L’INSTITUTFOURIER DE ANNALES

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A L

E S

E D L ’IN IT ST T U

F O U R

ANNALES

DE

L’INSTITUT FOURIER

Luis ARENAS-CARMONA

Representation fields for commutative orders Tome 62, n^o2 (2012), p. 807-819.

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REPRESENTATION FIELDS FOR COMMUTATIVE ORDERS

by Luis ARENAS-CARMONA (*)

Abstract. — A representation field for a non-maximal orderHin a central simple algebra is a subfield of the spinor class field of maximal orders which deter- mines the set of spinor genera of maximal orders containing a copy ofH. Not every non-maximal order has a representation field. In this work we prove that every commutative order has a representation field and give a formula for it. The main result is proved for central simple algebras over arbitrary global fields.

Résumé. — Un corps de représentation pour un ordre non maximal Hdans une algèbre centrale simple est un sous-corps du corps de classes spinoriel d’ordres maximaux qui détermine l’ensemble de genres spinoriels d’ordres maximaux qui contiennent un conjugué de H. Un ordre non maximal ne possède pas forcément un corps de représentation. Dans ce travail, nous montrons que chaque ordre com- mutatif a un corps de représentation F et nous donnons une formule pour F. Le résultat principal est prouvé pour des algèbres simples centrales sur des corps globaux arbitraires.

1. Introduction

Let K be a number field. Let A be either, a central simple K-algebra of degreen > 2, or a quaternion algebra satisfying Eichler condition [5].

The setOof maximal orders inAcan be split into isomorphism classes or, equivalently, conjugacy classes [8]. LetObe the set of isomorphism classes.

In [2] we found a field extension Σ/Kwhose Galois group Gal(Σ/K) yields a natural free and transitive action on O. For some families of orders H in Awe found a field F =F(H) between K and Σ such that the set OH

of classes of orders containing a conjugate of H is exactly one orbit of the subgroup Gal(Σ/F). In particular, the number of isomorphism classes

Keywords:maximal orders, central simple algebras, spinor genera, spinor class fields.

Math. classification:11R52, 11R56, 11R37, 16G30, 16G10.

(*) Supported by Fondecyt, Grant 1100127.

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of such orders is [Σ : F]. The first result in this direction seems to be due to Chevalley who considered the case when A is a matrix algebra andH is the maximal order of a maximal subfieldL [4]. In our notation, Chevalley result implies F(H) = L∩Σ. In [2] we extended this result to algebrasAwith no partial ramification. Chinburg and Friedman have given a similar result in whichHis an arbitrary rank-two commutative order in a quaternion algebraAsatisfying Eichler condition [5]. In case L=KHis a field, the representation field F is a subfield of L∩Σ that depends on the relative discriminant of the orderH. A more recent article by Linowitz and Shemanske [6] extends this result to central simple algebras of prime degree. Here we extend these results by proving the existence and providing a formula for the representation field for every commutative orderHin a central simple algebraAof arbitrary degree.

We use the language of spinor genera in all of this work for the sake of generality. In this general setting, the setOis not the set of conjugacy classes but the set of spinor genera of maximal orders, while OH is the set of spinor genera Φ with at least one order D ∈ Φ containing H as a sub-order. All of the above extends to this setting. A single spinor genera can have more than one class only whenAis a quaternion algebra failing to satisfy Eichler condition [2].

Theorem 1.1. — Let Abe a central simple algebra over the number field K. Let H be an arbitrary commutative order. Let D be a maximal order containingH. For every maximal ideal℘in the ring of integersOK

we letI℘be the only maximal two-sided ideal ofDcontaining℘1D, and let H℘be the image ofHinD/I_℘. LetE℘ be the center of the ringD/I_℘. Let t℘be the greatest common divisor of the dimensions of the irreducibleE℘- representations of the algebraE℘H℘. Then the representation field F(H) is the maximal subfieldF, of the spinor class fieldΣ, such that the inertia degreef℘(F/K)divides t℘ for every place℘.

