EISENSTEIN CONGRUENCE ON UNITARY GROUPS AND IWASAWA MAIN CONJECTURES FOR CM FIELDS
MING-LUN HSIEH
Abstract. The purpose of this article is to prove Iwasawa main conjecture for CM fields in certain cases through an extensive study on the divisibility re- lation between p-adic L-functions for CM fields and Eisenstein ideals of unitary groups of degree three.
Contents
Introduction 1
1. Notation and conventions 8
2. Shimura varieties for unitary groups 13
3. Modular forms on unitary groups 24
4. Hida theory for unitary groups 37
5. Ordinary p-adic Eisenstein series on U (2, 1) 51 6. Constant terms of the p-adic Eisenstein series 68
7. Eisenstein ideal and p-adic L-functions 76
8. Application to the main conjecture for CM fields 92
References 105
Introduction
The main conjecture for CM elliptic curves over totally real fields. Let M be an imaginary quadratic field and let E be an elliptic curve over a totally real field F with complex multiplication by the ring of integers OM of M. Let p be an odd prime split in M. Let F∞ be the cyclotomic Zp-extension of F and let ΛF := ZpJGal(F∞/F )K be an one-variable Iwasawa algebra. We study the cyclotomic main conjecture of Iwasawa theory for E which relates the size of Selmer groups to special values of p-adic L-function attached to E. Recall that the Selmer group SelF∞(E) is defined by
SelF∞(E) = ker (
H1(F∞, E[p∞]) →Y
v
H1(F∞,v, E) )
,
where v runs over all places of F∞. It is well known that SelF∞(E) is a cofinitely generated ΛF-module. Denote by charΛFSelF∞(E) the characteristic power series of SelF∞(E), which is an element in ΛF unique up to a ΛF-unit. Let Wp be the
Date: July 17, 2022.
2010 Mathematics Subject Classification. Primary 11R23 Secondary 11F33, 11F70.
1
p-adic completion of the ring of integers of the maximal unramified extension of Zp. On the other hand, the specialization of a suitable twist of a Katz p-adic L-function to the cyclotomic line yields a p-adic L-function Lp(E/F∞) ∈ ΛF ,Wp:= ΛF⊗ZpWp, which roughly interpolates the algebraic part of central L-values L(E/F ⊗ ν, 1) twisted by finite order characters ν : Gal(F∞/F ) → C× (See §8.6.2 for the precise definition). The main conjecture of Mazur and Swinnerton-Dyer for E predicts the following equality between ideals in ΛF ,Wp:
Conjecture 1 (The main conjecture for CM elliptic curves).
(charΛFSelF∞(E)) = (Lp(E/F∞)).
If F = Q and E has good ordinary reduction at p, the above conjecture is a theorem of K. Rubin [Rub91, Thm. 12.3]. Using the main result in this article, we obtain one-sided divisibility in the main conjecture for a class of CM elliptic curves over totally real fields. To state our theorem, we need some notation. Let K = F M and let DK/F be the relative discriminant of K/F . If L is a number field, let hL be the class number of L and DL be the absolute discriminant of L. Let h−K= hK/hF be the relative class number of K/F . One of our main results is as follows.
Theorem 1. Suppose that p - 6 · h−K· DF and that E has good ordinary reduction at all places above p. Then we have the inclusion between ideals in ΛF ,Wp
(charΛFSelF∞(E)) ⊂ (Lp(E/F∞)).
From the above theorem and control theorems (cf. [Gre99, Thm. 1.2, Thm. 4.1]), we can deduce the following consequence which provides evidence to Birch and Swinnerton-Dyer conjecture.
Corollary 1. Suppose that p - 6 · h−K· DF and that E has good ordinary reduction at all places above p. Then
(a) If L(E/F, 1) = 0, then the p-primary Selmer group SelF(E) has positive Zp-corank.
(b) If L(E/F, 1) 6= 0, then
lengthZp(SelF(E)) ≥ ordp(L(E/F, 1) ΩE ),
where ΩE is the period of a Néron differential of E over Z(p).
Remark. If W (E/F ) = −1, part (a) is a consequence of the Selmer parity conjec- ture proved by Nekovář [Nek06, Cor. 12.2.10]. Our different approach provides a constructive proof of this fact.
Iwasawa main conjecture for CM fields. We prove Theorem 1 by establishing a divisibility result towards Iwasawa main conjecture for CM fields, which we will describe after introducing some notation. Let K be a totally imaginary quadratic extension of F and assume that K is p-ordinary, namely every prime of F above p splits in K. Let Σ be a p-ordinary CM-type of K (See §1.4). Let SpK be the set of p-adic places of K. Fix an embedding ιp : Q ,→ Cp. Then Σ and ιp give rise to p-adic CM-type Σp, which is a subset of SpKsuch that Σpand its complex conjuga- tion Σpc give a partition of SpK. Let d = [F : Q]. Let K∞be the compositum of the cyclotomic Zp-extension and the anticyclotomic Zdp-extension of K. If we assumed
Leopoldt’s conjecture for K, then K∞would be the maximal Zd+1p -extension of K.
Let ΓK be the Galois group Gal(K∞/K), which is a free Zp-module of rank d + 1.
Let K0 ⊃ K(µp) be a finite abelian extension of K which is linear disjoint from K∞ and let ∆ = Gal(K0/K). Let ψ : ∆ → C×p be a character of ∆, which is called a branch character. Let Wp[ψ] be the ring generated by the values of ψ over Wp. Denote by Λ = Wp[ψ]JΓKK the (d + 1)-variable Iwasawa algebra over Wp[ψ].
