On Superspecial abelian surfaces over finite fields III

Jiangwei Xue (Xue) Collaborative Innovation Center of Mathematics, School of Mathematics and Statistics, Wuhan University, Luojiashan, 430072, Wuhan, Hubei, P.R. China (Xue) Hubei Key Laboratory of Computational Science (Wuhan University), Wuhan, Hubei, 430072, P.R. China. xue_j@whu.edu.cn , Chia-Fu Yu (Yu) Institute of Mathematics, Academia Sinica and NCTS, Astronomy-Mathematics Building, No. 1, Sec. 4, Roosevelt Road, Taipei 10617, TAIWAN. chiafu@math.sinica.edu.tw and Yuqiang Zheng (Zheng) School of Mathematics and Statistics, Wuhan University, Luojiashan, 430072, Wuhan, Hubei, P.R. China zhhhhxhq@whu.edu.cn (Zheng) Academy of Mathematics and Systems Science, Chinese Academy of Science, No. 55, Zhongguancun East Road, Beijing 100190, China zhengyq@amss.ac.cn

Abstract.

In the paper [On superspecial abelian surfaces over finite fields II. J. Math. Soc. Japan, 72(1):303–331, 2020], Tse-Chung Yang and the first two current authors computed explicitly the number $\lvert\mathrm{SSp}_{2}(\mathbb{F}_{q})\rvert$ of isomorphism classes of superspecial abelian surfaces over an arbitrary finite field $\mathbb{F}_{q}$ of even degree over the prime field $\mathbb{F}_{p}$ . There it was assumed that certain commutative $\mathbb{Z}_{p}$ -orders satisfy an étale condition that excludes the primes $p=2,3,5$ . We treat these remaining primes in the present paper, where the computations are more involved because of the ramifications. This completes the calculation of $\lvert\mathrm{SSp}_{2}(\mathbb{F}_{q})\rvert$ in the even degree case. The odd degree case was previous treated by Tse-Chung Yang and the first two current authors in [On superspecial abelian surfaces over finite fields.Doc. Math., 21:1607–1643, 2016]. Along the proof of our main theorem, we give the classification of lattices over local quaternion Bass orders, which is a new input to our previous works.

Key words and phrases:

superspecial abelian surfaces, quaternion algebra, Bass order, conjugacy classes of arithmetic subgroups

2020 Mathematics Subject Classification:

11R52, 11G10

1. Introduction

Throughout this paper, $p$ denotes a prime number, $q=p^{a}$ a power of $p$ , and $\mathbb{F}_{q}$ the finite field of $q$ -elements. We reserve ${\mathbb{N}}$ for the set of strictly positive integers. Let $k$ be a field of characteristic $p$ , and $\bar{k}$ an algebraic closure of $k$ . An abelian variety over $k$ is said to be supersingular if it is isogenous to a product of supersingular elliptic curves over $\bar{k}$ ; it is said to be superspecial if it is isomorphic to a product of supersingular elliptic curves over $\bar{k}$ . For any $d\in{\mathbb{N}}$ , denote by $\operatorname{SSp}_{d}(\mathbb{F}_{q})$ the set of $\mathbb{F}_{q}$ -isomorphism classes of $d$ -dimensional superspecial abelian varieties over $\mathbb{F}_{q}$ . The classification of supersingular elliptic curves (namely, the $d=1$ case) over finite fields were carried out by Deuring [8, 7], Eichler [10], Igusa[11], Waterhouse [24] and many others since the 1930s.

In a series of papers [26, 28, 27, 25], Tse-Chung Yang and the first two current authors attempt to calculate the cardinality $\lvert\operatorname{SSp}_{d}(\mathbb{F}_{q})\rvert$ explicitly in the case $d=2$ . More precisely, it is shown in [26] that for every fixed $d>1$ , $\lvert\operatorname{SSp}_{d}(\mathbb{F}_{q})\rvert$ depends only on the parity of the degree $a=[\mathbb{F}_{q}:\mathbb{F}_{p}]$ , and an explicit formula of $\lvert\operatorname{SSp}_{2}(\mathbb{F}_{q})\rvert$ is provided for the odd degree case. The most involving part of this explicit calculation is carried out prior in [27, 25], which counts the number of isomorphism classes of abelian surfaces over $\mathbb{F}_{p}$ within the simple isogeny class corresponding to the Weil $p$ -numbers $\pm\,\sqrt[]{p}\,$ . For the even degree case, an explicit formula of $\lvert\operatorname{SSp}_{2}(\mathbb{F}_{q})\rvert$ is obtained in [28] under a mild condition on $p$ (see Remark 3.7 of loc. cit.), which holds for all $p\geq 7$ . We treat the remaining primes $p\in\{2,3,5\}$ in the present paper, thus completing the calculation of $\lvert\operatorname{SSp}_{2}(\mathbb{F}_{q})\rvert$ in the even degree case.

For the rest of the paper, we assume that $q=p^{a}$ is an even power of $p$ . All isogenies and isomorphisms are over the base field $\mathbb{F}_{q}$ unless specified otherwise. The set $\operatorname{SSp}_{2}(\mathbb{F}_{q})$ naturally partitions into subsets by isogeny equivalence, which can be parametrized by (multiple) Weil numbers (see [26, §4.1]). For each integer $n\in{\mathbb{N}}$ , let $\zeta_{n}$ be a primitive $n$ -th root of unity, and $\pi_{n}$ be the Weil $q$ -number $(-p)^{a/2}\zeta_{n}$ . By the Honda-Tate theorem, there is a unique simple abelian variety $X_{n}/\mathbb{F}_{q}$ up to isogeny corresponding to the $\operatorname{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$ -conjugacy class of $\pi_{n}$ . Moreover, the $X_{n}$ ’s are mutually non-isogenous for distinct $n$ . Thanks to the Manin-Oort Theorem [29, Theorem 2.9], a simple abelian variety over $\mathbb{F}_{q}$ is supersingular if and only if it is isogenous to $X_{n}$ for some $n$ . Let $d(n)$ be the dimension of $X_{n}$ . The formula for $d(n)$ is given in [26, §3]. When $p\in\{2,3,5\}$ , we have

•

$d(n)=1$ if and only if $n\in\{1,2,3,6\}$ or $(n,p)\in\{(4,2),(4,3)\}$ ;
•

$d(n)=2$ if and only if $(n,p)=(4,5)$ or $n\in\{5,8,10,12\}$ .

Given a superspecial abelian surface $X/\mathbb{F}_{q}$ , we have two cases to consider:

(I)

the isotypic case where $X$ is isogenous to $X_{n}^{2/d(n)}$ for some $n\in{\mathbb{N}}$ with $d(n)\leq 2$ ;
(II)

the non-isotypic case where $X$ is isogenous to $X_{\underline{n}}:=X_{n_{1}}\times X_{n_{2}}$ for a pair ${\underline{n}}=(n_{1},n_{2})\in{\mathbb{N}}^{2}$ with $n_{1}<n_{2}$ and $d(n_{1})=d(n_{2})=1$ .

Let $o(n)$ (resp. $o({\underline{n}})$ ) denote the number of isomorphism classes of superspecial abelian surfaces over $\mathbb{F}_{q}$ that are isogenous to $X_{n}^{2/d(n)}$ (resp. $X_{\underline{n}}$ ). It was shown in [28, §3.2] that

(1.1)		$\displaystyle o(1)=o(2)=1,\ o(3)=o(6),\ o(5)=o(10);$
(1.2)		$\displaystyle o(1,3)=o(2,6),\ o(1,4)=o(2,4),\ o(1,6)=o(2,3),\ o(3,4)=o(4,6).$

Thus we have

(1.3)

\begin{split}\lvert\operatorname{SSp}_{2}(\mathbb{F}_{q})\rvert=&2+2o(3)+o(4)+2o(5)+o(8)+o(12)\\ &+o(1,2)+2o(2,3)+2o(2,4)+2o(2,6)+2o(3,4)+o(3,6).\end{split}

As mentioned before, the value of each $o(n)$ or $o({\underline{n}})$ in (1.3) has been worked out in [28] conditionally on $p$ . To make explicit this condition, we uniformize the notation. For each $r\in{\mathbb{N}}$ , let us denote

\breve{\mathbb{N}}^{r}:=\{{\underline{n}}=(n_{1},\cdots,n_{r})\in{\mathbb{N}}^{r}\mid 0<n_{1}<\cdots<n_{r}\}.

In particular, if $r=1$ , then $\breve{\mathbb{N}}^{r}={\mathbb{N}}$ and we drop the underline from ${\underline{n}}$ . For each ${\underline{n}}\in\breve{\mathbb{N}}^{r}$ with $r$ arbitrary, we define

(1.4)

A_{\underline{n}}:=\frac{\mathbb{Z}[T]}{\left(\prod_{i=1}^{r}\Phi_{n_{i}}(T)\right)},\qquad K_{\underline{n}}:=\frac{\mathbb{Q}[T]}{\left(\prod_{i=1}^{r}\Phi_{n_{i}}(T)\right)}\simeq\prod_{i=1}^{r}\mathbb{Q}(\zeta_{n_{i}}),

where $\Phi_{n}(T)\in\mathbb{Z}[T]$ is the $n$ -th cyclotomic polynomial. Clearly, $A_{\underline{n}}$ is a $\mathbb{Z}$ -order in $K_{\underline{n}}$ , so it is contained in the unique maximal order $O_{K_{\underline{n}}}:=\prod_{i=1}^{r}\mathbb{Z}[T]/(\Phi_{n_{i}}(T))$ . Let ${\underline{n}}\in\breve{\mathbb{N}}^{r}$ with $r\in\{1,2\}$ be an $r$ -tuple appearing in (1.3). In the proof of [28, Theorem 3.3], it is assumed that

(1.5)

A_{{\underline{n}},p}:=A_{\underline{n}}\otimes\mathbb{Z}_{p}\quad\text{is an \'{e}tale $\mathbb{Z}_{p}$-order}.

This condition fails precisely in the following two situations:

(C1)

$p$ is ramified in $\mathbb{Q}(\zeta_{n_{i}})$ for some $1\leq i\leq r$ , or
(C2)

$p$ divides the index $[O_{K_{\underline{n}}}:A_{\underline{n}}]$ .

If $r=1$ , then $A_{n}$ coincides with $O_{K_{n}}$ , so (C2) is possible only if $r=2$ . For the reader’s convenience, we list the indices $i({\underline{n}}):=[O_{K_{\underline{n}}}:A_{\underline{n}}]$ from [28, Table 1]:

${\underline{n}}$	$(1,2)$	$(2,3)$	$(2,4)$	$(2,6)$	$(3,4)$	$(3,6)$
$i({\underline{n}})$	$2$	$1$	$2$	$3$	$1$	$4$

Theorem 1.1.

Let ${\underline{n}}\in\breve{\mathbb{N}}^{r}$ be an $r$ -tuple appearing in (1.3), and $p$ be a prime satisfying (C1) or (C2). Then the values of $o({\underline{n}})$ for each $p$ are given by the following table

${\underline{n}}$	$3$	$4$	$5$	$8$	$12$	$(1,2)$	$(2,3)$	$(2,4)$	$(2,6)$	$(3,4)$	$(3,6)$
$p$	$3$	$2$	$5$	$2$	$2,3$	$2$	$3$	$2$	$3$	$2,3$	$2$	$3$
$o({\underline{n}})$	$2$	$2$	$1$	$1$	$3$	$3$	$1$	$2$	$3$	$2$	$8$	$2$

Moreover, the number of isomorphism classes of superspecial abelian surfaces over a finite field $\mathbb{F}_{q}$ of even degree over $\mathbb{F}_{p}$ with $p\in\{2,3,5\}$ is given by

(1.6)

\lvert\operatorname{SSp}_{2}(\mathbb{F}_{q})\rvert=\begin{cases}49&\text{if }p=2,\\ 45&\text{if }p=3,\\ 47&\text{if }p=5.\\ \end{cases}

Remark 1.2.

We provide an arithmetic interpretation of the values $o({\underline{n}})$ . Let $D=D_{p,\infty}$ be the unique quaternion $\mathbb{Q}$ -algebra up to isomorphism ramified precisely at $p$ and $\infty$ , and $\operatorname{Mat}_{2}(D)$ be the algebra of $2\times 2$ matrices over $D$ . Fix a maximal $\mathbb{Z}$ -order ${\mathcal{O}}$ in $D$ . As explained in [28, §1, p. 304], up to isomorphism, the arithmetic group $\operatorname{GL}_{2}({\mathcal{O}})$ depends only on $p$ and not on the choice of ${\mathcal{O}}$ . An element $x\in\operatorname{GL}_{2}({\mathcal{O}})$ of finite group order¹¹1Unfortunately, the word “order” plays double duties in this paragraph: for the order inside an algebra and also for the order of a group element. To make a distinction, we always insert the word “group” when the second meaning applies. is semisimple, so its minimal polynomial over $\mathbb{Q}$ in $\operatorname{Mat}_{2}(D)$ is of the form $P_{{\underline{n}}}(T):=\prod_{i=1}^{r}\Phi_{n_{i}}(T)$ for some ${\underline{n}}=(n_{1},\cdots,n_{r})\in\breve{\mathbb{N}}^{r}$ . It is not hard to show that $r\leq 2$ (see [28, §3.1]). A Galois cohomological argument shows that $o({\underline{n}})$ counts the number of conjugacy classes of elements of $\operatorname{GL}_{2}({\mathcal{O}})$ with minimal polynomial $P_{{\underline{n}}}(T)$ , and $\lvert\operatorname{SSp}_{2}(\mathbb{F}_{q})\rvert$ is equal to the total number of conjugacy classes of elements of finite group order in $\operatorname{GL}_{2}({\mathcal{O}})$ (see [28, Proposition 1.1]). Actually, this arithmetic interpretation works for $\operatorname{GL}_{d}({\mathcal{O}})$ with any $d\geq 2$ , not just for $d=2$ .

The proof of Theorem 1.1 will occupy the remaining part of the paper. In Section 2, we recall from [28, §3.1] the general strategy for computing $o({\underline{n}})$ . The isotypic case (i.e. $r=1$ ) will be treated in Section 3, and the non-isotypic case (i.e. $r=2$ ) will be treated in Section 4.

2. General strategy for computing $o({\underline{n}})$

Keep the notation and the assumptions of the previous section. We recall from [28, §3.1] and [26, §6.4] the general strategy for calculating $o({\underline{n}})$ with ${\underline{n}}\in\breve{\mathbb{N}}^{r}$ for $r\leq 2$ . Based on the arithmetic interpretation of $o({\underline{n}})$ in Remark 1.2, we further provide a lattice description of $o({\underline{n}})$ . Indeed, it is via this lattice description that the value of each $o({\underline{n}})$ is calculated.