In the above statement, it is implied that E^℘ is a field. In fact, E^℘ is the residue field of a local division algebraE℘ such that A℘ ∼=Mm(E℘).

We show in § 5 that Theorem 1.1 is indeed a generalization of the results in [2], [5], and [6]. For the sake of generality, we prove a generalization of Theorem 1.1 that includes also the function field case studied in [1]. In order to consider both cases simultaneously, we introduce the concept of A-curve in § 2.

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2. A-curves and Spinor class fields

In all that follows, we say that X is an A-curve with field of functions K, in any of the following cases:

Case 2.1. — X is a smooth irreducible curve over a finite field F and Kis the field of rational functions onX.

Case 2.2. — X = SpecOSwhereOS is the ring ofS-integers for a non- empty finite setSof places over a global fieldKcontaining the archimedean places.

In both cases X is provided with its Zariski Topology and its structure sheafOX. Furthermore, in Case 2.2 everyS-lattice Λ₀in a K-vector space VK defines a locally free sheaf Λ by Λ(U) =OX(U)Λ0, and the stalk at a place℘∈X is just the freeO_[℘]-lattice Λ_0[℘], whereO_[℘] and Λ_0[℘] denote the localizations at℘(as opposed to completions). The sheaf Λ thus defined is called anX-lattice. Analogously, we define an X-lattice in Case 2.1 as a locally free sheaf of lattices as in [1]. The spaceW spanned by Λ(U) is independent of the affine setU. We callW the space generated by Λ and denote itKΛ. The affine sub-case in Case 2.1 is also a sub-case of Case 2.2, if we letS be the complement of X in its smooth projective completion, and both definitions ofX-lattice coincide in this case. We say thatS =∅ whenX is a projective curve over a finite field.

Note that the completion Λ℘of the sheaf Λ at a place℘∈X is defined as the completion of the stalk Λ[℘], or equivalently the completion of the lattice Λ(U) for any affine setU ⊆X containing ℘. Since X-lattices can be defined by pasting lattices defined on the sets of an affine cover, some properties of usual lattices ([7] § 81) are inherited byX-lattices, namely:

• A lattice Λ is determined by it set of completions{Λ℘}℘, and it can be modified at a finite number of places to get a new lattice .

• The adelization GL_A(V) of the group GL(V) of K-linear maps on V acts on the set of lattices by acting on the set of completions at every place℘.

• IfV =Ais an algebra, an X-latticeD is an order (i.e., a sheaf of orders) if and only if every completion is an order. The same holds for maximal orders⁽¹⁾.

Since any two local maximal orders are conjugate in any completion A℘

with℘∈X, it follows that for any two global maximal ordersD and D⁰

(1)This is not in the reference, but follows easily from the previous results, since a lattice Dis an order if and only if 1∈DandDD=D.

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there exist an element a in the adelization A^∗

A satisfying D⁰

A = aD_Aa⁻¹, where

D_A= Y

℘∈S

A℘× Y

℘∈X

D℘

and D⁰_A is defined analogously. As usual we write simply D⁰ = aDa⁻¹, since no other action of adelic points on orders is considered in this work.

The ordersD andD⁰ are conjugate if we can choosea∈A^∗. We say that these orders are in the same spinor genus if we can choose a= bcwhere b∈ A^∗ and c℘ has reduced norm 1 for every place℘∈ X. The following result is needed in what follows. It is proved in [2] for Case 2.1 and follows easily from [8] (Theorem 33.4 and Theorem 33.15), or remark 2.5 in [1], for Case 2.2.

The set of spinor genera of maximal orders inA is in cor- respondence with the groupJK/K^∗H(D), where JK is the idele group of K and H(D) ⊆JK is the image under the reduced norm of the conjugation-stabilizer (A^∗_A)^D.

