On basic and Bass quaternion orders

By Sara Chari, Daniel Smertnig, and John Voight

Abstract

A quaternion order over a Dedekind domain is Bass if every -superorder is Gorenstein, and is basic if it contains an integrally closed quadratic -order. In this article, we show that these conditions are equivalent in local and global settings: a quaternion order is Bass if and only if it is basic. In particular, we show that the property of being basic is a local property of a quaternion order.

1. Introduction

Orders in quaternion algebras over number fields arise naturally in many contexts in algebra, number theory, and geometry—for example, in the study of modular forms and automorphic representations and as endomorphism rings of abelian varieties. In the veritable zoo of quaternion orders, authors have distinguished those orders having favorable properties, and as a consequence there has been a certain proliferation of terminology. In this article, we show that two important classes of orders coincide, tying up a few threads in the literature.

Setup

Let be a Dedekind domain and let be its field of fractions. Let be a quaternion algebra over , and let be an -order. We say that is Gorenstein if its codifferent is an invertible -lattice in , or equivalently is projective as a left or right -module. Gorenstein orders were studied by Brzezinski Reference 4, and they play a distinguished role in the taxonomy of quaternion orders—as Bass notes, Gorenstein rings are ubiquitous Reference 2. Subsequent to this work, and given the importance of the Gorenstein condition, we say is Bass if every -superorder in is Gorenstein. As Bass himself showed Reference 2, Bass orders enjoy good structural properties while also being quite general. A Bass order is Gorenstein, but not always conversely. Being Gorenstein or Bass is a local property over , because invertibility is so.

On the other hand, we say that is basic if there is a (commutative) quadratic -algebra such that is integrally closed in its total quotient ring . Basic orders were first introduced by Eichler Reference 8 over (who called them primitive), and studied more generally by Hijikata–Pizer–Shemanske Reference 12 (among their special orders), Brzezinski Reference 5, and more recently by Jun Reference 13. The embedded maximal quadratic -algebra allows one to work explicitly with them, since a basic order is locally free over of rank : for example, this facilitates the computation of the relevant quantities that arise in the trace formula Reference 11. Locally, basic orders also appear frequently: local Eichler orders are those that contain , and local Pizer (residually inert) orders Reference 14, §2 are those orders in a division quaternion algebra that contain the valuation ring of an unramified quadratic extension. It is not immediate from the definition that being basic is a local property.

Results

The main result of this article is to show these two notions of Bass and basic coincide, in both local and global settings. We first consider the local case.

Theorem 1.1.

Let be a discrete valuation ring (DVR) and let be a quaternion -order. Then is Bass if and only if is basic.

Theorem 1.1 was proven by Brzezinski Reference 5, Proposition 1.11 when is a complete DVR with and perfect residue field; the proof relies on a lengthy (but exhaustive) classification of Bass orders. Here, we present two essentially self-contained proofs that are uniform in the characteristic, one involving the manipulation of ternary quadratic forms and the second exploiting the structure of the radical.

Next, we turn to the global case.

Theorem 1.2.

Let be a Dedekind domain whose field of fractions is a number field, and let be a quaternion -order. The following statements hold.

(a)

is basic if and only if the localization is basic for all primes of .

(b)

is Bass if and only if is basic.

In fact, we show that if is Bass (equivalently, basic), then contains infinitely many nonisomorphic quadratic -algebras and moreover they can be taken to be free as -modules (Corollary 7.6). Theorem 1.2(b) over was proven by Eichler Reference 8, Satz 8 using a somewhat different method.

We also prove the conclusions of Theorem 1.2 in a large number of cases in which is a Dedekind domain whose field of fractions is a global function field: see Theorem 7.5. (We lack in the function field case a sufficiently general local–global result on representations by ternary quadratic forms, see section 6.)

Returning to the local situation, if is a DVR then several equivalent characterizations of Bass orders are known Reference 17, Proposition 24.5.3 and this list is further extended by our results. For the reader’s convenience we give a comprehensive list.

Corollary 1.3.

Let be a DVR with maximal ideal , and let be a quaternion -order. Then the following are equivalent.

(i)

is a Bass order;

(ii)

and the radical idealizer are Gorenstein;

(iii)

The Jacobson radical is generated by two elements (as left, respectively, right ideal);

(iv)

is a basic order;

(v)

Every -ideal is generated by two elements;

(vi)

Every -lattice is isomorphic to a direct sum of -ideals; and

(vii)

is not of the form with an integral -lattice.

