{M/F}{𝔩} )) χ(𝔩)N_{F/ℚ}(𝔩)^{−2s}L_{𝔩}^{0}(s, χ) = 1.  (4.10)

4.2. A generalization of a result of Hurwitz.

In the late 19th century, Hurwitz proved an analogue of the von Staut theorem for Hurwitz numbers ([Hz]). We shall give a generalization of his result to Hecke L– values of CM fields. The definition of Hilbert modular Eisenstein series is classical going back to Hecke; so, we first reproduce from [K1] and [HT] relevant results.

Let E_{𝔠} = E_{k}(ϕ, 𝔠) be the Eisenstein series of weight k defined in [HT] (2.3). Here we take ϕ : O_{p} × O_{p} → ℂ to be the constant function with value 1, and 𝔠 is a fixed polarization ideal. Over ℂ, having a pair (X, Λ) with 𝔠–polarization Λ is equivalent to having a lattice L ⊂ ℂ^{Σ} with X(ℂ) = ℂ^{Σ}/L and an alternating pairing L ∧ L ≅ 𝔠^{∗} with certain positivity (see [K1] I.1.4). Thus E_{𝔠} is a function of lattices L ⊂ F_{ℂ} with Λ : L ∧ L ≅ 𝔠^{∗} satisfying E_{α𝔠}(L, α^{−1}Λ) = E_{𝔠}(L, Λ). We then have

E_{𝔠}((L, Λ)) = \frac{(−1)^{k[F:ℚ]}Γ_{Σ}((k + s)Σ)}{\sqrt{|D|}} ∑_{w∈L−\{0\}} N(w)^{−k}|N(w)|^{−s}|_{s=0}, (4.11)

where N(w) is the norm of F_{ℂ} to ℂ induced by the norm map N_{F/ℚ} : F → ℚ. By this definition, E_{𝔠} only depends on the strict ideal class of 𝔠 (and E_{𝔠}((L, Λ)) depends only on (L, \overline{Λ}) for the polarization class \overline{Λ}). We write E = (E_{𝔠})_{𝔠∈Cl_{F}} ∈ G_{kI}(ℂ), where I is the sum of all embedding of F into \overline{ℚ} and k is a positive integer.

We may specify an O–lattice:

L = L_{z}^{𝔟} = 2πi(𝔟𝔠^{−1}z + 𝔟^{∗}) ⊂ ℂ^{Σ} = F ⊗_{ℚ} ℂ (z ∈ ℂ^{Σ}, Im(z_{σ}) > 0 ∀σ ∈ Σ) with Λ_{z} : L ∧_{O} L ≅ 𝔠^{∗} by the isomorphism induced by an alternating pairing

⟨2πi(az + b), 2πi(cz + d)⟩ = −(ad − bc).

By this, we may regard E_{𝔠} as a holomorphic function defined on ℌ^{Σ} for the upper half complex plane ℌ.

Since polarization is fixed as above, we often regard f ∈ G_{kI}(ℂ) as a function of O–lattices L with L∧L = 𝔠^{∗}, omitting Λ_{z} from the notation. For a given lattice L, we write 𝔠(L) for its polarization ideal. We define an operator for an ideal 𝔩

f|⟨𝔩⟩(L) = f(𝔩L).  (4.12)

Then f|⟨𝔩⟩ is defined over O–lattices L with 𝔠𝔩^{2}–polarization, because 𝔩L ∧ 𝔩L = 𝔩^{2}𝔠(L)^{∗} ⇔ 𝔠(𝔩L) = 𝔩^{−2}𝔠(L). Since f is of weight kI, if 𝔩 is a principal ideal generated by ϖ, we have f|⟨𝔩⟩(L) = f(ϖL) = N(ϖ)^{−k}f(L). Thus, if 𝔩 is principal, we have

E_{𝔠}|⟨𝔩⟩(L) = sgn(N(𝔩))^{k}N(𝔩)^{−k}E_{𝔠}(L).

Here sgn(N(𝔩)) = N(ϖ)/|N(ϖ)| for the generator ϖ of 𝔩. This operator ⟨𝔩⟩ is equal to the one introduced in 3.3 under the same symbol.

We shall compute the Hecke eigenvalue of E for T (𝔩). We consider the sum

(∗) = ∑_{L′} ∑_{w∈L′−{0}} N(w)^{−k}|N(w)|^{−s}|_{s=0},

where L′ runs over all O–lattices containing L with L′/L ≅ O/𝔩. Since there are 1+N(𝔩) such lattices L′, in the sum, each element in L contributes 1+N(𝔩) times, and each element in 𝔩^{−1}L outside L contribute once. Thus

(∗) = N(𝔩) ∑_{w∈L−{0}} N(w)^{−k}|N(w)|^{−s}|_{s=0} + ∑_{w∈𝔩^{−1}L−{0}} N(w)^{−k}|N(w)|^{−s}|_{s=0}.