By the strong approximation theorem, two orders in the same spinor genus are always conjugate if the automorphism group ofA is non-compact at some place℘∈S. WhenKis a number field and S=∞, this condition is equivalent to Eichler condition ifAis a quaternion algebra, and it is always satisfied for algebras of higher dimension. WhenX is a projective curve we haveS =∅, so the condition cannot hold. Note however that spinor genera of orders over projective curves still carry some global information that can be recovered in any affine subset ofX [1]. WhenX is an affine curve, the condition holds unless every infinite place ofX is totally ramified forA/K. Since we express our main result in terms of spinor genera, this result holds in full generality, but it is important to keep in mind the previous remark in the applications.

We let Σ be the class field corresponding to the group K^∗H(D) ([10],

§ XIII.9). For everyρ∈Gal(Σ/K), leta_ρ ∈A^∗

A be any element satisfying ρ = [n(aρ),Σ/K], where t 7→ [t,Σ/K] denotes the Artin maps on ideles andn:A^∗

A→J_K denotes the reduced norm. The action of Gal(Σ/K) on the setOof spinor genera of maximal orders is given by

ρ.spin(D) = spin(a_ρDa⁻¹_ρ ), ∀(ρ,D)∈Gal(Σ/K)×O.

Assume in all that follows thatHis a suborder ofD. A generator forD|H is an elementu∈A^∗_Asuch thatH⊆uDu⁻¹. Letd(D|H) the set of reduced norms of generators. There is no reason a priori for this set to be a group.

Note that if a normalizes D and b normalizes H, then for any generator

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u the element bua is a generator. In particular H(D)d(D|H) = d(D|H).

We conclude that the setK^∗d(D|H) is completely determined by its image G(D|H)⊆Gal(Σ/K) under the Artin map. IfG(D|H) is a group we define

F(H) =

a∈Σ|g(a) =a∀g∈ G(D|H) ,

and call F = F(H) the representation field for H. It follows from the definition that an order D⁰ = bDb⁻¹, for b ∈ A^∗_A, is in the same spinor genus than some maximal order containing a conjugate of H, if and only if n(b) ∈ n(u)K^∗H(D) for some generator u for D|H, or equivalently [n(b), F/K] = idF. We conclude that F satisfies the properties stated in

§ 1. IfG(D|H) fails to be a group we say that the representation field for His not defined. For examples of non-commutative orders for which the representation field is not defined, see [3] or Example 3.6 below.

3. Spinor image and representations

Let K be a local field of arbitrary characteristic. Assume in all of this section thatA=Mm(E) whereE is a central division algebra overKwith ring of integersOE, maximal idealmE, and residue fieldE=OE/mE. Let π_E and π_K be uniformizing parameters of E and K respectively. Recall that ifn:A^∗ →K^∗ is the reduced norm, then n(πE1A) =vπ_K^m for some unitv∈ O^∗_K.

LetH⊆Dbe orders inA. Then a local generatoru, or simply a generator whenever confusion is unlikely, is an elementu∈A^∗satisfyingH⊆uDu⁻¹. In the rest of this section, we assume thatDis maximal, but we make no assumption onHfor the sake of generality. The purpose of this section is to characterize the set of reduced norms of local generators in terms of the degrees of the representations of the reduction ofHmóduloπED.

For short, we let ^→1q and ^→0q denote vectors (1, . . . ,1) and (0, . . . ,0) of lengthq. For example (^→13,2·^→12,^→03) = (1,1,1,2,2,0,0,0). For any ringA we denote byA^mthe freeA-module withmgenerators, whileA^∗mdenotes the set ofm-powers of invertible elements inA.