The implications (v)(i)(vi) hold more generally Reference 17, Section 14.5. The implication (vi)(v) holds only in specific settings; for quaternion orders it follows from work of Drozd–Kiričenko–Roĭter Reference 7, Proposition 12.1, 12.5. While we do not give another proof of this implication, we provide a direct proof for (i)(v). With the exception of statement (vi), we therefore give a full proof of the equivalences in Corollary 1.3.

Outline

The paper is organized as follows. After introducing some background in section 2, we prove Theorem 1.1 and Corollary 1.3 in sections 34. In the remaining sections, we prove Theorem 1.2: in section 5 we treat the case when strong approximation applies, in section 6, we treat definite orders over rings of integers in a number field, and we conclude the proof in section 7.

2. Background

In this section, we briefly review the necessary background on orders and quadratic forms. For a general reference, see Voight Reference 17.

Properties of quaternion orders

Let be a Dedekind domain with . Let be a quaternion algebra over and let be an -order.

Definition 2.1.

We say that is Gorenstein if the codifferent

is invertible, and we say is Bass if every -superorder is Gorenstein.

For more detail and further references, see Voight Reference 17, Sections 24.2, 24.5. Being Gorenstein is a local property— is Gorenstein if and only if the localizations are Gorenstein for all primes of —so it follows that Bass is also a local property.

Definition 2.2.

We say that is basic if there is a (commutative) quadratic -algebra such that is integrally closed in its total quotient ring .

Remark 2.3.

The term primitive is also used (in place of basic), but it is potentially confusing: we will see below that a primitive ternary quadratic form corresponds to a Gorenstein order, not a “primitive” order.

Local properties

Now suppose is a local Dedekind domain, i.e., is a discrete valuation ring (DVR) with maximal ideal and residue field . The Jacobson radical of is the intersection of all maximal left (or equivalently right) ideals of . The semisimple -algebra is one of the following Reference 17, 24.3.1:

is a quaternion algebra (equivalently, is maximal);

, and we say that is residually split (or Eichler);

, and we say that is residually ramified; or

is a separable quadratic field extension of and we say that is residually inert.

The radical idealizer of is the left order .

Ternary quadratic forms

Still with a DVR, we review the correspondence between quaternion orders and ternary quadratic forms (see also Voight Reference 17, Chapters 5, 22 and Reference 17, Remark 22.6.20 for a full history).

We define a similarity of two ternary quadratic forms and to be a pair , where is an -module isomorphism and is such that for all .

Proposition 2.4 (Gross–Lucianovic Reference 10).

There is a discriminant-preserving bijection between quaternion -orders up to isomorphism and nondegenerate ternary quadratic forms over up to similarity. Moreover, an -order is Gorenstein if and only if the corresponding quadratic form is primitive.

We now briefly review the construction of the bijection in Proposition 2.4. Since is a PID, is free of rank as an -module. A good basis for an -order is an -basis with a multiplication table of the form

with . Every -basis of can be converted to a good basis in a direct manner. For all and , we find

Associated to and the good basis, we attach the ternary quadratic form defined by

The similarity class of is well-defined on the isomorphism class of . Conversely, given a nondegenerate ternary quadratic form , we associate to its even Clifford algebra , which is a quaternion -order. A change of good basis of induces a corresponding change of basis of , and conversely every such change of basis of arises from a change of good basis of .

3. Locally Bass orders are basic

In this section, we give our first proof of Theorem 1.1. To this end, in this section and the next let be a DVR with fraction field and maximal ideal . For , we write for and .

Let be a quaternion algebra over and an -order. According to the following remark, we could work equivalently in the completion of .

Remark 3.1.

The order is basic (or Bass) if and only if its completion is basic (or Bass). Indeed, invertibility and maximality can be checked in the completion.

We choose a good -basis for and let be the ternary quadratic form over associated to with respect to this basis, as in Equation 2.7.

Lemma 3.2.

The order is not basic if and only if for every there exists such that and .

Proof.

Let and consider the -algebra . Then fails to be integrally closed if and only if there exists , integral over , such that ; this holds if and only if there exists such that is integral over , which is equivalent to and , as claimed.

A slight reformulation gives a local version of the result of Eichler Reference 8, Satz 8. Recall that a semi-order is an integral -lattice with Reference 17, Section 16.6. Basic semi-orders are defined analogously to basic orders.

Lemma 3.3.

A semi-order is not basic if and only if it is of the form with an integral -lattice.