Let K∞0 = K0K∞and let MΣ be the maximal p-abelian Σp-ramified extension of K∞0 and let XΣ be the Galois group Gal(MΣ/K∞0 ). Then MΣ is also Galois over K, and ∆ acts on XΣ by the usual conjugation action. Define the Iwasawa module XΣ(ψ)to be the maximal ψ-isotypic quotient of XΣ. By [HT94, Thm. 1.2.2], XΣ(ψ) is a finitely generated and torsion Λ-module. Therefore, to XΣ(ψ) we can attached a characteristic power series Fψ,Σ ∈ Λ. On the other hand, it follows from Katz [Kat78] and Hida-Tilouine [HT93, Thm. II] that to (ψ, Σ) we attach a primitive (d + 1)-variable p-adic Hecke L-function Lψ,Σ ∈ Λ, which interpolates p-adically the algebraic part of critical Hecke L-values for ψ twisted by characters of ΓK and satisfies the functional equation. We can state the (d + 1)-variable main conjecture for CM fields as follows (cf. [HT94, Main conjecture, page 90]).
Conjecture 2 (Iwasawa main conjecture for CM fields). We have the following equality between ideals in Λ
(Fψ,Σ) = (Lψ,Σ).
The significance of the main conjecture originates from the applications to the Birch and Swinnerton-Dyer conjecture for CM elliptic curves (See [CG83] and [Rub91] for the case F = Q). When F = Q, this conjecture is a theorem of Rubin [Rub91] combined with Yager’s construction of p-adic L-functions for imagi- nary quadratic fields [Yag82]. Rubin uses the technique of Euler system constructed from elliptic units to bound the size of XΣ(ψ)in terms of L-values. In other words, he proves the divisibility (Fψ,Σ) ⊃ (Lψ,Σ), and then appeals to the class number formula to conclude the equality. For general CM fields, the Euler system tech- nique seems not applicable currently. In this article, instead of controlling the size of XΣ(ψ)we construct sufficiently many elements in XΣ(ψ)in terms of L-values with the technique of congruences among modular forms on the unitary group of degree three. Namely, we prove the reverse divisibility relation (Fψ,Σ) ⊂ (Lψ,Σ). To state our result precisely, we need to introduce some notation. Let mΛ be the maximal ideal of Λ. For a number field L, we let GL= Gal(Q/L) and let ωL: GL→ Z×p be the Teichmüller character. Let DK/F be the relative discriminant of K/F and let c(ψ) be the prime-to-p conductor of the branch character ψ. Let ψ+: GabF → C×p be the composition ψ ◦ V , where V : GabF → GabK is the Verschiebung map.
Theorem 1 is a consequence of the following divisibility result in the (d + 1)- variable main conjecture for CM fields and the control theorems of Selmer groups for CM elliptic curves due to Perrin-Riou [PR84].
Theorem 2. Suppose that (1) p - 3 · h−K· DF· ](∆),
(2) ψ is unramified at Σpc, and ψωK−a is unramified at Σp for some integer a 6≡ 2 (mod p − 1).
Then we have the following inclusion between ideals of Λ (Fψ,Σ) ⊂ (Lψ,Σ).
Remark. The assumption (2) is in particular used to make some twist of ψ unram- ified at p, which is necessary for the application of results on the non-vanishing modulo p of Hecke L-values in the proof of non-vanishing modulo p of Eisenstein series. It is possible to weaken this assumption if [Hid04a, Prop. 2.8] could be improved.
Thanks to the works of Hida and Rubin, Theorem 2 is sufficient to prove the main conjecture when ψ is anticyclotomic or obtained from the restriction of a Galois character of an imaginary quadratic field. By Hida’s solution to the aniticyclotomic main conjecture in [Hid06] and [Hid09b], Theorem 2 implies the main conjecture for certain anticyclotomic branch characters.
Corollary 2. Suppose that p - 3 · h−K· DF· ](∆) and that (H1) ψ is anticyclotomic,
(H2) the local character ψw is unramified and non-trivial for every w ∈ Σp, (H3) ψ|G
K(√
p∗ ) 6= 1, p∗= (−1)p−12 p.
Then we have the following equality between ideals of Λ (Fψ,Σ) = (Lψ,Σ).
Combined with Rubin’s two-variable main conjecture for imaginary quadratic fields [Rub91], Theorem 2 yields the following main conjecture by the same argu- ment in [Hid07, Thm. 5.7].
Corollary 3. In addition to the assumptions (1-2) in Theorem 2, suppose further that
(R1) K = F M, where M is an imaginary quadratic field in which p splits, (R2) Σ is the p-ordinary CM-type of K obtained by extending the inclusion ι∞:
M ,→ C,
(R3) K0 is abelian over M and p - [K0: M].
Then we have the following equality between ideals of Λ (Fψ,Σ) = (Lψ,Σ).
The main conjecture from Greenberg’s point of view. We give a different formulation of Conjecture 2 proposed by R. Greenberg in the context of Galois representations ([Gre94]). Let Sψ be the set of places dividing c(ψ) and let S ⊃ Sψ be a finite set of prime-to-p places in K. Let KSbe the maximal algebraic extension of K unramified outside S ∪ SpK. Define the tautological Λ-valued Galois character εΛ by
εΛ: Gal(KS/K) −→ ΓK,→ Λ× g −→ g|K∞.
Let Ψ : Gal(KS/K) → Λ× be the deformation of ψ defined by Ψ (g) = ψ(g)εΛ(g).
Let Λ∗= Homcont(Λ, Qp/Zp) be the Pontryagin dual of Λ. We make Λ∗ a discrete
Λ-module by λs(x) = s(λx) for all λ, x ∈ Λ and s ∈ Λ∗. The Λ-adic Selmer group associated to (Ψ, Σ) is defined by
SelK(Ψ, Σ) := ker{H1(KS/K, Ψ ⊗ΛΛ∗) → Y
w∈S∪Σcp
H1(Iw, Ψ ⊗ΛΛ∗)}, where Iw is the inertia group in a local decomposition group of GK at w. Define the S-imprimitive Selmer group by
SelSK(Ψ, Σ) := ker{H1(KS/K, Ψ ⊗ΛΛ∗) → Y
w∈Σcp
H1(Iw, Ψ ⊗ΛΛ∗)}.