Let $V=D^{2}$ be the unique simple left $\operatorname{Mat}_{2}(D)$ -module, which is at the same time a right $D$ -vector space of dimension $2$ . Let $M_{0}:={\mathcal{O}}^{2}$ be the standard right ${\mathcal{O}}$ -lattice in $V$ , whose endomorphism ring $\operatorname{End}_{{\mathcal{O}}}(M_{0})$ is just $\operatorname{Mat}_{2}({\mathcal{O}})$ . For each element $x\in\operatorname{GL}_{2}({\mathcal{O}})$ of finite group order with minimal polynomial $P_{{\underline{n}}}(T)\in\mathbb{Z}[T]$ , there is a canonical embedding $A_{\underline{n}}=\mathbb{Z}[T]/(P_{\underline{n}}(T))\hookrightarrow\operatorname{Mat}_{2}({\mathcal{O}})$ sending $T$ to $x$ . This embedding equips $M_{0}$ with an $(A_{\underline{n}},{\mathcal{O}})$ -bimodule structure, or equivalently, a faithful left $A_{\underline{n}}\otimes_{\mathbb{Z}}{\mathcal{O}}^{\mathrm{opp}}$ -module structure. Similarly, $V$ is equipped with a faithful left $K_{\underline{n}}\otimes_{\mathbb{Q}}D^{\mathrm{opp}}$ -module structure. The canonical involution induces an isomorphism between the opposite ring ${\mathcal{O}}^{\mathrm{opp}}$ and ${\mathcal{O}}$ itself (and similarly between $D^{\mathrm{opp}}$ and $D$ ), so we put

(2.1)

{\mathscr{A}}_{\underline{n}}:=A_{\underline{n}}\otimes_{\mathbb{Z}}{\mathcal{O}},\quad\text{and}\quad{\mathscr{K}}_{\underline{n}}:=K_{\underline{n}}\otimes_{\mathbb{Q}}D.

Clearly, ${\mathscr{A}}_{\underline{n}}$ is a $\mathbb{Z}$ -order in the semisimple $\mathbb{Q}$ -algebra ${\mathscr{K}}_{\underline{n}}$ . It has been shown in [28, p. 309] that the ${\mathscr{K}}_{\underline{n}}$ -module structure on $V$ is uniquely determined by the $r$ -tuple ${\underline{n}}$ . From [26, Theorem 6.11] (see also [28, Lemma 3.1]), the above construction induces a bijection between the following two finite sets:

(2.2)

\left\{\parbox{122.85876pt}{conjugacy classes of elements of $\operatorname{GL}_{2}({\mathcal{O}})$ with minimal polynomial $P_{\underline{n}}(x)$}\right\}\longleftrightarrow\left\{\parbox{86.72377pt}{isomorphism classes of ${\mathscr{A}}_{\underline{n}}$-lattices in the left ${\mathscr{K}}_{\underline{n}}$-module $V$}\right\}

Therefore, we have $o({\underline{n}})=\lvert{\mathscr{L}}({\underline{n}})\rvert$ , where ${\mathscr{L}}({\underline{n}})$ denote the set on the right.

Now fix a pair $({\underline{n}},p)$ in Theorem 1.1 and in turn a left ${\mathscr{K}}_{\underline{n}}$ -module $V$ . Given an ${\mathscr{A}}_{\underline{n}}$ -lattices $\Lambda\subset V$ , we write $[\Lambda]$ for its isomorphism class, and $O_{\Lambda}$ for its endomorphism ring $\operatorname{End}_{{\mathscr{A}}_{\underline{n}}}(\Lambda)\subset\operatorname{End}_{{\mathscr{K}}_{\underline{n}}}(V)$ . As a convention, the endomorphism algebra ${\mathscr{E}}_{\underline{n}}:=\operatorname{End}_{{\mathscr{K}}_{\underline{n}}}(V)$ acts on $V$ from the left, so it coincides with the centralizer of $K_{\underline{n}}$ in $\operatorname{Mat}_{2}(D)$ . Two ${\mathscr{A}}_{\underline{n}}$ -lattices $\Lambda_{1}$ and $\Lambda_{2}$ in $V$ are isomorphic if and only if there exists $g\in{\mathscr{E}}_{\underline{n}}^{\times}$ such that $\Lambda_{1}=g\Lambda_{2}$ .

For each prime $\ell\in{\mathbb{N}}$ , we use the subscript _ℓ to indicate $\ell$ -adic completion. For example, ${\mathscr{A}}_{{\underline{n}},\ell}$ (the $\ell$ -adic completion of ${\mathscr{A}}_{\underline{n}}$ ) is a $\mathbb{Z}_{\ell}$ -order in the semisimple $\mathbb{Q}_{\ell}$ -algebra ${\mathscr{K}}_{{\underline{n}},\ell}$ , and $\Lambda_{\ell}$ is an ${\mathscr{A}}_{{\underline{n}},\ell}$ -lattice in $V_{\ell}$ . For each prime $\ell$ , let ${\mathscr{L}}_{\ell}({\underline{n}})$ denote the set of isomorphism classes of ${\mathscr{A}}_{{\underline{n}},\ell}$ -lattices in the left ${\mathscr{K}}_{{\underline{n}},\ell}$ -module $V_{\ell}$ . For almost all primes $\ell$ , the $\mathbb{Z}_{\ell}$ -order ${\mathscr{A}}_{{\underline{n}},\ell}$ is maximal in ${\mathscr{K}}_{{\underline{n}},\ell}$ , in which case both of the following hold by [6, Theorem 26.24]:

(i)

$\Lambda_{\ell}$ is uniquely determined up to isomorphism (i.e. $\lvert{\mathscr{L}}_{\ell}({\underline{n}})\rvert=1$ ), and
(ii)

$O_{\Lambda,\ell}$ is maximal in ${\mathscr{E}}_{{\underline{n}},\ell}$ .

Let $S({\underline{n}},p)$ be the finite set of primes $\ell$ for which ${\mathscr{A}}_{{\underline{n}},\ell}$ is non-maximal. The profinite completion $\Lambda\mapsto\widehat{\Lambda}:=\prod_{\ell}\Lambda_{\ell}$ induces a surjective map

(2.3)

\Psi:{\mathscr{L}}({\underline{n}})\to\prod_{\ell}{\mathscr{L}}_{\ell}({\underline{n}})\simeq\prod_{\ell\in S({\underline{n}},p)}{\mathscr{L}}_{\ell}({\underline{n}}).

Two ${\mathscr{A}}_{\underline{n}}$ -lattices $\Lambda_{1}$ and $\Lambda_{2}$ in $V$ are said to be in the same genus if $\Psi([\Lambda_{1}])=\Psi([\Lambda_{2}])$ , or equivalently, $(\Lambda_{1})_{\ell}\simeq(\Lambda_{2})_{\ell}$ for every prime $\ell$ . The fibers of $\Psi$ partition ${\mathscr{L}}({\underline{n}})$ into a disjoint union of genera. Let ${\mathscr{G}}(\Lambda):=\Psi^{-1}(\Psi([\Lambda]))\subseteq{\mathscr{L}}({\underline{n}})$ be the fiber of $\Psi$ over $\Psi([\Lambda])$ , that is, the set of isomorphism classes of ${\mathscr{A}}_{\underline{n}}$ -lattices in the genus of $\Lambda$ . From [21, Proposition 1.4], we have

(2.4)

\lvert{\mathscr{G}}(\Lambda)\rvert=h(O_{\Lambda}),

where $h(O_{\Lambda})$ denote the class number of $O_{\Lambda}$ . In other words, $h(O_{\Lambda})$ is the number of locally principal right (or equivalently, left) ideal classes of $O_{\Lambda}$ .

Therefore, the computation of $o({\underline{n}})$ can be carried out in the following two steps:

(Step 1)

Classify the genera of ${\mathscr{A}}_{{\underline{n}}}$ -lattices in the left ${\mathscr{K}}_{{\underline{n}}}$ -module $V$ . Equivalently, classify the isomorphism classes of ${\mathscr{A}}_{{\underline{n}},\ell}$ -lattices in $V_{\ell}$ for each $\ell\in S({\underline{n}},p)$ .
(Step 2)

Pick a lattice $\Lambda$ in each genus and write down its endomorphism ring $O_{\Lambda}$ (at least locally at each prime $\ell$ ). The number $o({\underline{n}})$ is obtained by summing up the class numbers $h(O_{\Lambda})$ over all genera.

Remark 2.1.

The reason that condition (1.5) is assumed throughout the calculations in [28] is to make sure that the $\mathbb{Z}_{p}$ -order ${\mathscr{A}}_{{\underline{n}},p}$ is a product of Eichler orders [28, Remark 3.7]. In our setting, $p$ satisfies condition (C1) or (C2), so ${\mathscr{A}}_{{\underline{n}},p}$ becomes more complicated. This is precisely why the primes $p\in\{2,3,5\}$ are treated separately from the rest of the primes. Luckily for us, many ${\mathscr{A}}_{{\underline{n}},p}$ turn out to be Bass orders (see Definition 3.1 below), which makes the classification of ${\mathscr{A}}_{{\underline{n}},p}$ -lattices more manageable.

3. The isotypic case

In this section, we calculate the values of $o(n)$ for $n\in\{3,4,5,8,12\}$ and $p|n$ . Keep the notation of previous sections. In particular, $D=D_{p,\infty}$ is the unique quaternion $\mathbb{Q}$ -algebra ramified precisely at $p$ and $\infty$ , and ${\mathcal{O}}$ is a maximal $\mathbb{Z}$ -order in $D$ . Since $A_{n}$ is the maximal order in the $n$ -th cyclotomic field $K_{n}$ , and ${\mathcal{O}}$ has reduced discriminant $p$ , we have $S(n,p)=\{p\}$ . In other words, the $\ell$ -adic completion ${\mathscr{A}}_{n,\ell}$ is non-maximal in ${\mathscr{K}}_{n,\ell}$ if and only if $\ell=p$ . It turns out that ${\mathscr{A}}_{n,p}$ is always a Bass order in the quaternion $K_{n,p}$ -algebra ${\mathscr{K}}_{n,p}$ . Therefore, the classification of genera of the lattice set ${\mathscr{L}}(n)$ is then reduced to the classification of lattices over local quaternion Bass orders.

3.1. Classification of lattices over local quaternion Bass orders

The main references for this section are [2, 4] and [6, §37]. Let $F$ be a nonarchimedean local field, and $O_{F}$ be its ring of integers. Fix a uniformizer $\varpi$ of $F$ and denote the the finite residue field $O_{F}/\varpi O_{F}$ by ${\mathfrak{k}}$ . Let $B$ be a finite dimensional separable $F$ -algebra [6, Definition 7.1 and Corollary 7.6], and ${\mathscr{O}}$ be an $O_{F}$ -order (of full rank) in $B$ . We write $\operatorname{Ov}({\mathscr{O}})$ for the finite set of overorders of ${\mathscr{O}}$ , i.e. $O_{F}$ -orders in $B$ containing ${\mathscr{O}}$ . A minimal overorder of ${\mathscr{O}}$ is a minimal member of $\operatorname{Ov}({\mathscr{O}})\smallsetminus\{{\mathscr{O}}\}$ with respect to inclusion.

Definition 3.1.

An $O_{F}$ -order ${\mathscr{O}}$ in $B$ is Gorenstein if its dual lattice ${\mathscr{O}}^{\vee}:=\operatorname{Hom}_{O_{F}}({\mathscr{O}},O_{F})$ is projective as a left (or right) ${\mathscr{O}}$ -module. It is called a Bass order if every member of $\operatorname{Ov}({\mathscr{O}})$ is Gorenstein. It is called a hereditary order if every left ideal of ${\mathscr{O}}$ is projective as a left ${\mathscr{O}}$ -module. If ${\mathscr{O}}$ is the intersection of two maximal orders, then it is called an Eichler order.

We have the following inclusions of orders:

\text{(maximal)}\subset\text{(herediary)}\subset\text{(Eichler)}\subset\text{(Bass)}\subset\text{(Gorenstein)}.

If $B$ is division, then Eichler orders are also maximal. Bass notes in [1] that Gorenstein orders are ubiquitous.

Let $I$ be a fractional left ${\mathscr{O}}$ -ideal (of full rank) in $B$ . We say $I$ is proper over ${\mathscr{O}}$ if its associated left order $O_{l}(I):=\{x\in B\mid xI\subseteq I\}$ coincides with ${\mathscr{O}}$ . From [3, Example 2.6 and Corollary 2.7], the following lemma provides an equivalent characterization of Gorenstein orders in certain types of $F$ -algebras:

Lemma 3.2.

Suppose that $B$ is either a commutative algebra or a quaternion $F$ -algebra. Then ${\mathscr{O}}$ is Gorenstein if and only if every proper fractional left ${\mathscr{O}}$ -ideal $I\subset B$ is principal (i.e. there exists $x\in B^{\times}$ such that $I={\mathscr{O}}x$ ).

In the quaternion case, the above lemma can also be obtained by combining [9, Condition G4 or G4’, p. 1364] and [12, Theorem 1]. The lemma no longer holds in general for orders in more complicated algebras. If every proper fractional left ${\mathscr{O}}$ -ideal $I$ is principal, then ${\mathscr{O}}$ is Gorenstein, but the converse is not necessarily true. See [12, p. 220] and [3, p. 535] for some examples. Nevertheless, a proper fractional left ideal over a Gorenstein order is always left projective according to [18, Theorem 5.3, pp. 253–255]. However, unlike the situation over commutative rings, a projective module over a non-commutative ring may not be locally free. Brzezinski [3, Proposition 2.3] gave a precise characterization of the orders ${\mathscr{O}}$ such that every proper fractional left ${\mathscr{O}}$ -ideal $I\subset B$ is principal (Such orders are called strongly Gorenstein by him).

For the rest of Section 3.1, we assume that $\operatorname{char}(F)\neq 2$ and $B$ is a quaternion $F$ -algebra. The reduced trace and reduced norm maps of $B$ are denoted by $\operatorname{Tr}:B\to F$ and $\operatorname{Nr}:B\to F$ respectively. We write ${\mathfrak{d}}({\mathscr{O}})$ for the reduced discriminant of ${\mathscr{O}}$ , which is a nonzero integral ideal of $O_{F}$ . From [2, Proposition 1.2], ${\mathscr{O}}$ is hereditary if and only if ${\mathfrak{d}}({\mathscr{O}})$ is square-free. Thus if $B$ is division, then ${\mathscr{O}}$ is hereditary if and only if ${\mathscr{O}}$ is the unique maximal order of $B$ ; if $B\simeq\operatorname{Mat}_{2}(F)$ , then ${\mathscr{O}}$ is hereditary if and only if ${\mathscr{O}}$ is isomorphic to $\operatorname{Mat}_{2}(O_{F})$ or $\left[\begin{smallmatrix}O_{F}&O_{F}\\ \varpi O_{F}&O_{F}\end{smallmatrix}\right]$ .

Theorem 3.3.

The following are equivalent:

(a)

every left ${\mathscr{O}}$ -ideal is generated by at most 2 elements;
(b)

${\mathscr{O}}$ is Bass;
(c)

every indecomposable ${\mathscr{O}}$ -lattice is isomorphic to an ideal of ${\mathscr{O}}$ ;
(d)

${\mathscr{O}}\supseteq O_{L}$ for some semisimple quadratic $F$ -subalgebra $L\subseteq B$ .

Indeed, the implications $(a)\Rightarrow(b)\Rightarrow(c)$ hold in much more general settings according to [6, §37]. The implication $(c)\Rightarrow(a)$ is proved by Drozd, Kirichenko and Roiter [9] (see [6, p. 790]). Lastly, the equivalence $(b)\Leftrightarrow(d)$ is proved by Brzezinski [4, Proposition 1.12]. See Chari et al. [5] for more characterization of quaternion Bass orders.

We recall the notion of Eichler invariant following [2, Definition 1.8].

Definition 3.4.

Let ${\mathfrak{k}}^{\prime}/{\mathfrak{k}}$ be the unique quadratic field extension. When ${\mathscr{O}}\not\simeq\operatorname{Mat}_{2}(O_{F})$ , the quotient of ${\mathscr{O}}$ by its Jacobson radical ${\mathfrak{J}}({\mathscr{O}})$ falls into the following three cases:

{\mathscr{O}}/{\mathfrak{J}}({\mathscr{O}})\simeq{\mathfrak{k}}\times{\mathfrak{k}},\qquad{\mathfrak{k}},\quad\text{or}\quad{\mathfrak{k}}^{\prime},

and the Eichler invariant $e({\mathscr{O}})$ is defined to be $1,0,-1$ accordingly. As a convention, if ${\mathscr{O}}\simeq\operatorname{Mat}_{2}(O_{F})$ , then its Eichler invariant is defined to be $2$ .