Proof.

As in the previous lemma, if , then is not basic. Conversely, if is not basic, each is of the form with an integral . Take to be the -lattice generated by all these .

As an application of Lemma 3.2, we prove one implication in Theorem 1.1.

Proposition 3.4.

If is basic, then is Bass.

Proof.

Suppose is basic. Then every -superorder is also basic. So to show that is Bass, we may show that is Gorenstein. To do so, we prove the contrapositive. Suppose that is not Gorenstein. Then the quadratic form associated to has all coefficients . From Equation 2.6, we see that for all we have and . Therefore is not basic by Lemma 3.2.

Lemma 3.5.

If is maximal, residually inert, or residually split, then is basic and Bass.

Proof.

By the previous proposition it suffices to show that is basic. In each case, contains a separable quadratic algebra over which lifts to a valuation ring in . See also Voight Reference 17, 24.5.2, Proposition 24.5.5.

Remark 3.6.

It is not always possible to embed an integrally closed quadratic order that is a domain into a residually split (Eichler) order; this justifies the (more general) definition of basic orders allowing nondomains such as .

Lemma 3.7.

Suppose is Gorenstein with associated quadratic form in a good basis as in Equation 2.7 and that is not basic. Then the following statements hold.

(a)

If in , then .

(b)

Suppose that . Let and suppose . Then .

Proof.

For (a), to show that , by Lemma 3.2 there exists such that ; since , we have . Similarly, arguing with we have . For (b), without loss of generality we suppose and . By Lemma 3.2,

But , so . Thus , so , so ; since , we get .

Lemma 3.9.

Suppose is Gorenstein, not basic, and residually ramified. Then there exists a good basis of such that the associated quadratic form is given by

with and and one of the following conditions holds:

(i)

and ; or

(ii)

and and .

Proof.

As explained in section 2, a change of good basis of corresponds to a change of basis for , so we work with the latter. By a standard “normal form” argument (see e.g. Voight Reference 16, Proposition 3.10), there exists a basis such that becomes

with and not all in , and if . Let be the corresponding good basis for .

We modify this basis further to obtain the desired divisibility, as follows. First, suppose that . Then . Swapping basis vectors, we obtain the diagonal quadratic form with . If , then satisfies so is not residually ramified, a contradiction, so we must have and by symmetry . By Lemma 3.7, we get , and we are in case (i) (which becomes case (ii) after a basis swap).

Second, suppose that , so . By Lemma 3.7(a), we have . If , we keep the basis unchanged and pass all subscripts to . If , we take (swapping second and third basis elements); in this basis, we obtain the quadratic form

with , , , and , with . Otherwise, suppose . Since is residually ramified, we have . Moreover . Reducing modulo , we conclude that , so there exists such that . We take the new basis . In this basis, we obtain Equation 3.11 where now

Since and , we have . In all cases, we have . By Lemma 3.7, we immediately upgrade to . Finally, since is Gorenstein, either and then and as in case (ii), or we have .

To finish, we suppose that and we make one final change of basis to get us into case (i). As in the previous paragraph, we have , so there exists such that . We take the new basis , giving the quadratic form and

Now by construction, and so and we get to case (i).

We now prove Theorem 1.1.

Theorem 3.12.

The order is Bass if and only if is basic.

Proof.

We proved in Proposition 3.4. We prove by the contrapositive: we suppose that is not basic and show is not Bass by exhibiting a -superorder that is not Gorenstein. If is not Gorenstein then it is not Bass, so we are done. Suppose then that is Gorenstein. By Lemma 3.5, we must have residually ramified. Then by Lemma 3.9, there exists a good basis for such that the corresponding quadratic form satisfies either (i) or (ii) from that lemma.

We begin with case (i). We first claim that . By Lemma 3.2, there exists such that ; since we conclude ; then implies , and since we get . This gives us a (minimal) non-Gorenstein superorder, as follows. Let and let . Then and has the following multiplication table, with coefficients

in :

Thus is an -order with , all of whose coefficients are divisible by . We conclude is not Gorenstein and so is not Bass.

Case (ii) follows similarly, taking instead and , with associated quadratic form satisfying , , , , all of which are divisible by .

Remark 3.14.

If is a Gorenstein order that is neither residually split nor maximal, the radical idealizer is the unique minimal superorder by Reference 17, Proposition 24.4.12. In the previous proof , and hence necessarily . We have therefore proved that if and are both Gorenstein, then is basic. We return to this in the next section.

Remark 3.15.