Let ht1(Λ) be the set of height one prime ideals in Λ. If S is a cotorsion and cofinitely generated discrete Λ-module and P ∈ ht1(Λ), we denote by SP∗ = (S∗) ⊗ΛΛP the localization of the Pontryagin dual of S at P and put
`P(S) := lengthΛP(SP∗).
We write Lp(Ψ, Σ) for the Katz p-adic L-function Lψ,Σ. Conjecture 2 can be re- formulated as the following main conjecture for the one dimensional Λ-adic Galois representation Ψ .
Conjecture 3 (Main Conjecture). For every P ∈ ht1(Λ), we have the equality
`P(SelK(Ψ, Σ)) = ordP(Lp(Ψ, Σ)).
We shall consider the dual version of the above main conjecture, which has the advantage of incorporating imprimitive p-adic L-functions and Selmer groups.
Define the S-imprimitive p-adic L-function LSp(Ψ, Σ) for S by LSp(Ψ, Σ) = Lp(Ψ, Σ) · Y
w∈S−Sψ
(1 − Ψ (Frobw)).
Here Frobw is the geometric Frobenius. Let ε be the p-adic cyclotomic character of GK and let ΨD be the character defined by ΨD(g) = Ψ−1ε(g). The dual Selmer group for (Ψ, Σ) is the Selmer group associated to (ΨD, Σc) defined by
SelKS(ΨD, Σc) := ker{H1(KS/K, ΨD⊗ΛΛ∗) → Y
w∈Σp
H1(Iw, ΨD⊗ΛΛ∗)}.
We will prove the following theorem, which implies Theorem 2 by the functional equation of Selmer groups for CM fields [Hsi10].
Theorem 3. With the same assumptions (1-2) in Theorem 2, for every P ∈ ht1(Λ) we have the inequality
`P(SelSK(ΨD, Σc)) ≥ ordP(LSp(Ψ, Σ)).
Eisenstein congruence. Our main tool is the Eisenstein congruences on unitary groups. The application of Eisenstein congruences to Iwasawa theory was first introduced by Ribet in [Rib76], in which he uses the congruences between Eisen- stein series and cusp forms on GL(2) over Q to obtain a proof of the converse of Herbrand’s theorem. This approach was further exploited by Mazur and Wiles [MW84] in their proof of classical Iwaswa main conjecture, and by Wiles [Wil90]
in his elegant proof of Iwasawa main conjecture for totally real fields through the systematic use of Hida theory. Meanwhile, parallel to Eisenstein congruence, Hida and Tilouine studied CM congruences for GL(2) over totaly real fields extensively,
and in [HT94] they proved a divisibility result for the anti-cyclotomic main conjec- ture for CM fields. In [Urb01] and [Urb06], E. Urban proved the main conjecture for adjoint modular representation by Eisenstein congruences on GSp(4). In his joint work with C. Skinner [SU14], they prove the main conjecture for GL(2) by Eisenstein congruences on U (2, 2).
The proof of our Theorem 3 is based on a study of Eisenstein congruence on U (2, 1) the quasi-split unitary group of degree three. The use of Eisenstein con- gruence for U (2, 1) to study one-sided divisibility in the main conjecture for CM fields was initiated by F. Mainardi in his thesis [Mai04] under the supervision of J.
Tilouine and E. Urban. In [Mai08] he defines the Eisenstein ideal of the cuspidal ordinary topological Hecke algebra B, i.e. the Hecke algebra acting on the ordi- nary cuspidal Betti cohomology groups, and proves the characteristic power series of Selmer groups for CM fields is divisible by the Eisenstein ideal in B under some technical assumptions. In this article, we introduce the ideal of Eisenstein congru- ence Eis(Ψ, S) ⊂ Λ (Definition 7.19), which measures the congruence between Hida families of Eisenstein series and cusp forms on U (2, 1), and we first prove Theorem 3 for every P ∈ ht1(Λ) which is not exceptional (Definition 8.13) by establishing the following two inequalities:
(L|E) ordP(Eis(Ψ, S)) ≥ ordP(LSp(Ψ, Σ)), (E|S) `P(SelKS(ΨD, Σc)) ≥ ordP(Eis(Ψ, S)).
Let us explain the strategy briefly. To show the first inequality (L|E), we have to work with Hida families for unitary groups. The first step is to construct a Λ-adic Hida family of Eisenstein series Eord (a p-adic measure with values in the space of p-adic ordinary modular forms) on U (2, 1) with the optimal constant term, namely a product of the Katz p-adic L-function LSp(Ψ, Σ) and a Tate twist of the S-imprimitive Deligne-Ribet p-adic L-function LSDR associated to the char- acter ψ+τK/Fω−1F (τK/F is the quadratic character of K/F ). The idea of this construction is based on our previous work [Hsi11], which we describe briefly as follows. We begin with the construction of a good p-adic Siegel-Eisenstein series E2,2 on U (2, 2). Applying the pull-back formula, we obtain an Eisenstein series E on U(2, 1) by pulling back this Siegel-Eisenstein series to U(2, 1) via a suitable embedding U (2, 1) × U (1) ,→ U (2, 2). The desired ordinary Eisenstein series is constructed by taking the ordinary projection Eord of E. The idea of using the pull-back formula was suggested to the author by E. Urban, and has been used in [Urb06] and [SU14] to construct a Hida family of Eisenstein series on GSp(4) and U (2, 2) respectively.
