Semi-galois Categories III: Witt vectors by deformations of modular functions
aa r X i v : . [ m a t h . N T ] J u l Semi-galois Categories III:Witt vectors by deformations of modular functions
Takeo UramotoNagahama Institute of Bio-Science and Technology
Abstract
Based on our previous work on an arithmetic analogue of Christol’s theorem [15], this paperstudies in more detail the structure of the Λ-ring E K = K ⊗ W aO K ( O ¯ K ) of algebraic Witt vectorsfor number fields K . First developing general results concerning E K , we apply them to the casewhen K is an imaginary quadratic field. The main results include the “ modularity theorem ” foralgebraic Witt vectors, which claims that certain deformation families f : M ( b Z ) × H → C ofmodular functions of finite level always define algebraic Witt vectors b f by their special values,and conversely, every algebraic Witt vector ξ ∈ E K is realized in this way, that is, ξ = b f forsome deformation family f : M ( b Z ) × H → C . This gives a rather explicit description of theΛ-ring E K for imaginary quadratic fields K , which is stated as the identity E K = M K betweenthe Λ-ring E K and the K -algebra M K of modular vectors b f . This paper is a continuation of our previous work on arithmetic analogue of Christol’s theorem [15].This theorem claims that a (generalized)
Witt vector ξ ∈ W O K ( O ¯ K ) ( § O K of integers in a number field K if and only if the orbit of ξ under the action of the Frobenius lifts ψ p : W O K ( O ¯ K ) → W O K ( O ¯ K ) is finite (cf. Theorem 3.4, [15]); we then deduced that, with the aid ofthe work of Borger and de Smit [3], which heavily relies on class field theory, this is also preciselywhen the ghost components ξ a of ξ are periodic with respect to some modulus f of K (cf. Corollary2, [15]). With this background, the major goal of the current paper is then to study in more detailthe structure of the Λ-ring E K := K ⊗ W aO K ( O ¯ K ) (where W aO K ( O ¯ K ) is the Λ-ring of integral Wittvectors) in the case where K is an imaginary quadratic field; in particular, we prove that the Λ-ring E K coincides, as K -subalgebras of ( K ab ) I K , with the K -algebra M K of modular vectors — i.e. thosevectors b f ∈ ( K ab ) I K whose components b f a are given by special values of certain deformation families f : M ( b Z ) × H → C of modular functions— i.e. the fiber f m := f ( m, − ) : H → C at each m ∈ M ( b Z )is a modular function of finite level, and these f m satisfy certain correlation— prototypical examplesof such deformation families of modular functions are given by Fricke functions f a ( a ∈ Q / Z ); cf. § modularity theorem (cf. § Theorem 1.1 (modularity theorem) . We have the following identity as K -subalgebras of ( K ab ) I K : E K = M K . (1.1)1 ackground To be precise, for a flat O K -algebra A , recall that the ring W O K ( A ) of (generalized) Witt vectors with coefficients in A [1] is defined as the intersection W O K ( A ) := T n U n ( A ) of thefollowing O K -algebras U n ( A ) ⊆ A I K given by induction on n ≥ § § I K with respect to arithmeticderivations in the sense of Buium [6]): U ( A ) := A I K ; (1.2) U n +1 ( A ) := (cid:8) ξ ∈ U n ( A ) | ∀ p ∈ P K . ψ p ξ − ξ N p ∈ p U n ( A ) (cid:9) ; (1.3)where P K and I K denote the set of maximal ideals of O K and the monoid of nonzero ideals of O K respectively; A I K denotes the I K -times product of A ; N p denotes the absolute norm of p ∈ P K ; and ψ p : A I K → A I K denotes the shift ( ξ a ) ( ξ pa ). In the case when K is the rational number field Q ,say, this ring W O K ( A ) is isomorphic to the usual ring W Z ( A ) of big Witt vectors ; and the usual ring W p ( A ) of p -typical Witt vectors can be constructed in a similar way. While these rings of big and p -typical Witt vectors are conventionally constructed using Witt polynomials, Borger [1] recastedthese constructions of Witt vectors putting his focus on Frobenius lifts, and constructed the ring ofWitt vectors as the universal ring among those rings which are equipped with commuting family ofFrobenius lifts, or Λ -rings ; this construction naturally allows us to extend the base ring from Z toarbitrary Dedekind domains O with finite residue fields (or more): He proved the existence of sucha universal ring for this generalized setting by the above inductive construction. The basic theoryof these generalized Witt vectors was developed in [1]; our major concern in this paper is to studythe structure of the rings of these generalized Witt vectors.In particular, among generalized Witt vectors ξ ∈ W O K ( O ¯ K ) with coefficients in the ring O ¯ K of algebraic integers, those ξ ∈ W O K ( O ¯ K ) which are integral over O K — i.e. integral Witt vectors —were proved to be relevant to class field theory of the number field K , as discussed in our previouswork [15] based on [2, 3]; see also [4]. In fact, on the one hand, it was proved in [3] that the category C K of those (generalized) Λ-rings which are finite etale over K and have integral models ( § B f DR K of finite DR K -sets, where DR K is the profinite monoidcalled the Deligne-Ribet monoid and given by inverse limit of ray class monoids DR f ( f ∈ I K ); inthis proof, class field theory was used in an essential way, which suggests an inherent connectionbetween class field theory and (generalized) Λ-rings. Motivated by this duality [3], our previouswork [15] then related integral Witt vectors to the objects of C K ; to be precise, we proved that forany integral Witt vector ξ ∈ W aO K ( O ¯ K ), the Λ-ring X ξ = K ⊗ O K h ξ i generated by the orbit I K ξ of ξ under the action of the Frobenius lifts ψ p is finite etale over K and has an integral model, orin other words, forms an object of C K (cf. Proposition 4 [15])— indeed as further proved in thispaper, this type of objects X ξ is universal in C K (cf. § X ξ over K . (However, this theorem itself is of independent interest in that it givesa natural arithmetic (or F -) analogue of Christol’s theorem [7, 8] on formal power series ξ ∈ F q [[ t ]]over finite field F q ; cf. § Contribution
As a natural continuation of [15], the current paper then studies in more detail thestructure of the ring W aO K ( O ¯ K ) of integral Witt vectors and the K -algebra E K := K ⊗ W aO K ( O ¯ K )whose elements we call algebraic Witt vectors. In particular, after developing some general basicresults about the K -algebra E K ( § K is the rational numberfield Q ( § § §
4, where, as briefly summarized above, we relate algebraic Witt vectors ξ ∈ E K with certaindeformation families of modular functions (of finite level).To be more specific, the subject of § galois objects (cf. § C K , which provides a general basis for the study in §
4. In particular, we see that,for each finite set Ξ ⊆ W aO K ( O ¯ K ) of integral Witt vectors, we can construct a Λ-ring X Ξ , which areobjects of C K as in the case of singleton Ξ = { ξ } mentioned above. Concerning this construction,we prove in § X Ξ is a galois object of C K for every Ξ ⊆ W aO K ( O ¯ K ), and conversely, everygalois object of C K is of this form up to isomorphism. That is, the galois objects of C K are preciselythose of the form X Ξ for some finite Ξ ⊆ W aO K ( O ¯ K ). This particularly implies that the direct limitof the galois objects of C K is naturally isomorphic to our K -algebra E K = K ⊗ W aO K ( O ¯ K ); this givesa characterization of the K -algebra E K of algebraic Witt vectors as the universal ring among thoseΛ-rings finite etale over K and having integral models (i.e. universal with respect to the objects of C K ). For the purpose of §
4, the rest of § § § § C K ; by the result in § X Ξ for someΞ ⊆ W aO K ( O ¯ K ). In this subsection, we particularly determine the state complexity of integral Wittvectors ξ ∈ W aO K ( O ¯ K ), i.e. the minimum size c ξ of DFAO’s that generate ξ (cf. § c ξ for ξ ∈ W aO K ( O ¯ K ) is equal to the dimension dim K X ξ of the Λ-ring X ξ over K . (This is an F -analogue of Bridy’s result [5] on formal power series ξ ∈ F q [[ t ]]; cf. § § W aO K ( O ¯ K ) and E K for the case when K isthe rational number field Q as an immediate consequence of the result in § W a Z (¯ Z ) of integral Witt vectors is isomorphic to the group-ring Z [ Q / Z ],hence, E Q is isomorphic to the group-algebra Q [ Q / Z ]. In more elementary words, this isomorphism E Q ≃ Q [ Q / Z ] shows that the algebraic (resp. integral) Witt vectors ξ = ( ξ n ) n ∈ N ∈ E Q are preciselythe Q -linear (resp. Z -linear) combinations of the vectors ζ ( γ ) ∈ ( Q ab ) N of the form ζ ( γ ) = ( e πiγn ) n for γ ∈ Q / Z .The subject of § K is an imaginary quadraticfield. To be more specific, we prove in § ξ ∈ E K are precisely the modular vectors , i.e. the vectors b f ∈ ( K ab ) I K whose components b f a are defined by special valuesof certain deformation families f : M ( b Z ) × H → C of modular functions, which we shall call Wittdeformation families of modular functions (because their special values eventually define algebraicWitt vectors). Briefly, a Witt deformation (family) of modular functions is defined as a continuousfunction f : M ( b Z ) × H → C such that the fiber f m := f ( m, − ) : H → C at each m ∈ M ( b Z ) is amodular function of finite level, where the fibers f m satisfy a few conditions (cf. Definition 8, § f mγ ( γ − τ ) = f m ( τ ) for γ ∈ SL ( Z ) = Γ in particular. Concerningthis, we see that each Witt deformation f : M ( b Z ) × H → C of modular functions defines a modularvector b f ∈ ( K ab ) I K in a natural (but non-trivial) way that heavily relies on the work of Connes,Marcolli and Ramachandran [10], Connes and Marcolli [9], Laca, Larsen, and Neshveyev [12] andYalkinoglu [16]. Briefly speaking, this construction f b f is based on the isomorphisms betweenthe above-mentioned Deligne-Ribet monoid DR K and the (moduli) space Lat K of 1 -dimensional K -lattices [10, 12, 16], and between the (moduli) space Lat Q of 2 -dimensional Q -lattices and thequotient space Γ \ ( M ( b Z ) × H ) [9] (cf. § K is now imaginary quadratic, hence, of degree2 over Q , we have a natural embedding Lat K ֒ → Lat Q ; and then composed with the isomorphisms DR K ≃ Lat K and Lat Q ≃ Γ \ ( M ( b Z ) × H ), as well as the canonical embedding I K ֒ → DR K , each3itt deformation f : M ( b Z ) × H → C then induces a function b f : I K → C as follows: b f : I K ֒ → DR K ≃ −→ Lat K ֒ → Lat Q ≃ −→ Γ \ ( M ( b Z ) × H ) f −→ C . (1.4)We can see that b f takes its values in K ab , hence defines a vector b f ∈ ( K ab ) I K which is what we callthe modular vector associated to f ; the K -subalgebra M K ⊆ ( K ab ) I K is defined as that consistingof modular vectors in this sense. The major result of § modularity theorem ” claiming theidentity E K = M K as K -subalgebras of ( K ab ) I K ( § f : M ( b Z ) × H → C , the associated modular vector b f defies an algebraic Witt vector, i.e. b f ∈ E K ;and conversely, every algebraic Witt vector ξ ∈ E K is realized in this way, i.e. ξ = b f for some Wittdeformation f : M ( b Z ) × H → C . This identity E K = M K relates the geometry of Λ-rings in C K and that of modular functions. Acknowledgement
We are grateful to James Borger for fruitful discussions, which motivated usto prove Proposition 3.2 instead of considering the existence of cyclic Witt vectors, cf. Remark 6;and also, to Naoya Yamanaka and Hayato Saigo for their support and encouragement. This workwas supported by JSPS KAKENHI Grant number JP16K21115.