For example, if $B$ is division and ${\mathscr{O}}$ is the unique maximal order, then $e({\mathscr{O}})=-1$ . It is shown in [2, Proposition 2.1] that $e({\mathscr{O}})=1$ if and only if ${\mathscr{O}}$ is a non-maximal Eichler order. Note that $e({\mathscr{O}})=1$ can only occur when $B\simeq\operatorname{Mat}_{2}(F)$ . Moreover, if $e({\mathscr{O}})\neq 0$ , then ${\mathscr{O}}$ is automatically Bass by [2, Corollary 2.4 and Propoisition 3.1]. The classification of lattices over Eichler orders is well known (see [28, p. 315] for example), which we recall as follows.

Lemma 3.5.

Suppose $B\simeq\operatorname{Mat}_{2}(F)$ and let ${\mathscr{O}}\simeq\begin{bmatrix}O_{F}&O_{F}\\ \varpi^{e}O_{F}&O_{F}\end{bmatrix}$ be an Eichler order. Let $M$ be an ${\mathscr{O}}$ -lattice in a finite left $B$ -module $W$ . Then

(3.1)

W\simeq\begin{bmatrix}F\\ F\end{bmatrix}^{\oplus u},\quad\text{and}\quad M\simeq\bigoplus_{i=1}^{u}\begin{bmatrix}O_{F}\\ \varpi^{e_{i}}O_{F}\end{bmatrix},

where the $e_{i}$ ’s are integers such that $0\leq e_{i}\leq e$ and $e_{i}\leq e_{i+1}$ for all $i$ . Moreover, the isomorphism class of $M$ is uniquely determined by these $e_{i}$ ’s.

Henceforth we assume that $e({\mathscr{O}})\in\{0,-1\}$ . Let $n({\mathscr{O}})$ be the unique non-negative integer such that ${\mathfrak{d}}({\mathscr{O}})=(\varpi^{n({\mathscr{O}})})$ . Suppose that ${\mathscr{O}}$ is Bass but non-hereditary. From [2, Proposition 1.12], ${\mathscr{O}}$ has a unique minimal overorder ${\mathcal{M}}({\mathscr{O}})$ , which is also Bass by definition. According to [2, Propositions 3.1 and 4.1],

(3.2)

n({\mathcal{M}}({\mathscr{O}}))=\begin{cases}n({\mathscr{O}})-2&\text{if }e({\mathscr{O}})=-1,\\ n({\mathscr{O}})-1&\text{if }e({\mathscr{O}})=0,\end{cases}

and $e({\mathcal{M}}({\mathscr{O}}))=e({\mathscr{O}})$ if ${\mathcal{M}}({\mathscr{O}})$ is also non-hereditary. Thus starting from ${\mathcal{M}}^{0}({\mathscr{O}}):={\mathscr{O}}$ , we define ${\mathcal{M}}^{i}({\mathscr{O}}):={\mathcal{M}}({\mathcal{M}}^{i-1}({\mathscr{O}}))$ recursively to obtain a unique chain of Bass orders terminating at a hereditary order ${\mathcal{M}}^{m}({\mathscr{O}})$ :

(3.3)

{\mathscr{O}}={\mathcal{M}}^{0}({\mathscr{O}})\subset{\mathcal{M}}^{1}({\mathscr{O}})\subset{\mathcal{M}}^{2}({\mathscr{O}})\subset\cdots\subset{\mathcal{M}}^{m-1}({\mathscr{O}})\subset{\mathcal{M}}^{m}({\mathscr{O}}),

where $m$ is given as follows

•

$m=n({\mathscr{O}})-1$ if $e({\mathscr{O}})=0$ ; and
•

$m=\left\lfloor n({\mathscr{O}})/2\right\rfloor$ if $e({\mathscr{O}})=-1$ , where $x\mapsto\left\lfloor x\right\rfloor$ is the floor function on $\mathbb{R}$ .

The order ${\mathcal{M}}^{m}({\mathscr{O}})$ is called the hereditary closure of ${\mathscr{O}}$ and will henceforth be denoted by ${\mathcal{H}}({\mathscr{O}})$ . If $e({\mathscr{O}})=-1$ , then ${\mathcal{H}}({\mathscr{O}})$ is always a maximal order by [2, Proposition 3.1]. Thus when $e({\mathscr{O}})=-1$ , $n({\mathscr{O}})$ is even if $B\simeq\operatorname{Mat}_{2}(F)$ , and $n({\mathscr{O}})$ is odd if $B$ is division. If $e({\mathscr{O}})=0$ , then

•

${\mathcal{H}}({\mathscr{O}})\simeq\left[\begin{smallmatrix}O_{F}&O_{F}\\ \varpi O_{F}&O_{F}\end{smallmatrix}\right]$ if $B\simeq\operatorname{Mat}_{2}(F)$ , and
•

${\mathcal{H}}({\mathscr{O}})$ is the unique maximal order if $B$ is division.

Note that ${\mathscr{O}}$ is hereditary (i.e. $m=0$ ) if and only if $e({\mathscr{O}})=-1$ and $B$ is division, so $m$ is strictly positive in the remaining cases.

From [4, Proposition 1.12], there exists a quadratic field extension $L/F$ such that $O_{L}$ embeds²²2From the proof of [4, Theorem 3.3 and 3.10], any two embeddings of $O_{L}$ into ${\mathscr{O}}$ are conjugate by an element of the normalizer of ${\mathscr{O}}$ , thus expression (3.4) does not depend on the choice of the embedding $O_{L}\hookrightarrow{\mathscr{O}}$ . into ${\mathscr{O}}$ , and

(3.4)		$\displaystyle{\mathscr{O}}$	$\displaystyle=O_{L}+{\mathfrak{J}}({\mathcal{H}}({\mathscr{O}}))^{c},\qquad\text{where}$
(3.5)		$\displaystyle c$	$\displaystyle=\begin{cases}n({\mathscr{O}})/2&\text{if }e({\mathscr{O}})=-1\text{ and }B\simeq\operatorname{Mat}_{2}(F),\\ n({\mathscr{O}})-1&\text{otherwise.}\end{cases}$

In fact, $L/F$ is the unique unramified quadratic field extension if $e({\mathscr{O}})=-1$ , and it is a ramified quadratic field extension if $e({\mathscr{O}})=0$ . In the latter case, the ramified quadratic extension $L/F$ can be arbitrary if $n({\mathscr{O}})=2$ according to [4, (3.14)]; and it is uniquely determined by ${\mathscr{O}}$ if $n({\mathscr{O}})\geq 3$ and $F$ is nondyadic according to [17, Lemma 3.5].

Lemma 3.6.

Suppose that ${\mathscr{O}}\subset B$ is a Bass order with $e({\mathscr{O}})\in\{0,-1\}$ . Let $N$ be an indecomposable left ${\mathscr{O}}$ -lattice.

(1)

If $B$ is division, then

(3.6) $N\simeq{\mathcal{M}}^{i}({\mathscr{O}})\qquad\text{for some }0\leq i\leq m.$

(2)

Suppose that $B$ is split, i.e. $B\simeq\operatorname{Mat}_{2}(F)$ . Fix an identification of ${\mathcal{H}}({\mathscr{O}})$ with $\left[\begin{smallmatrix}O_{F}&O_{F}\\ \varpi O_{F}&O_{F}\end{smallmatrix}\right]$ (resp. $\operatorname{Mat}_{2}(O_{F})$ ) if $e({\mathscr{O}})=0$ (resp. $-1$ ).

(a)

If $e({\mathscr{O}})=0$ , then $N$ is isomorphic to one of the following ${\mathscr{O}}$ -lattices:

(3.7)

\begin{bmatrix}O_{F}\\ \varpi O_{F}\end{bmatrix},\quad\begin{bmatrix}O_{F}\\ O_{F}\end{bmatrix},\quad\text{or}\quad{\mathcal{M}}^{i}({\mathscr{O}})\quad\text{with}\quad 0\leq i\leq m-1.

(b)

If $e({\mathscr{O}})=-1$ , then $N$ is isomorphic to one of the following ${\mathscr{O}}$ -lattices:

(3.8) $\begin{bmatrix}O_{F}\\ O_{F}\end{bmatrix}\quad\text{or}\quad{\mathcal{M}}^{i}({\mathscr{O}})\quad\text{with}\quad 0\leq i\leq m-1.$

Proof.

According to the Drozd-Krichenko-Roiter Theorem [6, Theorem 37.16],

(3.9)

N\otimes_{O_{F}}F\simeq\begin{cases}B&\text{if $B$ is division},\\ F^{2}\text{ or }\operatorname{Mat}_{2}(F)&\text{if }B\simeq\operatorname{Mat}_{2}(F).\end{cases}

First, suppose that $B\simeq\operatorname{Mat}_{2}(F)$ and $N\otimes_{O_{F}}F\simeq F^{2}$ . Then $N\simeq O_{F}^{2}$ as an $O_{F}$ -module, and $\operatorname{End}_{O_{F}}(N)$ is a maximal order in $B$ containing ${\mathscr{O}}$ . It follows from (3.3) that $\operatorname{End}_{O_{F}}(N)$ contains ${\mathcal{H}}({\mathscr{O}})$ , which equips $N$ with a canonical ${\mathcal{H}}({\mathscr{O}})$ -module structure. Therefore, if $e({\mathscr{O}})=-1$ , then ${\mathcal{H}}({\mathscr{O}})=\operatorname{Mat}_{2}(O_{F})$ , and hence $N$ is homothetic to $\left[\begin{smallmatrix}O_{F}\\ O_{F}\end{smallmatrix}\right]$ . Similarly, if $e({\mathscr{O}})=0$ , then ${\mathcal{H}}({\mathscr{O}})=\left[\begin{smallmatrix}O_{F}&O_{F}\\ \varpi O_{F}&O_{F}\end{smallmatrix}\right]$ , and hence $N$ is homothetic to $\left[\begin{smallmatrix}O_{F}\\ O_{F}\end{smallmatrix}\right]$ or $\left[\begin{smallmatrix}O_{F}\\ \varpi O_{F}\end{smallmatrix}\right]$ .

Next, suppose that $N\otimes_{O_{F}}F\simeq B$ . Then we regard $N$ as a fractional left ideal of ${\mathscr{O}}$ . Let $O_{l}(N)=\{x\in B\mid xN\subseteq N\}$ be the associated left order of $N$ . Clearly, $O_{l}(N)$ contains ${\mathscr{O}}$ , so $O_{l}(N)={\mathcal{M}}^{i}({\mathscr{O}})$ for some $0\leq i\leq m$ . In particular, $O_{l}(N)$ is Gorenstein. It follows from Lemma 3.2 that $N\simeq O_{l}(N)$ as ${\mathscr{O}}$ -lattices.

Clearly, if $B$ is division, then ${\mathcal{M}}^{i}({\mathscr{O}})$ is indecomposable for every $0\leq i\leq m$ . On the other hand, if $B\simeq\operatorname{Mat}_{2}(F)$ , then the hereditary closure ${\mathcal{H}}({\mathscr{O}})={\mathcal{M}}^{m}({\mathscr{O}})$ is decomposable as an ${\mathscr{O}}$ -lattice. Thus $N\not\simeq{\mathcal{M}}^{m}({\mathscr{O}})$ in this case. It remains to show that ${\mathcal{M}}^{i}({\mathscr{O}})$ is indecomposable for the remaining $i$ ’s. Suppose otherwise so that ${\mathcal{M}}^{i}({\mathscr{O}})=N_{1}\oplus N_{2}$ , where each $N_{j}$ is an ${\mathscr{O}}$ -lattice in $N_{j}\otimes_{O_{F}}F\simeq F^{2}$ . Then

{\mathcal{M}}^{i}({\mathscr{O}})=O_{l}({\mathcal{M}}^{i}({\mathscr{O}}))=O_{l}(N_{1}\oplus N_{2})=O_{l}(N_{1})\cap O_{l}(N_{2}).

Since $O_{l}(N_{i})$ is a maximal order in $\operatorname{Mat}_{2}(F)$ for each $i$ , this would imply that ${\mathcal{M}}^{i}({\mathscr{O}}))$ is an Eichler order (i.e. $e({\mathcal{M}}^{i}({\mathscr{O}}))\in\{1,2\}$ ), contradicting to the fact that $e({\mathcal{M}}^{i}({\mathscr{O}}))=e({\mathscr{O}})\in\{0,-1\}$ for $0\leq i\leq m-1$ . ∎

Applying the Krull-Schmidt-Azumaya Theorem [6, Theorem 6.12], we immediately obtain the following proposition.

Proposition 3.7.

Suppose that ${\mathscr{O}}\subset B$ is a Bass order with $e({\mathscr{O}})\in\{0,-1\}$ . Let $M$ be an ${\mathscr{O}}$ -lattice in a finite left $B$ -module $W$ .

(1)

If $B$ is division, then $M\simeq\bigoplus_{i=0}^{m}{\mathcal{M}}^{i}({\mathscr{O}})^{\oplus t_{i}}$ with $(t_{0},\cdots,t_{m})\in\mathbb{Z}_{\geq 0}^{m+1}$ and $\sum_{i=0}^{m}t_{i}=\dim_{B}W$ .

(2)

If $B\simeq\operatorname{Mat}_{2}(F)$ , then $W\simeq\operatorname{Mat}_{2,u}(F)$ for some $u\geq 0$ . There are two cases to consider:

(2a)

if $e({\mathscr{O}})=0$ , then

(3.10)

M\simeq\begin{bmatrix}O_{F}\\ \varpi O_{F}\end{bmatrix}^{\oplus r}\bigoplus\begin{bmatrix}O_{F}\\ O_{F}\end{bmatrix}^{\oplus s}\bigoplus\bigoplus_{i=0}^{m-1}{\mathcal{M}}^{i}({\mathscr{O}})^{\oplus t_{i}}

with $(r,s,t_{0},\cdots,t_{m-1})\in\mathbb{Z}_{\geq 0}^{m+2}$ and $r+s+2\sum_{i=0}^{m-1}t_{i}=u$ ;

(2b)

if $e({\mathscr{O}})=-1$ , then

(3.11)

M\simeq\begin{bmatrix}O_{F}\\ O_{F}\end{bmatrix}^{\oplus s}\bigoplus\bigoplus_{i=0}^{m-1}{\mathcal{M}}^{i}({\mathscr{O}})^{\oplus t_{i}},

with $(s,t_{0},\cdots,t_{m-1})\in\mathbb{Z}_{\geq 0}^{m+1}$ and $s+2\sum_{i=0}^{m-1}t_{i}=u$ .

In all cases, the isomorphism class of $M$ is uniquely determined by the numerical invariants $r,s$ (if applicable) and the $t_{i}$ ’s.

3.2. Explicit computations

Recall that our goal is to compute the value of $o(n)$ for $n\in\{3,4,5,8,12\}$ and $p|n$ . As explained in Section 2, $o(n)$ coincides with the number of isomorphism classes of ${\mathscr{A}}_{n}$ -lattices in the left ${\mathscr{K}}_{n}$ -module $V=D^{2}$ (See (1.4) and (2.1) for the definition of ${\mathscr{A}}_{n}$ and ${\mathscr{K}}_{n}$ ). From [23, Theorem 11.1], the $n$ -th cyclotomic field $K_{n}$ has class number $1$ for each $n\in\{3,4,5,8,12\}$ .