When , the argument for Theorem 1.1 is quite simple Reference 17, Proposition 24.5.8: diagonalizing up to similarity, the ternary quadratic form associated to a Gorenstein order is with , and is Bass if and only if .

4. A second proof for local Bass orders being basic

In this section, we given a second proof of (the hard direction of) Theorem 1.1. We retain our notation from the previous section; in particular is a discrete valuation ring with maximal ideal .

By classification, we see that a quaternion -order is a local ring (has a unique maximal left [right] ideal, necessarily equal to its Jacobson radical ) if and only if is neither maximal nor residually split.

Lemma 4.1.

Suppose that is a local ring. Let . Then the following statements hold.

(a)

We have and .

(b)

If is not basic, then and .

Proof.

Since is Artinian, is nilpotent, so there exists such that . Thus the image of in the -algebra has reduced characteristic polynomial , so and , proving (a). Since satisfies its reduced characteristic polynomial , if , then is an Eisenstein polynomial so is a DVR and in particular integrally closed, contradicting that is not basic and proving the first part of (b). Finally, so .

Lemma 4.2.

Let be a local Artinian -algebra with via . If , …,  generate as ideal of , then they generate as -algebra.

Proof.

Let . Since is Artinian, there exists with . For , let . Since is commutative, it is easily seen that generates as -module. Using , we see that generates as -module.

Theorem 4.3.

Let be a residually ramified Bass -order and suppose that is generated by two elements as left [right] ideal. Then is basic.

Proof.

Let , and let , generate as left ideal (the other case being symmetric). Their images generate over , so .

Suppose to the contrary that is not basic. Observe since . Lemma 4.1 implies , , . Thus . By Lemma 4.2 the elements , generate as -algebra. Since they anticommute, we see that they are normal elements in . It follows that is generated by as -module.

Again using that and anticommute, we have . This implies that is in fact generated by as -module, and hence as -vector space.

Let be the length of an -module . Since we have . Because is residually ramified and we find . Now

yields . But gives , a contradiction.

The previous theorem together with the characterization of Bass orders Reference 17, Proposition 24.5.3 implies that every (residually ramified) Bass order is basic. Alternatively, it is easy to see directly that the assumption of Theorem 4.3 holds for Bass orders, as the next proposition shows.

Proposition 4.5.

If and are Gorenstein -orders, then is generated by two elements (as a left, respectively, right -ideal).

Proof.

If is hereditary, then is principal Reference 17, Main Theorems 21.1.4 and 16.6.1. If is Eichler, it is easily seen from an explicit description of that is generated by two elements Reference 17, 23.4.15. We thus suppose that is a local ring.

Let . Then with for some Reference 17, Proposition 24.4.12. (using that is Gorenstein). Since is the unique maximal left [right] -submodule of by the proof of the same proposition, dualizing implies that there is no right [left] -module properly between and Reference 17, Section 15.5. Hence is a cyclic right [left] -module. So with , . Since is also Gorenstein and , the ideal is invertible and hence principal Reference 17, Proposition 24.2.3 and Main Theorem 16.6.1. So . Altogether .

We now characterize local Bass orders.

Proof of Corollary 1.3.

(i)(ii) by definition; (ii)(iii) is Proposition 4.5; (iii)(iv) by Theorem 4.3 for residually ramified orders, in any other case is basic without any assumption on by Lemma 3.5. Propositon 3.4 shows (iv)(i).

(iv)(v): Let be a maximal order of a -quadratic algebra contained in . Any -ideal is an -lattice of rank . Since is local, is a free -lattice of rank two. Thus is generated by two elements over and also over (as left or right ideal). (v)(iii) is trivial.

(i)(vi) holds Reference 17, Proposition 24.5.3. The implications (v)(i)(vi) hold in large generality, whereas (vi)(v) for quaternion orders is a result of Drozd–Kiričenko–Roĭter. Finally, (iv)(vii) follows from Lemma 3.3.

5. Basic orders under strong approximation

In this section, we prove Theorem 1.2 when strong approximation applies. We start by showing that basic is a local property, i.e., an -order is basic if and only if its localization at every nonzero prime of is basic.

Setup

Moving now from the local to the global setting, we use the following notation. Let be a global field and let be the ring of -integers for a nonempty finite set of places of containing the archimedean places. Let be a quaternion algebra over , and let be an -order. For a prime , define the normalized valuation with valuation ring , and similarly define .