If 𝔩 is principal generated by ξ, we find the sum over 𝔩^{−1}L is just equal to sgn(N(𝔩))^{k}N(𝔩)^{k} = N(ξ)^{k} times the sum over L. The Hecke operator T (𝔩) is given by the sum divided by N(𝔩); so, we get

E_{𝔠}|T (𝔩) =(1 + sgn(N(𝔩))^{k}N(𝔩)^{k−1})E_{𝔠} if 𝔩 is principal.
E_{𝔠𝔩}|T (𝔩) =N(𝔩)^{−1}E_{𝔠𝔩^{2}}|⟨𝔩^{−1}⟩ + E_{𝔠} if 𝔩 is not principal.

(4.13)

Take an object (L, Λ, C) of level Γ_{0}(𝔩); so, C ≅ O/𝔩 is a subgroup of X(L). This is equivalent to choose L_{C} ⊃ L with L_{C}/L ≅ O/𝔩. For L = L_{z}^{𝔟}, we choose C_{z} = (𝔩𝔟)^{∗}/𝔟^{∗} ⊂ X(L_{z}^{𝔟}). Then we see X(L_{z}^{𝔟})/C_{z} ≅ X(L_{z}^{𝔩𝔟})), where L_{z}^{𝔩𝔟} on the right-hand-side has polarization ideal 𝔠𝔩; so, it is 𝔟𝔠^{−1}z + (𝔩𝔟)^{∗}. Then we define another operator [𝔩] by

f|[𝔩](L, C) = N(𝔩)^{−1}f(L_{C}).  (4.14)

Then f|[𝔩] is defined over O–lattices with 𝔠𝔩^{−1}–polarization. This definition is compatible with the operator [𝔩] in 3.3. By definition, we have

\begin{matrix}f & |T (𝔩)(L) = N(𝔩)^{−1} ∑_{L′:L/L≅O/𝔩} f(L′) \\ f & |U(𝔩)(L, C) = N(𝔩)^{−1} ∑_{L′:L′/L≅O/𝔩,L′≠L_{C}} f(L′, C′), \end{matrix}

where C′ = L_{C} + L′/L′. Again f|T (𝔩) and f|U(𝔩) is defined over lattices with 𝔠𝔩^{−1}-polarization. Then we find

E_{𝔠}|U(𝔩) = E_{𝔠}|T (𝔩) − E_{𝔠}|[𝔩].  (4.15)

From (4.14), we have

N(𝔩)(E_{𝔠}|[𝔩])|U(𝔩)(L, C) = ∑_{L′:L′/L≅O/𝔩,L≠L_{C}} E_{𝔠}|[𝔩](L′, C′) = E_{𝔠}|⟨𝔩^{−1}⟩(L, C). (4.16)

From (4.16), (4.15) and (4.13) combined, we get

\begin{matrix}(E_{𝔠} − N(𝔩)E_{𝔠𝔩^{−1}}|(⟨𝔩⟩[𝔩])|U(𝔩) = & (E_{𝔠𝔩} − N(𝔩)E_{𝔠}|⟨𝔩⟩[𝔩])| (N(𝔩)^{−1}⟨𝔩^{−1}⟩) \\ (E_{𝔠} − E_{𝔠𝔩}|[𝔩])|U(𝔩) = & (E_{𝔠𝔩^{−1}} − E_{𝔠}|[𝔩]). \end{matrix}  (4.17)

We write E_{1,𝔠} = E_{𝔠} − E_{𝔠𝔩}|[𝔩] and E_{𝔩^{k−1},𝔠} = E_{𝔠𝔩} − N(𝔩)E_{𝔠}|⟨𝔩⟩[𝔩]. Defining E_{1} = (E_{1,𝔠})_{𝔠} ∈ G_{kI}(ℂ) and E_{𝔩^{k−1}} = (E_{𝔩^{k−1},𝔠})_{𝔠} ∈ G_{kI}(ℂ), we have

E_{1}|U(𝔩) = E_{1} and E_{𝔩^{k−1}}|U(𝔩) = N(𝔩)^{k−1}E_{𝔩^{k−1}}.  (4.18)

We can compute Fourier expansion of the Eisenstein series (see [K1] III), and find

E_{𝔩^{k−1}} ∈ G_{kI}(ℤ_{(p)}), E_{1} ∈ G_{kI}(ℚ) removed constant term is p–integral. (4.19)

Let d^{κ} be the p–adic analytic differential operator of Dwork-Katz whose effect on q–expansions is given by a(ξ, d^{κ}f) = ξ^{κ}a(ξ, f) with ξ^{κ} = ∏_{σ} σ(ξ)^{κ_{σ}} ([K1] II). Under (unr), d^{κ} is an integral operator. Recall the Hecke character λ of conductor 1 and of infinity type kΣ + κ(1 − c). Then L_{A^{−1}}^{n}(0, λ) does not depend on the choice of A in its strict ideal class. We define f = d^{κ}E_{𝔩^{k−1}} ∈ V (W) and f′ = d^{κ}E_{1} ∈ V (W) ⊗_{ℤ} ℚ.