Since $K_{n}$ is totally imaginary and $p$ does not split completely in $K_{n}$ , we have ${\mathscr{K}}_{n}=K_{n}\otimes_{\mathbb{Q}}D=\operatorname{Mat}_{2}(K_{n})$ . Thus as a left $\operatorname{Mat}_{2}(K_{n})$ -module,

(3.12)

V\simeq\begin{cases}\operatorname{Mat}_{2}(K_{n})&\text{if }n\in\{3,4\},\\ K_{n}^{2}&\text{if }n\in\{5,8,12\}.\end{cases}

From this, we can easily write down its endomorphism algebra

(3.13)

{\mathscr{E}}_{n}=\operatorname{End}_{{\mathscr{K}}_{n}}(V)=\begin{cases}\operatorname{Mat}_{2}(K_{n})&\text{if }n\in\{3,4\},\\ K_{n}&\text{if }n\in\{5,8,12\}.\end{cases}

In particular, we see that every arithmetic subgroup of ${\mathscr{E}}_{n}^{\times}$ is infinite, and hence the abelian surfaces in these isogeny classes have infinite automorphism groups.

Given an ${\mathscr{A}}_{n}$ -lattice $\Lambda\subset V$ , its endomorphism ring $O_{\Lambda}=\operatorname{End}_{{\mathscr{A}}_{n}}(\Lambda)$ is an $A_{n}$ -order in ${\mathscr{E}}_{n}$ . Therefore, if $n\in\{5,8,12\}$ , then $O_{\Lambda}=A_{n}$ and $h(O_{\Lambda})=1$ . Now suppose that $n\in\{3,4\}$ . We are going to show in (3.19) that $\det(O_{\Lambda,p}^{\times})=A_{n,p}^{\times}$ . On the other hand, at each prime $\ell\neq p$ , we have $O_{\Lambda,\ell}\simeq\operatorname{Mat}_{2}(A_{n,\ell})$ since $O_{\Lambda,\ell}$ is maximal. Thus if we write $\widehat{O}_{\Lambda}$ (resp. $\widehat{A}_{n}$ ) for the profinite completion of $O_{\Lambda}$ (resp. $A_{n}$ ), then $\det(\widehat{O}_{\Lambda}^{\times})=\widehat{A}_{n}^{\times}$ . The same proof of [22, Corollaire III.5.7(1)] shows that $h(O_{\Lambda})=h(A_{n})=1$ . In conclusion, for every pair $(n,p)$ with $n\in\{3,4,5,8,12\}$ and $p|n$ , we have

(3.14)

o(n)=\lvert{\mathscr{L}}_{p}(n)\rvert,

which is consistent with [28, (4.3)].

Lemma 3.8.

Suppose that $n\in\{3,4,5,8\}$ and $p|n$ . Let $U_{p}:=K_{n,p}^{2}$ be the unique simple ${\mathscr{K}}_{n,p}$ -module. Then up to isomorphism, there is a unique ${\mathscr{A}}_{n,p}$ -lattice in $U_{p}$ .

Proof.

For each pair $(n,p)$ under consideration, $p$ is totally ramified in $K_{n}$ . It follows from [15, §2.4] that the Eichler invariants

(3.15)

e({\mathscr{A}}_{n,p})=e({\mathcal{O}}_{p})=-1.

From [2, Proposition 3.1], ${\mathscr{A}}_{n,p}$ is a Bass order. Thus the lemma is a direct application of Corollary 3.7. ∎

Lemma 3.9.

Suppose that $n\in\{3,4,5,8\}$ and $p|n$ . Then

(3.16)

o(n)=\begin{cases}2&\text{if }n\in\{3,4\},\\ 1&\text{if }n\in\{5,8\}.\\ \end{cases}

Proof.

First, suppose that $n\in\{5,8\}$ . Then $V_{p}\simeq K_{n,p}^{2}$ is a simple $\operatorname{Mat}_{2}(K_{n,p})$ -module. From Lemma 3.8, $\lvert{\mathscr{L}}_{p}(n)\rvert=1$ , and hence $o(n)=1$ by (3.14).

Next, suppose that $n\in\{3,4\}$ . We have already seen in the proof of Lemma 3.8 that ${\mathscr{A}}_{n,p}$ is a Bass order with Eichler invariant $-1$ . Let $\varpi_{n}=1-\zeta_{n}$ be the uniformizer of the local field $K_{n,p}$ . The reduced discriminant ${\mathfrak{d}}({\mathscr{A}}_{n,p})$ is given by

(3.17)

{\mathfrak{d}}({\mathscr{A}}_{n,p})={\mathfrak{d}}(A_{n,p}\otimes_{\mathbb{Z}_{p}}{\mathcal{O}}_{p})={\mathfrak{d}}({\mathcal{O}}_{p})A_{n,p}=pA_{n,p}=\varpi_{n}^{2}A_{n,p}.

Thus the chain of Bass orders in (3.3) reduces to ${\mathscr{A}}_{n,p}\subset{\mathcal{M}}({\mathscr{A}}_{n,p})$ , where ${\mathcal{M}}({\mathscr{A}}_{n,p})=\operatorname{Mat}_{2}(A_{n,p})$ under a suitable identification ${\mathscr{K}}_{n,p}=\operatorname{Mat}_{2}(K_{n,p})$ . In this case, $V_{p}$ is a free $\operatorname{Mat}_{2}(K_{n,p})$ -module of rank $1$ . From (3.11), every ${\mathscr{A}}_{n,p}$ -lattice $\Lambda_{p}$ in $V_{p}$ is isomorphic to either ${\mathscr{A}}_{n,p}$ or $\operatorname{Mat}_{2}(A_{n,p})$ . Correspondingly, the endomorphism ring $O_{\Lambda,p}$ is given by

(3.18)

O_{\Lambda,p}=\operatorname{End}_{{\mathscr{A}}_{n,p}}(\Lambda_{p})\simeq\begin{cases}{\mathscr{A}}_{n,p}&\text{if }\Lambda_{p}\simeq{\mathscr{A}}_{n,p},\\ \operatorname{Mat}_{2}(A_{n,p})&\text{if }\Lambda_{p}\simeq\operatorname{Mat}_{2}(A_{n,p}).\\ \end{cases}

In both cases, we have

(3.19)

\det(O_{\Lambda,p}^{\times})=A_{n,p}^{\times}.

Indeed, this is clear if $O_{\Lambda,p}\simeq\operatorname{Mat}_{2}(A_{n,p})$ . In the case $O_{\Lambda,p}\simeq{\mathscr{A}}_{n,p}$ , let $L$ be the unique unramified quadratic field extension of $K_{n,p}$ . From (3.4), $O_{\Lambda,p}$ contains a copy of $O_{L}$ , which implies that $\det(O_{\Lambda,p}^{\times})\supseteq\operatorname{N}_{L/K_{n,p}}(O_{L}^{\times})=A_{n,p}^{\times}$ . On the other hand, $\det(O_{\Lambda,p}^{\times})$ is obviously contained in $A_{n,p}^{\times}$ , so equality (3.19) holds in this case as well. We conclude that $o(n)=\lvert{\mathscr{L}}_{p}(n)\rvert=2$ if $n\in\{3,4\}$ and $p|n$ . ∎

Lemma 3.10.

$o(12)=3$ if $p\in\{2,3\}$ .

Proof.

Since $2$ and $3$ are ramified in $K_{n}$ , $2$ is inert in $\mathbb{Q}(\zeta_{3})$ and $3$ is inert in $\mathbb{Q}(\zeta_{4})$ , the $p$ -adic completion $K_{n,p}$ is a field extension of degree $4$ over $\mathbb{Q}_{p}$ with residue degree $2$ , so $e({\mathscr{A}}_{n,p})=e(A_{n,p}\otimes_{\mathbb{Z}_{p}}{\mathcal{O}}_{p})=1$ by [15, §2.4]. A similar calculation as (3.17) shows that ${\mathfrak{d}}({\mathscr{A}}_{n,p})=\varpi_{p}^{2}A_{n,p}$ , where $\varpi_{p}$ denotes a uniformizer of $K_{n,p}$ . From [2, Proposition 2.1], we may identify ${\mathscr{A}}_{n,p}$ with the Eichler order $\begin{bmatrix}A_{n,p}&A_{n,p}\\ \varpi_{p}^{2}A_{n,p}&A_{n,p}\end{bmatrix}$ . Since $V_{p}\simeq K_{n,p}^{2}$ is a simple $\operatorname{Mat}_{2}(K_{n,p})$ -module, every ${\mathscr{A}}_{n,p}$ -lattice in $V_{p}$ is isomorphic to one of the following lattices:

(3.20)

\begin{bmatrix}A_{n,p}\\ \varpi_{p}^{2}A_{n,p}\end{bmatrix},\qquad\begin{bmatrix}A_{n,p}\\ \varpi_{p}A_{n,p}\end{bmatrix},\qquad\begin{bmatrix}A_{n,p}\\ A_{n,p}\end{bmatrix}.

Therefore, $o(12)=\lvert{\mathscr{L}}_{p}(12)\rvert=3$ by (3.14). ∎

Remark 3.11.

We have proved (cf. (3.14)) that in the isotypic case, every genus in the set ${\mathscr{L}}(n)$ of lattice classes has class number one. This holds also in the case $p\nmid n$ according to [28, (4.3)].

4. The non-isotypic case

In this section, we compute the values of $o({\underline{n}})$ for ${\underline{n}}=(n_{1},n_{2})\in\breve{\mathbb{N}}^{2}$ and $p$ satisfying condition (C1) or (C2) (or both). More explicitly, the pairs $({\underline{n}},p)$ are listed in the following table:

${\underline{n}}=(n_{1},n_{2})$	$(1,2)$	$(2,3)$	$(2,4)$	$(2,6)$	$(3,4)$	$(3,6)$
$p$	$2$	$3$	$2$	$3$	$2,3$	$2,3$
$i({\underline{n}})$	$2$	$1$	$2$	$3$	$1$	$4$

Here we have also included the index $i({\underline{n}})=[O_{K_{\underline{n}}}:A_{{\underline{n}}}]$ , where $O_{K_{\underline{n}}}=A_{n_{1}}\times A_{n_{2}}$ is the unique maximal order of $K_{\underline{n}}$ .

Since $K_{{\underline{n}}}=K_{n_{1}}\times K_{n_{2}}$ , we have ${\mathscr{K}}_{\underline{n}}={\mathscr{K}}_{n_{1}}\times{\mathscr{K}}_{n_{2}}$ . Consequently, the left ${\mathscr{K}}_{\underline{n}}$ -module $V=D^{2}$ decomposes into a product $V_{n_{1}}\times V_{n_{2}}$ , where each $V_{n_{i}}$ is a simple left ${\mathscr{K}}_{n_{i}}$ -module with $\dim_{D}V_{n_{i}}=1$ . In turn, ${\mathscr{E}}_{\underline{n}}={\mathscr{E}}_{n_{1}}\times{\mathscr{E}}_{n_{2}}$ with ${\mathscr{E}}_{n_{i}}:=\operatorname{End}_{{\mathscr{K}}_{n_{i}}}(V_{n_{i}})$ . If $n_{i}\in\{1,2\}$ , then $K_{n_{i}}=\mathbb{Q}$ , so we have

(4.1)

{\mathscr{K}}_{n_{i}}=D,\quad V_{n_{i}}\simeq D,\quad{\mathscr{E}}_{n_{i}}\simeq D\quad\text{if}\quad n_{i}\in\{1,2\}.

If $n_{i}\in\{3,4,6\}$ , then $K_{n_{i}}$ is an imaginary quadratic extension of $\mathbb{Q}$ , and $p$ does not split completely in $K_{n_{i}}$ . Thus

(4.2)

{\mathscr{K}}_{n_{i}}\simeq\operatorname{Mat}_{2}(K_{n_{i}}),\quad V_{n_{i}}\simeq K_{n_{i}}^{2},\quad{\mathscr{E}}_{n_{i}}=K_{n_{i}}\quad\text{if}\quad n_{i}\in\{3,4,6\}.

To avoid conflict of notations between $V_{n_{i}}$ and $V_{\ell}$ , we will always write the full expression $V\otimes\mathbb{Q}_{\ell}$ instead of $V_{\ell}$ for the $\ell$ -adic completion of $V$ . On the other hand, the subscript ${}_{\underline{n}}$ will never be expanded out explicitly as ${}_{(n_{1},n_{2})}$ nor ${}_{n_{1},n_{2}}$ , so there should be no ambiguity about ${\mathscr{A}}_{n_{i},\ell}:={\mathscr{A}}_{n_{i}}\otimes_{\mathbb{Z}}\mathbb{Z}_{\ell}$ .

If $\ell\in{\mathbb{N}}$ is a prime with $\ell\nmid i({\underline{n}})$ and $\ell\neq p$ , then $A_{{\underline{n}},\ell}=O_{K_{\underline{n}},\ell}$ , and ${\mathcal{O}}_{\ell}\simeq\operatorname{Mat}_{2}(\mathbb{Z}_{\ell})$ , which implies that

{\mathscr{A}}_{{\underline{n}},\ell}=A_{{\underline{n}},\ell}\otimes_{\mathbb{Z}_{\ell}}{\mathcal{O}}_{\ell}\simeq\operatorname{Mat}_{2}(O_{K_{\underline{n}},\ell}).

Thus ${\mathscr{A}}_{{\underline{n}},\ell}$ is maximal in ${\mathscr{K}}_{{\underline{n}},\ell}$ for such an $\ell$ . From this, we can easily write the set $S({\underline{n}},p)$ of primes at which ${\mathscr{A}}_{\underline{n}}$ is non-maximal:

(4.3)

S({\underline{n}},p)=\begin{cases}\{2,3\}&\text{if }{\underline{n}}=(3,6)\text{ and }p=3;\\ \{p\}&\text{otherwise}.\end{cases}

Recall that the class number $h({\mathcal{O}})$ is given by the following formula [22, Proposition V.3.2]

(4.4)

h({\mathcal{O}})=\frac{p-1}{12}+\frac{1}{3}\left(1-\genfrac{(}{)}{}{}{-3}{p}\right)+\frac{1}{4}\left(1-\genfrac{(}{)}{}{}{-4}{p}\right),

where $\genfrac{(}{)}{}{}{\cdot}{p}$ denotes the Legendre symbol. In particular, $h({\mathcal{O}})=1$ if $p=2,3$ . We also note that $h(A_{n_{i}})=1$ for every $n_{i}\in\{3,4,6\}$ . Given $d\in{\mathbb{N}}$ , we write $\mathbb{Q}_{\ell^{d}}$ for the unique unramified extension of degree $d$ over $\mathbb{Q}_{\ell}$ , and $\mathbb{Z}_{\ell^{d}}$ for its ring of integers.

Lemma 4.1.

$o(2,3)=1$ if $p=3$ , and $o(3,4)=2$ if $p\in\{2,3\}$ .

Proof.

For ${\underline{n}}=(2,3)$ or $(3,4)$ , we have $A_{\underline{n}}=A_{n_{1}}\times A_{n_{2}}$ , and hence ${\mathscr{A}}_{\underline{n}}={\mathscr{A}}_{n_{1}}\times{\mathscr{A}}_{n_{2}}$ . Any ${\mathscr{A}}_{{\underline{n}}}$ -lattice $\Lambda\subset V$ decomposes into a product $\Lambda_{n_{1}}\times\Lambda_{n_{2}}$ , where each $\Lambda_{n_{i}}$ is an ${\mathscr{A}}_{n_{i}}$ -lattice in $V_{n_{i}}$ . Thus the techniques developed in Section 3 applies here.