Lemma 3.1. — LetHbe a suborder of the maximal orderD=Mm(OE), and let H be its image in Mm(E). Assume that the reduced norm of a generatoru for D|H spans the principal ideal

n(u)

= (π^s_K) = π^s_KOK. Then there exists a flag ofH-modules

{0}=V0⊆V1⊆ · · · ⊆Vt=E^m

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and integersr₁, . . . , r_t satisfying s=

t

X

i=1

ridim_E(Vi/V_i−1).

Proof. — Let P, Q ∈Mm(O_E)^∗ be such that u= P DQ for a diagonal matrix D = diag(π_E^r¹ ^→1q1, . . . , π^r_E^t ^→1qt), where r₁ < r₂ < · · · < r_t and q1+· · ·+qt=m. The condition H⊆uDu⁻¹ is equivalent to H(uO_E^m)⊆ uO^m_E, or equivalently

(P⁻¹HP)DO^m_E ⊆DO_E^m.

Replacing Hby P⁻¹HP if needed, we can assume thatP is the identity.

Now we defineVi ⊆Eⁿ as the image of the matrix

diag(^→1q₁, . . . ,^→1q_i,^→0q_i+1, . . . ,^→0q_t)∈Mm(E).

Certainly the flagV0⊆ · · · ⊆Vt satisfies the last condition. We claim that every V_i is a submodule. Otherwise, there exists an element v ∈ V_i, for some i, and an element h ∈ H such that hv /∈ Vi. Now take pre-images h∈Handv ∈Ve_i ofhand v respectively, whereVe_i ⊆ O_E^m is the image of the matrix

diag(^→1q₁, . . . ,^→1q_i,^→0q_i+1, . . . ,^→0q_t)∈Mm(OE).

Then π_E^rⁱv ∈ DO^m_E, whence h(π_E^rⁱv) ∈ DO^m_E. Since conjugation by πE

leavesD invariant, then π_E^−rⁱhπ_E^rⁱ is an integral matrix with reduction ˆh.

Sinceh(π^r_Eⁱv)∈DO^m_E, then

π_E^−rⁱh(π_E^rⁱv)∈π_E^−rⁱ(DO_E^m)∩ O_E^m⊆Vei+πEO^m_E,

whence ˆh(v)∈V_i. SinceV_i is generated byV_i_K=V_i∩K^m, whereKis the residue field ofK, we may assume that v∈K^m. Note that ˆhis the image of hunder an automorphism σ ∈Gal(E/K), and sincev ∈K^m, we have ˆh(v) =σ[h(v)]. SinceσfixesViKpoint-wise, then necessarilyσ(Vi) =Vi. It follows that ˆh(v)∈V_i if and only ifh(v)∈V_i. This contradicts the choice

ofv.

Lemma 3.2. — LetHbe a suborder of the maximal orderD=Mm(O_E), and letHbe its image inMm(E). Assume that there is anH-moduleV ⊆ E^m of dimension r. Then there exists a local generator ufor D|H whose reduced norm generates the ideal(π^m−r_K ).

Proof. — LetM ∈Mm(E) be a matrix of rankr. We claim thatM has a pre-image ˜M inMm(OE), satisfying the following conditions:

• M˜ is invertible inMm(E).

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• The reduced normn( ˜M) is a generator of the ideal (π_K^m−r).

• πEM˜⁻¹∈Mm(OE) and its reductionL∈Mm(E) satisfies Im(M) = ker(L).

To define the lifting ˜M we writeM =P DQ, whereP andQare invertible inMm(E) andD= diag(^→1r,^→0m−r), then chose arbitrary liftings ˜P and ˜Q inMm(OE)^∗, and finally set ˜M = ˜PD˜Q, where ˜˜ D= diag(^→1r, π_E^→1m−r)∈ Mm(OE). Then

πEM˜⁻¹= (πEQ˜⁻¹π⁻¹_E )

diag(πE

→

1r,^→1m−r)

P˜⁻¹∈Mm(OE).