The construction of the measures φ_{f} and φ_{f′} makes sense for f and f′. Decompose Cl_{∞} = Γ_{f} × Δ. We fix a character ψ of Δ and define again two measures E_{ψ} = φ_{f}^{ψ} and E′_{ψ} = φ_{f′}^{ψ} by (3.14) for the above f and f′. From the definition: E_{1,𝔠} = E_{𝔠} − E_{𝔠𝔩}|[𝔩] and E_{𝔩^{k−1},𝔠} = E_{𝔠𝔩} − N(𝔩)E_{𝔠}|⟨𝔩⟩[𝔩], we find, for n > 1,

\begin{aligned}f(x(A)) & = c_{0} (λ(A)L_{A^{−1}}^{n}(0, λ) − λ(𝔩(AR_{n−1}))L_{(𝔩AR_{n−1})^{−1}}^{n−1}(0, λ)) \\ f′(x(A)) & = c_{0} (λ(A)L_{A^{−1}}^{n}(0, λ) − N(𝔩)^{−1}λ(AR_{n−1})L_{(AR_{n−1})^{−1}}^{n−1}(0, λ)) , \end{aligned} (4.20)

where c_{0} = \frac{(−1)^{k[F:ℚ]}Γ_{Σ}(kΣ+κ)π^{κ}}{\sqrt{|D|} Im(δ)^{κ}Ω^{kΣ+2κ}} for the discriminant D of F. For a character ψχ

of Γ_{f} with conductor 𝔩^{n} (n > 0), we have for λ of infinity type kΣ + κ(1 − c),

∫_{Γ_{f}} χdE_{ψ} = N(𝔩)^{n(1−k)} ∫_{Γ_{f}} χdE′_{ψ} = \frac{π^{κ}Γ_{Σ}(kΣ + κ)L(0, ψ^{−1}χ^{−1}λ)}{\sqrt{|D|} Im(δ)^{κ}Ω^{kΣ+2κ}}  (4.21)

where L(s, α) is the primitive L–function of α. If the conductor of χψ is 1, by (4.9), we have slightly different expressions:

∫_{Γ_{f}} χdE_{ψ} = \begin{cases}\frac{(1−ψ^{−1}λ(𝔏))(1−ψ^{−1}λ(\overline{𝔏}))π^{κ}Γ_{Σ}(kΣ+κ)L(0,ψ^{−1}λ)}{\sqrt{|D|} Im(δ)^{κ}Ω^{kΣ+2κ}} & if 𝔩 = 𝔏\overline{𝔏}, \\ (1 − ψ^{−1}λ(𝔩))\frac{π^{κ}Γ_{Σ}(kΣ+κ)L(0,ψ^{−1}λ)}{\sqrt{|D|} Im(δ)^{κ}Ω^{kΣ+2κ}} & if 𝔩 is inert, \\ (1 − ψ^{−1}λ_{k}(𝔏))\frac{π^{κ}Γ_{Σ}(kΣ+κ)L(0,ψ^{−1}λ)}{\sqrt{|D|} Im(δ)^{κ}Ω^{kΣ+2κ}} & if 𝔩 = 𝔏^{2}, \end{cases} (4.22)

and

N(𝔩)^{1−k} ∫_{Γ_{f}} χdE′_{ψ}

= \begin{cases}N(𝔩)^{−1} \frac{(1−N(𝔩)ψ^{−1}λ(𝔏))(1−N(𝔩)ψ^{−1}λ(\overline{𝔏}))π^{κ}Γ_{Σ}(kΣ+κ)L(0,ψ^{−1}λ)}{\sqrt{|D|} Im(δ)^{κ}Ω^{kΣ+2κ}} & if 𝔩 = 𝔏\overline{𝔏}, \\ −N(𝔩)^{−1}(1 − ψ^{−1}λ(𝔩)N(𝔩)^{2})\frac{π^{κ}Γ_{Σ}(kΣ+κ)L(0,ψ^{−1}λ)}{\sqrt{|D|} Im(δ)^{κ}Ω^{kΣ+2κ}} & if 𝔩 is inert, \\ −ψ^{−1}λ(𝔏)(1 − ψ^{−1}λ(𝔏)N(𝔩))\frac{π^{κ}Γ_{Σ}(kΣ+κ)L(0,ψ^{−1}λ)}{\sqrt{|D|} Im(δ)^{κ}Ω^{kΣ+2κ}} & if 𝔩 = 𝔏^{2}, \end{cases} (4.23)

Now we get the following generalization of Hurwitz’s theorem, which has been proven in [H96] Theorem 8.4.1 by a different method.