This section summarizes necessary terminology and results from [15, 9, 10, 12, 16]. But in order toavoid duplications, we refer the reader to § § Witt vectors (Definition 1, pp. 543), Λ -rings (Definition 2, pp. 544), integralmodels of finite etale Λ -rings over K (Remark 4, pp. 544) and our arithmetic analogue of Christol’stheorem (Theorem 3.4, pp. 557) in particular. Also, for the categorical concepts and results on semi-galois categories , we refer the reader to [14]. C K Here we recall some basic facts concerning the semi-galois category C K constructed by Borger andde Smit [3] for a number field K . Let K be a number field and O K the ring of integers in K . Wedenote by P K and I K the set of non-zero prime ideals of O K and the monoid of non-zero ideals of O K respectively. For each p ∈ P K , we denote by k p the residue field k p := O K / p and by N p theabsolute norm N p := k p . Definition 1 (the category C K ) . The objects of the category C K [3] are the Λ-rings that are finiteetale over K and have integral models (cf. Remark 4, pp.544 [15]); the arrows of C K are the Λ-ringhomomorphisms over K between them.The category C K is equipped with a functor F K : C opK → Sets that assigns to each object X ∈ C K the finite set F K ( X ) := Hom K ( X, ¯ K ) of the K -algebra homomorphisms from the underlying K -algebra X to the algebraic closure ¯ K of K ; and to each arrow f : X → Y of Λ-rings the pullback f ∗ : Hom K ( Y, ¯ K ) → Hom K ( X, ¯ K ). With this functor F K , we can see that the pair h C opK , F K i formsa semi-galois category. Therefore, by the general results developed in [14], the semi-galois category h C opK , F K i should be canonically equivalent to the semi-galois category h B f M, F M i of finite M -setswith M being the fundamental monoid M = π ( C opK , F K ) (cf. Definition 4, § X ∈ C K is a finite etale Λ-ring over K , i.e., a finite etale K -algebra equipped with commuting family of Frobenius lifts ψ p : X → X foreach p ∈ P K ; thus, the finite set F K ( X ) = Hom K ( X, ¯ K ) is equipped with a continuous action ofthe absolute galois group G K of K as well as the action of the monoid I K by natural pullbacks of ψ a : X → X for a ∈ I K (see Remark 2, § ψ a ); and the actions of G K and I K commute with each other. Therefore, the finite set F K ( X ) forms a finite ( G K × I K )-set withrespect to this action. To be specific, the action of the monoid G K × I K on the finite set F K ( X ) isgiven as follows: F K ( X ) × ( G K × I K ) → F K ( X )( s, ( σ, a )) σ ◦ s ◦ ψ a . Of course, the objects of C K are not just finite etale Λ-rings over K but have integral models (cf.Remark 4, § X has an integral model in terms of the corresponding finite ( G K × I K )-set F K ( X ). As proved there,the Deligne-Ribet monoid DR K (after their work [11] where this profinite monoid appeared) comesinto play for this characterization: As we recall below, we have a canonical map G K × I K → DR K ;and it was proved in [3] that a finite etale Λ-ring X over K has an integral model (i.e. an objectof C K ) if and only if the ( G K × I K )-action on F K ( X ) factors through this map G K × I K → DR K (Theorem 1.2 [3]).To be precise, the Deligne-Ribet monoid DR K is defined as the inverse limit of the ray classmonoids , i.e. the finite monoids DR f given for each f ∈ I K as the quotient monoid DR f := I K / ∼ f of the monoid I K by the following congruence relation ∼ f on I K : for a , b ∈ I K , a ∼ f b ⇔ ab − = ( t ) for ∃ t ∈ K + ∩ (1 + fb − ); (2.1)where K + denotes the totally positive elements of K . This congruence relation ∼ f is of finite index,and thus, the ray class monoid DR f is a finite monoid. To form an inverse system of the ray classmonoids DR f , note that if f | f ′ then a ∼ f ′ b implies a ∼ f b ; therefore, we have a canonical monoidsurjection DR f ′ ։ DR f by the assignment [ a ] f ′ [ a ] f where [ a ] f ∈ DR f denotes the equivalenceclass of a ∈ I K in DR f . With respect to this surjections DR f ′ ։ DR f , the ray class monoids DR f ( f ∈ I K ) constitute an inverse system of finite monoids; and the Deligne-Ribet monoid DR K is thendefined as the inverse limit of this system: Definition 2 (the Deligne-Ribet monoid DR K ) . The
Deligne-Ribet monoid DR K is the profinitemonoid defined as the following inverse limit of the above inverse system of the ray class monoids DR f : DR K := lim f ∈ I K DR f . (2.2) Remark 1.
By definition, DR K is commutative; also, for each f ∈ I K , we can identify DR × f = C f ,where DR × f is the unit group of DR f and C f is the strict ray class group with the conductor f · ( ∞ )(cf. 2.6, [11]). Taking inverse limit, we then have an isomorphism DR × K ≃ lim f C f ≃ G abK , where thesecond isomorphism is the class field isomorphism.The above-mentioned map G K × I K → DR K is then given, on the second factor, as the canonicalmap I K → DR K ; and on the first factor, as the composition G K ։ G abK ≃ lim f C f ≃ DR × K ⊆ DR K .With this, we restate here the result of Borger and de Smit [3]:5 heorem 2.1 (Theorem 1.2 [3]) . A finite etale Λ -ring X over K has an integral model if and onlyif the ( G K × I K ) -action on the finite set F K ( X ) factors through the map G K × I K → DR K givenabove; in other words, this means that the fundamental monoid π ( C opK , F K ) is isomorphic to DR K and we have the following equivalence of categories: C opK ≃ −→ B f DR K X F K ( X ) Remark 2 (every component of X ∈ C K is abelian) . Recall that, since the underlying K -algebraof each object X ∈ C K is finite etale over K , we have an isomorphism X ≃ L × · · · × L n for somefinite extensions L i /K of number fields. As described above, moreover, since the action of G K ontoF K ( X ) factors through the abelianization G K ։ G abK , it follows that every component L i of X mustbe abelian over K . (See § without integral models.) We shall use this fact throughout this paper. Remark 3 (the case of K = Q ) . In [2], preceding [3], the authors studied the case when K is therational number field Q . In this case, it is shown that the Deligne-Ribet monoid DR Q is isomorphicto the multiplicative monoid b Z of profinite integers. In particular, the monoid I Q is identified withthe multiplicative monoid N of positive integers; and we can see that DR N for N ∈ N is isomorphicto the multiplicative monoid Z /N Z . Remark 4 (galois objects) . The major subject of § C K in termsof integral Witt vectors. For this reason, we recall here a few basic facts about galois objects of thesemi-galois category C K ; nevertheless, instead of recalling the most general facts on galois objects(cf. § B f DR K .In this category B f DR K , the galois objects are precisely the rooted DR K -sets , i.e. those DR K -sets S ∈ B f DR K which have some s ∈ S such that S = s · DR K ; in this case, s ∈ S is called a root of S . (In general, rooted objects are not galois; but in the current situation, DR K is commutative;from this, it follows that rooted objects in B f DR K are always galois.) In terms of Λ-rings X ∈ C K ,this means that X ∈ C K is galois if and only if there exists s ∈ F K ( X ) = Hom K ( X, ¯ K ) such thatevery s ∈ F K ( X ) can be written as s = σ ◦ s ◦ ψ a for some σ ∈ G K and a ∈ I K . We shall use thisfact for the study in § The Deligne-Ribet monoid DR K was defined as the inverse limit lim f DR f of the ray class monoids DR f ( f ∈ I K ); but as proved by Yalkinoglu [16], this profinite monoid DR K has yet another aspectas the “moduli space” of 1 -dimensional K -lattices up to scaling [10, 12]. In the case when K isan imaginary quadratic field in particular, such K -lattices are naturally 2 -dimensional Q -lattices ;therefore, DR K then has a natural embedding to the space of 2-dimensional Q -lattices Lat Q up toscaling, which is further proved isomorphic to a certain quotient space Γ \ ( M ( b Z ) × H ) [9]. In thissubsection, we recall these facts essentially from [9, 