Note that the matrix π_EQ˜⁻¹π_E⁻¹ is invertible in Mm(O_E), whence the reduction L of πEM˜⁻¹ has the right rank. The last condition follows now if we prove LM = 0, but this is clear since it is the reduction of πEM˜⁻¹M˜ =πEId.

Let M be a projection matrix with image V and let L be as above.

Then M has rank r and satisfies LXM = 0 for every X ∈ H, since X leavesV = Im(M) invariant. We claim that ˜M is a generator forD|H, this concludes the proof. Since every elementX ∈Hleaves theE-vector space V =M(E^m) invariant, then every elementxofHmust leave invariant its pre-image,i.e.,the module ˜M(O_E^m) +πEO^m_E. Now we have

π_EO^m_E = ˜M( ˜M⁻¹π_EO_E^m)⊆M˜O^m_E,

since ˜M⁻¹πE =π⁻¹_E (πEM˜⁻¹)πE ∈Mm(OE). As this implies x( ˜MO^m_E)⊆ M˜O^m_E for anyx∈H, we conclude that ˜M is a generator. The result follows.

Lemma 3.3. — LetHbe a suborder of the maximal orderD=Mm(OE) and letHbe its image in M^m(E). Assume that every irreducible representation of the E-algebra EH has dimension d. Then there exists a local generator for D|H whose reduced norm generates (π_K^s) if and only if d dividess.

Proof. — On one hand, assume d divides s. Since conjugation by any power of πE leaves D invariant and the reduced norm of the diagonal matrixπ_E1_Aspans the ideal (π_K^m), we may assume 06s < m. It suffices to show that there exists aH-submodule of E^m of dimension m−s. By the representation theory for finite dimensional algebras, there exists a composition series

{0}=V0⊆V1⊆ · · · ⊆V_m/d=E^m

where the submoduleV_i has dimensiondi, and the result follows. On the other hand, if there exists a local generator forD|H whose reduced norm

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spans (π_K^s), then there exists a sequence of submodules V₁, . . . , V_t and integers r1, . . . , rt as in Lemma 3.1 satisfying s = P

iridim_E(Vi/V_i−1).

Refining the sequence if needed, we may assume it is a composition series, whence dim_E(Vi/Vi−1) =dfor everyi, and thereforeddivides s.

Next result is now immediate from the previous lemma:

Lemma 3.4. — Let H be a suborder of D = Mm(O_E) and let H be its image in Mm(E). Assume that any irreducible representation of the E-algebra EH has the same degree d. Then the set of reduced norms of generators forD|Hisd(D|H) =K^∗dO_K^∗.

The results in this section seem to indicate that the existence of a representation field depends only on the dimensions of the representations of the algebraEH. This is not so, as illustrated by Example 3.6. It holds, however, whenHis the largest order with reductionHas next lemma shows.

Lemma 3.5. — LetH=p⁻¹(H)be a suborder ofD=M^m(OE), where His a subalgebra of Mm(E) and p: Mm(OE) → Mm(E) the usual projection. Then there exists a local generator forD|Hwhose reduced norm generates(π_K^s)if and only if there exists anH-moduleV ⊆E^mof dimension t, wherem−tis the remainder ofswhen divided bym.

Proof. — It suffices to prove that ifuis a generator whose reduced norm generates (π^s_K) then there exists an H-moduleV of dimension t. Since u is a generator, we have H(uO^m_E) ⊆ uO_E^m. Post-multiplying u by a power of πE we may assume uO_E^m ⊆ O_E^m, but uO^m_E is not contained inπEO^m_E. SinceH contains every matrix of the form π_Ea with a ∈D, we conclude thatπEO_E^m⊆uO^m_E. It follows thatu=P DQwhereP andQare units in M^m(OE) andD = diag(πE

→

1r,^→1d). Let V be the image of uO_E^m in E^m. It follows that dim_EV =dwhile on the other hand the reduced norm ofu generates (π_K^r), so thatr=s. It follows thatd=m−r=t, since they are

congruent modulom and 0< d, t6m.