Building global quadratic orders

Using discriminants, we combine local (embedded) quadratic orders to construct a candidate global quadratic order which we may try to embed in . Recall that free quadratic -orders are, via the discriminant, in bijection with elements that are squares in .

Lemma 5.1.

Suppose that is basic for all . Then there exist infinitely many , corresponding to integrally closed quadratic -orders (up to isomorphism), such that embeds in .

Proof.

For each , let be an integrally closed quadratic -order in and let . For each , let . If , then by maximality of . Define

By the Chebotarev density theorem applied to the Hilbert class field of , there exist infinitely many prime ideals such that and is principal. Let

and let

By the Chinese Remainder Theorem, there is an element such that for each . By the Chebotarev density theorem applied to the ray class field of of conductor , there exist infinitely many prime elements such that .

Define , so . Then for , we have , where . Because , the element is a square in . Let be the (free) quadratic -order of discriminant . Then for , which is integrally closed. For , we have , and is integrally closed because . Therefore, is integrally closed for each prime , so is integrally closed. Since there were infinitely many choices for primes and , the same is true for .

Selectivity conditions

We must now show that we can choose in Lemma 5.1 such that . To reach this conclusion, we now invoke the hypothesis that is -indefinite, so that strong approximation Reference 17, Chapter 28 applies.

Lemma 5.4.

Suppose that is -indefinite. Then for all but finitely many integrally closed quadratic -orders we have if and only if for all primes of .

Proof.

Let be the set of integrally closed quadratic orders (up to isomorphism) such that for all . We refer to Voight Reference 17, Main Theorem 31.1.7: under the hypothesis that is -indefinite, there exists a finite extension with the property that embeds in whenever is not a subfield of . As there are only finitely many subfields , only finitely many will not embed in .

Lemma 5.5.

Suppose that is -indefinite, and suppose is basic for every prime of . Then contains infinitely many nonisomorphic integrally closed quadratic -orders.

Proof.

Suppose that is basic for every prime of . Then, contains a maximal commutative -order for every prime . By Lemma 5.1, there exist infinitely many such that the corresponding quadratic order is integrally closed and for all . For all but finitely many such choices of , we have an embedding .

Proof of theorem

With these lemmas in hand, we now prove Theorem 1.2 under the hypothesis that is -indefinite and .

Proof of Theorem 1.2, is -indefinite and .

First, part (a). If is basic for every prime of , then contains an integrally closed quadratic -order by Lemma 5.5. Conversely, if is basic, then it contains a maximal commutative -order . Then, the localization at every prime is a maximal -order in by the local-global dictionary for lattices, so is basic for every prime of . Being Bass is a local property, and local orders are basic if and only if they are Bass by Theorem 3.12, so (b) follows from (a).

This proof gives in fact a bit more.

Corollary 5.6.

Suppose that is -indefinite. If is basic, then contains infinitely many nonisomorphic integrally closed quadratic -orders.

Proof.

Combine Theorem 1.2(a) with Lemma 5.5.

6. Basic orders and definite ternary theta series

In this section, we finish the proof of Theorem 1.2 in the remaining case of a -definite quaternion algebra under some hypotheses. For this purpose, we replace the application of strong approximation with a statement on representations of ternary quadratic forms.

Ternary representations

As above, let be a global field, let be a nonempty finite set of places of containing the archimedean places, and let be the ring of -integers in . For nonzero , we write for the absolute norm of .

Conjecture 6.1 (Ternary representation).

Let be a nondegenerate ternary quadratic form over such that is anisotropic for all . Then there exists such that every squarefree with is represented by if and only if is represented by the completion for all places of .

For further reading, see Schulze–Pillot Reference 15 and Cogdell Reference 6. We now present results in the cases where the conjecture holds.

Theorem 6.2 (Blomer–Harcos).

When is a number field, the ternary representation conjecture (Conjecture 6.1) holds for the set of archimedean places with an ineffective constant .

Proof.

This is almost the statement given by Blomer–Harcos Reference 3, Corollary 2, but where it is assumed that is positive definite: we recover the result for definite by multiplying by two different prime elements with appropriate signs.

Remark 6.3.

Using Theorem 6.2, one can show that Conjecture 6.1 holds for all (finite sets) , but we do not need this result in what follows.

In the case where is a function field, we know of the following partial result.

Theorem 6.4 (Altuǧ–Tsimerman, Reference 1, Corollary 1.1).

The ternary representation conjecture holds with an effective constant when and and .

Discriminants

Define the discriminant quadratic form on by

We define similarly for each prime .