10, 16, 12] which will provide a key for relatingWitt vectors to modular functions.Throughout this subsection, let us suppose that K is an imaginary quadratic field, and also let O K = Z τ + Z with τ ∈ H , where H denotes the upper half plane in C . For a number field K , wedenote by A K the ring of adeles of K ; by A K,f the ring of finite adeles; and also, by b O K the ring6f finite integral adeles. In general, for a ring R , we denote by R ∗ the invertible elements. Finally,[ − ] : A ∗ K → G abK denotes the Artin map.We start with relating DR K with 1 -dimensional K -lattices , which is defined as follows: Definition 3 ( K -lattices; [10]) . A (1-dimensional) K -lattice is a pair (Λ , φ ) of a finitely generated O K -submodule Λ of C such that K ⊗ O K Λ ≃ K and a homomorphism φ : K/O K → K Λ / Λ. Example 1 (cf. Lemma 2.4 [10]) . Fractional ideals of K are “prototypical” examples of K -latticesin the sense that, up to scaling by some λ ∈ C ∗ , every K -lattice Λ ⊆ C becomes a fractional idealas λ Λ ⊆ K . Such a scaling factor λ ∈ C ∗ is unique modulo K ∗ . Proposition 2.1 (cf. Proposition 2.6 [10]) . There are bijective correspondences (1) between b O K × b O ∗ K ( A ∗ K /K ∗ ) and the set of K -lattices, and (2) between b O K × b O ∗ K ( A ∗ K,f /K ∗ ) and the set of K -latticesup to scaling.Proof. For this proof, the reader is referred to [10]; we just describe the correspondence. For [ ρ, t ] ∈ b O K × b O ∗ K ( A ∗ K /K ∗ ), the correspondence is given by [ ρ, t ] (Λ t , φ ( ρ,t ) ), where Λ t := t − ∞ ( t f b O K ∩ K )with t f , t ∞ denoting the non-archimedian and archimedian components of t = ( t f , t ∞ ) ∈ A ∗ K and φ ( ρ,t ) : K/O K → K Λ t / Λ t is given by the following composition (the upper row): K/O K / / ≃ (cid:15) (cid:15) K/O K / / ≃ (cid:15) (cid:15) K Λ t f / Λ t f t − ∞ / / ≃ (cid:15) (cid:15) K Λ t / Λ t A K,f / b O K ρ / / A K,f / b O K t f / / A K,f /t f b O K (2.3)where ρ, t f and t − ∞ denote the maps given by the straightforward multiplications. Based on this,the second correspondence is given juts by forgetting the scaling factor t ∞ . Proposition 2.2 ( DR K as moduli space) . There is a bijective correspondence between DR K andthe set of K -lattices up to scaling.Proof. This is given as a combination of the above results of [10] and that of [16] together with theclass field theory isomorphism [ − ] : A ∗ K,f /K ∗ ≃ −→ G abK . That is, by Proposition 8.2 [16], we havean isomorphism DR K ≃ b O K × b O ∗ K G abK ; then, by combining with G abK ≃ A ∗ K,f /K ∗ , this induces anisomorphism DR K ≃ b O K × b O ∗ K ( A ∗ K,f /K ∗ ). As proved above, the latter corresponds to the set of K -lattices up to scaling, hence the claim.In what follows, we shall denote by Lat K the set of K -lattices up to scaling; for each K -lattice(Λ , φ ), we shall denote by [Λ , φ ] ∈ Lat K the equivalence class (up to scaling) of (Λ , φ ). So far, weproved the following isomorphisms: DR K ≃ b O K × b O ∗ K ( A ∗ K,f /K ∗ ) ≃ Lat K . (2.4)Moreover, since K is now imaginary quadratic, hence, of degree 2 over Q , we can then think of K -lattices as 2 -dimensional Q -lattices in the following sense:7 efinition 4 (2-dimensional Q -lattice) . A (2-dimensional) Q -lattice is a pair (Λ , φ ) of a lattice Λin C and a homomorphism φ : Q / Z → Q Λ / Λ.Recall that we now have O K = Z τ + Z for τ ∈ H , hence, K = Q τ + Q ; this choice of the basis( τ, t of K over Q gives an isomorphism K ≃ Q , with O K ≃ Z . Also, note that for a K -latticeΛ ⊆ C , we have K Λ = Q Λ ⊆ C . By these identifications, each K -lattice (Λ , φ ) naturally defines a(2-dimensional) Q -lattice (Λ , φ ′ ), where φ ′ : Q / Z → Q Λ / Λ is given by: φ ′ : Q / Z ≃ K/O
K φ −→ K Λ / Λ = Q Λ / Λ . (2.5)In this sense, we identify K -lattices (Λ , φ ) with (2-dimensional) Q -lattices; and denote by the samesymbol (Λ , φ ).In general, as in the case of K -lattices, 2-dimensional Q -lattices can be classified with a certain“moduli space”; the following proposition essentially due to [9] gives a description of the space: Proposition 2.3 ([9]) . There is a bijective correspondence between Γ \ ( M ( b Z ) × GL +2 ( R )) and theset of Q -lattices. By this correspondence, we have a bijective correspondence between Γ \ ( M ( b Z ) × H ) and the set of Q -lattices up to scaling by C ∗ as well.Proof. The proof is essentially due to [9]; but we need modify the constructions there so that ourconstructions below get compatible with the conventions in [13]. With this slight modification, theproof in § m, α ) ∈ M ( b Z ) × GL +2 ( R ), the corresponding Q -lattice is defined by ( m, α ) (Λ α , mα ), where Λ α := Z · α · ( i, t regarding Z as consisting ofrow vectors, and mα : Q / Z → Q Λ α / Λ α is given by a a · mα · ( i, t for a = ( a , a ) ∈ Q / Z .(Here M ( b Z ) acts on Q / Z as Q / Z ≃ A Q ,f / b Z .) The group Γ = SL ( Z ) acts on M ( b Z ) × GL +2 ( R )by ( γ, ( m, α )) ( mγ − , γα ). With this definition, the above correspondence ( m, α ) (Λ α , mα )induces a bijection from Γ \ ( M ( b Z ) × GL +2 ( R )) to the set of Q -lattices.Concerning the second claim, note that the action of scaling λ = s + it ∈ C ∗ on Λ α correspondsto the action of the following matrix on the right of α ∈ GL +2 ( R ): λ = (cid:18) s − tt s (cid:19) . (2.6)Also the quotient GL +2 ( R ) / C ∗ by this action of C ∗ can be identified with the upper half plane H via the following correspondence GL +2 ( R ) → H : α = (cid:18) a bc d (cid:19) α ( i ) := ai + bci + d . (2.7)By this identification, we eventually obtain a bijection from Γ \ ( M ( b Z ) × H ) to the set of Q -latticesup to scaling. Remark 5 (basis of Λ α ) . Suppose that α ∈ GL +2 ( R ) is given as follows: α = (cid:18) a bc d (cid:19) . (2.8)Then the basis of Λ α is given by ( ai + b, ci + d ); that is, Λ α = Z ( ai + b ) + Z ( ci + d ). Also, note that τ α := α ( i ) = ( ai + b ) / ( ci + d ) corresponds to this lattice Λ α up to scaling.8et Lat Q denote the set of 2-dimensional Q -lattices up to scaling. In the above lemma, we haveconstructed the isomorphism Lat Q ≃ Γ \ ( M ( b Z ) × H ); and, we also have a natural embedding ofsets Lat K ֒ → Lat Q in the sense mentioned above. Consequently, we have constructed the followingsequence of maps: DR K ≃ b O K × b O ∗ K ( A ∗ K,f /K ∗ ) ≃ Lat K ֒ → Lat Q ≃ Γ \ ( M ( b Z ) × H ) . (2.9)We will use this sequence of maps in § C K In this section, before proceeding to our major subject in §
4, we develop some general facts aboutthe Λ-ring K ⊗ W aO K ( O ¯ K ) of algebraic Witt vectors. The first subsection ( § C K , from which we deduce the universality of K ⊗ W aO K ( O ¯ K ) with respect tothe Λ-rings in C K ; the second subsection ( § C K ,where we particularly determine the state complexity of integral Witt vectors; the last subsection( § W aO K ( O ¯ K ) and K ⊗ W aO K ( O ¯ K ) when K is the rational number field Q ,which are proved isomorphic to the group-ring Z [ Q / Z ] and Q [ Q / Z ] respectively. After developingthese basic results, the next section ( §
4) then proceeds to our major subject, where we study theserings when K is an imaginary quadratic field. The subject of this subsection is to classify the galois objects in C K in terms of integral Witt vectors;in this section, let E K denote the direct limit of the galois objects of C K (see § inverse system of galois objects of C opK ). We prove that E K is naturally isomorphic to K ⊗ W aO K ( O ¯ K ); thisgives a characterization of our Λ-ring K ⊗ W aO K ( O ¯ K ) of algebraic Witt vectors, which gives a basisof our study in § Definition 5 (Λ-ring X Ξ ) . Let Ξ ⊆ W aO K ( O ¯ K ) be a finite subset. Then the associated Λ -ring X Ξ is defined as follows: X Ξ := K ⊗ O K [ I K Ξ] , (3.1)where I K Ξ is the orbit of Ξ under the action of ψ a s ( a ∈ I K ).We show that these Λ-rings are always galois objects of C K ; and conversely, every galois objectof C K is of this form up to isomorphism: Proposition 3.1.
