A quaternion order $\mathcal{O}$ over a Dedekind domain $R$ is Bass if every $R$-superorder is Gorenstein, and $\mathcal{O}$ is basic if it contains an integrally closed quadratic $R$-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 $R$ be a Dedekind domain and let $F$ be its field of fractions. Let $B$ be a quaternion algebra over $F$, and let $\mathcal{O} \subseteq B$ be an $R$-order. We say that $\mathcal{O}$ is Gorenstein if its codifferent is an invertible $R$-lattice in $B$, or equivalently $\operatorname {Hom}_R(\mathcal{O},R)$ is projective as a left or right $\mathcal{O}$-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 $\mathcal{O}$ is Bass if every $R$-superorder$\mathcal{O}' \supseteq \mathcal{O}$ in $B$ 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 $R$, because invertibility is so.
On the other hand, we say that $\mathcal{O}$ is basic if there is a (commutative) quadratic $R$-algebra$S \subseteq \mathcal{O}$ such that $S$ is integrally closed in its total quotient ring $FS$. Basic orders were first introduced by Eichler Reference 8 over $R=\mathbb{Z}$ (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 $R$-algebra$S$ allows one to work explicitly with them, since a basic order $\mathcal{O}$ is locally free over $S$ of rank $2$: 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 $R \times R$, 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 was proven by Brzezinski Reference 5, Proposition 1.11 when $R$ is a complete DVR with $\operatorname {char}R \neq 2$ 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.
In fact, we show that if $\mathcal{O}$ is Bass (equivalently, basic), then $\mathcal{O}$ contains infinitely many nonisomorphic quadratic $R$-algebras$S$ and moreover they can be taken to be free as $R$-modules (Corollary 7.6). Theorem 1.2(b) over $R=\mathbb{Z}$ 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 $R$ 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 $R$ 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.
The implications (v)$\,\Rightarrow \,$(i)$\,\Rightarrow \,$(vi) hold more generally Reference 17, Section 14.5. The implication (vi)$\,\Rightarrow \,$(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)$\,\Rightarrow \,$(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 3–4. 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 $R$ be a Dedekind domain with $\operatorname {Frac}(R)=F$. Let $B$ be a quaternion algebra over $F$ and let $\mathcal{O} \subseteq B$ be an $R$-order.
For more detail and further references, see Voight Reference 17, Sections 24.2, 24.5. Being Gorenstein is a local property—$\mathcal{O}$ is Gorenstein if and only if the localizations $\mathcal{O}_{(\mathfrak{p})} \coloneq \mathcal{O} \otimes _R R_{(\mathfrak{p})}$ are Gorenstein for all primes $\mathfrak{p}$ of $R$—so it follows that Bass is also a local property.
Local properties
Now suppose $R$ is a local Dedekind domain, i.e., $R$ is a discrete valuation ring (DVR) with maximal ideal $\mathfrak{p}$ and residue field $\kappa \coloneq R/\mathfrak{p}$. The Jacobson radical of $\mathcal{O}$ is the intersection of all maximal left (or equivalently right) ideals of $\mathcal{O}$. The semisimple $\kappa$-algebra$\mathcal{O}/\!\operatorname {rad}\mathcal{O}$ is one of the following Reference 17, 24.3.1:
•
$\mathcal{O}/\!\operatorname {rad}\mathcal{O}$ is a quaternion algebra (equivalently, $\mathcal{O}$ is maximal);
•
$\mathcal{O}/\!\operatorname {rad}\mathcal{O} \simeq \kappa \times \kappa$, and we say that $\mathcal{O}$ is residually split (or Eichler);
•
$\mathcal{O}/\!\operatorname {rad}\mathcal{O} \simeq \kappa$, and we say that $\mathcal{O}$ is residually ramified; or
•
$\mathcal{O}/\!\operatorname {rad}\mathcal{O}$ is a separable quadratic field extension of $\kappa$ and we say that $\mathcal{O}$ is residually inert.
The radical idealizer of $\mathcal{O}$ is the left order $\mathcal{O}^\natural \coloneq \mathcal{O}_{\mathsf{L}}(\operatorname {rad}\mathcal{O})$.