Theorem 4.3. Let the notation and the assumption be as in Theorem 1.1. Fix a Hecke character λ of conductor 1 with λ((a)) = a^{−kΣ} for all principal ideal (a) of M with a positive integer k. Then for a finite order character χ of Cl_{∞}, \frac{Γ_{Σ}(kΣ)L(0,λχ)}{Ω^{kΣ}} is 𝔓–integral and belongs to W unless p−1|k and the conductor of χ is equal to 1. If (p − 1)|k and p^{m}|k exactly, writing L_{𝔩}(s, λ) for the Euler 𝔩–factor of L(s, λ), \frac{L(0,λ)}{Ω^{kΣ}} belongs to the field of fractions of W, and we have

p^{m}L_{𝔩}(−1, λ)\frac{Γ_{Σ}(kΣ)L(0, λ)}{Ω^{kΣ}}

≡ ε(1 − N(𝔩))\frac{(1 − (\frac{M/F}{𝔩} ) N(𝔩))hR_{p}}{w\sqrt{D}} ∏_{𝔭∈Σ_{p}} (1 − \frac{1}{N(𝔭)}) mod 𝔓^{m},

where h is the class number of M, w is the number of roots of unity in M, D is the discriminant of F/ℚ and R_{p} is the p–adic regulator of F and ε = 1 if 𝔩 is unramified in M/F and if 𝔩 ramifies, ε = λ(𝔏)^{−1} for the ramified prime 𝔏|ℓ.

As in [Cz], the ratio R_{p}/\sqrt{D} is canonically defined although the numerator and denominator depends on the choice of ordering of embedding of F into \overline{ℚ}_{p}.

Proof. Since the rationality of the L–values follows from Shimura’s rationality result [Sh2] combined with [K1] 5.3.0, we only need to prove the last congruence formula. We assume that F ≠ ℚ for simplicity (because the case of k = 2 and F = ℚ needs an extra care). We need to look into the q–expansion at the cusp (𝔟, 𝔟𝔠^{−1}) given by, if k ≥ 2 is even,

N(𝔟)^{−1}E_{𝔠}| ( \begin{matrix}b & 0 \\ 0 & b^{−1} \end{matrix} ) (q) = 2^{−d}N(𝔟)^{k−1}ζ_{𝔟^{−1}}(1 − k)

+ ∑_{0≪ξ∈𝔟^{2}𝔠^{−1}} ∑_{(α,β)∈(𝔟×𝔟𝔠^{−1}), αβ=ξ} |N(α)|^{k−1}q^{ξ}, (4.24)

where d = [F : ℚ] and ζ_{𝔟^{−1}}(s) = ∑_{𝔵∼𝔟^{−1}, 𝔵⊂O} N(𝔵)^{−s} is the partial zeta function of the ideal class 𝔟^{−1}. Here b ∈ F_{𝔸}^{×} is an adelic generator of the ideal 𝔟 with b_{∞} = 1, and f| ( \begin{matrix}b & 0 \\ 0 & b^{−1} \end{matrix} ) is as in [DR] 5.8 (see also [H88] Theorem 4.9). We can rewrite the expansion as

N(𝔟)^{−k}E_{𝔠}| ( \begin{matrix}b & 0 \\ 0 & b^{−1} \end{matrix} ) (q)

= 2^{−d}ζ_{𝔟^{−1}}(1 − k) + ∑_{0≪ξ∈𝔟^{2}𝔠^{−1}} ∑_{A|ξ𝔟^{−2}𝔠, A∼𝔟^{−1}} N(A)^{k−1}q^{ξ}, (4.25)

where A runs over all integral factors of ξ𝔟^{−2}𝔠 in the ideal class of 𝔟^{−1}.

Taking 𝔟 = O, we have a(0, E_{1,𝔠}) = (1 − N(𝔩)^{k−1})2^{−d}ζ_{O}(1 − k). Then we find from a result of Colmez [Cz] that

p^{m}2^{−d}ζ_{O}(1 − k) ≡ \frac{R_{p}}{2\sqrt{D}} ∏_{𝔭∈Σ_{p}} (1 − \frac{1}{N(𝔭)}) mod 𝔓^{m}.

Then the desired formula follows from (4.23) by the q–expansion principle ([K1] (1.2.16)). ⊓ ⊔

4.3. Proof of Theorem 1.1.

By (4.25) and the rationality of the differential operator d (cf. [K1] II), we have 𝔽_{p}[f] = 𝔽_{p} for f = d^{κ}E_{𝔩^{k−1}}. By Theorems 3.2 and 3.3, we need to verify the condition (H) in Theorem 3.2 for f (which implies the condition (h) in Theorem 3.3). For a given q–expansion g(q) = ∑_{ξ} a(ξ, g)q^{ξ} ∈ 𝔽[[q^{ξ}]]_{ξ∈𝔠^{−1}} at the infinity cusp (O, 𝔠^{−1}), we know that, for u ∈ O_{𝔩} ⊂ F_{𝔸},

a(ξ, g|α_{u}) = e_{F} (uξ)a(ξ, g) for α_{u} = ( \begin{matrix}1 & u \\ 0 & 1 \end{matrix} ) (e_{F} (x) = exp(2πiTr_{F/ℚ}(x)).