Lemma 6.6.

Let be prime with an integrally closed quadratic order. Let be such that . Then there exists a submodule such that

(i)

for all ;

(ii)

for ; and

(iii)

.

Proof.

First, we have that contains , which is necessarily integrally closed. Then, is an -module. Moreover, is principal. Define . For any , we have . Then, we define . Since for all , we have , and . Also, , so in particular, we have .

Lemma 6.7.

Suppose is basic for all primes . Then there exists an -lattice , a totally negative , and for every prime elements such that is integrally closed and the following conditions hold:

(i)

is a positive definite quadratic form;

(ii)

is squarefree for every prime ; and

(iii)

for all but finitely many .

Proof.

For , let be such that is minimal and let be the largest integer such that . Similarly, for , let be as in Lemma 6.6 with for all . Define

By the Chebotarev density theorem applied to the narrow class field, there exists a prime such that is principal and is totally negative. Since , we have , so there exists with . Let be a uniformizer for , define to be the -suborder with basis

all of whose discriminants are divisible by . Define . Then . We also have that for all since .

For the remaining primes , let be such that is minimal and let . Define

By construction we have for all . Checking locally we have for all . We also have that for all and for all . Now, is positive definite (because was negative definite and was totally negative), so (i) holds.

To conclude, we check (ii) and (iii). Let for a prime . If , then so by construction (we removed the square part). If , by construction . Otherwise, since is basic and , we have . In particular, for all but finitely many , so (iii) holds.

We give a final lemma before proving the theorem.

Lemma 6.8.

Suppose is -definite and that Conjecture 6.1 holds over . Let an -order such that is basic for every prime of . Then contains infinitely many nonisomorphic integrally closed free quadratic -orders.

Proof.

By Lemma 6.7, we obtain the following: an -lattice , a totally negative , and for every prime elements such that is integrally closed and the conditions (i)–(iii) hold.

For each , let and . Define . Note that if then , so . Therefore, .

By the Chebotarev density theorem applied to the narrow Hilbert class field, there exists a prime such that is principal and is totally negative. In particular, . Define as in Equation 5.2 and as in Equation 5.3. Applying the Chebotarev density theorem again, this time to the ray class field with conductor , there exist totally positive prime elements with arbitrarily large absolute norm such that for all . Let . Then is totally positive and squarefree by construction, and there are infinitely many such choices.

Let be such a discriminant. We claim that is locally represented by . Indeed, we have for all by construction. For , we have for some , so . For , we have , so , so is surjective; in particular represents .

Therefore is locally represented by . Therefore, if the conclusion of Conjecture 6.1 holds, taking to be of sufficiently large norm, there is an element with , so .

Finally, let . For , we have that is maximal in its field of fractions by construction. For , we have , so again is maximal in its field of fractions. Therefore, is maximal in its field of fractions and so is basic.

We now prove Theorem 1.2 in the definite case for the ring of integers of a number field.

Proof of Theorem 1.2, definite, the ring of integers of a number field.

For part (a), if is basic for every prime of , then contains an integrally closed quadratic -order by Lemma 6.8 using Theorem 6.2. The converse is exactly as in the proof of Theorem 1.2 in the indefinite case, as given in Section 5.

Being Bass is a local property, and local orders are basic if and only if they are Bass by Theorem 3.12, so (b) follows from (a).

Corollary 6.9.

Suppose that is -definite and let be an -order. If is basic, then contains infinitely many nonisomorphic integrally closed quadratic -orders.

Proof.

Combine Theorem 1.2 in the definite case with Lemma 6.8.

7. Localizations

We conclude the proof of Theorem 1.2 by deducing the basic property of an order over a Dedekind domain from that of its localizations. Throughout, let be a Dedekind domain with and let be a quaternion -order.

Lemma 7.1.

Let be Dedekind domains such that is a global field. Let be an -order. Then there is an -order such that and

for every prime of with ,

is a maximal order for every prime of with .

In particular, if Bass, then is Bass.

Proof.

Since and are necessarily overrings of a global ring, their class groups are finite. It follows that there exists a multiplicative set such that Reference 9, Theorem 5.5. Let , …,  be generators for the -module . There exists (a common denominator) such that

This implies . Thus , …,  generate an -order with . In particular, for every prime of with . Let be the set of prime ideals of with for which is not maximal. Since any has , the set is finite. By the local–global dictionary for lattices, there exists an -order with such that for all and is maximal for . Since for all primes of with , we still have .