We define a similarity of two ternary quadratic forms $Q \colon R^3 \rightarrow R$ and $Q' \colon R^3 \rightarrow R$ to be a pair $(f,u)$, where $f \colon R^3 \rightarrow R^3$ is an $R$-module isomorphism and $u \in R^\times$ is such that $Q'(f(x))=uQ(x)$ for all $x \in \mathcal{O}$.
We now briefly review the construction of the bijection in Proposition 2.4. Since $R$ is a PID, $\mathcal{O}$ is free of rank $4$ as an $\mathcal{O}$-module. A good basis$1,i,j,k$ for an $R$-order$\mathcal{O}$ is an $R$-basis with a multiplication table of the form
with $a,b,c,u,v,w \in R$. Every $R$-basis of $\mathcal{O}$ can be converted to a good basis in a direct manner. For all $x,y,z \in R$ and $\alpha =xi+yj+zk \in \mathcal{O}$, we find
The similarity class of $Q$ is well-defined on the isomorphism class of $\mathcal{O}$. Conversely, given a nondegenerate ternary quadratic form $Q\colon R^3 \to R$, we associate to $Q$ its even Clifford algebra $\mathcal{O}=\operatorname {Clf}^0(Q)$, which is a quaternion $R$-order. A change of good basis of $\mathcal{O}$ induces a corresponding change of basis of $Q$, and conversely every such change of basis of $Q$ arises from a change of good basis of $\mathcal{O}$.
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 $R$ be a DVR with fraction field $F \coloneq \operatorname {Frac}(R)$ and maximal ideal $\mathfrak{p}=\pi R$. For $x,y \in R$, we write $\pi \mid x,y$ for $\pi \mid x$ and $\pi \mid y$.
Let $B$ be a quaternion algebra over $F$ and $\mathcal{O} \subseteq B$ an $R$-order. According to the following remark, we could work equivalently in the completion of $R$.
We choose a good $R$-basis$1,i,j,k$ for $\mathcal{O}$ and let $Q$ be the ternary quadratic form over $R$ associated to $\mathcal{O}$ with respect to this basis, as in Equation 2.7.
A slight reformulation gives a local version of the result of Eichler Reference 8, Satz 8. Recall that a semi-order$I \subseteq B$ is an integral $R$-lattice with $1 \in I$Reference 17, Section 16.6. Basic semi-orders are defined analogously to basic orders.
As an application of Lemma 3.2, we prove one implication in Theorem 1.1.
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 $R$ is a discrete valuation ring with maximal ideal $\mathfrak{p} = \pi R$.
By classification, we see that a quaternion $R$-order$\mathcal{O}$ is a local ring (has a unique maximal left [right] ideal, necessarily equal to its Jacobson radical $\operatorname {rad}\mathcal{O}$) if and only if $\mathcal{O}$ is neither maximal nor residually split.
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.
We now characterize local Bass orders.
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 $R$-order$\mathcal{O}$ is basic if and only if its localization at every nonzero prime $\mathfrak{p}$ of $R$ is basic.
Setup
Moving now from the local to the global setting, we use the following notation. Let $F$ be a global field and let $R=R_{(T)} \subseteq F$ be the ring of $T$-integers for a nonempty finite set $T$ of places of $F$ containing the archimedean places. Let $B$ be a quaternion algebra over $F$, and let $\mathcal{O} \subseteq B$ be an $R$-order. For a prime $\mathfrak{p} \subseteq R$, define the normalized valuation $v_\mathfrak{p}$ with valuation ring $R_{(\mathfrak{p})} \subseteq F$, and similarly define $\mathcal{O}_{(\mathfrak{p})} \coloneq \mathcal{O} \otimes _R R_{(\mathfrak{p})} \subseteq B$.
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 $\mathcal{O}$. Recall that free quadratic $R$-orders are, via the discriminant, in bijection with elements $d \in R/R^{\times 2}$ that are squares in $R/4R$.