For any finite set Ξ ⊆ W aO K ( O ¯ K ) , the associated Λ -ring X Ξ is a galois objectwith its roots given by X Ξ ∋ η η ∈ ¯ K .Proof. The proof of the first claim that X Ξ has an integral model (i.e. is an object of C K ) followssimilarly to that of Proposition 4, [15]. We prove that X Ξ is a galois object with the homomorphism s : X Ξ ∋ ξ ξ ∈ ¯ K being its root, where 1 = O K is the trivial ideal of O K . To be more specific,by definition of galois objects, it suffices to see that every homomorphism s : X Ξ → ¯ K is given by s = σ ◦ s ◦ ψ a for some a ∈ I K and σ ∈ G K (cf. Remark 4, § ≡ Ξ on I K defined by a ≡ Ξ b if and onlyif ψ a ξ = ψ b ξ for every ξ ∈ Ξ, which is of finite index since the orbit of Ξ under the action of ψ a ’s( a ∈ I K ) is finite (cf. Theorem 3.4 [15]). Let I K = J ⊔ · · · ⊔ J N be the ≡ Ξ -class decomposition; andchoose their representatives, say a i ∈ J i for i = 1 , · · · , N . Also, since X is finite over K with eachcomponent being abelian, we can take a finite abelian subfield ¯ K/L/K so that the image s ( X ) forevery s : X → ¯ K is contained in L . Then let us define a homomorphism t : X → L × · · · × L by t ( η ) := ( η a , · · · , η a N ), which is injective. In fact, note first that if a ≡ Ξ b then η a = η b for every η ∈ X Ξ by the fact that X Ξ is generated by the orbit I K Ξ and by definition of ≡ Ξ . Therefore,the values of η a are determined by those of η a i for i = 1 , · · · , N ; hence t : X → L × · · · × L isinjective. This means that the G K -set Hom K ( X, ¯ K ) is a quotient of F Hom K ( L, ¯ K ) induced from t : X → L × · · · × L ; and by construction, this proves the claim that every s ∈ Hom K ( X, ¯ K ) is givenby s = σ ◦ s ◦ ψ a i for some i . Proposition 3.2.
Every galois object X ∈ C K is isomorphic to X Ξ for some finite Ξ ⊆ W aO K ( O ¯ K ) .Proof. Let s : X → ¯ K be a root of X and A ≤ X be its integral model. For each ξ ∈ X we define ξ s = ( ξ s a ) ∈ O I K ¯ K by ξ s a := s ( ψ a ξ ). It is not difficult to see that ξ s ∈ W aO K ( O ¯ K ) for every ξ ∈ A because A is an integral model. Also, since X = K ⊗ A and A is finite over O K , there exist some ξ , · · · , ξ n ∈ A that generate X over K . Let Ξ := { ξ s , · · · , ξ sn } ⊆ W aO K ( O ¯ K ). We prove that X isisomorphic to X Ξ . Notice that the assignment ξ ξ s for ξ ∈ X defines a Λ-ring homomorphism X → X Ξ , which is clearly surjective. To show its injectivity, note that since s is a root of the galoisobject X , every homomorphism s ′ : X → ¯ K is a composition s ′ = s ◦ f for some f ∈ End( X ). Also,by the fact that End( X ) is generated by ψ p ’s, this implies that the values ξ s a = s ( ψ a ξ ) determines ξ ∈ X ; hence, ξ ξ s is indeed injective.Therefore, the galois objects in C K are precisely of the form X Ξ for some finite Ξ ⊆ W aO K ( O ¯ K ).Consequently, we obtain the following presentation of the K -algebra E K in terms of integral Wittvectors: Theorem 3.1.
We have a canonical isomorphism of Λ -rings: E K ≃ K ⊗ W aO K ( O ¯ K ) . In particular, W aO K ( O ¯ K ) is isomorphic to the direct limit of the maximal integral models of galoisobjects of C K .Proof. This isomorphism E K → K ⊗ W aO K ( O ¯ K ) is given as (the direct limit of) the isomorphisms X → X Ξ constructed in the above proposition for galois objects X . (Recall also that E K is a directlimit of galois objects X .) Corollary 3.1.
The K -algebra K ⊗ W aO K ( O ¯ K ) is isomorphic to the K -algebra of all locally constant K ab -valued G K -equivariant functions on DR K , namely, E K = Hom G K ( DR K , ¯ K ) .Proof. This is a direct consequence of Theorem 3.1 above and Theorem 10.1 [16] due to Neshveyev:To be more precise, Theorem 10.1 [16] claims that E K is isomorphic to the K -algebra of suchfunctions. Since we proved that E K is also isomorphic to K ⊗ W aO K ( O ¯ K ), the composition of theseisomorphisms shows the claim of this corollary. 10 emark 6 (cyclic Witt vector) . We are concerned with whether every galois object X ∈ C K isactually isomorphic to X ξ for some ξ ∈ W aO K ( O ¯ K ), that is, whether we can take Ξ ⊆ W aO K ( O ¯ K ) inProposition 3.2 as a singleton Ξ = { ξ } . This is true for e.g. the (cofinal) galois objects of the form Q [ x ] / ( x n −
1) of C Q ; but we do not know whether every galois object X has such a ξ in general,which we shall call a cyclic Witt vector for X (a la, cyclic vectors for differential modules known indifferential galois theory). Remark 7 (galois correspondence) . Although we do not give a proof here, it would be meaningfulto mention a certain galois correspondence that naturally extends the usual galois correspondenceof galois theory for number fields. To be specific, our galois correspondence is the one between thefollowing objects:1. Λ-subalgebras of E K ;2. profinite quotients of DR K ;3. semi-galois full subcategories of C K .This follows from the presentation of E K = Hom G K ( DR K , ¯ K ) and DR K = Hom K ( E K , ¯ K ) as wellas the duality between DR K and C K . As we proved in the above subsection, the galois objects of C K are precisely those of the form X Ξ for some finite Ξ ⊆ W aO K ( O ¯ K ). In this subsection, we then study the structure of the galois objects X Ξ and represent X Ξ in terms of ξ ∈ Ξ. Starting from some general facts about galois objects in C K , we describe the components L i of X Ξ ≃ L × · · · × L r and also determine the state complexity of integral Witt vectors ξ ∈ W aO K ( O ¯ K ), which is a natural analogue of Bridy’s result [5] on formalpower series ξ ∈ F q [[ t ]] algebraic over F q [ t ].To this end, we need to prepare some general lemmas: Lemma 3.1.
Let X ∈ C K . If ψ p is an automorphism on X , then the action of ψ p on F K ( X ) isequal to that of some σ ∈ G K .Proof. The proof is done by localization and Theorem1.1, [4]. Let A ≤ X be the maximal integralmodel of X and K p be the completion of K at p ; also let X p = K p ⊗ X and A p = O K p ⊗ A . Then X p together with ψ p is a Λ p -ring with integral model A p . Since ψ p is an automorphism on X p , theset S unr := T ∞ n =0 p n S , where S := F K p ( X p ), is equal to S itself. This implies that, by Theorem 1.1,[4], the action of p on S unr = S is equal to the Frobenius σ p ∈ G K p /I p (where I p ≤ G K p denotesthe inertia subgroup). Lemma 3.2.
Let X ∈ C K be a galois object. If f ∈ End( X ) is an automorphism of the Λ -ring X ,then the action of f onto F K ( X ) by pullback is equal to the action of some σ ∈ G K .Proof. Let f ∈ End( X ) be an automorphism. Since X is galois and I K is dense in DR K , the actionof f on F K ( X ) by pullback is equal to that of ψ a for some a ∈ I K . (To see this, recall the definitionof galois objects, Definition 12, § B f DR K are those DR K -sets which are of the form of finite quotients DR K ։ H , cf. Lemma 17, § f is an automorphism, so is the action of ψ a . Hence, if a = p · · · p n , the actions of ψ p i ’s are allautomorphisms as well. By the above lemma, the actions of ψ p i ’s come from some σ ∈ G K ; hence,so is the action of ψ a = ψ p · · · ψ p n , which is equal to that of f .11o proceed further, let us recall the following notion from semigroup theory: Definition 6 ( J -equivalence) . Let M be a commutative monoid. For two elements s, s ′ ∈ M , wedenote as s ≤ J s ′ if we have an inclusion of the (both-sided) ideals sM ⊆ s ′ M , or in other words,there exists m ∈ M such that s = s ′ m . Furthermore, we denote as s ∼ J s ′ and say that s and s ′ are J -equivalent if s ≤ J s ′ and s ′ ≤ J s . The set of J -equivalent classes is denoted as M/ J . Lemma 3.3.
Let X ∈ C K be galois with s : X → ¯ K ∈ F K ( X ) its root. Then, f, f ′ ∈ End( X ) are J -equivalent in the (commutative) monoid End( X ) if and only if the corresponding s ◦ f, s ◦ f ′ ∈ F K ( X ) belong to the same G K -component.Proof. For the reason mentioned in the above lemma, we may put f = ψ a and f ′ = ψ a ′ for some a , a ′ ∈ I K . First suppose that s a := s ◦ ψ a and s a ′ := s ◦ ψ a ′ belong to the same G K -componentin F K ( X ); that is, s a = σ ◦ s a ′ for some σ ∈ G K . By the density of I K in DR K , the action of σ isequal to the action of ψ b for some b ∈ I K , which means that s a = s a ′ b , hence we have ψ a ≤ J ψ ′ a inEnd( X ). The converse inequality ψ ′ a ≤ J ψ a is similar, thus, ψ a and ψ a ′ are J -equivalent. Secondsuppose that ψ a and ψ a ′ are J -equivalent in End( X ), whence ψ a = ψ a ′ b and ψ a ′ = ψ ab ′ for some b , b ′ ∈ I K . Dually, i.e. in terms of the DR K -set F K ( X ), this means that s a and s a ′ ∈ F K ( X ) aremutually accessible by the action of DR K , that is, s a · DR K = s a ′ · DR K . We put S := s a · DR K = s a ′ · DR K . Then, note that this S forms a rooted DR K -set, hence, galois in B f DR K (cf. Remark4, § ψ a = ψ a ′ b and ψ a ′ = ψ ab ′ , one sees that the actions of ψ b , ψ b ′ on S are(mutually inverse) automorphisms. Thus, by Lemma 3.2 applied to (the Λ-ring in C K dual to) thisgalois object S ∈ B f DR K , it follows that the actions of ψ b , ψ b ′ on S are equal to those of some σ ∈ G K , which implies that s a = s a ′ b and s a ′ = s ab ′ belong to the same G K -component. Thiscompletes the proof.Now we study the structure of the galois objects X ξ for ξ ∈ W aO K ( O ¯ K ). (For simplicity, we firststudy the structure of X Ξ only for singleton Ξ = { ξ } , but the similar result holds in general.) Since X ξ is finite etale over K , we have an isomorphism X ξ ≃ L × · · · × L r for some finite extensions L i /K . However, we do not yet quite know what and how many components L i each X ξ has. In thefollowing, we first discuss this problem. In this relation, let us denote by M ξ the quotient monoid I K / ≡ ξ , where a ≡ ξ b if and only if ψ a ξ = ψ b ξ . (By Theorem 3.2, [15], M ξ is a finite monoid.) Proposition 3.3.