Since being Bass is a local property, and at all of we have either maximal or equal to , the order is Bass.

Lemma 7.3.

Suppose is a global field, and let be the (nonempty) set of places of such that . Suppose . If is Bass, there exist infinitely many nonisomorphic maximal quadratic -orders that embed into .

Proof.

Since is infinite, there exists a place such that is unramified. Let be a finite set of places containing and all archimedean places of . By Lemma 7.1 there exists an -order such that and is Bass. Thus is locally Bass and hence locally basic by Theorem 1.1. Since is -indefinite, Lemma 5.5 implies that there are infinitely many nonisomorphic maximal quadratic -orders , with each a maximal, quadratic -order that embeds in . Thus there are infinitely many nonisomorphic such orders .

Lemma 7.4.

Let be Dedekind domains with a global field. Suppose that every -order that is Bass is basic. Then every -order that is Bass is basic.

Proof.

As in Lemma 7.3.

Theorem 7.5.

Suppose that is a global field. Let be the nonempty (possibly infinite) set of places such that . Let be an -order. Suppose that one of the following conditions holds:

(i)

is a number field;

(ii)

and is -indefinite; or

(iii)

.

Then the following statements hold.

(a)

is basic if and only if the localization is basic for all primes of .

(b)

is Bass if and only if is basic.

Proof.

Being Bass is a local property, and local orders are basic if and only if they are Bass by Theorem 3.12. Thus it suffices to show (b). Basic orders are Bass by Proposition 3.4, and we are left to show that an -order that is Bass is basic.

Suppose first that . If is -indefinite, the claim follows from Theorem 1.2 in the indefinite case, as proved in Section 5. Suppose that is a number field and is -definite. Let be the set of all archimedean places of . Then is the ring of integers of , and the claim holds by the proof of Theorem 1.2 for the definite case in Section 6 together with Theorem 6.2. Lemma 7.4 shows that the result also holds for .

Finally, if , apply Lemma 7.3.

Proof of Theorem 1.2.

Restrict Theorem 7.5 to the case is a number field.

Corollary 7.6.

If one of the conditions in Theorem 7.5(i)(iii) holds, and is basic, then contains infinitely many nonisomorphic integrally closed free quadratic -orders.

Proof.

Corollaries 5.6 and 6.9 for and Lemma 7.3 for .

Acknowledgments

The authors would like to thank Wai Kiu Chan and Tom Shemanske for helpful conversations.

Mathematical Fragments

Theorem 1.1.

Let be a discrete valuation ring (DVR) and let be a quaternion -order. Then is Bass if and only if is basic.

Theorem 1.2.

Let be a Dedekind domain whose field of fractions is a number field, and let be a quaternion -order. The following statements hold.

(a)

is basic if and only if the localization is basic for all primes of .

(b)

is Bass if and only if is basic.

Corollary 1.3.

Let be a DVR with maximal ideal , and let be a quaternion -order. Then the following are equivalent.

(i)

is a Bass order;

(ii)

and the radical idealizer are Gorenstein;

(iii)

The Jacobson radical is generated by two elements (as left, respectively, right ideal);

(iv)

is a basic order;

(v)

Every -ideal is generated by two elements;

(vi)

Every -lattice is isomorphic to a direct sum of -ideals; and

(vii)

is not of the form with an integral -lattice.

Proposition 2.4 (Gross–Lucianovic Reference 10).

There is a discriminant-preserving bijection between quaternion -orders up to isomorphism and nondegenerate ternary quadratic forms over up to similarity. Moreover, an -order is Gorenstein if and only if the corresponding quadratic form is primitive.

Equation (2.6)
Equation (2.7)
Lemma 3.2.

The order is not basic if and only if for every there exists such that and .

Lemma 3.3.

A semi-order is not basic if and only if it is of the form with an integral -lattice.

Proposition 3.4.

If is basic, then is Bass.

Lemma 3.5.

If is maximal, residually inert, or residually split, then is basic and Bass.

Lemma 3.7.

Suppose is Gorenstein with associated quadratic form in a good basis as in Equation 2.7 and that is not basic. Then the following statements hold.

(a)

If in , then .

(b)

Suppose that . Let and suppose . Then .

Lemma 3.9.

Suppose is Gorenstein, not basic, and residually ramified. Then there exists a good basis of such that the associated quadratic form is given by

with and and one of the following conditions holds:

(i)

and ; or

(ii)

and and .

Equation (3.11)
Theorem 3.12.