Selectivity conditions
We must now show that we can choose $S$ in Lemma 5.1 such that $S \hookrightarrow \mathcal{O}$. To reach this conclusion, we now invoke the hypothesis that $B$ is $T$-indefinite, so that strong approximation Reference 17, Chapter 28 applies.
Proof of theorem
With these lemmas in hand, we now prove Theorem 1.2 under the hypothesis that $B$ is $T$-indefinite and $\# T < \infty$.
This proof gives in fact a bit more.
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 $T$-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 $F$ be a global field, let $T$ be a nonempty finite set of places of $F$ containing the archimedean places, and let $R=R_{(T)} \subseteq F$ be the ring of $T$-integers in $F$. For nonzero $a \in R$, we write $\mathsf{N}(a)\coloneq \#(R/aR)$ for the absolute norm of $a$.
For further reading, see Schulze–Pillot Reference 15 and Cogdell Reference 6. We now present results in the cases where the conjecture holds.
In the case where $F$ is a function field, we know of the following partial result.
Discriminants
Define the discriminant quadratic form on $\mathcal{O}$ by
We define similarly $\operatorname {disc}_\mathfrak{p} \colon \mathcal{O}_{(\mathfrak{p})} \rightarrow R_{(\mathfrak{p})}$ for each prime $\mathfrak{p}$.
We give a final lemma before proving the theorem.
We now prove Theorem 1.2 in the definite case for $R$ the ring of integers of a number field.
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 $R$ be a Dedekind domain with $F=\operatorname {Frac}R$ and let $\mathcal{O}$ be a quaternion $R$-order.
Acknowledgments
The authors would like to thank Wai Kiu Chan and Tom Shemanske for helpful conversations.
Salim Ali Altuğ and Jacob Tsimerman, Metaplectic Ramanujan conjecture over function fields with applications to quadratic forms, Int. Math. Res. Not. IMRN 13 (2014), 3465–3558, DOI 10.1093/imrn/rnt047. MR3229761, Show rawAMSref\bib{AltugTsimerman}{article}{
author={Altu\u {g}, Salim Ali},
author={Tsimerman, Jacob},
title={Metaplectic Ramanujan conjecture over function fields with applications to quadratic forms},
journal={Int. Math. Res. Not. IMRN},
date={2014},
number={13},
pages={3465--3558},
issn={1073-7928},
review={\MR {3229761}},
doi={10.1093/imrn/rnt047},
}
Reference [2]
Hyman Bass, On the ubiquity of Gorenstein rings, Math. Z. 82 (1963), 8–28, DOI 10.1007/BF01112819. MR153708, Show rawAMSref\bib{Bass:ubiquity}{article}{
author={Bass, Hyman},
title={On the ubiquity of Gorenstein rings},
journal={Math. Z.},
volume={82},
date={1963},
pages={8--28},
issn={0025-5874},
review={\MR {153708}},
doi={10.1007/BF01112819},
}
Reference [3]
Valentin Blomer and Gergely Harcos, Twisted $L$-functions over number fields and Hilbert’s eleventh problem, Geom. Funct. Anal. 20 (2010), no. 1, 1–52, DOI 10.1007/s00039-010-0063-x. MR2647133, Show rawAMSref\bib{BlomerHarcos}{article}{
author={Blomer, Valentin},
author={Harcos, Gergely},
title={Twisted $L$-functions over number fields and Hilbert's eleventh problem},
journal={Geom. Funct. Anal.},
volume={20},
date={2010},
number={1},
pages={1--52},
issn={1016-443X},
review={\MR {2647133}},
doi={10.1007/s00039-010-0063-x},
}
Reference [4]
Juliusz Brzeziński, A characterization of Gorenstein orders in quaternion algebras, Math. Scand. 50 (1982), no. 1, 19–24, DOI 10.7146/math.scand.a-11940. MR664504, Show rawAMSref\bib{Brzezinski:Gororder}{article}{