First suppose that ${\underline{n}}=(2,3)$ and $p=3$ . In this case, ${\mathscr{A}}_{n_{1}}=\mathbb{Z}\otimes_{\mathbb{Z}}{\mathcal{O}}={\mathcal{O}}$ , and $V_{n_{1}}\simeq D$ by (4.1). Since $h({\mathcal{O}})=1$ , there is a unique isomorphism class of ${\mathcal{O}}$ -lattices in $V_{n_{1}}$ . As $S({\underline{n}},p)=\{3\}$ in this case, we consider the ${\mathscr{A}}_{n_{2},3}$ -lattices in $V_{n_{2}}\otimes\mathbb{Q}_{3}\simeq K_{n_{2},3}^{2}$ . From Lemma 3.8, there is a unique ${\mathscr{A}}_{n_{2},3}$ -lattice up to isomorphism in $V_{n_{2}}\otimes\mathbb{Q}_{3}$ . This implies that there is only a single genus of ${\mathscr{A}}_{n_{2}}$ -lattices in $V_{n_{2}}$ . Note that $\operatorname{End}_{{\mathscr{A}}_{n_{2}}}(\Lambda_{n_{2}})=A_{n_{2}}\simeq\mathbb{Z}[\zeta_{3}]$ , which has class number $1$ . Thus there is a unique isomorphism class of ${\mathscr{A}}_{n_{2}}$ -lattices in $V_{n_{2}}$ . As a result, $o(2,3)=1\cdot 1=1$ if $p=3$ .

Next, suppose that ${\underline{n}}=(3,4)$ and $p=2$ . Since $2$ is ramified in $K_{n_{2}}=\mathbb{Q}(\zeta_{4})$ , the same proof as above shows that there is a unique isomorphism class of ${\mathscr{A}}_{n_{2}}$ -lattices in $V_{n_{2}}$ in this case. On the other hand, $2$ is inert in $K_{n_{1}}=\mathbb{Q}(\zeta_{3})$ , so $A_{n_{1},2}=\mathbb{Z}_{4}$ . It follows from [15, Lemma 2.10] that

(4.5)

{\mathscr{A}}_{n_{1},2}=A_{n_{1},2}\otimes_{\mathbb{Z}_{2}}{\mathcal{O}}_{2}=\mathbb{Z}_{4}\otimes_{\mathbb{Z}_{2}}{\mathcal{O}}_{2}\simeq\begin{bmatrix}\mathbb{Z}_{4}&\mathbb{Z}_{4}\\ 2\mathbb{Z}_{4}&\mathbb{Z}_{4}\end{bmatrix}.

Hence every ${\mathscr{A}}_{n_{1},2}$ -lattices in $V_{n_{1}}\otimes_{\mathbb{Q}}\mathbb{Q}_{2}$ is isomorphic to either $\left[\begin{smallmatrix}\mathbb{Z}_{4}\\ 2\mathbb{Z}_{4}\end{smallmatrix}\right]$ or $\left[\begin{smallmatrix}\mathbb{Z}_{4}\\ \mathbb{Z}_{4}\end{smallmatrix}\right]$ . We find that there are two genera of ${\mathscr{A}}_{n_{1}}$ -lattices in $V_{n_{1}}$ , each consisting of a unique isomorphism class since $h(A_{n_{1}})=h(\mathbb{Z}[\zeta_{3}])=1$ . Therefore, $o(3,4)=(1+1)\cdot 1=2$ if $p=2$ .

Lastly, the value of $o(3,4)$ for $p=3$ can be computed in exactly the same way as above, since $3$ is ramified in $K_{n_{1}}$ and inert in $K_{n_{2}}$ . ∎

Lemma 4.2.

$o(1,2)=3$ if $p=2$ .

Proof.

Set ${\underline{n}}=(1,2)$ throughout this proof. In this case, ${\mathscr{A}}_{{\underline{n}}}$ is non-maximal only at $p=2$ , and ${\mathcal{O}}_{2}$ is the unique maximal order in the division quaternion $\mathbb{Q}_{2}$ -algebra $D_{2}$ . From [28, (5.4)], we have

A_{\underline{n}}=\{(a,b)\in\mathbb{Z}\times\mathbb{Z}\mid a\equiv b\pmod{2}\},

which implies that

(4.6)

{\mathscr{A}}_{{\underline{n}},2}=A_{\underline{n}}\otimes_{\mathbb{Z}}{\mathcal{O}}_{2}=\{(x,y)\in{\mathcal{O}}_{2}\times{\mathcal{O}}_{2}\mid x\equiv y\pmod{2{\mathcal{O}}_{2}}\}.

In particular, ${\mathscr{A}}_{{\underline{n}},2}$ is a subdirect sum³³3Let $\{R_{i}\mid 1\leq i\leq n\}$ be a finite set of (unital) rings, and $R:=\prod_{i=1}^{n}R_{i}$ be their direct product. A ring $T$ is called a subdirect sum of the $R_{i}$ ’s if there exists an embedding $\rho:T\to R$ such that every canonical projection $\operatorname{pr}_{i}:R\to R_{i}$ maps $\rho(T)$ surjectively onto $R_{i}$ . of two copies of ${\mathcal{O}}_{2}$ , so it is a Bass order by [9, Proposition 12.3].

Let ${\mathfrak{P}}$ be the unique two-sided prime ideal of ${\mathcal{O}}$ above $p=2$ . We put

(4.7)

{\mathscr{R}}:=\{(x,y)\in{\mathcal{O}}\times{\mathcal{O}}\mid x\equiv y\pmod{{\mathfrak{P}}}\},

which has index $4$ in the maximal order ${\mathbb{O}}:={\mathcal{O}}\times{\mathcal{O}}$ in ${\mathscr{K}}_{{\underline{n}}}=D\times D$ . Indeed, ${\mathscr{R}}/({\mathfrak{P}}\times{\mathfrak{P}})\simeq{\mathcal{O}}/{\mathfrak{P}}\simeq\mathbb{F}_{4}$ while ${\mathbb{O}}/({\mathfrak{P}}\times{\mathfrak{P}})\simeq\mathbb{F}_{4}\times\mathbb{F}_{4}$ . Clearly, both ${\mathscr{R}}$ and ${\mathbb{O}}$ are overorders of ${\mathscr{A}}_{{\underline{n}}}$ . We claim that there are no other overorders except ${\mathscr{A}}_{{\underline{n}}}$ itself. It is enough to prove this locally at $p=2$ . From ${\mathscr{A}}_{{\underline{n}},2}/(2{\mathcal{O}}_{2}\times 2{\mathcal{O}}_{2})\simeq{\mathcal{O}}_{2}/2{\mathcal{O}}_{2}$ , we find that

(4.8)

{\mathscr{A}}_{{\underline{n}},2}/{\mathfrak{J}}({\mathscr{A}}_{{\underline{n}},2})\simeq{\mathcal{O}}_{2}/{\mathfrak{J}}({\mathcal{O}}_{2})\simeq\mathbb{F}_{4},

where ${\mathfrak{J}}(\cdot)$ denotes the Jacboson radical. Hence ${\mathscr{A}}_{{\underline{n}},2}$ is completely primary in the sense of [18, p. 262]. Now according to [18, Lemma 6.6], every non-maximal completely primary Gorenstein order has a unique minimal overorder. As ${\mathscr{A}}_{{\underline{n}},2}$ has index $4$ in ${\mathscr{R}}_{2}$ , the left ${\mathscr{A}}_{{\underline{n}},2}$ -module ${\mathscr{R}}_{2}/{\mathscr{A}}_{{\underline{n}},2}$ is isomorphic to the unique simple left ${\mathscr{A}}_{{\underline{n}},2}$ -module $\mathbb{F}_{4}$ . Thus ${\mathscr{R}}_{2}$ coincides with the unique minimal overorder of ${\mathscr{A}}_{{\underline{n}},2}$ . Similarly, ${\mathscr{R}}_{2}$ is also completely primary, and ${\mathbb{O}}_{2}$ is the unique minimal overorder of ${\mathscr{R}}_{2}$ by the same argument. This verifies the claim about the overorders of ${\mathscr{A}}_{{\underline{n}}}$ .

In this case, $V\otimes\mathbb{Q}_{2}$ is a free left module of rank $1$ over ${\mathscr{K}}_{{\underline{n}},2}$ . For any ${\mathscr{A}}_{{\underline{n}},2}$ -lattice in $V\otimes\mathbb{Q}_{2}$ , its associated left order necessarily coincides with one of the three overorders of ${\mathscr{A}}_{{\underline{n}},2}$ . Since ${\mathscr{A}}_{{\underline{n}},2}$ is completely primary, it is indecomposable as a left module over itself. Taking into account that ${\mathscr{A}}_{{\underline{n}},2}$ is Gorenstein, it follows from [3, Proposition 2.3] that every proper ${\mathscr{A}}_{{\underline{n}},2}$ -lattice in the $V\otimes\mathbb{Q}_{2}$ is principal. This holds for ${\mathscr{R}}_{2}$ as well by the same token. On the other hand, every proper ${\mathbb{O}}_{2}$ -lattice in the $V\otimes\mathbb{Q}_{2}$ is principal since ${\mathbb{O}}_{2}$ is maximal. Therefore, every ${\mathscr{A}}_{{\underline{n}},2}$ -lattice in $V\otimes\mathbb{Q}_{2}$ is isomorphic to one of the following

(4.9)

{\mathscr{A}}_{{\underline{n}},2},\qquad{\mathscr{R}}_{2},\qquad{\mathbb{O}}_{2}.

We find that there are three genera of ${\mathscr{A}}_{\underline{n}}$ -lattices in $V$ represented by ${\mathscr{A}}_{\underline{n}},{\mathscr{R}}$ and ${\mathbb{O}}$ respectively. Clearly,

\operatorname{End}_{{\mathscr{A}}_{\underline{n}}}({\mathscr{A}}_{\underline{n}})={\mathscr{A}}_{\underline{n}}^{\mathrm{opp}},\quad\operatorname{End}_{{\mathscr{A}}_{\underline{n}}}({\mathscr{R}})={\mathscr{R}}^{\mathrm{opp}},\quad\operatorname{End}_{{\mathscr{A}}_{\underline{n}}}({\mathbb{O}})={\mathbb{O}}^{\mathrm{opp}}.

In each case, the opposite ring can be canonically identified with the original ring itself, so we drop the superscript ^opp henceforth.

It remains to show that $h({\mathscr{A}}_{\underline{n}})=h({\mathscr{R}})=h({\mathbb{O}})=1$ . This clearly holds true for the maximal order ${\mathbb{O}}={\mathcal{O}}\times{\mathcal{O}}$ since $h({\mathbb{O}})=h({\mathcal{O}})\cdot h({\mathcal{O}})=1\cdot 1=1$ . If we prove $h({\mathscr{A}}_{\underline{n}})=1$ , then $h({\mathscr{R}})=1$ since $h({\mathscr{A}}_{\underline{n}})\geq h({\mathscr{R}})$ by [28, (6.1)]. Let $\widehat{\mathbb{O}}$ (resp. $\widehat{\mathscr{A}}_{\underline{n}}$ ) be the profinite completion of ${\mathbb{O}}$ (resp. ${\mathscr{A}}_{\underline{n}}$ ). Since $h({\mathbb{O}})=1$ , it follows from [28, (6.3)] that

(4.10)

h({\mathscr{A}}_{\underline{n}})=\lvert{\mathbb{O}}^{\times}\backslash\widehat{\mathbb{O}}^{\times}/(\widehat{\mathscr{A}}_{\underline{n}})^{\times}\rvert.

Clearly, ${\mathscr{A}}_{{\underline{n}},2}^{\times}\supseteq 1+2{\mathbb{O}}_{2}$ and ${\mathscr{A}}_{{\underline{n}},\ell}^{\times}=\widehat{\mathbb{O}}_{\ell}^{\times}$ for every prime $\ell\neq 2$ , so there is an $\widehat{\mathbb{O}}^{\times}$ -equivariant projection

({\mathbb{O}}/2{\mathbb{O}})^{\times}\simeq{\mathbb{O}}_{2}^{\times}/(1+2{\mathbb{O}}_{2})\twoheadrightarrow\widehat{\mathbb{O}}^{\times}/(\widehat{\mathscr{A}}_{\underline{n}})^{\times}.

Hence to prove $h({\mathscr{A}}_{\underline{n}})=1$ , it is enough to show that the canonical projection ${\mathbb{O}}^{\times}\to({\mathbb{O}}/2{\mathbb{O}})^{\times}$ is surjective. Since ${\mathbb{O}}={\mathcal{O}}\times{\mathcal{O}}$ , this amounts to show that

(4.11)

\varphi:{\mathcal{O}}^{\times}\to({\mathcal{O}}/2{\mathcal{O}})^{\times}\quad\text{is surjective}.

From [22, Proposition V.3.1], ${\mathcal{O}}^{\times}\simeq\operatorname{SL}_{2}(\mathbb{F}_{3})$ , which has order $24$ . On the other hand,

\lvert({\mathcal{O}}/2{\mathcal{O}})^{\times}\rvert=\lvert({\mathcal{O}}/{\mathfrak{P}}^{2})^{\times}\rvert=\lvert\mathbb{F}_{4}^{\times}\rvert\cdot\lvert\mathbb{F}_{4}\rvert=3\cdot 4=12.

Thus to prove the surjectivity of $\varphi$ , it suffices to show that $\ker(\varphi)=\{\pm 1\}$ . Since every $\alpha\in{\mathcal{O}}^{\times}$ has finite group order, this follows from a well-known lemma of Serre [19, Theorem, p. 17–19] (See also [16, Lemma, p. 192], [20] and [13, Lemma 7.2] for some variations and generalizations). Therefore, $h({\mathscr{A}}_{\underline{n}})=1$ as claimed.

In conclusion, we have

o(1,2)=h({\mathscr{A}}_{\underline{n}})+h({\mathscr{R}})+h({\mathbb{O}})=1+1+1=3.\qed

For ${\underline{n}}=(3,6)$ , the values of $o({\underline{n}})$ for $p\in\{2,3\}$ will be calculated in a few steps.

Lemma 4.3.

$o(3,6)=\prod_{\ell\in S({\underline{n}},p)}\lvert{\mathscr{L}}_{\ell}({\underline{n}})\rvert$ for both $p\in\{2,3\}$ , where the set $S({\underline{n}},p)$ is given in (4.3).

Proof.

We identify both $A_{3}$ and $A_{6}$ with $\mathbb{Z}[\zeta_{3}]$ via the following maps:

\mathbb{Z}[T]/(T^{2}+T+1)\to\mathbb{Z}[\zeta_{3}],\quad T\mapsto\zeta_{3},\quad\text{and}\quad\mathbb{Z}[T]/(T^{2}-T+1)\to\mathbb{Z}[\zeta_{3}],\quad T\mapsto-\zeta_{3}.

As a result, the maximal order $O_{K_{\underline{n}}}$ of $K_{\underline{n}}$ is identified with $\mathbb{Z}[\zeta_{3}]\times\mathbb{Z}[\zeta_{3}]$ . From [28, (5.7)], we have

(4.12)

A_{\underline{n}}=\{(a,b)\in O_{K_{\underline{n}}}\mid a\equiv b\pmod{2\mathbb{Z}[\zeta_{3}]}\}.