Example 3.6. — Let A = Mⁿ(K) and let L = {a ∈ A|ae1 ∈ Ke1}, where{e1, . . . , e_n}is the canonical basis ofKⁿ. LetI be an integral ideal, i.e.,a sub-lattice ofOX, and letH=H(I) be the order

H(I) =

L∩Mⁿ(OX)

+Mⁿ(I),

whereMn(I) is the sheaf defined byMn(I)(U) =Mn

I(U)

for any affine setU andMn(OX) is defined analogously.

The set of reductions H℘depends only on the primes℘such that I_℘6=

O℘, but this is not so for the set of norms of generators. In fact, if I℘ is

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the maximal ideal ofO℘, then Lemma 3.5 applies and any local generator at℘must have reduced norm inO_℘^∗K_℘^∗n or π_℘ⁿ⁻¹O_℘^∗K_℘^∗n. In particular, if I=℘and if [℘,Σ/K] has order at least 3 in the Galois group Gal(Σ/K), then the representation field does not exist. However, ifI℘ = (π_℘^t), then diag

π_℘^−s,^→1n−1

= diag(π_℘^−s,1, . . . ,1) is a generator for any s 6 t. In particular, ift=t(℘) is big enough for every prime℘dividingI, then the representation fields is defined.

4. Proof of Theorem 1.1

Lemma 4.1. — Let R be a commutative ring that is complete with respect to an idealJ. Then any idempotent P ∈R/J can be lifted to an idempotentP˜∈R.

Proof. — Let P₁ an arbitrary lifting of P to R and define recursively Pn+1=Pn+ (1−2Pn)(P_n²−Pn). Note that

P_n+1² −P_n+1= (P_n²−P_n)h

1−(1−2P_n)²[1−(P_n²−P_n)]i .

Since (1−2Pn)²= 1 + 4(P_n²−Pn)≡1 (moduloJ), we prove by recursion thatP_n²−P_n∈Jⁿ and the result follows by completeness.

Lemma 4.2. — LetHbe an commutative order in a local central simple algebraAand letPbe the sub-order generated by the idempotents ofH. Let C be the centralizer ofP in A. Write Has a direct productH=Q

i∈IHi

with H_i connected. Then we have a decomposition C = Q

i∈IC_i, where every Ci is a central simple algebra. Furthermore, there exist a family of maximal orders{Di}_i∈I, withHi ⊆Di ⊆Ci and a maximal order D⊆A such that

d(D|H)⊇Y

i∈I

d(D_i|H_i).

Proof. — If H=Q

i∈IHi as above, we letP_i be the idempotent corresponding to the factor Hi. Then Pi is a central idempotent in C, whence there exists a decomposition of C as above where C_i ∼= P_iC. Assume A = Mm(E), where E is a division algebra. Note that E^m = Q

iPiE^m as either leftC-modules or rightE-modules. By taking anE-basis of each PiE^m, we obtain a basis ofE^min which the orderPis the ring of diagonal

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matrices of the form







a1Iq₁ 0 · · · 0 0 a₂I_q₂ · · · 0 ... ... . .. ... 0 0 ... arIq_r







, a1, . . . , ar∈ OK,

whereIq is the q byq identity matrix. It follows thatC is the algebra of matrices of the form

A=







A₁ 0 · · · 0 0 A2 · · · 0 ... ... . .. ... 0 0 ... A_r







, A_i∈Mqi(E).

We can identify Ci ∼=M^qi(E) with the set of matrices A, as above, such that Aj = 0 for j 6= i. In particular Ci is central simple. By a suitable change of basis in everyPiE^m, we can assumeHi⊆Mq_i(OE). Now we can chooseD=Mm(OE) andD_i=Mqi(OE) fori= 1, . . . , r.