author={Brzezi\'{n}ski, Juliusz},
title={A characterization of Gorenstein orders in quaternion algebras},
journal={Math. Scand.},
volume={50},
date={1982},
number={1},
pages={19--24},
issn={0025-5521},
review={\MR {664504}},
doi={10.7146/math.scand.a-11940},
}
Reference [5]
J. Brzezinski, On automorphisms of quaternion orders, J. Reine Angew. Math. 403 (1990), 166–186, DOI 10.1515/crll.1990.403.166. MR1030414, Show rawAMSref\bib{Brzezinski:onaut}{article}{
author={Brzezinski, J.},
title={On automorphisms of quaternion orders},
journal={J. Reine Angew. Math.},
volume={403},
date={1990},
pages={166--186},
issn={0075-4102},
review={\MR {1030414}},
doi={10.1515/crll.1990.403.166},
}
Reference [6]
James W. Cogdell, On sums of three squares(English, with English and French summaries), J. Théor. Nombres Bordeaux 15 (2003), no. 1, 33–44. Les XXIIèmes Journées Arithmetiques (Lille, 2001). MR2018999, Show rawAMSref\bib{Cogdell}{article}{
author={Cogdell, James W.},
title={On sums of three squares},
language={English, with English and French summaries},
note={Les XXII\`emes Journ\'{e}es Arithmetiques (Lille, 2001)},
journal={J. Th\'{e}or. Nombres Bordeaux},
volume={15},
date={2003},
number={1},
pages={33--44},
issn={1246-7405},
review={\MR {2018999}},
}
Reference [7]
Ju. A. Drozd, V. V. Kiričenko, and A. V. Roĭter, Hereditary and Bass orders(Russian), Izv. Akad. Nauk SSSR Ser. Mat. 31 (1967), 1415–1436. MR0219527, Show rawAMSref\bib{DKR}{article}{
author={Drozd, Ju. A.},
author={Kiri\v {c}enko, V. V.},
author={Ro\u {\i }ter, A. V.},
title={Hereditary and Bass orders},
language={Russian},
journal={Izv. Akad. Nauk SSSR Ser. Mat.},
volume={31},
date={1967},
pages={1415--1436},
issn={0373-2436},
review={\MR {0219527}},
}
Reference [8]
Martin Eichler, Untersuchungen in der Zahlentheorie der rationalen Quaternionenalgebren(German), J. Reine Angew. Math. 174 (1936), 129–159, DOI 10.1515/crll.1936.174.129. MR1581481, Show rawAMSref\bib{Eichler:untersuch}{article}{
author={Eichler, Martin},
title={Untersuchungen in der Zahlentheorie der rationalen Quaternionenalgebren},
language={German},
journal={J. Reine Angew. Math.},
volume={174},
date={1936},
pages={129--159},
issn={0075-4102},
review={\MR {1581481}},
doi={10.1515/crll.1936.174.129},
}
Reference [9]
Stefania Gabelli, Generalized Dedekind domains, Multiplicative ideal theory in commutative algebra, Springer, New York, 2006, pp. 189–206, DOI 10.1007/978-0-387-36717-0_12. MR2265809, Show rawAMSref\bib{Gabelli:dedekind}{article}{
author={Gabelli, Stefania},
title={Generalized Dedekind domains},
conference={ title={Multiplicative ideal theory in commutative algebra}, },
book={ publisher={Springer, New York}, },
date={2006},
pages={189--206},
review={\MR {2265809}},
doi={10.1007/978-0-387-36717-0\_12},
}
Reference [10]
Benedict H. Gross and Mark W. Lucianovic, On cubic rings and quaternion rings, J. Number Theory 129 (2009), no. 6, 1468–1478, DOI 10.1016/j.jnt.2008.06.003. MR2521487, Show rawAMSref\bib{GL:quatrings}{article}{
author={Gross, Benedict H.},
author={Lucianovic, Mark W.},
title={On cubic rings and quaternion rings},
journal={J. Number Theory},
volume={129},
date={2009},
number={6},
pages={1468--1478},
issn={0022-314X},
review={\MR {2521487}},
doi={10.1016/j.jnt.2008.06.003},
}
Reference [11]