For any ξ ∈ W aO K ( O ¯ K ) we have the following isomorphism: X ξ ≃ Y [ a ] ∈ M ξ / J K ( ξ ab ; b ∈ I K ) . In particular, the number of components is equal to M ξ / J ) .Proof. For short let us put the right hand side as Y ξ := Q [ a ] ∈ M ξ / J K ( ξ ab ; b ∈ I K ); we construct thetarget isomorphism f : X ξ → Y ξ . Let M ξ / J = { [ a ] , · · · , [ a r ] } , and for each [ a i ] ∈ M ξ / J , denoteits representative as a i ∈ I K . Then we can define f : X ξ → Y ξ by f ( η ) := ( η a i ) ri =1 . To prove thelemma, note first that this f is injective: If f ( η ) = f ( η ′ ) then η a i = η ′ a i for every i = 1 , · · · , r . Butfor each a ∈ I K we have [ a ] = [ a i ] for some i ; this means that, for every ζ ∈ X ξ , we have ζ a = ζ σ a i for some σ ∈ G K . (In fact [ a ] = [ a i ] implies a = a i b and a i = ab ′ for some b , b ′ ∈ I K in M ξ , whence ψ a = ψ a i ψ b and ψ a i = ψ a ψ b ′ on X ξ ; then apply Lemma 3.3 to the root s : X ξ ∋ ζ ζ ∈ ¯ K where1 is the unit in I K .) Thus η a = η σ a i = η ′ a i σ = η ′ a for every a ∈ I K , which implies η = η ′ as requested.12inally we see that f is surjective. First note that, since X ξ is finite etale over K , we have anisomorphism X ξ ≃ L × · · · × L m for some finite extensions L i /K ; and each L i is obtained as theimage of X ξ under some s i : X ξ → ¯ K ∈ F K ( X ξ ). As shown in Proposition 3.1, each s i ∈ F K ( X ξ )is given as s i ( η ) = η σ i a i ( ∀ η ∈ X ξ ) for some a i ∈ I K and σ i ∈ G K , namely, s i is G K -equivalent to s ′ i ∈ F K ( X ξ ) given by s ′ i ( η ) = η a i . So, the image L i of X ξ under s i is isomorphic to K ( η a i ; η ∈ X ξ ),which is further isomorphic to K ( ξ a i b ; b ∈ I K ) because X ξ is generated by ψ b ξ ’s ( b ∈ I K ) over K .By Lemma 3.3, the two maps s ′ i and s ′ j are G K -equivalent if and only if [ a i ] = [ a j ]. This means thatthe G K -equivalent classes of s ∈ F K ( X ξ ) are classified precisely by the set M ξ / J = { [ a ] , · · · , [ a r ] } with s ′ i : η η a i ∈ F K ( X ξ ) for each [ a i ] ∈ M ξ / J being representative. Therefore, one concludesthat m = r and L i ≃ K ( ξ a i b ; b ∈ I K ), which completes the proof.Finally, we determine the state complexity of integral Witt vectors ξ ∈ W aO K ( O ¯ K ) (cf. Definition7) as a natural analogue of the result of Bridy [5] for formal power series ξ ∈ F q [[ t ]], where usingRiemann-Roch theorem he gave a sharp estimate of the state complexity of a formal power series ξ ∈ F q [[ t ]] algebraic over F q [ t ] in terms of the dimension dim F q ( t ) X ξ of the function field X ξ of thecurve generated by ξ over F q ( t ). Analogously, we now show that the state complexity of an integralWitt vector ξ ∈ W aO K ( O ¯ K ) is equal to the dimension dim K X ξ of the Λ-ring X ξ . In fact, this holdsfor general Ξ ⊆ W aO K ( O ¯ K ).To be precise, the state complexity of a finite set Ξ ⊆ W aO K ( O ¯ K ) is defined as follows: (See § Definition 7 (state complexity) . Let Ξ ⊆ W aO K ( O ¯ K ) be any finite set of integral Witt vectors. Wesay that a DFA A generates Ξ if for every ξ ∈ Ξ there exists an output function τ : S A → O ¯ K suchthat A τ generates ξ . The state complexity of Ξ, denoted by c Ξ , is then defined as the minimum sizemin S A of the state set S A of those DFA A which can generate Ξ. Proposition 3.4 (estimate of state complexity) . For any finite Ξ ⊆ W aO K ( O ¯ K ) , we have the fol-lowing identity: c Ξ = dim K X Ξ . (3.2) In particular, c ξ = dim K X ξ for ξ ∈ W aO K ( O ¯ K ) .Proof. Firstly note that we have dim K X Ξ = K ( X Ξ ). Also it is easy to see that F K ( X Ξ ) forms(the state set of) a DFA over P K that generates Ξ; thus we have the one-side inequality: c Ξ ≤ K ( X Ξ )= dim K X Ξ . To prove the inverse inequality, note that by Proposition 3.1, we know that X Ξ is a galois object;hence, K ( X Ξ ) = X Ξ ). Therefore, it suffices to prove: c Ξ ≥ X Ξ ) . To this end, let A = (Ω , δ, s ) be the minimum DFA generating Ξ. Then, by the minimality, every s ∈ Ω is represented as s = s · u for some u = p · · · p n ∈ P ∗ K . Also, we can see that if s · u = s · v for u, v ∈ P ∗ K , then ψ a u ξ = ψ a v ξ for every ξ ∈ Ξ, where a w = p · · · p n ∈ I K for w = p · · · p n ∈ P ∗ K .13n fact, for each ξ ∈ S there exists an output function τ : Ω → O ¯ K such that for every w ∈ P ∗ K :( ψ a u ξ ) a w = ξ a uw = τ (( s · u ) · w )= τ (( s · v ) · w )= ( ψ a v ξ ) a w , which means that ψ a u ξ = ψ a v ξ . Since X Ξ is generated by I K Ξ, this then implies that ψ a u = ψ a v on X Ξ ; hence, the assignment Ω ∋ s = s · u ψ a u ∈ End( X Ξ ) is well-defined. Furthermore, by thefact that I K ֒ → DR K is dense, this assignment Ω → End( X Ξ ) is in fact surjective, which proves thedesired inequality c Ξ = ≥ X Ξ ). Remark 8.
Note that the identity c ξ = dim K X ξ relates seemingly unrelated quantities, namelythe state complexity c ξ and the dimension dim K X ξ of the K -algebra X ξ . In fact, while the former isjust the size of the orbit I K ξ under the action of the Frobenius ψ p , the latter is the quantity relevantto the algebraic degree of its coefficients ξ a ’s. As we discuss below, this identity seems related tothe difficulty of actually constructing integral Witt vectors; to our thought, this is because theremust be some geometric reason for why the coefficients ξ a ’s of integral Witt vectors ξ distribute asthey do. Concerning this, it will be meaningful to remark that the j -invariants j ( a ) in the theory ofcomplex multiplication actually constitute an example of integral Witt vector; this is proved fromtheir reciprocity law (cf. § W a Z ( ¯ Z ) As an immediate consequence of the above general development, applied to the case K = Q , thissection concludes with a presentation of the ring W a Z (¯ Z ), which assets that W a Z (¯ Z ) is isomorphicto the group-ring Z [ Q / Z ]. After this presentation we proceed to the case when K is an imaginaryquadratic field. Corollary 3.2.
The ring W a Z (¯ Z ) is isomorphic to the group-ring Z [ Q / Z ] .Proof. We first see that E Q = Q ⊗ W a Z (¯ Z ) is isomorphic to the group-ring Q [ Q / Z ]. In fact, by theresult of Borger and de Smit [2] combined with our result in § E Q is isomorphicto the Q -algebra given by the direct limit lim n Q [ x ] / ( x n −
1) = S n Q [ x ] / ( x n − Q -algebras Q [ x ] / ( x n −
1) is given by the following embeddings, for positive integers n, m : Q [ x ] / ( x n − → Q [ x ] / ( x nm − x x m . The group-ring Q [ Q / Z ], on the other hand, is also isomorphic to this Q -algebra S n Q [ x ] / ( x n − Q [ Q / Z ] → [ n Q [ x ] / ( x n − /N x ∈ Q [ x ] / ( x N − . Hence we have E Q ≃ Q [ Q / Z ]. Finally we see that, in this isomorphism, the subring W a Z (¯ Z ) ⊆ E Q corresponds to Z [ Q / Z ] ⊆ Q [ Q / Z ]. Indeed, as proved in Theorem 3.4 [2], the maximal integral model14f the Λ-ring Q [ x ] / ( x n −
1) is Z [ x ] / ( x n −
1) for each n ; and thus, the subring S n Z [ x ] / ( x n − ⊆ S n Q [ x ] / ( x n −
1) corresponds to W a Z (¯ Z ) ⊆ E Q . On the other hand, we see that S n Z [ x ] / ( x n − Z [ Q / Z ] under the above isomorphism Q [ Q / Z ] ≃ S n Q [ x ] / ( x n − W a Z (¯ Z ) ≃ Z [ Q / Z ]. This completes the proof. Remark 9.