The construction of our degree two Siegel-Eisenstein seriesE2,2 is inspired by the construction of Harris, Li and Skinner [HLS06]. However, their Eisenstein series does not quite fit for our purpose since the ordinary projection of the pull-back of this Eisenstein series is zero. A heuristic reason is that the Fourier coefficients of their Eisenstein series are only supported in the integral Hermitian matrices which are non-degenerate modulo p. Therefore, we have to modify their construction to meet the ordinary condition, which complicates the computation of the constant term of the ordinary projection of of the pull-back ofE2,2. To resolve this, we apply the techniques in [Hsi11, §6].
Second, to make congruences between Hecke eigenvalues of Eisenstein series and cusp forms for U (2, 1) modulo the constant term, we have to show that the Hida
family of Eisenstein series Eord is co-prime to the constant term (or even non- vanishing modulo mΛ). This is usually the most difficult step in the approach of Eisenstein congruence. In the case of U (2, 2) [SU14], this is achieved by showing the non-vanishing modulo p of a very clever linear combination of Fourier coefficients of certain Klingen-Eisenstein series. In our situation, we show directly that some Fourier-Jacobi coefficient of Eord is non-vanishing modulo mΛ, making heavy use of the theory of Shintani on primitive Fourier-Jacobi coefficients of automorphic forms on U (2, 1) [Shi79]. Our idea is to introduce an auxiliary Eisenstein series E◦, whose Fourier-Jacobi coefficients are manageable, and show that the non-vanishing modulo p of p-primitive Fourier-Jacobi coefficients of E◦ and Eord are equivalent (See Proposition 7.8). The Fourier-Jacobi coefficients of E◦are essentially a product of two Hecke L-values for CM fields thanks to the works of Murase and Sugano ([MS00], [MS02]) generalizing Shintani’s theory and the calculation of Tonghai Yang on period integrals of theta functions. The problem is thus reduced to the non- vanishing modulo p of these Hecke L-values, which is addressed in [Hid04a] and our previous work [Hsi12]. Using the non-vanishing of Eord(mod mΛ) combined with Hida theory for unitary groups, we are able to construct a non-trivial Hida family of cusp forms congruent to the Eisenstein series Eord modulo p-adic L-functions, which leads to the inequality (L|E).
To show the second inequality (E|S), we need to construct sufficiently many elements in the Selmer group in terms of the Eisenstein ideal. This can be done by the technique of lattice construction due to E. Urban in [Urb01]. In our case, a variant of the inequality (E|S) in the context of the topological cuspidal ordinary Hecke algebra has been worked out by Mainardi [Mai08]. Working with the coherent cuspidal ordinary Hecke algebra instead, we achieve the inequality (E|S).
There remains the case when P is exceptional. These are precisely common divisors of the Katz p-adic L-function Lψ,Σ and the S-imprimitive Deligne-Ribet p-adic L-function LSDR. We have trouble proving (E|S) for exceptional primes P in general unless ordP(Lψ,Σ) ≤ 1. Nonetheless, if ∆ has order prime to p, then results of Hida [Hid10] and the author [Hsi14] on the vanishing of the µ-invariant of anticyclotomic p-adic L-functions for CM fields imply that there is no exceptional prime unless ψ+= τK/FωF and WΣ(ψ) ≡ −1 (mod mΛ), where WΣ(ψ) ∈ Λ×is the root number in the functional equation of the Katz p-adic function Lψ,Σ, and in this particular case, the only possible exceptional prime is the pole Peof LDR and ordPe(Lψ,Σ) = 1 thanks to a recent result of A. Burungale [Bur14] on the vanishing of µ-invariant of the cyclotomic derivative of Katz p-adic L-functions attached to self-dual characters with root number −1.
Structure of this article. This paper is organized as follows.
In §1, we fix notation and definitions through this article. In §2 and §3, we review the general theory of Shimura varieties associated to unitary groups over totally real fields and the theory of Katz-Hida geometric modular forms. In §4, we ex- tend Hida theory to include Eisenstein series on unitary groups U (r, 1) over totally real fields. We prove the classicality and the control theorem for p-adic ordinary modular forms. Hida theory provides the framework for the study of the con- gruence among modular forms. In particular, the fundamental exact sequence in Theorem 4.26 (cf. [Urb06, Thm. 2.4.19] and [SU14, Thm. 6.3.10]) is crucial to make congruence between Eisenstein series and cusp forms modulo constant terms in §7.
Hida developed his theory for cusp forms in great generality ([Hid02] and [Hid04b]), and moreover he establishes an axiomatic control theorem for automorphic sheaves on Shimura varieties of PEL-type using several standard results from the theory of the minimal and toroidal compactifications of Shimura varities of PEL-type which has been worked out by K.-W. Lan [Lan08] in full details. Hida theory for modular forms on U (n, n)/Q has been carried out in [SU14], and the proofs therein work for U (r, 1)/Q as well. However, due to the appearance of non-torsion units in OF×, one requires a slight modification for the base change property in the totally real case (See §4.1).
In §5 and §6, we construct the desired ordinary p-adic Eisenstein series on U (2, 1) and make an explicit calculation on its constant terms (Proposition 6.6). In §7, we show our ordinary p-adic Eisenstein series is non-vanishing modulo p (Proposi- tion 7.13) and construct congruences between cusp forms and Eisenstein series on U (2, 1), which leads to our first inequality (L|S) (Corollary 7.20).
Finally in §8, by the technique of lattice construction we prove the second in- equality (E|S) (Theorem 8.14). A variant of Mainardi’s work in the context of the coherent Hecke algebras is carried out in §8.3 and §8.4. The applications to the main conjectures for CM elliptic curves and CM fields are given in §8.6.
Acknowledgments. The author learnt the fundamental ideas in this article from a preprint [Urb06] of Eric Urban and the oral communication by him when the author was his Ph.D. student in Columbia University. This work is impossible without his insight and guidance.