Hiroaki Hijikata, Arnold K. Pizer, and Thomas R. Shemanske, The basis problem for modular forms on $\Gamma _0(N)$, Mem. Amer. Math. Soc. 82 (1989), no. 418, vi+159, DOI 10.1090/memo/0418. MR960090, Show rawAMSref\bib{HPS}{article}{
author={Hijikata, Hiroaki},
author={Pizer, Arnold K.},
author={Shemanske, Thomas R.},
title={The basis problem for modular forms on $\Gamma _0(N)$},
journal={Mem. Amer. Math. Soc.},
volume={82},
date={1989},
number={418},
pages={vi+159},
issn={0065-9266},
review={\MR {960090}},
doi={10.1090/memo/0418},
}
Reference [12]
H. Hijikata, A. Pizer, and T. Shemanske, Orders in quaternion algebras, J. Reine Angew. Math. 394 (1989), 59–106. MR977435, Show rawAMSref\bib{HPS:orders}{article}{
author={Hijikata, H.},
author={Pizer, A.},
author={Shemanske, T.},
title={Orders in quaternion algebras},
journal={J. Reine Angew. Math.},
volume={394},
date={1989},
pages={59--106},
issn={0075-4102},
review={\MR {977435}},
}
Reference [13]
Sungtae Jun, On the certain primitive orders, J. Korean Math. Soc. 34 (1997), no. 4, 791–807. MR1485952, Show rawAMSref\bib{Jun}{article}{
author={Jun, Sungtae},
title={On the certain primitive orders},
journal={J. Korean Math. Soc.},
volume={34},
date={1997},
number={4},
pages={791--807},
issn={0304-9914},
review={\MR {1485952}},
}
Reference [14]
Arnold Pizer, The action of the canonical involution on modular forms of weight $2$ on $\Gamma _{0}(M)$, Math. Ann. 226 (1977), no. 2, 99–116, DOI 10.1007/BF01360861. MR437463, Show rawAMSref\bib{Pizer}{article}{
author={Pizer, Arnold},
title={The action of the canonical involution on modular forms of weight $2$ on $\Gamma _{0}(M)$},
journal={Math. Ann.},
volume={226},
date={1977},
number={2},
pages={99--116},
issn={0025-5831},
review={\MR {437463}},
doi={10.1007/BF01360861},
}
Reference [15]
Rainer Schulze-Pillot, Representation by integral quadratic forms—a survey, Algebraic and arithmetic theory of quadratic forms, Contemp. Math., vol. 344, Amer. Math. Soc., Providence, RI, 2004, pp. 303–321, DOI 10.1090/conm/344/06226. MR2060206, Show rawAMSref\bib{SP}{article}{
author={Schulze-Pillot, Rainer},
title={Representation by integral quadratic forms---a survey},
conference={ title={Algebraic and arithmetic theory of quadratic forms}, },
book={ series={Contemp. Math.}, volume={344}, publisher={Amer. Math. Soc., Providence, RI}, },
date={2004},
pages={303--321},
review={\MR {2060206}},
doi={10.1090/conm/344/06226},
}
Reference [16]
John Voight, Identifying the matrix ring: algorithms for quaternion algebras and quadratic forms, Quadratic and higher degree forms, Dev. Math., vol. 31, Springer, New York, 2013, pp. 255–298, DOI 10.1007/978-1-4614-7488-3_10. MR3156561, Show rawAMSref\bib{Voight:idenmat}{article}{
author={Voight, John},
title={Identifying the matrix ring: algorithms for quaternion algebras and quadratic forms},
conference={ title={Quadratic and higher degree forms}, },
book={ series={Dev. Math.}, volume={31}, publisher={Springer, New York}, },
date={2013},
pages={255--298},
review={\MR {3156561}},
doi={10.1007/978-1-4614-7488-3\_10},
}
Reference [17]
John Voight, Quaternion algebras, Grad. Texts. in Math., vol. 288, Springer-Verlag, New York, 2020.
Article Information
Author Information
Sara Chari
Department of Mathematics, Dartmouth College, 6188 Kemeny Hall, Hanover, New Hampshire 03755
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