More explicitly, in the above isomorphism W a Z (¯ Z ) ≃ Z [ Q / Z ], each γ ∈ Q / Z ⊆ Z [ Q / Z ]corresponds to the following Witt vector ζ ( γ ) ∈ W a Z (¯ Z ) ⊆ ( Q ab ) N : ζ ( γ ) := ( e πiγn ) n ∈ N . (3.3)Therefore we can conclude that, in general, the integral Witt vectors ξ ∈ W a Z (¯ Z ) are precisely the Z -linear combinations of these ζ ( γ ) , γ ∈ Q / Z (while the algebraic Witt vectors ξ ∈ E Q are the Q -linear combinations of ζ ( γ ) ’s). This provides a complete classification of the integral Witt vectorsover Z . In some sense, the above isomorphism E Q ≃ Q [ Q / Z ] clarifies how the coefficients ξ n of an algebraicWitt vector ξ over Q are correlated; they are essentially correlated as special values e πiγn for fixed γ ∈ Q / Z of the function e z .The purpose of this section is to prove an analogue of this result in the case where K is animaginary quadratic field. To be precise, this section proves that the K -algebra E K = K ⊗ W aO K ( O ¯ K )for an imaginary quadratic field K is isomorphic to (or actually coincides with) the K -algebra M K that consists of modular vectors ; technically speaking, modular vectors are defined by specialvalues of certain deformations of modular functions ( § modularity theorem , exhibits that such vectors arising from deformations of modular functionsalways define algebraic Witt vectors, and conversely, every algebraic Witt vector arises in this way( § j -invariant function As a demonstration of what integral Witt vectors can be, this section is devoted to describing thestructure of the Λ-ring of Witt vectors, W O K ( O H K ), where H K denotes the Hilbert class field of K in particular; with this presentation, we deduce that the j -invariants j ( a ), a proto-typical modularfunction, constitute an example of integral Witt vectors. A general principle relating algebraic Wittvectors and modular functions in the case where K is an imaginary quadratic field is then discussedin § Lemma 4.1.
Let
L/K be a finite galois extension, p ∈ P K be unramified in L , and also P ∈ P L beabove p . Then, for any ξ ∈ W O K ( O L ) and a ∈ I K , we have the following congruence: ξ pa ≡ ξ (cid:0) L | K P (cid:1) a mod P v p ( a ) , (4.1) where (cid:0) L | K P (cid:1) ∈ Gal ( L/K ) denotes the Frobenius automorphism on L for P | p . roof. The proof uses the same method as Lemma 3.1 [15], i.e. by induction on v p ( a ) via arithmeticderivation. Denote as σ P = (cid:0) L | K P (cid:1) for short. For the base case let v p ( a ) = 0. Since ξ ∈ W O K ( O L )is a Witt vector, we have ξ pa ≡ ξ N pa mod p for any a ∈ I K , hence ξ pa ≡ ξ N pa mod P in particular.But since p is now unramified in L , we also have ξ N pa ≡ ξ σ P a mod P , which then implies ξ pa ≡ ξ σ P a mod P as requested.For induction assume that the claim is true up to v p ( a ) ≤ n ; and let v p ( a ) = n + 1 whence we canwrite as a = pa ′ for some a ′ ∈ I K with v p ( a ′ ) = n . By ξ ∈ W O K ( O L ), there exist η ( i ) ∈ W O K ( O L )and r i ∈ p ( i = 1 , · · · , m ) such that: ψ p ξ − ξ N p = m X i =1 r i η ( i ) . (4.2)Looking at the a -th and a ′ -th components, we have the following identities: ξ pa − ξ N pa = m X i =1 r i η ( i ) a , (4.3) ξ a − ξ N pa ′ = m X i =1 r i η ( i ) a ′ . (4.4)The slight difference from the proof of Lemma 3.1, [15] is here: Before subtracting the first from thesecond, apply σ P to the second equation. Then by subtracting them, we obtain: ξ pa − ξ σ P a = ξ N ppa ′ − ( ξ σ P a ′ ) N p + m X i =1 r i ( η ( i ) pa ′ − η ( i ) σ P a ′ ) . (4.5)Under the induction hypothesis ξ pa ′ − ξ σ P a ′ ∈ P n and η ( i ) pa ′ − η ( i ) σ P a ′ ∈ P n , one obtains ξ N ppa ′ − ( ξ σ P a ′ ) N p ∈ P n (by ξ pa ′ − ξ σ P a ′ ∈ P n in particular) in the the same way as Lemma 3.1 [15]. Thisproves ξ pa − ξ σ P a ∈ P n , which is our target congruence. This completes the proof.With this lemma, we now describe the structure of the Λ-ring W O K ( O H K ) for the Hilbert classfield H K , which is a natural extension of Proposition 3, [15]. In general, the description of W O K ( O L )seems difficult particularly due to the existence of ramified primes; in the case of the Hilbert classfield H K , however, this obstruction disappears because every prime p ∈ P K is unramified in H K .Since H K is maximal among such unramified extensions, the description of W O K ( O H K ) is generalenough to classify those Witt vectors whose coefficients generate unramified extensions of K . ThisΛ-ring W O K ( O H K ) is then proved to be the right place in which the classical j -invariant j ( a ) shouldlive (in the sense discussed below). Proposition 4.1.
Let H K be the Hilbert class field of K . Then we have the following presentationof the ring W O K ( O H K ) of Witt vectors: W O K ( O H K ) = (cid:26) ξ ∈ O I K H K | ξ pa ≡ ξ (cid:0) L | K p (cid:1) a mod p v p ( a ) ( ∀ p , ∀ a ) (cid:27) . (4.6) Proof.
The proof of this proposition is similar to that of Proposition 3, [15] but needs some slightmodification. Since H K is abelian over K , we denote by σ p ∈ Gal ( H K /K ), rather than σ P , theFrobenius automorphism for p in H K . 16ut the right hand side as V ( O H K ) for short. The above lemma applied to L = H K implies theinclusion W O K ( O H K ) ⊆ V ( O H K ); so we need prove the inverse inclusion V ( O H K ) ⊆ W O K ( O H K ).To this end, it suffices to prove that V ( O H K ) itself forms a Λ-ring (cf. Remark 3, [15]), namely, forany ξ ∈ V ( O H K ) and p ∈ P K , we should have ψ p ξ − ξ N p ∈ p V ( O H K ). Again, as the localizationargument in Proposition 3, [15], we only need to prove ψ p ξ − ξ N p ∈ πO K, p ⊗ V ( O H K ) where π ∈ p is a uniformizer.By ξ ∈ V ( O H K ), i.e. ξ pa − ξ N pa ∈ p v p ( a ) for each a ∈ I K , we have ψ p ξ − ξ N p ∈ πO K, p ⊗ O I K H K ;thus ψ p ξ − ξ N p = πu · η for some η ∈ O I K H K and u ∈ O × K, p of the form u = 1 /b with b ∈ O K \ p . Weshow η ∈ V ( O H K ). For this purpose, first put ζ = ψ p ξ − ξ N p . Then: ζ pa − ζ σ p a = ( ξ p a − ξ N ppa ) − ( ξ pa − ξ N pa ) σ p = ( ξ p a − ξ σ p pa ) − ( ξ N ppa − ( ξ σ p a ) N p )By definition of V ( O H K ), we have ξ p a − ξ σ p pa ∈ p v p ( a ) and ξ pa − ξ σ p a ∈ p v p ( a ) . Then, by the sameargument as that of Lemma 3.1, [15], the latter implies ξ N ppa − ( ξ σ p a ) N p ∈ p v p ( a ) ; thus one obtains πu ( η pa − η σ p a ) = ζ pa − ζ σ a a ∈ p v p ( a ) . Hence, η pa ≡ η σ p a mod p v p ( a ) . Moreover, for any q = p as well, one indeed has η qa ≡ η σ q a mod q v q ( a ) because η pa − η σ p a = ( ζ pa − ζ σ a a ) b/π , ζ ∈ V ( O H K )and π / ∈ q . (See the proof of Proposition 3, [15].) These prove that η ∈ V ( O H K ) as requested. Thiscompletes the proof.With this presentation of W O K ( O H K ) applied to imaginary quadratic K , we can now prove thatthe j -invariant function j can be interpreted as a function generating a (proto-typical) example ofintegral Witt vector: Corollary 4.1.
Let K be an imaginary quadratic field; and define ι ∈ O I K H K by ι a := j ( a − ) for each a ∈ I K with the j -invariant function. Then ι is an integral Witt vector: ι = ( ι a ) a ∈ I K ∈ W aO K ( O H K ) . (4.7) Moreover, ι generates H K as a Λ -ring in the sense of §
2, namely X ι ≃ H K whose maximal integralmodel is the ring O H K of integers in H K .Proof. The first claim follows from the integrality and Hasse’s reciprocity law of the j -invariants j ( a ) (cf. [13]) combined with Proposition 4.1. The last claim follows from the conjugacy of j ( a − )’sand the fact that K ( j ( a − )) = H K . Corollary 4.2.