The influence of Hida’s ideas and works in this article is evident. The author would like to thank him for sharing his mathematical ideas by writing so many ex- cellent books and also for answering questions with patience during the preparation of this article. He also thanks Prof. Coates and Prof. Tilouine for their encour- agement and helpful suggestions and A. Burungale for sending his preprint to the author, which leads to the removal of the global root assumption in the previous version of this manuscript.
Part of the material in §8 grew from the courses in the summer school for Iwasawa theory held in McMaster University during August 19-23, 2007. The author would like to thank the organizers Prof. Kolster and Prof. Sharifi for such a wonderful summer school.
Finally, the author is very grateful to the referees for a careful reading of the manuscript and valuable suggestions for the improvement of the exposition.
1. Notation and conventions
1.1. Throughout F is a totally real field of degree d over Q and K is a totally imag- inary quadratic extension of F . Denote by c the complex conjugation, the unique non-trivial element in Gal(K/F ). Denote by a = Hom(F , C) the set of archimedean places of F and by h the set of finite places of F . Let DK/F (resp. DK/F) be the relative discriminant (resp. different) of K/F and SK/F =v ∈ h | v|DK/F .
Henceforth we fix an odd rational prime p which is unramified in F and assume the following ordinary condition:
(ord) Every prime of F above p splits in K.
This condition implies that p is unramified in K. We denote by Spthe set of places in F lying above p. We choose once and for all an embedding ι∞: Q ,→ C and an isomorphism ι : C→ C∼ p, where Cpis the completion of an algebraic closure of Qp. Let ιp = ιι∞ : Q ,→ Cp be their composition. Let Zp be the p-adic completion of the ring Z of algebraic integers in Cp and let mp be the maximal ideal of Zp and let m := ι−1(mp). Every number field L will be regarded as a subfield in C (resp.
Cp) via ι∞ (resp. ιp) and Hom(L,Q) = Hom(L, Cp).
If L is a finite extension of Qp or a number field, we denote by OL the ring of integers of L. If L0 is a quadratic extension of L, denote by τL0/L the quadratic character associated to L0/L. If L is number field, let GL = Gal(Q/L) be the absolute Galois group. Let AL be the adele ring of L and AL,f be the finite part of AL. For brevity, we shall write O = OF throughout this article.
We often write w for a place of K and v for a place of F . Write Kw (resp. Fv) for the completion of K (resp. F ) at w (resp. v) and Fp for F ⊗QQp. Denote by
$w (resp. $v) a uniformizer of Kw(resp. Fv). We also write Kv for K ⊗FFv and Ov (resp. Op) for OFv (resp. O ⊗ZZp).
Denote by bZ the finite completion of Z. If M is an abelian group, let cM = M ⊗ZZ. If L is a number field, bb L = AL,f and bOL=Q
v<∞OLv. 1.2. For a finite set of rational primes, we define Z() by
Z() =na
b ∈ Q | bZ + qZ = Z for all q ∈ o .
By definition, Z() = Q if is empty. If R is a Z-algebra, we let R()= R⊗ZZ(). When = {p}, we write R() as R(p). Denote by Z+ the set of positive integers.
Put
O(),+=a ∈ O ⊗ZZ() | a is totally positive .
If R is a ring, we denote by SCH/R the category of R-schemes and by SET S the category of sets. If X is a scheme over F (resp. O), RF /QX (resp. RO/ZX) is the restriction of scalar of X from F (resp. O) to Q (resp. Z).
If R is an O-algebra, the complex conjugation c induces an involution on R ⊗OK by r ⊗ x 7→ r ⊗ c(x). Define the n×n Hermitian matricesHn(R) over R ⊗OK by
Hn(R) = {g ∈ Mn(R ⊗OK) | g = g∗} ,
where g∗ = c(gt) and gt is the transpose of g. When n = 1 and g ∈ R ⊗OK, we sometimes write g for g∗. If g ∈ GLn(R ⊗OK), we write g−∗and g−tfor (g∗)−1and (gt)−1respectively. We denote by Bn(R) the upper triangular subgroup of GLn(R).
Let Tn(R) be the diagonal torus of Bn(R) and let Nn(R) be the unipotent radical of Bn(R).
1.3. Characters. Let IK= Hom(K, Q). For w ∈ SpK, we put
(1.1) Iw= {σp∈ Hom(K, Cp) | σp induces w} = Hom(Kw, Cp).
We shall identity IK with twIw by σ 7→ σp := ιp◦ σ.
Let χ : A×K/K×→ C× be an arithmetic Hecke character of K×. We say that χ has infinity type κ =P
σ∈IKκσσ ∈ Z[IK] if χ∞(α) = ι∞(ακ) := Y
σ∈IK
σ(α)κσ for all α ∈ K×.
We can associate χ to a unique p-adic Hecke character χ : Ab ×K/K× → Z×p defined by
χ(z) = ιb p(χ(z) Y
σ∈IK
z−κσ σ)Y
w|p
Y
σp∈Iw
σp(zw)κσ.
We callχ the p-adic avatar of χ, whereas χ the complex avatar ofb χ.b
Denote by recK : A×K/K× → GabK the geometrically normalized reciprocity law which takes recK($w) = Frobw, where Frobw is the geometric Frobenius at w.
Throughout this article, every character of GK implicitly will be regarded as a Hecke character of K× via recK.
1.4. CM-types. Let Σ be a CM-type of K, i.e. Σ is a subset of IK such that Σ ∩ Σc = ∅ and Σ t Σc = IK.
Denote by Σpthe set of places of K above p induced by ιp◦ σ for σ ∈ Σ. We further assume Σ is a p-ordinary CM-type. Namely,
Σp∩ Σpc = ∅ and Σpt Σpc = SpK.
The existence of a p-ordinary CM type is assured by the assumption (ord).
Henceforth, we simply write Σc = Σc and Σpc = Σpc and identify the CM-type Σ (resp. the p-adic CM-type Σp) with the set a of archimedean places of F (resp.
the set Sp of places of F above p).
