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\title[Selective Orders in Central Simple Algebras] {Selective Orders in Central Simple Algebras}

\author {Benjamin Linowitz}

\address{Department of Mathematics\\ 
6188 Kemeny Hall\\
Dartmouth College\\
Hanover, NH 03755}

\email[] {benjamin.linowitz@dartmouth.edu}
\urladdr{http://www.math.dartmouth.edu/~linowitz/ }



\author {Thomas R. Shemanske}

\address{Department of Mathematics\\ 
6188 Kemeny Hall\\
Dartmouth College\\
Hanover, NH 03755}

\email[] {thomas.r.shemanske@dartmouth.edu}
\urladdr{http://www.math.dartmouth.edu/~trs/ }

\date{\today}



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\begin{document}
\subjclass[2010] {Primary 11R54; Secondary 11S45, 20E42}

\keywords{Order, central simple algebra, affine building, embedding}


\begin{abstract}
  Let $B$ be a central simple algebra of degree $n$ over a number
  field $K$, and let $L/K$ be a field extension of degree $n$ which
  embeds in $B$.  The question of which of the isomorphism classes of
  maximal orders in $B$ admit an embedding of an $\O_K$-suborder
  $\Omega$ of $\O_L$ is the question of selectivity of the order
  $\Omega$. In this paper we are concerned with algebras of degree $n \ge 3$,
  and continue the work of several authors to characterize the degree
  to which and conditions under which selectivity will occur.  We take
  a local approach via the theory of affine buildings which provides
  explicit information about the structure of local embeddings, and
  leverage that information to produce a global characterization.  We
  clarify that selectivity can never occur in an algebra which is a
  division algebra at a finite prime as well as construct
  representatives of the isomorphism classes of maximal orders which do admit
  embeddings of $\Omega$.  An example of selectivity in an algebra
  with partial ramification is provided.
\end{abstract}




\maketitle


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\section{Introduction}


Let $K$ be a number field and $B$ a central simple $K$-algebra. For
each prime $\nu$ of $K$ the Wedderburn structure theorem implies that
there is an isomorphism of local algebras $B_\nu = B\otimes_K K_\nu
\cong M_{\kappa_\nu}(D_\nu)$, where $D_\nu$ is a central simple
division algebra over $K_\nu$ with index $m_\nu$.  The
Albert-Brauer-Hasse-Noether theorem allows one to easily determine
when a field extension of $K$ embeds into $B$:

\begin{nnthm} (ABNH)\label{thm:ABNH} Let $K$ be a number field and $B$
  a central simple algebra of dimension $n^2$ over its center
  $K$. Suppose that $[L:K] = n$.  There is an embedding of $L/K$ into
  $B$ if and only if for each prime $\nu$ of $K$ and for all primes
  $\fP$ of $L$ lying above $\nu$, $m_\nu \mid [L_\fP:K_\nu]$.
\end{nnthm}

It is natural to seek an integral analogue of the
Albert-Brauer-Hasse-Noether theorem.  Let $K,L$ be as above with rings
of integers $\O_K$ and $\O_L$ respectively.  Suppose that $K \subset L
\subset B$.  For any $\O_K$-order $\Omega \subseteq \O_L$, there
exists an $\O_K$-order $\R$ of rank $n$ in $B$ which contains $\Omega$
(see page 131 of \cite{Reiner-book}), and it is clear that any order
in the same isomorphism class of $\R$ admits an embedding of $\Omega$.
An integral analogue of the Albert-Brauer-Hasse-Noether theorem should
address the question of whether the orders in $B$ which are not
isomorphic to $\R$ but lie in the same genus (locally isomorphic at
all finite primes) also admit embeddings of $\Omega$.

Since it is clear that either every order in an isomorphism class of
orders in $B$ admits an embedding of $\Omega$ or none at all do, it is
a convenient abuse of language to say that an isomorphism class either
admits or does not admit an embedding of $\Omega$.  In the case that
not all isomorphism classes of maximal orders admit an embedding of
$\Omega$, we follow Chinburg and Friedman \cite{Chinburg-Friedman} and
call $\Omega$ selective.