The condition (h) for g concerns about the linear independence of g|α_{u} for u ∈ 𝔩^{−r}O_{𝔩}/O_{𝔩}. For any function ϕ : 𝔠^{−1}/𝔩^{r}𝔠^{−1} = O/𝔩^{r} → 𝔽, put g|ϕ = ∑_{ξ} ϕ(ξ)a(ξ, g)q^{ξ}. By definition, we have g|R_{ϕ} = ∑_{u∈O/𝔩^{r}} ϕ(u)g|α_{u} = g|ϕ^{∗} for the Fourier transform ϕ^{∗}(v) = ∑_{u} ϕ(u)e_{F} (uv). For the characteristic function χ_{v} of a singleton {v} ⊂ 𝔠^{−1}/𝔩^{r}𝔠, its Fourier transform is given by u ↦ e_{F} (vu). By the Fourier inversion formula (and the q–expansion principle: [K1] 1.9.17), the linear independence of {g|α_{u} = g|χ_{u}^{∗}}_{u} is equivalent to the linear independence of {g|χ_{u}}_{u}.

We recall that f_{ψ} is a tuple (f_{ψ,𝔠})_{𝔠} ∈ ⨁_{𝔠} V (𝔠; W). Thus we need to prove that there exists 𝔠 such that for a given congruence class u ∈ 𝔠^{−1}/𝔩^{r}𝔠^{−1}

a(ξ, f_{ψ,𝔠}) ≢ 0 mod 𝔪_{W} for at least one ξ ∈ u.  (4.26)

Since a(ξ, d^{κ}h) = ξ^{κ}a(ξ, h), (4.26) is achieved if

a(ξ, g_{ψ,𝔠}) ≢ 0 mod 𝔪_{W} for at least one ξ ∈ u prime to p  (4.27)

holds for g = (E_{𝔠} − N(𝔩)E_{𝔠𝔩^{−1}}|⟨𝔩⟩|[𝔩])_{𝔠}, because 𝔩 ∤ p. Recall (3.13):

g_{ψ} = ∑_{𝔯∈R} λψ^{−1}(ℜ) (∑_{𝔰∈S} ψλ^{−1}(𝔰)g|⟨𝔰⟩) |[𝔯].  (4.28)

We described the effect on the q–expansion of the operators [𝔩] and ⟨𝔩⟩ in 3.3. This combined with the q–expansion of the Eisenstein series described in the proof of Theorem 4.3 gives us the following q–expansion g_{𝔞,𝔟}(q) of g at the cusp (𝔞, 𝔟) (carrying Tate_{𝔞,𝔟}(q) = (𝔾_{m} ⊗ 𝔞^{∗})/q^{𝔟}):

a(ξ, g_{𝔞,𝔟}) = N(𝔞) ∑_{\begin{matrix}(a,b)∈(𝔞×𝔟)/O^{×} \\ ab=ξ \end{matrix}} \frac{N(a)^{k}}{|N(a)|} − N(𝔞) ∑_{\begin{matrix}(a,b)∈(𝔞×𝔩𝔟)/O^{×} \\ ab=ξ \end{matrix}} \frac{N(a)^{k}}{|N(a)|}

= N(𝔞) ∑_{\begin{matrix}(a,b)∈(𝔞×(𝔟−𝔩𝔟))/O^{×} \\ ab=ξ \end{matrix}} \frac{N(a)^{k}}{|N(a)|} (4.29)

We now study the question (4.27) using this explicit formula (4.29). Our evaluation points x(A) have polarization ideals 𝔠(A) which cover all classes in the coset N_{M/F} (Cl_{M})𝔠(R). Therefore, we only need to look into the contribution of all g|⟨𝔰⟩|[𝔯]|[𝔮] belonging to a single class 𝔠 in N_{M/F} (Cl_{M})𝔠(R). In the class 𝔠^{−1}, we pick a prime 𝔱 outside p𝔩. We take an element 0 ≪ ξ ∈ u for u ∈ O/𝔩^{r} so that (ξ) = 𝔱𝔫𝔩^{e} for an integral ideal 𝔫 ∤ p𝔩ℭ prime to the relative discriminant D of M/F and 0 ≤ e ≤ r. Since we have a freedom of choosing ξ modulo 𝔩^{r}, the ideal 𝔫 varies freely in a given ray class modulo 𝔩^{r−e}.