The state complexity of ι ∈ W aO K ( O H K ) is equal to the class number of K .Proof. This follows from the second claim of the above corollary and Proposition 3.4 that provesthe equality c ι = dim K X ι = dim K H K .This corollary strengthens Hasse’s reciprocity law in the sense that the j -invariants j ( a ) are notonly reciprocal (i.e. j ( p − a ) = j ( a ) σ p ), but they are arithmetically smooth as well. The reciprocitylaw itself is rephrased as the second statement of Corollary 4.1; and the fact that ι forms an integralWitt vector gives us a new way to look at the j -invariant modular function as well as what integralWitt vectors can be. 17 .2 Modular vectors We see that the above result on the j -invariant function is a special case of a more general relationshipbetween algebraic Witt vectors and modular functions. To be precise, the following concept of Wittdeformation family of modular functions is a key to explain this general relationship. In whatfollows, let us denote by F = S N F N the field of modular functions of finite level rational over Q ab ,where F N is the field of modular functions of level N rational over Q ( e πi/N ) in the sense of § F N = Q ( j, f a | a ∈ N − Z \ Z ) with f a Fricke functions (cf. Example 3). (Recallthat the field of all modular functions of finite level is equal to C ⊗ Q F ; cf. Proposition 6.1, [13].) Definition 8 (Witt deformation family of modular functions) . A Witt deformation family ofmodular functions (or simply, a
Witt deformation of modular functions ) is a continuous function f : M ( b Z ) × H → C satisfying the following axioms:1. for each m ∈ M ( b Z ), the fiber f m := f ( m, − ) : H → C is a modular function in F ;2. for each m ∈ M ( b Z ) and u ∈ GL ( b Z ), we have: f mu = f um ; (4.8)where f m f um denotes the action of u ∈ GL ( b Z ) onto the modular field F (cf. § f : M ( b Z ) × H → C factors through the projection p N : M ( b Z ) ։ M ( Z /N Z ) forsome N as follows: M ( b Z ) × H p N × id H (cid:15) (cid:15) f / / C M ( Z /N Z ) × H rrrrrrrrrrrr (4.9) Remark 10.
In relation to the
Bost-Connes system [10], the concept of Witt deformation is basedon a slight modification of the arithmetic subalgebra constructed in Definition 2.22 [10]. Modulo thismodification, our modularity theorem in § § Example 2 ( j -invariant function) . The j -invariant function j naturally defines the simplest Wittdeformation family of modular functions, denoted by the same symbol j : M ( b Z ) × H → C : for each( m, τ ) ∈ M ( b Z ) × H , j ( m, τ ) := j ( τ ) . (4.10)The fact that this j : M ( b Z ) × H → C indeed defines a Witt deformation follows readily from thedefinition of the action of GL ( A Q ,f ) = GL ( b Z ) GL +2 ( Q ) on j , in particular, j u = j for u ∈ GL ( b Z )(cf. § xample 3 (Fricke functions) . The
Fricke functions are modular funcitons f a : H → C indexed byrow vectors a ∈ Q / Z , and defined as follows (cf. pp.133 [13]): for w , w with w /w ∈ H , f a ( w /w ) = g ( w , w ) g ( w , w )∆( w , w ) ℘ ( a w + a w ; w , w ) . (4.11)Each Fricke function f a naturally defines a Witt deformation family χ a of modular functions by: χ a : M ( b Z ) × H → C ( m, τ ) f am ( τ )(In some sense this χ a “deforms” the Fricke functions f a by the action of m ∈ M ( b Z ) on the index a ∈ Q / Z .) The fact that this χ a : M ( b Z ) × H → C indeed defines a Witt deformation of modularfunctions can be proved, again, using the definition of the action of GL ( A Q ,f ) = GL ( b Z ) GL +2 ( Q )on f a ’s: That is, for uα ∈ GL ( b Z ) GL +2 ( Q ) = GL ( A Q ,f ), one has f uαa = f au ◦ α (cf. § Definition 9 (modular vector) . Let f : M ( b Z ) × H → C be a Witt deformation family of modularfunctions. Then the modular vector b f ∈ ( K ab ) I K associated to f is the function b f : I K → K ab defined by the following composition: b f : I K ֒ → DR K ≃ −→ b O K × b O ∗ K ( A ∗ K,f /K ∗ ) ≃ −→ Lat K ֒ → Lat Q ≃ −→ Γ \ ( M ( b Z ) × H ) f −→ C . (4.12)Note that f induces a function on Γ \ ( M ( b Z ) × H ) thanks to the second condition of Witt deformation,and b f indeed takes its values in K ab , which follows from Lemma 4.2 below. Modular vectors in thissense themselves constitute a Q -algebra; to make it a K -algebra, we define the K -algebra M K ofmodular vectors as follows (and abusively, we shall call the elements of M K as well modular vectors ): M K := K ⊗ Q (cid:8) b f ∈ ( K ab ) I K | f : M ( b Z ) × H → C is a Witt deformation. (cid:9) (4.13) Remark 11.
Instead, we may allow the fibers f m of Witt deformations f : M ( b Z ) × H → C tobe modular functions in K ⊗ Q F ; then the modular vectors b f constitute a K -algebra in themselves,which is equal to (4.13). We may use the term “Witt deformations” in this extended sense too.Concerning modular vectors, we first see more explicitly the coefficients b f a of modular vectors b f = ( b f a ) in terms of the values of the Witt deformation f : M ( b Z ) × H → C of modular functions: Lemma 4.2.
Let f : M ( b Z ) × H → C be a Witt deformation of modular functions. Then, for each a ∈ I K , the component b f a ∈ K ab at a of the associated modular vector b f = ( b f a ) ∈ ( K ab ) I K is givenby the following equation: b f a = f ( m a , τ a − ); (4.14) where, on one hand, if we denote as a − = Z w + Z w with w /w ∈ H , we define τ a − := w /w ; onthe other hand, m a ∈ M ( b Z ) is such that the following diagram commutes with some α ∈ GL +2 ( R ) (i.e. such that Z · α · ( i, t = a − ; cf. § Q / Z q τ (cid:15) (cid:15) m a / / Q / Z α · ( i, t / / Q a − / a − (cid:15) (cid:15) K/O K / / K a − / a − (4.15)19 here K/O K → K a − / a − is a natural inclusion.Proof. The proof is just based on a careful chase of the composition (4.12). Firstly, the composition I K ֒ → DR K ≃ b O K × b O ∗ K ( A ∗ K,f /K ∗ ) sends the ideal a ∈ I K (denoting a = ρ b O K ∩ K for some ρ ∈ b O K )to [ ρ, ρ − ] ∈ b O K × b O ∗ K ( A ∗ K,f /K ∗ ) (cf. Proposition 4.2 and Proposition 8.2, [16]); secondly, one seesthat the further composition with the maps b O K × b O ∗ K ( A ∗ K,f /K ∗ ) → Lat K → Lat Q sends [ ρ, ρ − ] to[ a − , φ a ] ∈ Lat Q , where φ a : Q / Z → Q a − / a − is given by the following composition (cf. § φ a : Q / Z q τ / / K/O K / / K a − / a − / / Q a − / a − ; (4.16)where the second arrow K/O K → K a − / a − is a natural inclusion. Therefore, by the constructionof the isomorphism Lat Q ≃ Γ \ ( M ( b Z ) × H ) (cf. § a − , φ a ] ∈ Lat Q indeed corresponds to[ m a , τ a − ] ∈ Γ \ ( M ( b Z ) × H ) for the m a ∈ M ( b Z ) and τ a − ∈ H described above. This completes theproof. Example 4 (the modular vector for j ) . Let j : M ( b Z ) × H → C be the Witt deformation of modularfunctions given in Example 2. Then Lemma 4.2 shows that the associated modular vector b j is givenby b j a = j ( a − ). Therefore, this b j is equal to ι constructed in § b j = ι is a member of W aO K ( O ¯ K ), hence, of the K -algebra E K in particular. This example showsthat some modular vector (i.e. b j ∈ M K ) indeed defines an algebraic Witt vector (i.e. b j = ι ∈ E K ).The goal of the next subsection is to show that this example is a special case of a more generalrelationship between modular vectors and algebraic Witt vectors, where we prove that these twoconcepts actually coincide. This section proves that the K -algebra M K of modular vectors coincides with the K -algebra E K ofalgebraic Witt vectors, which we call the modularity theorem . In other words, this theorem claimsthat every modular vector b f defines an algebraic Witt vector, i.e. b f ∈ E K , and conversely, everyalgebraic Witt vector ξ ∈ E K is realized as a modular vector, i.e. ξ = b f for some Witt deformation f : M ( b Z ) × H → C of modular functions. To be precise: Theorem 4.1 (modularity theorem) . We have the following identity as K -subalgebras of ( K ab ) I K : E K = M K . (4.17)Our proof of this theorem consists of two steps. We first prove the inclusion M K ⊆ E K , whichis based on Corollary 3.1 and Shimura’s reciprocity law. After that, we prove the converse inclusion E K ⊆ M K , which is based on the characterization of E K by simple conditions due to Neshveyev,Theorem 10.1, §
10 [16].To be specific, recall that Corollary 3.1 proved that the K -algebra E K is identified with that oflocally constant K ab -valued G abK -equivariant functions on DR K , i.e. E K = Hom G abK ( DR K , K ab ); inthe proof of the inclusion M K ⊆ E K , we see that every modular vector b f defines a locally constant( K ab -valued) G abK -equivariant function on DR K , by which we deduce the inclusion M K ⊆ E K . Onthe other hand, the K -algebra E K was characterized by Neshveyev, §
10 [16] as the K -subalgebra E C ( DR K )— the C -algebra of continuous functions on DR K — that satisfies the following conditions(cf. pp.408, [16]; note that DR K is homeomprhic to Y K in [16]):1. every function in E is locally constant on DR K ;2. E separates the points of DR K ;3. E contains the idempotents ρ a for each a ∈ I K (cf. Lemma 4.5);4. every function in E is K ab -valued and G abK -equivariant.In other words, if E is a K -subalgebra of E K and satisfies the second and the third conditions here,one actually has the identity E = E K . Therefore, after the proof of the inclusion M K ⊆ E K , theremained task for the proof of the identity E K = M K is only to prove that M K satisfies the secondand the third conditions here. Our proof of the target identity E K = M K will follow this line ofargument.Now let us start proving the inclusion M K ⊆ E K . To this end, we prepare the following lemma,which is used to prove that modular vectors b f define K ab -valued G abK -equivariant functions on DR K . Lemma 4.3.
Let (Λ , φ ) ∈ Lat K be a K -lattice such that Λ is a fractional ideal of K and correspondsto [ m, τ ] ∈ Γ \ ( M ( b Z ) × H ) ; also let s ∈ A ∗ K,f /K ∗ and ( J − s Λ , s − φ ) ∈ Lat K corresponds to [ m s , τ s ] .Moreover, let f : M ( b Z ) × H → C be a Witt deformation of modular functions. Then f m ( τ ) ∈ K ab ,and the following identity holds: (cid:0) f m ( τ ) (cid:1) [ s ] = f m s ( τ s ) . (4.18) Proof.