The first integral refinement of the Albert-Brauer-Hasse-Noether
theorem was due to Chevalley \cite{Chevalley-book}. Let $K$ be a
number field, $B=M_n(K)$ and $L/K$ be a degree $n$ extension of
fields. Chevalley's elegant result is:

\begin{nnthm} (Chevalley)\label{thm:Chevalley} The ratio of the number
  of isomorphism classes of maximal orders in $B=M_n(K)$ into which
  $\O_L$ can be embedded to the total number of isomorphism classes of
  maximal orders is $[\widetilde{K}\cap L:K]^{-1}$ where
  $\widetilde{K}$ is the Hilbert class field of $K$.
\end{nnthm}



In 1999, Chinburg and Friedman \cite{Chinburg-Friedman} considered
quaternion algebras $B$ satisfying the Eichler condition and proved
that the proportion of isomorphism classes of maximal orders admitting
an embedding of a commutative, quadratic $\O_K$-order $\Omega$ contained
in an embedded quadratic extension of the center is either $1/2$ or
$1$. Their result was later extended to Eichler orders of arbitrary
level by Chan and Xu \cite{Chan-Xu} and independently by Guo and Qin
\cite{Guo-Qin}. Maclachlan \cite{Maclachlan} considered Eichler orders
of square-free level and proved the analogous theorem for optimal
embeddings. The first author of this paper gave a number of criteria
\cite{Linowitz-selectivity} which imply that $\Omega$ is not selective
with respect to the genus of any fixed order $\R$ of $B$.


Before continuing with the progress on this problem, we note that
selectivity has many interesting applications to hyperbolic
geometry. Suppose that $K\neq \Q$ is a totally real number field, $B$
is a quaternion algebra over $K$ in which a unique real prime splits,
and that $\R_1$ and $\R_2$ are maximal orders of $B$ representing
distinct isomorphism classes. Vigneras \cite{Vigneras-isospectral}
used $\R_1$ and $\R_2$ to construct non-isometric compact hyperbolic
$2$-manifolds $M_1$ and $M_2$. She further showed that a sufficient
condition for $M_1$ and $M_2$ to be isospectral with respect to the
Laplace-Beltrami operator was the nonexistence of a selective order
$\Omega$ which embedded into exactly one of $\{
\R_1,\R_2\}$. Vigneras' original examples of isospectral but not
isometric hyperbolic $2$-manifolds have enormous genus:
100,801. Recently Doyle, Voight and the first author of this paper
\cite{Doyle-Linowitz-Voight} have applied the results of
\cite{Chinburg-Friedman},\cite{Chan-Xu},\cite{Guo-Qin} in order to
produce isospectral but not isometric hyperbolic $2$-orbifolds with
underlying surface of genus $0$, and have proven that these orbifolds
are minimal among all isospectral but not isometric surfaces arising
from maximal arithmetic Fuchsian groups. Although we do not consider
applications to hyperbolic geometry in this paper, it is likely our
results can be applied in order to prove the isospectrality of
lattices in more general symmetric spaces.

The first work beyond Chevalley's in the non-quaternion setting was by
Arenas-Carmona \cite{Arenas-Carmona-2003}.  The setting was a central
simple algebra $B$ over a number field $K$ of dimension $n^2$, $n\ge
3$ with the proviso that at each finite prime $\nu$ of $K$, $B_\nu$ is
either $M_n(K_\nu)$ or a division algebra, that is $\nu$ either splits
in $B$ or is totally ramified.  He considered embeddings of the ring
of integers $\O_L$ of an extension $L/K$ of degree $n$ into maximal
orders of $B$ and establishes a proportion analogous to Chevalley's:
his Theorem 1 says that the ratio of the number of isomorphism classes
of maximal orders in $B$ into which $\O_L$ can be embedded to the
total number of isomorphism classes of maximal orders is $[\Sigma\cap
L:K]^{-1}$ where $\Sigma$ is a spinor class field.  In
Corollary~\ref{cor:fullyramified}, we show that if $B_\nu$ is a
division algebra for any prime $\nu$, this proportion is always one,
meaning (that restricted to algebras with no partial ramification), we
must have $B_\nu = M_n(K_\nu)$ for every finite prime $\nu$ in order
to realize selective orders.