Letui∈C_i^∗be a generator forDi|Hi as defined in § 2, for everyi. Then u=P

iu_i∈C^∗ is a generator forD|H, since H=Y

i

Hi ⊆Y

i

(uiDiu⁻¹_i ) =u Y

i

Di

!

u⁻¹⊆uDu⁻¹.

Furthermore, from the explicit description of C given above we see that the reduced norm ofuisn(u) =Q

in(ui). Now the conclusion follows.

Now we are ready to prove the following generalization of Theorem 1.1.

Proposition 4.3. — Let X be an A-curve and let K be the field of functions ofX. LetAbe a central simpleK-algebra. LetHbe an arbitrary commutative order in A. Let D be a maximal order containing H. For every place℘ ∈X we let I℘ be the only maximal two-sided ideal of the completion D_℘, and let H℘ be the image of H in D_℘/I_℘. Let E℘ be the center of the ring D℘/I℘. Let t℘ be the greatest common divisor of the dimensions of the irreducibleE^℘-representations of the algebraE^℘H^℘. Then the representation fieldF(H)is the maximal subfieldF of the spinor class fieldΣsuch that the inertia degree f℘(F/K)dividest℘, for every place℘.

Proof. — We letH℘ be the quotient ofH℘by its radical. In particular, H℘=H℘/J for an idealJ such thatJⁿ⊆πEDfor somen. It follows that H_℘ is complete with respect to J, so any idempotent in H℘ can be lifted to an idempotent inH℘by Lemma 4.1. We conclude that the imageHi of

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everyH_iinH℘, defined as in last lemma, has no idempotents and therefore Hi is a field since H℘ is semisimple. It follows from Lemma 3.4 that the image of the local generators for each H_i is d_℘(D_i|H_i) = K_℘^∗dⁱO^∗_℘, where everydiis the dimension of the corresponding representation. On the other hand, by Lemma 3.1 we get the contentiond℘(D|H)⊆K℘^∗t^℘O^∗_℘. Note that

K_℘^∗t^℘O^∗_℘=Y

i

K_℘^∗dⁱO_℘^∗ =Y

i

d℘(Di|Hi)⊆ d℘(D|H)⊆K_℘^∗t^℘O_℘^∗,

whence equality follows. We conclude thatd℘(D|H) =K℘^∗t^℘O_℘^∗ as claimed.

5. Applications and examples

Example 5.1. — Let A=Mn(K) and let Tn be the order of all upper triangular matrices of the form







a1 a2 · · · an

0 a₁ · · · a_n−1 ... ... . .. ... 0 0 · · · a1







, a₁, . . . , a_n∈ O_K.

ThenH℘has only one irreducible representation of dimension 1 for all ℘.

It follows thatt℘= 1 for all℘and thereforeF(Tn) =K. We conclude that every maximal order inMn(K) contains a copy of the orderT_n.

More generally, for an arbitrary commutative suborderH, the dimensions of the irreducible representations ofH are the same as the dimensions of the irreducible representations of the maximal semi-simple suborderHs of H. Next result follows:

Corollary 5.2. — If H is an arbitrary commutative order in the al- gebraA, andHs is the maximal semi-simple suborder ofH, thenF(Hs) = F(H). In particular, if a maximal orderDcontains a conjugate ofH_s, then there exists an orderD⁰ in the spinor genus of DsatisfyingH⊆D⁰.

Recall that the inertia degree f℘(F/K) of an unramified abelian extension F/K divides the inertia degree f_℘(L/K) of an arbitrary extension L/K, at every ℘, if and only if F ⊆ L. Next result follows (compare to Theorem 4.3.4 in [2]):

Corollary 5.3. — If Ahas no partial ramification outside of S and H=Q

iH_i, where eachH_i is the maximal order of a fieldL_i, thenF(H) = Σ∩(T

iLi).

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Example 5.4. — IfHis the maximal order in a semi-simple commutative algebra of prime dimension that is not a field, then the greatest common divisor of the degrees [L_i:K] is 1, and thereforeHis contained in at least one order in every spinor genus.