1.5. Let Kncbe the normal closure of K inQ. To every σ ∈ IK we can attach an idempotent eσ∈ OK⊗ZOKnc such that aeσ= σ(a)eσ for a ∈ OKand
OK⊗ZOKnc = M
σ∈IK
OKnc· eσ.
For a subset J of IK, we put eJ = P
σ∈Jeσ. Let e+ = eΣ and e− = eΣc. For w ∈ SKp, let ew= eIw. Since p is unramified in K, ew belongs to OK⊗ZZp and
ew(OK⊗ZZp) = Ov,
where v is the place of F lying below w. By the definition of Σp, we have e+ = P
w∈Σpew(resp. e− =P
w∈Σcpew) and e+(OK⊗ Zp) = e−(OK⊗ Zp) = Op. If M is an OK-module, we put Mp= M ⊗ Zp and
MΣ= e+(Mp) and MΣc = e−(Mp).
For σ ∈ IK, denote by C(σ) the K ⊗QC-module whose underlying space is C with K-action via ι∞σ. Put
C(J ) =M
σ∈J
C(σ).
Let Op(Σ) (resp. Op(Σc)) denote the OK⊗ZZp-module Op on which OK acts through e+ (resp. e−).
1.6. Unitary groups. Let r and s be two nonnegative integers such that r ≥ s ≥ 0.
Let (W, ϑ) be a skew-Hermitian K-space of dimension r − s with a skew-Hermitian form ϑ. We fix a K-basis w1, . . . , wr−s of W and assume that ϑ(wi, wj) = aiδi,j. We assume further that√
−1 · σ(ai) > 0 and ιp(σ(ai)) are p-adic units for all i and σ ∈ Σ. Let XK =
s
L
i=1
Kxi and YK =
s
L
i=1
Kyi be K-vector spaces of dimension s and let (V, ϑr,s) be the skew-Hermitian space with the underlying K-vector space V = YK⊕ W ⊕ XKand ϑr,s the skew-Hermitian form on V defined by
ϑr,s=
−1s
ϑ 1s
with respect to the decomposition YK⊕ W ⊕ XK. Let h , ir,s : V ×V → Q be the alternating pairing defined by hv, v0ir,s = TrK/Q(ϑr,s(v, v0)).
Let G = GU (V ) be the group of unitary similitudes attached to the quadratic space (V, ϑr,s). As an algebraic group over F , the R-points of G for a F -algebra R are
G(R) = {g ∈ GLr+s(K ⊗FR) | gϑr,sg∗= ν(g)ϑr,s for some ν(g) ∈ R×}.
The morphism ν : G → Gm/F is called the similitude map. The unitary group U (V ) is defined by
U (V )(R) := {g ∈ GU (V )(R) | ν(g) = 1} .
We let GU (0, r − s) be the group of unitary similitudes attached to (W, −ϑ).
For g ∈ EndKV , we let g∨denote the element in EndKV such that ϑr,s(vg, v0) = ϑr,s(v, v0g∨), v, v0∈ V . Then g∨= g−1ν(g) if g ∈ G.
1.7. Standard basis. The basis yi, xi
i=1,...,s and wi
i=1,...,r−s is called the standard basis of V . The basisewyi, ewwi, ewxi
w∈SKp (resp. eσyi.eσwi, eσxi
w∈IK) is called the standard p-adic (resp. complex) basis of V ⊗QQp(resp. V ⊗QC). We identify GL(VΣ) (resp. GL(VΣc) with GLr+s(Fp) =Q
v∈SpGLr+s(Fv) with respect to the standard p-adic basis of VΣ (resp. VΣc). Consider the following embedding:
RF /QG(Qp) = Y
v∈Sp
G(Fv) → GL(VΣ) × GL(VΣc) × Fp× g 7→ (g|VΣ, g|VΣc, ν(g)).
For each v ∈ Sp, the above embedding gives rises an identification
GU (V )(Fv) = G(Fv) '(x, x0, ν) ∈ GLr+s(Fv) × GLr+s(Fv) × Fv×| x0= ϑtr,sx−tϑ−tr,s· ν . If w is the place above p in Σp, then we have the identifications:
(1.2) GU (Fv) = GLr+s(Fv) × Fv×, g → (g|ewV, ν(g)), U (V )(Fv) = GLr+s(Fv), g → g|ewV.
Thus, (g, ν) ∈ GU (Fv) has the similitude ν. Similarly, the standard complex basis gives rise to the identification:
RF /QG(R) = GLr+s(C(Σ))×C×(Σ).
We shall fix these identifications throughout this article.
1.8. Lattices and polarization. In what follows we make a choice of the lattice M in V . Denote by dKthe absolute different of K/Q. Let X∨= d−1K x1⊕· · · d−1K xs= (d−1K )sand Y = OKy1⊕ · · · OKys= OKs be the standard OK-lattices in XK and YK respectively. We choose an OK-lattice L in W such that TrK/Q(ϑ(L, L)) ⊂ Z and Lp = L ⊗ZZp=Pr−s
i=1(OK⊗ZZp)wi = (OK⊗ZZp)r−s. Define the OK-lattice M in V by
(1.3) M := Y ⊕ L ⊕ X∨.
Then hM, M ir,s ⊂ Z and Mp := M ⊗ZZp is self-dual with respect to h , ir,s as p is unramified in K. A pair of sublattices Polp =N−1, N0 of Mp is called an ordered polarization of Mpif N−1and N0are maximal isotropic submodules in Mp
and they are dual to each other with respect to h , ir,s. Moreover, we require that rank NΣ−1 = rank NΣ0c = r, rank NΣ−1c = rank NΣ0 = s.
We now endow M with the standard ordered polarization as follows. Put M−1= YΣ⊕ LΣ⊕ YΣc and M0= X∨Σc⊕ LΣc⊕ X∨Σ.