We pick a pair (a, b) ∈ F^{2} with ab = ξ with a ∈ 𝔰^{−1} and b ∈ 𝔱𝔰. Then (a) = 𝔰^{−1}𝔩^{α}𝔵 for an integral ideal 𝔵 prime to 𝔩 and (b) = 𝔰𝔱𝔩^{e−α}𝔵′ for an integral ideal 𝔵′ prime to 𝔩. Since (ab) = 𝔱𝔫𝔩^{e}, we find that 𝔵𝔵′ = 𝔫. By (4.29), b has to be prime to 𝔩; so, we find α = e. Since 𝔵𝔵′ = 𝔫 and hence 𝔯 = O because 𝔫 is prime to D. Thus for each factor 𝔵 of 𝔫, we could have two possible pairs (a_{𝔵}, b_{𝔵}) with contribution to the q–expansion such that

a_{𝔵}b_{𝔵} = ξ and ((a_{𝔵}) = 𝔰_{𝔵}^{−1}𝔩^{e}𝔵, (b_{𝔵}) = (ξa_{𝔵}^{−1}) = 𝔰_{𝔵}𝔱𝔫𝔵^{−1})

for 𝔰_{𝔵} ∈ S representing the ideal class of the ideal 𝔩^{e}𝔵. We then write down the q–expansion coefficient of q^{ξ} at the cusp (O, 𝔱) of g_{ψ}:

\begin{matrix}a & (ξ, g_{ψ}) = ∑_{𝔵|𝔫} N(𝔰_{𝔵})^{−1}ψ^{−1}λ(𝔰_{𝔵})\frac{N(a_{𝔵})^{k}}{|N(a_{𝔵})|} \\ = ψ^{−1}λ(𝔩)^{−e}N(𝔩)^{−e} ∑_{𝔵|𝔫} \frac{1}{ψ^{−1}λ(𝔵)N(𝔵)} \\ = ψ^{−1}λ(𝔫𝔩^{e})^{−1}N(𝔫𝔩^{e})^{−1} ∏_{𝔶|𝔫} \frac{1 − (ψ^{−1}λ(𝔶)N(𝔶))^{e(𝔶)+1}}{1 − ψ^{−1}λ(𝔶)N(𝔶)} , \end{matrix}  (4.30)

where 𝔫 = ∏_{𝔶|𝔫} 𝔶^{e(𝔶)} is the prime factorization of 𝔫.

We define, for the valuation v of W (normalized so that v(p) = 1)

μ_{C}(ψ) = Inf_{𝔫}v (∏_{𝔶|𝔫} \frac{1 − (ψ^{−1}λ(𝔶)N(𝔶))^{e(𝔶)+1}}{1 − ψ^{−1}λ(𝔶)N(𝔶)} ) ,  (4.31)

where 𝔫 runs over a ray class C modulo 𝔩^{r−e} made of all integral ideals prime to D𝔩 of the form 𝔱^{−1}ξ𝔩^{−e}, 0 ≪ ξ ∈ u. If μ_{C}(ψ) = 0 for one C, we get the desired non-vanishing. Since μ_{C}(ψ) only depends on the class C, we may assume (and will assume) that e = 0 without losing generality; thus ξ is prime to 𝔩, and C is the class of u[𝔱^{−1}].

Suppose that 𝔫 is a prime 𝔶. Then by (4.30), we have

a(ξ, g_{ψ}) = 1 + (ψ^{−1}λ(𝔶)N(𝔶))^{−1}.

If ψ^{−1}λ(𝔶)N(𝔶) ≡ −1 mod 𝔪_{W} for all prime ideals 𝔶 in the ray class C modulo 𝔩^{r}, the character A ↦ (ψ^{−1}λ(A)N(A) mod 𝔪_{W}) is of conductor 𝔩^{r}. We write ψ̃ for the character: A ↦ (ψ^{−1}λ(A)N(A) mod 𝔪_{W}) of the ideal group of F with values in 𝔽^{×}. This character therefore has conductor ℭ̃|𝔩^{r}. Since ψ is anticyclotomic, its restriction to F_{𝔸}^{×} has conductor 1. Since λ has conductor 1, the conductor of ψ̃ is a factor of the conductor of λ mod 𝔪_{W}, which is a factor of p. Thus ℭ̃|p. Since 𝔩 ∤ p, we find that ℭ̃ = 1.

We shall show that if μ_{C}(ψ) > 0, M/F is unramified and ψ̃ ≡ ^{(\underline{M/F} )} mod 𝔪_{W}. We now choose two prime ideals 𝔶 and 𝔶′ so that 𝔱𝔶𝔶′ = (ξ) with ξ ∈ u. Then by (4.30), we have

a(ξ, g_{ψ}) = (1 + \frac{1}{ψ^{−1}λ(𝔶)N(𝔶)}) (1 + \frac{1}{ψ^{−1}λ(𝔶′)N(𝔶′)}) .  (4.32)

Since ψ̃(𝔶𝔶′) = ψ̃(u[𝔱^{−1}]) = ψ̃(C) = −1, we find that if a(ξ, g_{ψ}) ≡ 0 mod 𝔪_{W},

−1 = ψ̃(𝔶/𝔶′) = ψ̃(𝔩^{−1})ψ̃(𝔶^{2}) = −ψ̃(𝔶^{2}).