In particular, $A_{\underline{n}}/(2O_{K_{\underline{n}}})\simeq\mathbb{F}_{4}$ . Hence $[O_{K_{\underline{n}}}:A_{\underline{n}}]=4$ , and the overorders of $A_{\underline{n}}$ are precisely $O_{K_{\underline{n}}}$ and $A_{\underline{n}}$ itself. From (4.2), ${\mathscr{E}}_{\underline{n}}=K_{{\underline{n}}}$ , so the endomorphism ring $O_{\Lambda}$ of any ${\mathscr{A}}_{\underline{n}}$ -lattice $\Lambda\subset V$ is an overorder of $A_{\underline{n}}$ . Thus $O_{\Lambda}$ is equal to either $A_{\underline{n}}$ or $O_{K_{\underline{n}}}$ . Since $h(A_{\underline{n}})=h(O_{K_{{\underline{n}}}})=1$ by [28, (5.8)], we conclude that

(4.13)

o(3,6)=\prod_{\ell\in S({\underline{n}},p)}\lvert{\mathscr{L}}_{\ell}({\underline{n}})\rvert.\qed

Lemma 4.4.

$o(3,6)=2$ if $p=3$ .

Proof.

From (4.3), $S({\underline{n}},3)=\{2,3\}$ , so we need to classify ${\mathscr{A}}_{{\underline{n}},\ell}$ -lattices in $V\otimes\mathbb{Q}_{\ell}$ for both $\ell=2,3$ . For $\ell=2$ , we have ${\mathcal{O}}_{2}=\operatorname{Mat}_{2}(\mathbb{Z}_{2})$ . Hence

(4.14)

{\mathscr{A}}_{{\underline{n}},2}=A_{{\underline{n}},2}\otimes_{\mathbb{Z}_{2}}{\mathcal{O}}_{2}=\operatorname{Mat}_{2}(A_{{\underline{n}},2}).

Recall that $V=K_{\underline{n}}^{2}$ by (4.2). Applying Morita’s equivalence, we reduce the classification of ${\mathscr{A}}_{{\underline{n}},2}$ -lattices in $V\otimes\mathbb{Q}_{2}$ to that of $A_{{\underline{n}},2}$ -lattices in $K_{{\underline{n}},2}$ . Now $A_{{\underline{n}},2}$ is a commutative Bass order over $\mathbb{Z}_{2}$ , so it follows from Lemma 3.2 that each $A_{{\underline{n}},2}$ -lattice in $K_{{\underline{n}},2}$ is isomorphic to an overorder of $A_{{\underline{n}},2}$ (namely, either $A_{{\underline{n}},2}$ or $O_{K_{\underline{n}},2}$ ). Therefore, $\lvert{\mathscr{L}}_{2}({\underline{n}})\rvert=2$ .

Next, we compute $\lvert{\mathscr{L}}_{3}({\underline{n}})\rvert$ . Since $3$ is coprime to $[O_{K_{\underline{n}}}:A_{\underline{n}}]=4$ , we have

(4.15)		$\displaystyle A_{{\underline{n}},3}$	$\displaystyle=O_{K_{\underline{n}}}\otimes_{\mathbb{Z}}\mathbb{Z}_{3}=A_{n_{1},3}\times A_{n_{2},3},\quad\text{and hence}$
(4.16)		$\displaystyle{\mathscr{A}}_{{\underline{n}},3}$	$\displaystyle=A_{{\underline{n}},3}\otimes_{\mathbb{Z}_{3}}{\mathcal{O}}_{3}={\mathscr{A}}_{n_{1},3}\times{\mathscr{A}}_{n_{2},3}.$

Here ${\mathscr{A}}_{n_{i},3}=\mathbb{Z}_{3}[\zeta_{3}]\otimes_{\mathbb{Z}_{3}}{\mathcal{O}}_{3}$ for each $i=1,2$ . On the other hand, $V_{n_{i}}\otimes\mathbb{Q}_{3}\simeq K_{n_{i},3}^{2}$ by (4.2). It follows from Lemma 3.8 that $\lvert{\mathscr{L}}_{3}({\underline{n}})\rvert=1$ .

We conclude from Lemma 4.3 that $o(3,6)=2\cdot 1=2$ when $p=3$ . ∎

Proposition 4.5.

$o(3,6)=8$ if $p=2$ .

Proof.

In this case, $S({\underline{n}},2)=\{2\}$ by (4.3). Let $\mathbb{Q}_{4}$ be the unique unramified quadratic extension of $\mathbb{Q}_{2}$ , and $\mathbb{Z}_{4}$ be its ring of integers. Then $A_{n_{i},2}=\mathbb{Z}_{2}[\zeta_{3}]=\mathbb{Z}_{4}$ for both $i=1,2$ . The same calculation as in (4.5) shows that

(4.17)

O_{K_{\underline{n}},2}\otimes_{\mathbb{Z}_{2}}{\mathcal{O}}_{2}=(\mathbb{Z}_{4}\times\mathbb{Z}_{4})\otimes_{\mathbb{Z}_{2}}{\mathcal{O}}_{2}\simeq\begin{bmatrix}\mathbb{Z}_{4}&\mathbb{Z}_{4}\\ 2\mathbb{Z}_{4}&\mathbb{Z}_{4}\end{bmatrix}\times\begin{bmatrix}\mathbb{Z}_{4}&\mathbb{Z}_{4}\\ 2\mathbb{Z}_{4}&\mathbb{Z}_{4}\end{bmatrix}.

For simplicity, put ${\mathbb{E}}:=\left[\begin{smallmatrix}\mathbb{Z}_{4}&\mathbb{Z}_{4}\\ 2\mathbb{Z}_{4}&\mathbb{Z}_{4}\end{smallmatrix}\right]$ and identify $(O_{K_{\underline{n}},2}\otimes_{\mathbb{Z}_{2}}{\mathcal{O}}_{2})$ with ${\mathscr{B}}:={\mathbb{E}}\times{\mathbb{E}}$ . Then it follows from (4.12) that

(4.18)

{\mathscr{A}}_{{\underline{n}},2}=\{(x,y)\in{\mathscr{B}}\mid a\equiv b\pmod{2{\mathbb{E}}}\}.

In particular, $2{\mathscr{B}}$ is a two-sided ideal of ${\mathscr{A}}_{{\underline{n}},2}$ contained in ${\mathfrak{J}}({\mathscr{A}}_{{\underline{n}},2})$ . We put

(4.19)

\bar{\mathscr{A}}_{{\underline{n}},2}:={\mathscr{A}}_{{\underline{n}},2}/(2{\mathscr{B}})={\mathbb{E}}/2{\mathbb{E}},\qquad\bar{\mathscr{B}}:={\mathscr{B}}/2{\mathscr{B}}=({\mathbb{E}}/2{\mathbb{E}})\times({\mathbb{E}}/2{\mathbb{E}}),

where $\bar{\mathscr{A}}_{{\underline{n}},2}$ embeds into $\bar{\mathscr{B}}$ diagonally.

From (4.2), $V_{n_{i}}\otimes\mathbb{Q}_{2}=K_{n_{i},2}^{2}=\left[\begin{smallmatrix}\mathbb{Q}_{4}\\ \mathbb{Q}_{4}\end{smallmatrix}\right]$ for each $i=1,2$ . For simplicity, we identify $V\otimes\mathbb{Q}_{2}$ with $\operatorname{Mat}_{2}(\mathbb{Q}_{4})$ and regard the $i$ -th column as a ${\mathscr{K}}_{n_{i},2}$ -module for each $i$ . Any ${\mathscr{B}}$ -lattice in $V\otimes\mathbb{Q}_{2}$ is isomorphic to one of the following

(4.20)

\begin{bmatrix}\mathbb{Z}_{4}&\mathbb{Z}_{4}\\ 2\mathbb{Z}_{4}&2\mathbb{Z}_{4}\end{bmatrix},\qquad\begin{bmatrix}\mathbb{Z}_{4}&\mathbb{Z}_{4}\\ \mathbb{Z}_{4}&\mathbb{Z}_{4}\end{bmatrix},\qquad\begin{bmatrix}\mathbb{Z}_{4}&\mathbb{Z}_{4}\\ 2\mathbb{Z}_{4}&\mathbb{Z}_{4}\end{bmatrix},\qquad\begin{bmatrix}\mathbb{Z}_{4}&\mathbb{Z}_{4}\\ \mathbb{Z}_{4}&2\mathbb{Z}_{4}\end{bmatrix}.

From Lemma 4.3, we are only concerned with ${\mathscr{A}}_{{\underline{n}},2}$ -lattices in $V\otimes\mathbb{Q}_{2}$ , so for ease of notation we drop the subscript ₂ from $\Lambda_{2}$ and write $\Lambda$ for an ${\mathscr{A}}_{{\underline{n}},2}$ -lattice in $V\otimes\mathbb{Q}_{2}$ . Replacing $\Lambda$ by $g\Lambda$ for a suitable $g\in{\mathscr{E}}_{{\underline{n}},2}^{\times}$ if necessary, we assume that ${\mathscr{B}}\Lambda$ is equal to one of the ${\mathscr{B}}$ -lattices $\Delta$ in (4.20). Clearly $2\Delta\subseteq\Lambda\subseteq\Delta$ since $2{\mathscr{B}}\subseteq{\mathscr{A}}_{{\underline{n}},2}$ . Thus $\bar{\Lambda}:=\Lambda/(2\Delta)$ is an $\bar{\mathscr{A}}_{{\underline{n}},2}$ -submodule of $\bar{\Delta}:=\Delta/2\Delta$ . Moreover, as a $\bar{\mathscr{B}}$ -module, $\bar{\Delta}$ is spanned by $\bar{\Lambda}$ . Fix one $\Delta$ in (4.20). Let ${\mathfrak{S}}(\Delta)$ be the set of ${\mathscr{A}}_{{\underline{n}},2}$ -sublattices $\Lambda$ of $\Delta$ satisfying ${\mathscr{B}}\Lambda=\Delta$ , and ${\mathfrak{S}}(\bar{\Delta})$ be the set of $\bar{\mathscr{A}}_{{\underline{n}},2}$ -submodules $\bar{M}$ of $\bar{\Delta}$ satisfying $\bar{\mathscr{B}}\bar{M}=\bar{\Delta}$ . For any $\bar{M}\in{\mathfrak{S}}(\bar{\Delta})$ , there is a unique ${\mathscr{A}}_{{\underline{n}},2}$ -sublattice $\Lambda$ of $\Delta$ satisfying $\Lambda\supseteq 2\Delta$ and $\bar{\Lambda}=\bar{M}$ . Moreover, ${\mathscr{B}}\Lambda=\Delta$ by Nakayama’s lemma. Hence the association $\Lambda\mapsto\bar{\Lambda}$ induces a bijective map:

(4.21)

{\mathfrak{S}}(\Delta)\xrightarrow{\sim}{\mathfrak{S}}(\bar{\Delta}),\qquad\Lambda\mapsto\bar{\Lambda}.

The group $\operatorname{End}_{{\mathscr{B}}}(\Delta)^{\times}$ acts on both ${\mathfrak{S}}(\Delta)$ and ${\mathfrak{S}}(\bar{\Delta})$ , and the map in (4.21) is $\operatorname{End}_{{\mathscr{B}}}(\Delta)^{\times}$ -equivariant as well. Two members $\Lambda,\Lambda^{\prime}$ of ${\mathfrak{S}}(\Delta)$ are ${\mathscr{A}}_{{\underline{n}},2}$ -isomorphic if and only if there exists an $\alpha\in\operatorname{End}_{{\mathscr{B}}}(\Delta)^{\times}$ such that $\alpha\Lambda=\Lambda^{\prime}$ . Therefore, we have established the following bijection

(4.22)

\left\{\parbox{115.63243pt}{Isomorphism classes of ${\mathscr{A}}_{{\underline{n}},2}$-lattices $\Lambda$ in $V\otimes\mathbb{Q}_{2}$ with ${\mathscr{B}}\Lambda\simeq\Delta$}\right\}\longleftrightarrow\left\{\parbox{130.08621pt}{$\operatorname{End}_{{\mathscr{B}}}(\Delta)^{\times}$-equivalent classes of $\bar{\mathscr{A}}_{{\underline{n}},2}$-submodules $\bar{M}$ in $\bar{\Delta}$ such that $\bar{\mathscr{B}}\bar{M}=\bar{\Delta}$}\right\}.

Note that $\operatorname{End}_{{\mathscr{B}}}(\Delta)=\mathbb{Z}_{4}\times\mathbb{Z}_{4}$ for every $\Delta$ in (4.20), so the action of $\operatorname{End}_{{\mathscr{B}}}(\Delta)^{\times}$ on ${\mathfrak{S}}(\bar{\Delta})$ factors through $\mathbb{F}_{4}^{\times}\times\mathbb{F}_{4}^{\times}$ .

Recall from (4.19) that $\bar{\mathscr{A}}_{{\underline{n}},2}={\mathbb{E}}/2{\mathbb{E}}$ , and $\bar{\mathscr{B}}=({\mathbb{E}}/2{\mathbb{E}})^{2}$ . We describe the $4$ -dimensional $\mathbb{F}_{4}$ -algebra ${\mathbb{E}}/2{\mathbb{E}}$ more concretely. Let $R$ be the commutative $\mathbb{F}_{4}$ -algebra $\mathbb{F}_{4}\times\mathbb{F}_{4}$ . Regard $Q:=R$ as an $(R,R)$ -bimodule such that for each $(a,d)\in R$ and $(b,c)\in Q$ , the multiplications are given by the following rules:

(4.23)

(a,d)\cdot(b,c)=(ab,dc),\qquad(b,c)\cdot(a,d)=(bd,ca).

We form the trivial extension [14, Example 1.14] of $R$ by $Q$ and denote it as $\bar{{\mathbb{E}}}:=\left<\begin{smallmatrix}\mathbb{F}_{4}&\mathbb{F}_{4}\\ \mathbb{F}_{4}&\mathbb{F}_{4}\end{smallmatrix}\right>$ , where $R$ is identified with the diagonal of $\left<\begin{smallmatrix}\mathbb{F}_{4}&\mathbb{F}_{4}\\ \mathbb{F}_{4}&\mathbb{F}_{4}\end{smallmatrix}\right>$ and $Q$ is identified with the anti-diagonal. More explicitly, the product in $\bar{\mathbb{E}}$ is defined by the following rule:

(4.24)

\left<\begin{matrix}a&b\\ c&d\end{matrix}\right>\cdot\left<\begin{matrix}a^{\prime}&b^{\prime}\\ c^{\prime}&d^{\prime}\end{matrix}\right>:=\left<\begin{matrix}aa^{\prime}&ab^{\prime}+bd^{\prime}\\ ca^{\prime}+dc^{\prime}&dd^{\prime}\end{matrix}\right>.

For each $x\in\mathbb{Z}_{4}$ , let $\bar{x}$ be its canonical image in $\mathbb{F}_{4}=\mathbb{Z}_{4}/2\mathbb{Z}_{4}$ . One easily checks that the following map induces an isomorphism between ${\mathbb{E}}/2{\mathbb{E}}$ and $\bar{\mathbb{E}}$ :

\begin{bmatrix}\mathbb{Z}_{4}&\mathbb{Z}_{4}\\ 2\mathbb{Z}_{4}&\mathbb{Z}_{4}\end{bmatrix}\to\left<\begin{matrix}\mathbb{F}_{4}&\mathbb{F}_{4}\\ \mathbb{F}_{4}&\mathbb{F}_{4}\end{matrix}\right>,\qquad\begin{bmatrix}x&y\\ 2z&w\end{bmatrix}\mapsto\left<\begin{matrix}\bar{x}&\bar{y}\\ \bar{z}&\bar{w}\end{matrix}\right>.