Throughout this proof, let us denote Λ s := J − s Λ and φ s := s − φ for short. Moreover letΛ = Z w + Z w = Z · α · ( i, t and Λ s = Z w ′ + Z w ′ = Z · α s · ( i, t with τ = w /w , τ s = w ′ /w ′ ∈ H and α, α s ∈ GL +2 ( R ); also let β ∈ GL +2 ( Q ) be such that ( w , w ) t = β ( w ′ , w ′ ) t , whence α s = β − α and τ = α ( i ) , τ s = α s ( i ). Define the embedding q Λ : A K,f → M ( A Q ,f ) with respect to the basis ofΛ in the sense of § q Λ ( µ )( w , w ) t = ( µw , µw ) t for µ ∈ K . Then, for each prime p , wehave: Z p (cid:18) w ′ w ′ (cid:19) = (Λ s ) p = Λ p s − p = Z p (cid:18) w s − p w s − p (cid:19) = Z p q Λ ( s p ) − (cid:18) w w (cid:19) = Z p q Λ ( s p ) − β (cid:18) w ′ w ′ (cid:19) . Since this holds for each prime p , we have some u ∈ GL ( b Z ) such that q Λ ( s ) − β = u ; therefore, q Λ ( s ) − = uβ − . With this in mind, we first compute ( f m ( τ )) [ s ] . By Shimura’s reciprocity law and q Λ ( s ) − = uβ − , we have: ( f m ( τ )) [ s ] = f q Λ ( s ) − m ( τ )= f mu ◦ β − ( τ ) .
21o prove that the last one is equal to f m s ( τ s ), consider the following composition: Q / Z m / / Q / Z α · ( i, t / / Q Λ / Λ s − / / Q Λ s / Λ s ; (4.19)here the last arrow is defined by the following; suppose Λ = t b O K ∩ K for t ∈ A K,f . Then: Q Λ / Λ = (cid:15) (cid:15) s − / / Q Λ s / Λ s = (cid:15) (cid:15) K Λ / Λ ≃ (cid:15) (cid:15) K Λ s / Λ s ≃ (cid:15) (cid:15) A K,f /t b O K s − / / A K,f /s − t b O K (4.20)Recall that q Λ ( s ) is defined with respect to the basis of Λ = Z · α · ( i, t . Therefore, we have thefollowing commutative diagram: (recall also that Λ s = Z · α s · ( i, t with α s = β − α ; and we writejust by α : Q / Z → Q Λ / Λ to mean the composition α · ( i, t : Q / Z → Q α/ Z α → Q Λ / Λ.) Q Λ / Λ s − / / α − (cid:15) (cid:15) Q Λ s / Λ s Q / Z q Λ ( s ) − / / Q / Z β − α O O (4.21)This diagram shows that the composition (4.19) is equal to the following: Q / Z m / / Q / Z q Λ ( s ) − / / Q / Z β − α / / Q Λ s / Λ s . (4.22)Therefore, m s α s = mq Λ ( s ) − α = muβ − α ; and hence we have: (note that τ = α ( i ) and τ s = α s ( i )by definition above.) f mu ◦ β − ( τ ) = f mu ◦ β − α ( i )= f m s ( α s ( i ))= f m s ( τ s ) . Consequently, we proved that ( f m ( τ )) [ s ] = f m s ( τ s ) as requested. This completes the proof.With this lemma, we first prove the one-side inclusion for the target equality E K = M K : Lemma 4.4.
We have the following inclusion: M K ⊆ E K . (4.23)22 roof. Let f : M ( b Z ) × H → C be a Witt deformation of modular functions with b f ∈ M K beingthe associated modular vector. We can extend b f : I K → C to a continuous function on DR K bythe density of I K in DR K , and identify b f with this extended function on DR K . (Actually, recallthat b f ∈ ( K ab ) I K itself was originally defined as the restriction of a function on DR K onto I K ; cf.Definition 9.) By Corollary 3.1, in order to see that b f ∈ E K , it suffices to show that the extendedfunction b f : DR K → C is locally constant ( K ab -valued), and G abK -equivariant.First, to see that this b f : DR K → C is locally constant, note that b f : I K → C takes only finitelymany K ab -values on I K because b f a = f ( m a , τ a − ), hence the second operand τ a − takes only finitelymany values in H ; and also f : M ( b Z ) × H → C factors through p N : M ( b Z ) ։ M ( Z /N Z ) for some N , hence the first operand of f ( m a , τ a − ) as well can essentially take finitely many values. Since,furthermore, I K is dense in DR K , it follows that the extended function b f : DR K → C is indeedlocally constant (and K ab -valued).Finally, to see that b f : DR K → C is G abK -equivariant, we remark that the action of [ s ] ∈ G abK for s ∈ A ∗ K,f on the element of DR K that corresponds to [ ρ, t ] ∈ b O K × b O ∗ K ( A ∗ K,f /K ∗ ) under thehomeomorphism ψ : DR K → b O K × b O ∗ K ( A ∗ K,f /K ∗ ) corresponds to [ ρ, s − t ] (i.e. not [ ρ, st ]); that is,we have (cid:0) ψ − ( ρ, t ) (cid:1) [ s ] = ψ − ( ρ, s − t ); this follows from the construction of the homeomorphism Ψ : b O K × b O ∗ K G abK → DR K of § ψ − ( ρ, t ) ∈ DR K corresponds to [ m, τ ] ∈ Γ \ ( M ( b Z ) × H )under the correspondence DR K → Γ \ ( M ( b Z ) × H ) (cf. (4.12)), then (cid:0) ψ − ( ρ, t ) (cid:1) [ s ] = ψ − ( ρ, s − t ) ∈ DR K corresponds to [ m s , τ s ] (in the sense of the notation in Lemma 4.3). Therefore, we have: b f ( (cid:0) ψ − ( ρ, t ) (cid:1) [ s ] ) = b f ( ψ − ( ρ, s − t ))= f m s ( τ s )= (cid:0) f m ( τ ) (cid:1) [ s ] = b f ( ψ − ( ρ, t )) [ s ] . In the third equality, we used Lemma 4.3; this shows that b f : DR K → C is indeed G abK -equivariant.Therefore b f is K ab -valued locally constant G K -equivariant function on DR K , hence b f ∈ E K . Thiscompletes the proof.Now we have the inclusion M K ⊆ E K . Thus, in order to prove our target identity E K = M K , itsuffices to prove that M K satisfies the above-mentioned two conditions (2) and (3). In the followingwe first prove the third condition (3), and then prove the second (2), which will complete the proofof our target theorem. Lemma 4.5.
For each a ∈ I K , let us define ρ a ∈ ( K ab ) I K as follows: ρ a ( b ) := (cid:26) if a | b otherwise (4.24) Then ρ a ∈ M K for every a ∈ I K .Proof. This function ρ a corresponds to the characteristic function of a DR K , which we now showbelongs to M K . To this end, let us fix uniformizing elements π p ∈ p for each prime ideal p ∈ P K throughout this proof. Under the homeomorphism DR K ≃ b O K × b O ∗ K ( A ∗ K,f /K ∗ ) of [16], the subspace23 DR K ⊆ DR K corresponds to a b O K × b O ∗ K ( A ∗ K,f /K ∗ ); and note that, under the further composition b O K × b O ∗ K ( A ∗ K,f /K ∗ ) → Lat K → Lat Q → Γ \ ( M ( b Z ) × H ), this subspace is eventually mapped intoΓ \ (cid:0)Q p q τ ( π p ) v p ( a ) M ( b Z ) × H (cid:1) , where q τ : A K,f → M ( A Q ,f ) denotes the embedding determined by O K = Z τ + Z . Let us define the function f a : M ( b Z ) × H → C as the characteristic function of thesubspace Q p q τ ( π p ) v p ( a ) M ( b Z ) × H ⊆ M ( b Z ) × H . It is clear that f a satisfies the conditions (1), (2)of Witt deformations of modular functions. In order to see that f a also satisfies the condition (3),we note that f a factors through p N ( a ) : M ( b Z ) ։ M ( Z /N ( a ) Z ), where N ( a ) is the absolute normof a . In fact, this follows from the fact that the kernel of p N ( a ) is N ( a ) M ( b Z ), which is included in Q p q τ ( π p ) v p ( a ) M ( b Z ). From these facts, we find that f a is in fact a Witt deformation of modularfunctions; and by construction, it is clear that b f a = ρ a . This implies that ρ a ∈ M K as requested.This completes the proof. Lemma 4.6.
The functions in M K are enough to separate the points of DR K .Proof. Let x = y ∈ DR K be distinct points. Since DR K is the inverse limit of DR f ’s for f ∈ I K ,there exists f ∈ I K such that x and y still represent distinct elements of DR f under the projection DR K ։ DR f . Here recall the decomposition DR f = ` d | f C f / d (cf. (4.4), pp.394 [16]). If x and y respectively belong to distinct components, then the characteristic functions defined above areenough to separate x, y ; if x, y belong to the same component C f / d but represent distinct elements,they can be separated by some modular function because C f / d is isomorphic to the galois group Gal ( K f / d /K ) of the strict ray class field K f / d over K with conductor f / d and the special values ofmodular functions generate K ab over K . Using this modular function and characteristic functionsgiven above, one can define a modular vector that separates x and y . This completes the proof. Corollary 4.3.
We have the following inclusion: E K ⊆ M K ; (4.25) hence E K = M K .Proof. In the above developments, we have proved that M K is a K -subalgebra of E K , separates thepoints of DR K , and contains the idempotents ρ a for a ∈ I K . Therefore, by Theorem 10.1 [16], weconclude the target identity E K = M K . References [1] J. Borger. The basic geometry of Witt vectors I, the affine case.
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