Since we can choose 𝔶 arbitrary, we find that ψ̃ is quadratic. Thus μ_{C}(ψ) > 0 if and only if ψ̃(𝔠) = −1, which is independent of the choice of u. We now move the strict ideal class 𝔠 in 𝔠(R)N_{M/F} (Cl_{M}). By class field theory, assuming that ψ̃ has conductor 1, we have

ψ̃(𝔠) = −1 for all 𝔠 ∈ 𝔠(R)N_{M/F} (Cl_{M})

if and only if ψ̃(𝔠(R)) = −1 and ψ̃(A) = (\frac{M/F}{A} ) for all A ∈ Cl_{F}. (4.33)

If M/F is unramified, by definition, 2δ𝔠^{∗} = 2δ𝔡^{−1}𝔠^{−1} = R. Taking square, we find that (𝔡𝔠)^{2} = 4δ^{2} ≪ 0. Thus 1 = ψ̃(𝔡^{−2}𝔠^{−2}) = (−1)^{[F:ℚ]}, and this never happens when [F : ℚ] is odd. Thus (4.33) is equivalent to the following three conditions:

(M1) M/F is unramified everywhere (so the strict class number of F and [F : ℚ] are even);

(M2) The strict ideal class of the polarization ideal 𝔠 of X(R) in F is not a norm class of an ideal class of M (⇔ (\frac{M/F}{𝔠} ) = −1);

(M3) ψ̃ is the character of M/F.

Thus the non-vanishing result stated in the theorem is the case where ψ is the identity character. The condition (M1) and (M3) combined is equivalent to ψ̃^{∗} ≡ ψ̃ mod 𝔪_{W}, where the dual character ϵ^{∗} is defined by ϵ^{∗}(x) = ϵ(x^{−c})N(x)^{−1}. The vanishing under (M1-3) of L(0, χ^{−1}ψ^{−1}λ) ≡ 0 for all anti-cyclotomic χψ follows from the functional equation of the p-adic Katz measure interpolating the p–adic Hecke L–values (see [HT] Theorem II), because the constant term of the functional equation is given by ψ̃(𝔠) = −1. This finishes the proof. ⊓ ⊔

References

Books

[AAF] G. Shimura, Arithmeticity in the Theory of Automorphic Forms, Mathematical Surveys and Monographs 82, AMS, 2000

[ABV] D. Mumford, Abelian Varieties, TIFR Studies in Mathematics, Oxford University Press, 1994

[ACM] G. Shimura, Abelian Varieties with Complex Multiplication and Modular Functions, Princeton University Press, 1998

[ALR] J.-P. Serre, Abelian l-Adic Representations, Benjamin, 1968

[AME] N. M. Katz and B. Mazur, Arithmetic Moduli of Elliptic Curves, Ann. of Math. Studies 108, Princeton University Press, 1985

[CRT] H. Matsumura, Commutative Ring Theory, Cambridge studies in advanced mathematics 8, Cambridge Univ. Press, 1986

[DAV] G. Faltings and C.-L. Chai, Degeneration of Abelian Varieties, Springer, [GIT] D. Mumford, Geometric Invariant Theory, Ergebnisse 34, Springer, 1965

[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

[LAP] H. Yoshida, Lectures on Absolute CM period, Mathematical Surveys and Monographs, AMS, 106, 2003

[PAF] H. Hida, p–Adic Automorphic Forms on Shimura Varieties, Springer Monographs in Mathematics, 2004

Articles

[B] J. Boxall, Autour d’un probleme de Coleman, C. R. Acad. Sci. Paris 315 (1992), Series I: 1063–1066

[C] C.-L. Chai, Arithmetic minimal compactification of the Hilbert-Blumenthal moduli spaces, Ann. of Math. 131 (1990), 541–554

[C1] C.-L. Chai, Every ordinary symplectic isogeny class in positive characteristic is dense in the moduli, Inventiones Math. 121 (1995), 439–479

[C2] C.-L. Chai, Local monodromy for deformation of one-dimensional formal groups, J. reine angew. Math. 524 (2000), 227-238.

[C3] C.-L. Chai, A rigidity result for p–divisible formal groups, preprint, 2003 (downloadable at: www.math.upenn.edu/˜ chai/papers.html)

[C4] C.-L. Chai, Families of ordinary abelian varieties: canonical coordinates, padic monodromy, Tate-linear subvarieties and Hecke orbits, preprint, 2003 (downloadable at: www.math.upenn.edu/˜ chai/papers.html)

[COU] L. Clozel, H. Oh and E. Ullmo, Hecke operators and equidistribution of Hecke points, Invent. Math. 144 (2001), no. 2, 327–351

[Cz] P. Colmez, Résidu en s = 1 des fonctions zêta p–adiques, Inventiones Math. 91 (1988), 371-389

[D1] P. Deligne, Variétés abéliennes ordinaires sur un corps fini, Inventiones Math. 8 (1969), 238–243

[D2] P. Deligne, Travaux de Shimura, Sem. Bourbaki, Exp. 389, Lecture notes in Math. 244 (1971), 123–165

[D3] P. Deligne, Variétés de Shimura: interprétation modulaire, et techniques de construction de modéles canoniques, Proc. Symp. Pure Math. 33.2 (1979), 247–290