We call Pol0p =M−1, M0 the standard (ordered) polarization of Mp. We make the following identification according to the standard p-adic basis.
(1.4) MΣ0 = X∨Σ = Op(Σ)s
MΣ−1= YΣ⊕ LΣ = Op(Σ)r and MΣ−1c = YΣc= Op(Σc)s
MΣ0c = X∨Σc⊕ LΣc = Op(Σc)r. 1.9. Filtration. Let R be a ring and N be a free R-module of rank l with an or- dered basisv1, . . . , vl , define the standard (decreasing) filtration Fil•st(N,v1, . . . , vl )) of N by Filnst(N,v1, . . . , vl ) = Pl−n+1i=1 Rvi. We endow the Op-modules MΣ0 and MΣ0c with the filtration defined by
(1.5)
FiliMΣ0 = Filist(XΣ∨,e+xs, . . . , e+x1 ) if 1 ≤ i ≤ s, FiliMΣ0c=
(Fili−sst (LΣc,e−w1, . . . , e−wr−s ) if s + 1 ≤ i ≤ r, LΣc⊕ Filist(XΣ∨c,e−x1, . . . , e−xr ) if 1 ≤ i ≤ s.
Define the filtration Fil•MΣ of MΣ as follows.
(1.6)
FiliMΣ =
Fili−rst (XΣ∨,e+xs, . . . , e+x1 ) if r + 1 ≤ i ≤ r + s, Fili−sst (LΣ,e+wr−s, . . . , e+w1 ) ⊕ MΣ0 if s + 1 ≤ i ≤ r, Filist(YΣ,e+ys, . . . , e+y1 ) ⊕ LΣ⊕ MΣ0 if 1 ≤ i ≤ s.
1.10. Open compact subgroups. For v ∈ h, put
(1.7) Kv0:= {g ∈ G(Fv) | (M ⊗OOv)g = M ⊗OOv} . and K0 =Q
v∈hKv0. Let N0≥ 3 be a prime-to-p positive integer and fix an open compact subgroup K such that Kv = Kv0 for all v|p. We assume K is neat in the following sense:
(neat)
K ⊂ {g ∈ G(AF ,f) | M (g − 1) ⊂ N0· M } ν(K) ∩ O+×⊂ (K ∩ O×)2
The first condition assures that K is torsion-free ([Shi97, Lemma 24.3 (2)]), and the second one is possible by choosing sufficiently large N0 (cf. [Hid04b, page 136]
). Under the identification (1.2), we have
Kv0= GL(MΣ)×O×v → GL∼ r+s(Ov)×Ov× for v|p.
For gp= (g(1)p , ν(gp)) ∈ Kp0, we write g(1)p =A B
C D
according to the decomposi- tion MΣ = MΣ−1⊕ MΣ0. For n ∈ Z+, put
I1(pn) =g ∈ Kp0| gp∈ Nr+s(Op) × {1} (mod pn) and define
Kn=
g ∈ K | gp≡1r ∗ 0 1s
× {1} mod pn
, K1n=n
g ∈ K | gp(1)|gr•MΣ ≡ 1 (mod pn), ν(gp) ≡ 1 (mod pn)o
= K(p)I1(pn), K0n=n
g ∈ K | gp(1)∈ Br+s(Op) (mod pn)o .
2. Shimura varieties for unitary groups
One approach to explore the arithmetic of modular forms on unitary groups is to study the associated Shimura varieties and understand its structure as a moduli space of certain abelian schemes with additional structures. We review these objects in this section, following the exposition in [Hid04b, Chapter7]. In what follows, let K be a finite extension of Knc(e2πi/N0) which is unramified at p and let p be the prime ideal of K induced by ιp : K ,→ Cp. Let O := OK,(p) be the localization of OK at p and let Op be p-adic completion of O. We shall identify Op with the p-adic closure of ιp(O) in Cp via ιp. An O-algebra R is called a base ring, and similarly a scheme S over SpecO is called a base scheme.
2.1. Shimura varieties associated to GU (V ). Let be a finite set of ra- tional primes not dividing N0. Let U ⊂ K be an open compact subgroup in RF /QG(AQ,f).
Definition 2.1 (S-quadruple). Let S be a locally noetherian connectedO-scheme and let ¯s be a geometric point of S. A S-quadruple of level U() is a quadruple A = (A, ¯λ, ι, ¯η())S consisting of the following data:
• A is an abelian scheme of dimension (r + s)d over S.
• ι : OK,→ EndSA ⊗ZZ().
• λ is a prime-to- polarization of A over S and ¯λ is the O(),+-orbit of λ.
Namely
λ = O¯ (),+λ :=λ0∈ Hom(A, At) ⊗ZZ() | λ0= λ ◦ a, a ∈ O(),+ .
• ¯η() = η()U is a π1(S, ¯s)-invariant U -orbit of the isomorphism of OK- modules η(): M ⊗ bZ() ∼→ T()(A¯s) := H1(A¯s, bZ()). Here we define η()g(x) = η(xg∨) for g ∈ U .
Furthermore, the quadruple (A, ¯λ, ι, η()U )S satisfies the following conditions (K1)- (K3):
(K1) Lettdenote the Rosati involution induced by λ on EndSA ⊗ Z(). Then ι(b)t= ι(c(b)), ∀ b ∈ OK.
(K2) Let eλ be the Weil pairing induced λ. Lifting the isomorphism Z/N0Z ' Z/N0Z(1) induced by e2πi/N0 to an isomorphism ζ : bZ ' bZ(1), we can re- gard eλas a skew-Hermitian form eλ: T()(A¯s) × T()(As¯) → d−1K ⊗ZZb(). Let eη denote the skew-Hermitian form on T()(A) induced by eη(x, x0) = ϑr,s(η(x), η(x0)). We require that
eλ= u · eη for some u ∈ A()F ,f. (K3) The determinant condition:
(2.1) det(X − ι(b)| Lie A) = Y
σ∈Σ
(X − (σc)(b))r(X − σ(b))s∈ OS[X], ∀ b ∈ OK.