Henceforth ${\mathbb{E}}/2{\mathbb{E}}$ will be identified with $\bar{\mathbb{E}}$ via this induced isomorphism. Each column of $\bar{\mathbb{E}}$ admits a canonical $\bar{\mathbb{E}}$ -module structure. These two $\bar{\mathbb{E}}$ -modules will be denoted by $\left[\begin{smallmatrix}\mathbb{F}_{4}\\ \mathbb{F}_{4}\end{smallmatrix}\right]^{\dagger}$ and $\left[\begin{smallmatrix}\mathbb{F}_{4}\\ \mathbb{F}_{4}\end{smallmatrix}\right]^{\ddagger}$ respectively. It is clear from (4.24) that $\left[\begin{smallmatrix}0\\ \mathbb{F}_{4}\end{smallmatrix}\right]^{\dagger}$ (resp. $\left[\begin{smallmatrix}\mathbb{F}_{4}\\ 0\end{smallmatrix}\right]^{\ddagger}$ ) is the unique $1$ -dimensional (as an $\mathbb{F}_{4}$ -vector space) $\bar{\mathbb{E}}$ -submodule of $\left[\begin{smallmatrix}\mathbb{F}_{4}\\ \mathbb{F}_{4}\end{smallmatrix}\right]^{\dagger}$ (resp. $\left[\begin{smallmatrix}\mathbb{F}_{4}\\ \mathbb{F}_{4}\end{smallmatrix}\right]^{\ddagger}$ ). We leave it as a simple exercise to check that $\left[\begin{smallmatrix}\mathbb{F}_{4}\\ \mathbb{F}_{4}\end{smallmatrix}\right]^{\dagger}$ and $\left[\begin{smallmatrix}\mathbb{F}_{4}\\ \mathbb{F}_{4}\end{smallmatrix}\right]^{\ddagger}$ are non-isomorphic $\bar{\mathbb{E}}$ -modules.

Now for each $\Delta$ in (4.20), we classify the $(\mathbb{F}_{4}^{\times}\times\mathbb{F}_{4}^{\times})$ -orbits of ${\mathfrak{S}}(\bar{\Delta})$ . For $i=1,2$ , let $\Delta_{i}$ be the $i$ -th column of $\Delta$ so that $\bar{\Delta}=\bar{\Delta}_{1}\times\bar{\Delta}_{2}$ . Let $\bar{M}\subseteq\bar{\Delta}$ be an $\bar{\mathbb{E}}$ -submodule, and $\operatorname{pr}_{i}:\bar{M}\to\bar{\Delta}_{i}$ be the projection map to the factor $\bar{\Delta}_{i}$ . Then $(\bar{\mathbb{E}}\times\bar{\mathbb{E}})\bar{M}=\bar{\Delta}$ if and only if both $\operatorname{pr}_{i}$ are surjective. Thus

(4.25)

\dim_{\mathbb{F}_{4}}\bar{M}\geq 2\quad\text{for every}\quad\bar{M}\in{\mathfrak{S}}(\bar{\Delta}).

Suppose that $\bar{M}\in{\mathfrak{S}}(\bar{\Delta})$ from now on.

If $\dim_{\mathbb{F}_{4}}\bar{M}=2$ , then both $\operatorname{pr}_{i}$ are $\bar{\mathbb{E}}$ -isomorphisms, and $\bar{M}$ is the graph of the isomorphism $\operatorname{pr}_{2}\circ\operatorname{pr}_{1}^{-1}:\bar{\Delta}_{1}\to\bar{\Delta}_{2}$ . Necessarily, $\Delta$ is equal to either $\left[\begin{smallmatrix}\mathbb{Z}_{4}&\mathbb{Z}_{4}\\ 2\mathbb{Z}_{4}&2\mathbb{Z}_{4}\end{smallmatrix}\right]$ or $\left[\begin{smallmatrix}\mathbb{Z}_{4}&\mathbb{Z}_{4}\\ \mathbb{Z}_{4}&\mathbb{Z}_{4}\end{smallmatrix}\right]$ . In these two cases, any isomorphism $\bar{\Delta}_{1}\to\bar{\Delta}_{2}$ is a scalar multiplication by $\mathbb{F}_{4}^{\times}$ . After a suitable multiplication by an element of $\mathbb{F}_{4}^{\times}\times\mathbb{F}_{4}^{\times}$ , we may identify $\bar{M}$ with the diagonal of $\bar{\Delta}_{1}\times\bar{\Delta}_{2}$ .

Next, suppose that $\dim_{\mathbb{F}_{4}}\bar{M}=3$ . Then $\ker(\operatorname{pr}_{1})$ is a $1$ -dimensional submodule of $\bar{M}\cap\bar{\Delta}_{2}$ , so it must coincide with the unique $1$ -dimensional submodule of $\bar{\Delta}_{2}$ . A similar result holds for $\ker(\operatorname{pr}_{2})$ . We claim that $\bar{\Delta}_{1}\simeq\bar{\Delta}_{2}$ in this case as well. If not, without lose of generality, we may assume that $\bar{\Delta}_{1}=\left[\begin{smallmatrix}\mathbb{F}_{4}\\ \mathbb{F}_{4}\end{smallmatrix}\right]^{\dagger}$ and $\bar{\Delta}_{2}=\left[\begin{smallmatrix}\mathbb{F}_{4}\\ \mathbb{F}_{4}\end{smallmatrix}\right]^{\ddagger}$ so that $\bar{\Delta}=\bar{\mathbb{E}}$ . By the above discussion, $\bar{M}\supseteq\left<\begin{smallmatrix}0&\mathbb{F}_{4}\\ \mathbb{F}_{4}&0\end{smallmatrix}\right>$ . Since $\dim_{\mathbb{F}_{4}}\bar{M}=3$ , there exists $u,v\in\mathbb{F}_{4}^{\times}$ such that

\bar{M}=\mathbb{F}_{4}\left<\begin{matrix}u&0\\ 0&v\end{matrix}\right>+\mathbb{F}_{4}\left<\begin{matrix}0&0\\ 1&0\end{matrix}\right>+\mathbb{F}_{4}\left<\begin{matrix}0&1\\ 0&0\end{matrix}\right>.

Indeed, both $u$ and $v$ have to be nonzero since $\operatorname{pr}_{i}$ is surjective for each $i=1,2$ . However, $\left<\begin{smallmatrix}u&0\\ 0&v\end{smallmatrix}\right>\in\bar{\mathbb{E}}^{\times}$ , which implies that $\bar{M}=\bar{\mathbb{E}}=\bar{\Delta}$ . This contradicts $\dim_{\mathbb{F}_{4}}\bar{M}=3$ and verifies our claim. It remains to consider the cases $\bar{\Delta}=\left[\begin{smallmatrix}\mathbb{F}_{4}\\ \mathbb{F}_{4}\end{smallmatrix}\right]^{\dagger}\times\left[\begin{smallmatrix}\mathbb{F}_{4}\\ \mathbb{F}_{4}\end{smallmatrix}\right]^{\dagger}$ or $\bar{\Delta}=\left[\begin{smallmatrix}\mathbb{F}_{4}\\ \mathbb{F}_{4}\end{smallmatrix}\right]^{\ddagger}\times\left[\begin{smallmatrix}\mathbb{F}_{4}\\ \mathbb{F}_{4}\end{smallmatrix}\right]^{\ddagger}$ . For simplicity, we write them as $\operatorname{Mat}_{2}(\mathbb{F}_{4})^{\dagger}$ and $\operatorname{Mat}_{2}(\mathbb{F}_{4})^{\ddagger}$ respectively. Suppose that $\bar{\Delta}=\operatorname{Mat}_{2}(\mathbb{F}_{4})^{\dagger}$ . Then there exists $u,v\in\mathbb{F}_{4}^{\times}$ such that

\bar{M}=\mathbb{F}_{4}\begin{bmatrix}u&v\\ 0&0\end{bmatrix}+\mathbb{F}_{4}\begin{bmatrix}0&0\\ 1&0\end{bmatrix}+\mathbb{F}_{4}\begin{bmatrix}0&0\\ 0&1\end{bmatrix}.

Multiplication by $(u^{-1},v^{-1})\in\mathbb{F}_{4}^{\times}\times\mathbb{F}_{4}^{\times}$ sends $\bar{M}$ to

\bar{M}_{0}:=\left\{\begin{bmatrix}a&b\\ c&d\end{bmatrix}\in\operatorname{Mat}_{2}(\mathbb{F}_{4})^{\dagger}\,\middle|\,a=b\right\}.

Thus all 3-dimensional members of ${\mathfrak{S}}(\operatorname{Mat}_{2}(\mathbb{F}_{4})^{\dagger})$ are in the same $(\mathbb{F}_{4}^{\times}\times\mathbb{F}_{4}^{\times})$ -orbit. A similar result holds for $\bar{\Delta}=\operatorname{Mat}_{2}(\mathbb{F}_{4})^{\ddagger}$ .

Lastly, if $\dim_{\mathbb{F}_{4}}\bar{M}=4$ , then $\bar{M}=\bar{\Delta}$ .

Combining (4.20), (4.22) and the above classifications, we find that

\lvert{\mathscr{L}}_{2}({\underline{n}})\rvert=\sum_{\Delta}\lvert(\mathbb{F}_{4}^{\times}\times\mathbb{F}_{4}^{\times})\backslash{\mathfrak{S}}(\bar{\Delta})\rvert=3+3+1+1=8.

Thus $o(3,6)=8$ if $p=2$ according to Lemma 4.3. ∎

Proposition 4.6.

Suppose that $p\in\{2,3\}$ . Then

(4.26)

o(2,2p)=\begin{cases}2&\text{if }p=2,\\ 3&\text{if }p=3.\end{cases}

Proof.

Let ${\underline{n}}=(n_{1},n_{2})=(2,2p)$ for $p\in\{2,3\}$ . Then $K_{\underline{n}}=\mathbb{Q}\times K_{n_{2}}$ , and ${\mathscr{K}}_{\underline{n}}\simeq D\times\operatorname{Mat}_{2}(K_{n_{2}})$ by (4.1) and (4.2). Here $K_{n_{2}}$ is equal to $\mathbb{Q}(\,\sqrt[]{-1}\,)$ or $\mathbb{Q}(\,\sqrt[]{-3}\,)$ according to whether $p=2$ or $p=3$ . Let ${\mathfrak{p}}$ be the unique ramified prime ideal of $A_{n_{2}}$ above $p$ . From [28, (5.5) and (5.6)], we have

(4.27)

A_{\underline{n}}=\{(a,b)\in\mathbb{Z}\times A_{n_{2}}\mid a\equiv b\pmod{{\mathfrak{p}}}\},

which is a suborder of index $p$ in $O_{K_{\underline{n}}}=\mathbb{Z}\times A_{n_{2}}$ .

From (4.3), $S({\underline{n}},p)=\{p\}$ . Clearly, ${\mathcal{O}}_{p}$ is canonically a subring of ${\mathscr{A}}_{n_{2},p}=A_{n_{2}}\otimes_{\mathbb{Z}}{\mathcal{O}}_{p}$ , and $({\mathfrak{p}}{\mathscr{A}}_{n_{2},p})\cap{\mathcal{O}}_{p}=p{\mathcal{O}}_{p}$ . It follows from (4.27) that

(4.28)

{\mathscr{A}}_{{\underline{n}},p}=\{(x,y)\in{\mathcal{O}}_{p}\times{\mathscr{A}}_{n_{2},p}\mid x\equiv y\pmod{{\mathfrak{p}}{\mathscr{A}}_{n_{2},p}}\}.

For simplicity, let us put ${\mathscr{C}}:={\mathcal{O}}_{p}\times{\mathscr{A}}_{n_{2},p}$ , and $J:={\mathfrak{J}}(O_{K_{\underline{n}},p})=p\mathbb{Z}_{p}\times{\mathfrak{p}}$ , where ${\mathfrak{p}}$ denotes the maximal ideal of $A_{n_{2},p}$ by an abuse of notation. Then $J{\mathscr{C}}\subseteq{\mathscr{A}}_{{\underline{n}},p}$ , so we further define three quotient rings

(4.29)	$\displaystyle\bar{\mathcal{O}}_{p}$	$\displaystyle:={\mathcal{O}}_{p}/p{\mathcal{O}}_{p},$
(4.30)	$\displaystyle\bar{\mathscr{C}}$	$\displaystyle:={\mathscr{C}}/(J{\mathscr{C}})=({\mathcal{O}}_{p}/p{\mathcal{O}}_{p})\times({\mathscr{A}}_{n_{2},p}/{\mathfrak{p}}{\mathscr{A}}_{n_{2},p})=\bar{\mathcal{O}}_{p}\times\bar{\mathcal{O}}_{p},$
(4.31)	$\displaystyle\bar{\mathscr{A}}_{{\underline{n}},p}$	$\displaystyle:={\mathscr{A}}_{{\underline{n}},p}/(J{\mathscr{C}})=\bar{\mathcal{O}}_{p},$

where $\bar{\mathscr{A}}_{{\underline{n}},p}$ embeds into $\bar{\mathscr{C}}$ diagonally by (4.28). From [22, Corollary II.1.7], ${\mathcal{O}}_{p}$ contains a copy of $\mathbb{Z}_{p^{2}}$ , and there exists $\eta\in{\mathcal{O}}_{p}$ such that

(4.32)

{\mathcal{O}}_{p}=\mathbb{Z}_{p^{2}}+\mathbb{Z}_{p^{2}}\eta,\qquad\eta^{2}=p,\quad\text{and}\quad x\eta=\eta\tilde{x},\quad\forall x\in\mathbb{Z}_{p^{2}}.

Here $x\mapsto\tilde{x}$ denotes the unique nontrivial $\mathbb{Q}_{p}$ -automorphism of $\mathbb{Q}_{p^{2}}$ . If we write $\bar{\eta}$ for the canonical image of $\eta$ in $\bar{\mathcal{O}}_{p}$ , then

(4.33)

\bar{\mathcal{O}}_{p}=\mathbb{F}_{p^{2}}+\mathbb{F}_{p^{2}}\bar{\eta}.

In particular, $\dim_{\mathbb{F}_{p^{2}}}\bar{\mathcal{O}}_{p}=2$ , and the Jacobson radical ${\mathfrak{J}}(\bar{\mathcal{O}}_{p})=\mathbb{F}_{p^{2}}\bar{\eta}$ is the unique $1$ -dimensional submodule of $\bar{\mathcal{O}}_{p}$ .

From (4.1) and (4.2), $V_{n_{1}}$ is a free module of rank $1$ over ${\mathscr{K}}_{n_{1}}\simeq D$ , and $V_{n_{2}}=K_{n_{2}}^{2}$ is a simple module over ${\mathscr{K}}_{n_{2}}\simeq\operatorname{Mat}_{2}(K_{n_{2}})$ . Fix a suitable identification of ${\mathscr{K}}_{n_{2},p}=\operatorname{Mat}_{2}(K_{n_{2},p})$ so that the hereditary closure ${\mathcal{H}}({\mathscr{A}}_{n_{2},p})$ is equal to $\operatorname{Mat}_{2}(A_{n_{2},p})$ . From Lemma 3.8, every ${\mathscr{C}}$ -lattice in $V\otimes\mathbb{Q}_{p}$ is isomorphic to

(4.34)

\Delta:=\Delta_{1}\times\Delta_{2}={\mathcal{O}}_{p}\times\begin{bmatrix}A_{n_{2},p}\\ A_{n_{2},p}\end{bmatrix}.

Put $\bar{\Delta}:=\Delta/J\Delta=\bar{\Delta}_{1}\times\bar{\Delta}_{2}$ , where

(4.35)

\bar{\Delta}_{1}={\mathcal{O}}_{p}/p{\mathcal{O}}_{p}=\bar{\mathcal{O}}_{p},\quad\text{and}\quad\bar{\Delta}_{2}=\Delta_{2}/{\mathfrak{p}}\Delta_{2}.