[DR] P. Deligne and K. A. Ribet, Values of abelian L–functions at negative integers over totally real fields, Inventiones Math. 59 (1980), 227–286

[DT] M. Dimitrov and J. Tilouine, Variété de Hilbert et arithmétique des formes modulaires de Hilbert pour Γ_{1}(𝔠, N), in this volume: “Geometric Aspects of Dwork’s Theory, A Volume in memory of Bernard Dwork” (edited by Alan Adolphson, Francesco Baldassarri, Pierre Berthelot, Nicholas Katz, and Francois Loeser), Walter de Gruyter, 2003 [FW] B. Ferrero and L. Washington, The Iwasawa invariant μ_{p} vanishes for abelian

number fields, Ann. of Math. 109 (1979), 377–395

[F] T. Finis, Divisibility of anticyclotomic L-functions and theta functions with complex multiplication, preprint, 2001/2002

[G] R. Gillard, Croissance du nombre de classes dans des ℤ_{l}-extensions liées aux corps quadratiques imaginaires, Math. Ann. 279 (1988), 349-372

[H88] H. Hida, On p-adic Hecke algebras for GL_{2} over totally real fields, Ann. of Math. 128 (1988), 295–384

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

[H03a] H. Hida, p–Adic automorphic forms on reductive groups, to appear in the proceedings of the automorphic semester of the year 2000 at l’institut Henri Poincaré (preprint downloadable at www.math.ucla.edu/ hida)

[H03b] H. Hida, The Iwasawa μ–invariant of p–adic Hecke L–functions, preprint, 2003 (preprint downloadable at www.math.ucla.edu/ hida)

[H03c] H. Hida, Non-vanishing modulo p of Hecke L–values and application, preprint in preparation

[H03d] H. Hida, Automorphism groups of Shimura varieties of PEL type, preprint, 2003 (preprint downloadable at www.math.ucla.edu/ hida)

[HT] H. Hida and J. Tilouine, Anticyclotomic Katz p–adic L–functions and congruence modules, Ann. Sci. Ec. Norm. Sup. 4-th series 26 (1993), 189–259

[Hz] A. Hurwitz, Über die Entwicklungkoeffizienten der lemniskatischen Funktionen, Math. Ann. 51 (1899), 196–226 (Werke II, 342–373)

[I] J. Igusa, Kroneckerian model of fields of elliptic modular functions, Amer. J. Math. 81 (1959), 561–577

[K] N. M. Katz, P –adic properties of modular schemes and modular forms, Lecture notes in Math 350 (1973), 69–190

[K1] N. M. Katz, p-adic L-functions for CM fields, Inventiones Math. 49 (1978), 199–297

[K2] N. M. Katz, Serre-Tate local moduli, In “Surfaces Algébriques”, Lec. Notes in Math. 868 (1981), 138–202

[Ko] R. Kottwitz, Points on Shimura varieties over finite fields, J. Amer. Math. Soc. 5 (1992), 373–444

[M] J. Milne, Canonical models of (mixed) Shimura varieties and automorphic vector bundles, Perspective Math. 10 (1990), 283–414

[MS] J. Milne and K-y. Shih, Automorphism groups of Shimura varieties and reciprocity laws, Amer. J. Math. 103 (1981), 911-935

[O] F. Oort, A stratification of a moduli space of abelian varieties, in “Moduli of abelian varieties” (Texel Island, 1999), Progress in Math. 195 (2001), 245–416

[R] M. Rapoport, Compactifications de l’espace de modules de Hilbert-Blumenthal, Comp. Math. 36 (1978), 255–335 [ST] J.-P. Serre and J. Tate, Good reduction of abelian varieties, Ann. of Math.

88 (1968), 452–517

[Sh] G. Shimura, Moduli and fibre system of abelian varieties, Ann. of Math. 83 (1966), 294–338 (Collected papers I, 707–751)

[Sh1] G. Shimura, On canonical models of arithmetic quotients of bounded symmetric domains, Ann. of Math. 91 (1970), 144-222; II, 92 (1970), 528-549 (Collected papers II, 299–377 and 378–399)

[Sh2] G. Shimura, On some arithmetic properties of modular forms of one and several variables, Ann. of Math. 102 (1975), 491–515 (Collected papers II, 683–707)

[Si] W. Sinnott, On a theorem of L. Washington, Astérisque 147-148 (1987), 209–224

[T] J. Tate, Endomorphisms of abelian varieties of finite fields, Inventiones Math. 2 (1966), 134–144

[V] V. Vatsal, Uniform distribution of Heegner points, Invent. Math. 148 (2002), 1–46

[V1] V. Vatsal, Special values of anticyclotomic L–functions, Duke Math. J 116 (2003), 219–261

[W] L. Washington, The non-p–part of the class number in a cyclotomic ℤ_{p}– extension, Inventiones Math. 49 (1978), 87–97

[Z] J. G. Zarhin, Endomorphisms of abelian varieties over fields of finite characteristics, Math. USSR Izvestija 9 (2) (1975), 255-260