Define the fibered category C()U over SCHOK,() as follows. Objects over S are S-quadruples. For A = (A, ¯λ, ι, ¯η())S and A0 = (A0, ¯λ0, ι0, ¯η0())S, we define the morphism by
HomC()U (A, A0) =n
φ ∈ HomOK(A, A0) | φ∗¯λ0 = ¯λ, φ(¯η0()) = ¯η()o . We say A ' A0 if there exists an isomorphism in Hom
C()U (A, A0).
If = ∅ is the empty set, we define the functor SU : SCH/K→ SET S by SU(S) =A = (A, ¯λ, ι, ηU )S ∈ CU(S) / ' .
By the theory of Shimura-Deligne, SU is represented by a quasi-projective scheme SG(U )/K over K. We call SG(U )/K the Shimura variety attached to G = GU (V ) overK.
2.2. Kottwitz model. Suppose that = {p}. Let K be an open compact sub- group such that Kp= Kp0. Define the functor S(p)K : SCH/O→ SET S by
S(p)K (S) =n
A = (A, ¯λ, ι, ¯η(p))S ∈ C(p)K (S)o / ' .
In [Kot92], Kottwitz shows that S(p)K is represented by a quasi-projective scheme SG(K)/OoverO if K is neat.
2.3. Igusa schemes associated to GU (V ). Let Polp =N−1, N0 be a polar- ization of Mp = M ⊗ZZp, where M is the OK-lattice defined in (1.3).
Definition 2.2 (S-quintuples). Let n be a positive integer. Define the fibered category C(p)K,n,Pol
p whose objects over a base scheme S are S-quintuples (A, j)S = (A, ¯λ, ι, ¯η(p), j)S of level Kn, where A ∈ C(p)K (S) is a S-quadruple and
j : µpn⊗ZN0,→ A[pn]
is a monomorphism as OK-group schemes over S. We call j a level-pn structure of A. Morphisms between S-quintuples are
HomC(p) K,n,Polp
((A, j), (A0, j0)) =n
φ ∈ HomC(p) K
(A, A0) | φj = j0o .
Define the functor I(p)K,n,Pol
p: SCH/O→ SET S by I(p)K,n,Pol
p(S) =n
(A, j) = (A, ¯λ, ι, ¯η(p), j)S ∈ C(p)K,n(S)o / ' . It is known that I(p)K,n,Pol
p are relatively representable over SG(K)/O (cf. [HLS06, Lemma(2.1.6.4)] and [SGA64, Prop. 3.12]), and thus it is represented by a scheme.
Denote by IG0(Kn)/O the scheme that represents I(p)K,n,Pol0 p
for the standard polar- ization Pol0p =M−1, M0 defined in (1.4). LetA be the universal quadruple of level K(p) over SG(K). Then we have
IG0(Kn) = Inj
OK(µpn⊗ZM0,A).
In addition, IG0(Kn) is a model of SG(Kn) overO in view of the following lemma.
Lemma 2.3. Let L ⊃K(e2πi/pn) be a field. There is a non-canonical isomorphism over Spec L.
IG0(Kn)/L
→ S∼ G(Kn)/L.
Proof. Since MΣ = MΣ0 ⊕ MΣ−1, we have a natural exact sequence 0−→MΣ0 i
0
−→MΣ i−1
−→MΣ−1−→0.
We fix an isomorphism ζpn: Zn' µpn, which induces an isomorphism ζp−1n : µpn⊗ MΣ0 ' Zn⊗MΣ0. Let S be a scheme over L and A= (A, ¯λ, η(p)K)Sbe a S-quadruple of level K(p). A level-pn structure j of A is equivalent to a class ηp(j)K(pn), where ηp(j) : M−1⊕ M0= Mp
→ T∼ p(A). We can define the isomorphism (dependent on the choice of ζpn) IG0(Kn)/L→ S∼ G(Kn)/L by
h
(A, ¯λ, ι, ¯η(p), j)i
→h
(A, ¯λ, ι, (η(p)×ηp(j))Kn)i
.
2.3.1. Change of the polarization. It is clear that the notion of level-pn structures depends on the choice of the polarization of Mp. Choose γ ∈ Kp0 such that N−1= M−1γ and N0= M0γ. Then we see that j 7→ γj is an isomorphism from the level- pn structures with respect to Pol0p to those of Polp. Therefore the map [(A, j)] → [(A, γj)] induces an isomorphism between I(p)K,n,Pol0
p
and I(p)K,n,Pol
p.
2.3.2. p-adic one forms. Suppose p is nilpotent in R and pmR = 0 for some m ≥ 1.
Let (A, j) be a R-quintuple of level Kn, n ≥ m. Identify M0= MΣ−1⊕ MΣ0c with the basis in (1.4). Then the level pn-structure j over R induces a trivialization of Lie A:
j∗+: MΣ0 ⊗ R→ e∼ +Lie A[pn] = e+Lie A ; j∗−: MΣ0c⊗ R→ e∼ −Lie A[pn] = e−Lie A.
Let ωA = Hom(Lie A, R) be the R-module of invariant one forms of A. Taking the duality HomOp(−, Op) of the identification in (1.4), we obtain an isomorphism induced by j∗:
(2.2)
ω(j)+= ω(j−) : Op(Σ)r⊗ R→ e∼ +ωA; ω(j)−= ω(j+) : Op(Σc)s⊗ R→ e∼ −ωA. 2.4. Complex uniformization. Let U ⊂ K0 be an open compact subgroup in G(AF ,f). We recall the description of the complex points SG(U )(C) following [Shi98, Chap.VI].