The same proof as (4.22) shows that there is a bijection

(4.36)

\left\{\parbox{86.72377pt}{Isomorphism classes of ${\mathscr{A}}_{{\underline{n}},p}$-lattices in $V\otimes\mathbb{Q}_{p}$}\right\}\longleftrightarrow\left\{\parbox{130.08621pt}{$\operatorname{End}_{{\mathscr{C}}}(\Delta)^{\times}$-equivalent classes of $\bar{\mathscr{A}}_{{\underline{n}},p}$-submodules $\bar{M}$ in $\bar{\Delta}$ such that $\bar{\mathscr{C}}\bar{M}=\bar{\Delta}$}\right\}.

Clearly, $\dim_{\mathbb{F}_{p}}\bar{\Delta}_{2}=2$ . When regarded as an ${\mathscr{A}}_{n_{2},p}$ -module, $\bar{\Delta}_{2}$ is isomorphic to the unique simple ${\mathscr{A}}_{n_{2},p}$ -module $\mathbb{F}_{p^{2}}$ by (4.30) and (4.33). We fix such an isomorphism and write $\bar{\Delta}_{2}=\mathbb{F}_{p^{2}}$ . Note that $\operatorname{End}_{{\mathscr{C}}}(\Delta)={\mathcal{O}}_{p}\times A_{n_{2},p}$ , so the action of $\operatorname{End}_{{\mathscr{C}}}(\Delta)^{\times}$ on $\bar{\Delta}$ factors through $\bar{\mathcal{O}}_{p}^{\times}\times\mathbb{F}_{p}^{\times}$ . Recall that $\bar{\mathscr{C}}=\bar{\mathcal{O}}_{p}\times\bar{\mathcal{O}}_{p}$ and $\bar{\mathscr{A}}_{{\underline{n}},p}=\bar{\mathcal{O}}_{p}$ by (4.30) and (4.31).

Let $\bar{M}\subseteq\bar{\Delta}$ be an $\bar{\mathcal{O}}_{p}$ -submodule, and $\operatorname{pr}_{i}:\bar{M}\to\Delta_{i}$ be the canonical projections for $i=1,2$ . Then $(\bar{\mathcal{O}}_{p}\times\bar{\mathcal{O}}_{p})\bar{M}=\bar{\Delta}$ if and only if both $\operatorname{pr}_{i}$ are surjective. Suppose that this is the case. Then necessarily

\dim_{\mathbb{F}_{p^{2}}}\bar{M}\geq\dim_{\mathbb{F}_{p^{2}}}\bar{\Delta}_{1}=2.

If $\dim_{\mathbb{F}_{p^{2}}}\bar{M}=3$ , then $\bar{M}=\bar{\Delta}$ .

Now suppose that $\dim_{\mathbb{F}_{p^{2}}}\bar{M}=2$ . Then $\ker(\operatorname{pr}_{2})$ is a $1$ -dimensional submodule of $\bar{\Delta}_{1}$ , so $\ker(\operatorname{pr}_{2})=\mathbb{F}_{p^{2}}\bar{\eta}$ . Therefore, there exists $a,c\in\mathbb{F}_{p^{2}}^{\times}$ such that

(4.37)

\bar{M}=\mathbb{F}_{p^{2}}(\bar{\eta},0)+\mathbb{F}_{p^{2}}(a,c)\subseteq\bar{\Delta}=(\mathbb{F}_{p^{2}}+\mathbb{F}_{p^{2}}\bar{\eta})\times\mathbb{F}_{p^{2}}.

Indeed, both $a,c$ have to be nonzero since $\operatorname{pr}_{i}$ is surjective for each $i=1,2$ . Multiplication by $(ca^{-1},1)\in\bar{\mathcal{O}}_{p}^{\times}\times\mathbb{F}_{p}^{\times}$ sends $\bar{M}$ to the following submodule of $\bar{\Delta}$ :

(4.38)

\bar{\Gamma}:=\{(a+b\bar{\eta},c)\in\bar{\Delta}\mid a=c\}.

Let $\Gamma$ be the unique ${\mathscr{A}}_{{\underline{n}},p}$ -sublattice of $\Delta$ such that $\Gamma\supseteq J\Delta$ and $\Gamma/J\Delta=\bar{\Gamma}$ . Then every ${\mathscr{A}}_{{\underline{n}},p}$ -lattice in $V\otimes\mathbb{Q}_{p}$ is isomorphic to either $\Delta$ or $\Gamma$ . Let ${\mathfrak{P}}$ be the unique two-sided prime ideal of ${\mathcal{O}}_{p}$ . We compute

(4.39)

\begin{split}\operatorname{End}_{{\mathscr{A}}_{{\underline{n}},p}}(\Gamma)&=\{(x,y)\in{\mathcal{O}}_{p}\times A_{n_{2},p}\mid(x,y)\bar{\Gamma}\subseteq\bar{\Gamma}\}\\ &=\{(x,y)\in{\mathcal{O}}_{p}\times A_{n_{2},p}\mid x\equiv y\pmod{{\mathfrak{P}}}\}.\end{split}

Here ${\mathfrak{P}}\cap A_{n_{2},p}={\mathfrak{p}}$ for any embedding of $A_{n_{2},p}$ into ${\mathcal{O}}_{p}$ , so the congruence relation does not depend on the choice of such an embedding.

We have seen from above that there are two genera of ${\mathscr{A}}_{\underline{n}}$ -lattices in $V$ . According to [22, Proposition V.3.1], $h({\mathcal{O}})=1$ for $p\in\{2,3\}$ , so ${\mathcal{O}}$ is the unique maximal order in $D$ up to $D^{\times}$ -conjugation. If $\widetilde{\Delta}$ is an ${\mathscr{A}}_{\underline{n}}$ -lattices in $V$ with $\widetilde{\Delta}\otimes\mathbb{Z}_{p}=\Delta$ , then

(4.40)

\operatorname{End}_{{\mathscr{A}}_{\underline{n}}}(\widetilde{\Delta})\simeq{\mathcal{O}}\times A_{n_{2}}.

Similarly, if $\widetilde{\Gamma}$ is the unique ${\mathscr{A}}_{\underline{n}}$ -sublattice of $\widetilde{\Delta}$ such that $\widetilde{\Gamma}\otimes\mathbb{Z}_{p}=\Gamma$ , then $\operatorname{End}_{{\mathscr{A}}_{\underline{n}}}(\widetilde{\Gamma})$ is the unique suborder of $\operatorname{End}_{{\mathscr{A}}_{\underline{n}}}(\widetilde{\Delta})$ satisfying

(4.41)

\operatorname{End}_{{\mathscr{A}}_{\underline{n}}}(\widetilde{\Gamma})\otimes\mathbb{Z}_{\ell}=\begin{cases}\operatorname{End}_{{\mathscr{A}}_{\underline{n}},p}(\Gamma)&\text{if }\ell=p,\\ \operatorname{End}_{{\mathscr{A}}_{\underline{n}}}(\widetilde{\Delta})\otimes\mathbb{Z}_{\ell}&\text{otherwise}.\end{cases}

For simplicity, let us put ${\mathfrak{O}}:=\operatorname{End}_{{\mathscr{A}}_{\underline{n}}}(\widetilde{\Delta})$ and ${\mathfrak{R}}=\operatorname{End}_{{\mathscr{A}}_{\underline{n}},p}(\Gamma)$ . From the general strategy explained in Section 2, we have

(4.42)

o(2,2p)=h({\mathfrak{O}})+h({\mathfrak{R}})\qquad\text{for }p=2,3.

Here $h({\mathfrak{O}})=h({\mathcal{O}})h(A_{n_{2}})=1\cdot 1=1$ . It remains to compute $h({\mathfrak{R}})$ .

Replacing $\widetilde{\Delta}$ by $g\widetilde{\Delta}$ for a suitable $g\in{\mathscr{E}}_{\underline{n}}^{\times}$ if necessary, we assume that ${\mathfrak{O}}={\mathcal{O}}\times A_{n_{2}}$ . Let $\widehat{\mathfrak{O}}$ (resp. $\widehat{\mathfrak{R}}$ ) be the profinite completion of ${\mathfrak{O}}$ (resp. ${\mathfrak{R}}$ ). We apply [28, (6.3)] to obtain

(4.43)

h({\mathfrak{R}})=\lvert{\mathfrak{O}}^{\times}\backslash\widehat{\mathfrak{O}}^{\times}/\widehat{\mathfrak{R}}^{\times}\rvert=\lvert{\mathfrak{O}}^{\times}\backslash{\mathfrak{O}}_{p}^{\times}/{\mathfrak{R}}_{p}^{\times}\rvert.

From (4.39), ${\mathfrak{P}}\times{\mathfrak{p}}\subseteq{\mathfrak{R}}_{p}$ , and

{\mathfrak{O}}_{p}/({\mathfrak{P}}\times{\mathfrak{p}})=\mathbb{F}_{p^{2}}\times\mathbb{F}_{p},\qquad{\mathfrak{R}}_{p}/({\mathfrak{P}}\times{\mathfrak{p}})=\mathbb{F}_{p}.

where $\mathbb{F}_{p}$ embeds into $\mathbb{F}_{p^{2}}\times\mathbb{F}_{p}$ diagonally. It follows that ${\mathfrak{O}}_{p}^{\times}/{\mathfrak{R}}_{p}^{\times}$ can be further simplified into $(\mathbb{F}_{p^{2}}^{\times}\times\mathbb{F}_{p}^{\times})/\operatorname{diag}(\mathbb{F}_{p}^{\times})$ . On the other hand, ${\mathfrak{O}}^{\times}={\mathcal{O}}^{\times}\times A_{n_{2}}^{\times}$ . Since $p\in\{2,3\}$ , the natural map ${\mathfrak{O}}^{\times}\to\mathbb{F}_{p^{2}}^{\times}\times\mathbb{F}_{p}^{\times}$ sends $\{\pm 1\}\times A_{n_{2}}^{\times}$ onto $\mathbb{F}_{p}^{\times}\times\mathbb{F}_{p}^{\times}$ , which contains $\operatorname{diag}(\mathbb{F}_{p}^{\times})$ . Therefore,

(4.44)

\begin{split}h({\mathfrak{R}})&=\lvert{\mathfrak{O}}^{\times}\backslash{\mathfrak{O}}_{p}^{\times}/{\mathfrak{R}}_{p}^{\times}\rvert=\lvert({\mathcal{O}}^{\times}\times A_{n_{2}}^{\times})\backslash(\mathbb{F}_{p^{2}}^{\times}\times\mathbb{F}_{p}^{\times})/\operatorname{diag}(\mathbb{F}_{p}^{\times})\rvert\\ &=\lvert({\mathcal{O}}^{\times}\times\{1\})\backslash(\mathbb{F}_{p^{2}}^{\times}\times\mathbb{F}_{p}^{\times})/(\mathbb{F}_{p}^{\times}\times\mathbb{F}_{p}^{\times})\rvert=\lvert{\mathcal{O}}^{\times}\backslash\mathbb{F}_{p^{2}}^{\times}/\mathbb{F}_{p}^{\times}\rvert=\lvert{\mathcal{O}}^{\times}\backslash\mathbb{F}_{p^{2}}^{\times}\rvert.\end{split}

Here we have used freely the commutativity of $\mathbb{F}_{p^{2}}^{\times}\times\mathbb{F}_{p}^{\times}$ .

If $p=2$ , we have already shown in the proof of Lemma 4.2 that the canonical map ${\mathcal{O}}^{\times}\to({\mathcal{O}}/2{\mathcal{O}})^{\times}$ is surjective (see (4.11)). Consequently, ${\mathcal{O}}^{\times}\to({\mathcal{O}}_{p}/{\mathfrak{P}})^{\times}=\mathbb{F}_{4}^{\times}$ is surjective as well. Thus $h({\mathfrak{R}})=1$ if $p=2$ .

Next, suppose that $p=3$ . According to [22, Exercise III.5.2],

D=\left(\frac{-1,-3}{\mathbb{Q}}\right),\quad\text{and}\quad{\mathcal{O}}=\mathbb{Z}+\mathbb{Z}i+\mathbb{Z}\frac{1+j}{2}+\mathbb{Z}\frac{i(1+j)}{2}.

Here $\{1,i,j,ij\}$ is the standard $\mathbb{Q}$ -basis of $D$ . We have

{\mathcal{O}}^{\times}=\left\{\pm 1,\quad\pm i,\quad\pm\frac{1\pm j}{2},\quad\pm\frac{i(1\pm j)}{2}\right\}.

Note that $j\in{\mathfrak{P}}$ , so the image of ${\mathcal{O}}^{\times}$ in $({\mathcal{O}}_{p}/{\mathfrak{P}})^{\times}=\mathbb{F}_{9}^{\times}$ is equal to $\{\pm\bar{1},\pm\bar{i}\}$ , which has index $2$ in $\mathbb{F}_{9}^{\times}$ . Therefore, $h({\mathfrak{R}})=2$ if $p=3$ .

In conclusion, we find that $o(2,4)=1+1=2$ if $p=2$ , and $o(2,6)=1+2=3$ if $p=3$ . ∎

Remark 4.7.

Assume that $p\in\{2,3\}$ . We have shown that except for $({\underline{n}},p)=((2,6),3)$ (and $((1,3),3)$ by [28, Remark 3.2(i)]), every genus in the set ${\mathscr{L}}({\underline{n}})$ of lattice classes has class number one. For the exceptional cases, ${\mathscr{L}}({\underline{n}})$ consists of two genera; one genus has class number one while the other one has class number two.

Concluding the Proof of Theorem 1.1.

For $p=2,3,5$ , the value of $o({\underline{n}})$ is listed in the following table (see [28] for the values not covered in the present paper):

${\underline{n}}$	$3$	$4$	$5$	$8$	$12$	$(1,2)$	$(2,3)$	$(2,4)$	$(2,6)$	$(3,4)$	$(3,6)$
$p=2$	$3$	$2$	$2$	$1$	$3$	$3$	$2$	$2$	$4$	$2$	$8$
$p=3$	$2$	$3$	$2$	$4$	$3$	$3$	$1$	$4$	$3$	$2$	$2$
$p=5$	$3$	$1$	$1$	$4$	$4$	$4$	$2$	$0$	$6$	$0$	$8$

Formula (1.6) is obtained by plugging in the above values in (1.3). ∎

Acknowledgments

J. Xue is partially supported by Natural Science Foundation of China grant #11601395. C.-F. Yu is partially supported by the MoST grant 109-2115-M-001-002-MY3.

References

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On Superspecial abelian surfaces over finite fields III

Abstract.

Key words and phrases:

2020 Mathematics Subject Classification:

1. Introduction

Theorem 1.1.

Remark 1.2.

2. General strategy for computing o​(n¯)o({\underline{n}})

Remark 2.1.

3. The isotypic case

3.1. Classification of lattices over local quaternion Bass orders

Definition 3.1.

Lemma 3.2.

Theorem 3.3.

Definition 3.4.

Lemma 3.5.

Lemma 3.6.

Proof.

Proposition 3.7.

3.2. Explicit computations

Lemma 3.8.

Proof.

Lemma 3.9.

Proof.

Lemma 3.10.

Proof.

Remark 3.11.

4. The non-isotypic case

Lemma 4.1.

Proof.

Lemma 4.2.

Proof.

Lemma 4.3.

Proof.

Lemma 4.4.

Proof.

Proposition 4.5.

Proof.

Proposition 4.6.

Proof.

Remark 4.7.

Concluding the Proof of Theorem 1.1.

Acknowledgments

References

2. General strategy for computing $o({\underline{n}})$