Example 5.5. — If H=OK×H1, then His contained in at least one order in every spinor genus.

Recall that a split order in Mⁿ(K) is an order containing a copy of O_Kⁿ =OK× · · · × OK [9]. By applying last example toH=O_Kⁿ, we obtain a simple proof of the following generalization of Theorem 5 in [4].

Corollary 5.6. — Every spinor genera of maximalX-orders inMn(K) contains an split order. IfX is affine, then every maximal order is split.

An order H in A is called non-selective if for every spinor genus Φ of maximal orders, there exists an orderD∈Φ containingH. An orderH⊆A is selective if it is not non-selective. This is the natural generalization of the concept of selective order defined in [5] and [6]. Assume now [L:K] =pis a prime, andAhas no partial ramification, so ifLis the maximal order of L, Corollary 5.3 applies andF(L) = Σ∩L. Since [L:K] is a prime, the orderLis selective if and only ifL⊆Σ. In this case the order is contained in 1/pof all conjugacy classes. Assume this is the case and letH⊆Lbe an arbitrary suborder. ThenH is non-selective if and only if there is a place

℘that is inert for L/K such that the image of Hin the residue field L℘

coincide with the residue fieldK℘ofK, since in this case t℘= 1, soF(H) splits at℘, and thereforeF(H) =K. Since the condition is equivalent to℘ dividing the relative discriminant ofH/OK, next result follows:

Corollary 5.7. — LetA be a central simple algebra without partial ramification and let Lbe the maximal order of a field L of prime degree overK. ThenLis selective inAif and only ifL⊆Σ. In this case a suborder H⊆Lis selective unless there is a inert place ofL/K dividing the relative discriminant ofHoverOK.

Note that we no longer require that dimKA=p²in the above result (see [5] and [6]). More generally, the fieldLalways containsF, whence we have next result:

Corollary 5.8. — LetA be a central simple algebra without partial ramification and let Hbe an order contained in a field L ⊆A. Let N be the total number of spinor genera. Then

]n Φ∈O

∃D∈Φ such thatH⊆Do

=N d,

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for some divisordof[L:K].

BIBLIOGRAPHY

[1] L. Arenas-Carmona, “Spinor class fields for sheaves of lattices”, Preprint, arXiv:1009.3280v1 [math.NT] 16 Sep 2010.

[2] ——— , “Applications of spinor class fields: embeddings of orders and quaternionic lattices”,Ann. Inst. Fourier (Grenoble)53(2003), no. 7, p. 2021-2038.

[3] ——— , “Relative spinor class fields: a counterexample”,Arch. Math. (Basel)91 (2008), no. 6, p. 486-491.

[4] C. Chevalley,L’arithmétique sur les algèbres de matrices, Herman, Paris, 1936.

[5] T. Chinburg&E. Friedman, “An embedding theorem for quaternion algebras”, J. London Math. Soc.60(1999), no. 2, p. 33-44.

[6] B. Linowitz&T. R. Shemanske, “Embedding orders into central simple algebras”, Preprint, arXiv:1006.3683v1 [math.NT] 18 Jun 2010.

[7] O. T. O’Meara,Introduction to quadratic forms, Academic press, New York, 1963.

[8] I. Reiner,Maximal orders, Academic press, London, 1975.

[9] T. R. Shemanske, “Split orders and convex polytopes in buildings”,J. Number Theory130(2010), no. 1, p. 101-115.

[10] A. Weil,Basic Number Theory, second ed., Springer Verlag, Berlin, 1973.

Manuscrit reçu le 17 octobre 2010, accepté le 9 février 2011.

Luis ARENAS-CARMONA Universidad de Chile

Departamento de matematicas Facultad de ciencia

Casilla 653 Santiago (Chile) learenas@u.uchile.cl