The order is Bass if and only if is basic.

Lemma 4.1.

Suppose that is a local ring. Let . Then the following statements hold.

(a)

We have and .

(b)

If is not basic, then and .

Lemma 4.2.

Let be a local Artinian -algebra with via . If , …,  generate as ideal of , then they generate as -algebra.

Theorem 4.3.

Let be a residually ramified Bass -order and suppose that is generated by two elements as left [right] ideal. Then is basic.

Proposition 4.5.

If and are Gorenstein -orders, then is generated by two elements (as a left, respectively, right -ideal).

Lemma 5.1.

Suppose that is basic for all . Then there exist infinitely many , corresponding to integrally closed quadratic -orders (up to isomorphism), such that embeds in .

Equation (5.2)
Equation (5.3)
Lemma 5.5.

Suppose that is -indefinite, and suppose is basic for every prime of . Then contains infinitely many nonisomorphic integrally closed quadratic -orders.

Corollary 5.6.

Suppose that is -indefinite. If is basic, then contains infinitely many nonisomorphic integrally closed quadratic -orders.

Conjecture 6.1 (Ternary representation).

Let be a nondegenerate ternary quadratic form over such that is anisotropic for all . Then there exists such that every squarefree with is represented by if and only if is represented by the completion for all places of .

Theorem 6.2 (Blomer–Harcos).

When is a number field, the ternary representation conjecture (Conjecture 6.1) holds for the set of archimedean places with an ineffective constant .

Lemma 6.6.

Let be prime with an integrally closed quadratic order. Let be such that . Then there exists a submodule such that

(i)

for all ;

(ii)

for ; and

(iii)

.

Lemma 6.7.

Suppose is basic for all primes . Then there exists an -lattice , a totally negative , and for every prime elements such that is integrally closed and the following conditions hold:

(i)

is a positive definite quadratic form;

(ii)

is squarefree for every prime ; and

(iii)

for all but finitely many .

Lemma 6.8.

Suppose is -definite and that Conjecture 6.1 holds over . Let an -order such that is basic for every prime of . Then contains infinitely many nonisomorphic integrally closed free quadratic -orders.

Corollary 6.9.

Suppose that is -definite and let be an -order. If is basic, then contains infinitely many nonisomorphic integrally closed quadratic -orders.

Lemma 7.1.

Let be Dedekind domains such that is a global field. Let be an -order. Then there is an -order such that and

for every prime of with ,

is a maximal order for every prime of with .

In particular, if Bass, then is Bass.

Lemma 7.3.

Suppose is a global field, and let be the (nonempty) set of places of such that . Suppose . If is Bass, there exist infinitely many nonisomorphic maximal quadratic -orders that embed into .

Lemma 7.4.

Let be Dedekind domains with a global field. Suppose that every -order that is Bass is basic. Then every -order that is Bass is basic.

Theorem 7.5.

Suppose that is a global field. Let be the nonempty (possibly infinite) set of places such that . Let be an -order. Suppose that one of the following conditions holds:

(i)

is a number field;

(ii)

and is -indefinite; or

(iii)

.

Then the following statements hold.

(a)

is basic if and only if the localization is basic for all primes of .

(b)

is Bass if and only if is basic.

Corollary 7.6.

If one of the conditions in Theorem 7.5(i)(iii) holds, and is basic, then contains infinitely many nonisomorphic integrally closed free quadratic -orders.

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Article Information

Author Information
Sara Chari
Department of Mathematics, Dartmouth College, 6188 Kemeny Hall, Hanover, New Hampshire 03755
schari0301@gmail.com
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Daniel Smertnig
Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, N2L 3G1 Canada
dsmertni@uwaterloo.ca
Homepage
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John Voight
Department of Mathematics, Dartmouth College, 6188 Kemeny Hall, Hanover, New Hampshire 03755
jvoight@gmail.com
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Additional Notes

The second author was supported by the Austrian Science Fund (FWF) project J4079-N32.

The third author was supported by an NSF CAREER Award (DMS-1151047) and a Simons Collaboration Grant (550029).

Communicated by
Benjamin Brubaker
Journal Information
Proceedings of the American Mathematical Society, Series B, Volume 8, Issue 2, ISSN 2330-1511, published by the American Mathematical Society, Providence, Rhode Island.
Publication History
This article was received on , revised on , and published on .
Copyright Information
Copyright 2021 by the authors under Creative Commons Attribution 3.0 License (CC BY 3.0)
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