In \cite{Linowitz-Shemanske-EmbeddingPrimeDegree}, we considered
central simple algebras $B$ of odd prime degree $p$, but considered
embedding nonmaximal orders $\Omega \subset \O_L$ into maximal orders
in $B$ and proved analogous results (Theorem 3.3 of
\cite{Linowitz-Shemanske-EmbeddingPrimeDegree}) refined to indicate
how the conductor of $\Omega$ affects selectivity.  In that paper we
noted the first instance of Corollary~\ref{cor:fullyramified},
precluding selectivity when $B$ was a degree $p$ division algebra.



Continuing with the history, in a recent preprint
\cite{Arenas-Carmona-2011-Representation-Fields}, Carmona extends his
results on spinor class fields to:
\begin{nnthm}
  Let $\mathfrak A$ be a central simple algebra over the number field
  $K$.  Let $\mathfrak H$ be an arbitrary commutative order, Let
  $\mathfrak D$ be a maximal order containing $\mathfrak H$.  For
  every maximal ideal $\wp$ in the ring of integers $\O_K$ we let
  $I_\wp$ be the only maximal two-sided ideal of $\mathfrak D$
  containing $\wp 1_{\mathfrak D}$, and let ${\mathbb H}_\wp$ be the
  image of $\mathfrak H$ in ${\mathfrak D}/I_\wp$.  Let $\mathbb
  E_\wp$ be the center of the ring ${\mathfrak D}/I_\wp$.  Let $t_\wp$
  be the greatest common divisor of the dimensions of the irreducible
  $\mathbb E_\wp$-representations of the algebra $\mathbb E_\wp
  \mathbb H_\wp$.  Then the representation field $F(\mathfrak H)$ is
  the maximal subfield $F$, of the spinor class field $\Sigma$, such
  that the inertia degree $f_\wp(F/K)$ divides $t_\wp$ for every place
  $\wp$.
\end{nnthm}






In this paper we have a number of goals.  While we work in a setting
slightly less general than
\cite{Arenas-Carmona-2011-Representation-Fields}, we produce results
which nonetheless refine several of those in
\cite{Arenas-Carmona-2011-Representation-Fields}, give an explicit
blueprint of how to realize the embeddings locally, and have
applications both to and which extend beyond the specific question of
selective orders.  We first note that
Corollary~\ref{cor:fullyramified} and
Example~\ref{ex:partialramification} provide significant clarification
to the circumstances in which selectivity can occur, in particular
refining the final corollaries of
\cite{Arenas-Carmona-2011-Representation-Fields}.  The key issue is
one of ramification in the algebra.

A central simple algebra $B$ of prime degree $p \ge 2$ over a number
field $K$ can only split or be a division algebra at any finite prime
of $K$, that is to say there is no partial ramification at the finite
primes.  When considering general central simple algebras of degree
$n\ge 3$ over $K$, partial ramification can occur.  The results of
\cite{Arenas-Carmona-2003} explicitly assume no partial ramification as
do the final corollaries of
\cite{Arenas-Carmona-2011-Representation-Fields} in stating results
about selectivity.  However, Corollary~\ref{cor:fullyramified} says
that selectivity never occurs if there is a finite prime $\nu$ of $K$
for which $B_\nu$ is a division algebra, which is to say that an
assumption of no partial ramification at the finite primes (in the
context of selectivity) is exactly the statement that the algebra must
split at all finite primes.  Example~\ref{ex:partialramification}
confirms selectivity is still possible in the case of partial
ramification.

We begin the paper with a detailed investigation of a local embedding
problem.  Since a central simple algebra $B/K$ is split at almost all
primes of $K$, we consider the generic case of embedding the global
ring of integers $\O_L$ into maximal orders in $M_n(k)$ where $k$ is a
non-archimedean local field.
Corollary~\ref{cor:embeddingisalocalquestion} shows that this is really a
local question:

\begin{nncor}Let $B$ be a central simple algebra over a number field
  $K$ of dimension $n^2$, and $L$ a degree $n$ field extension of $K$
  which embeds into $B$. Let $\fp$ be a prime of $K$ which splits in
  $B$ and is unramified in $L$, and let $\E_\fp$ a maximal order in
  $B_\fp$.  Then the following are equivalent:
  \begin{compactenum}
  \item $\O_L \subset \E_\fp$
  \item $\O_{L_\fP} \subset \E_\fp$ for some prime $\fP$ of $L$
    dividing $\fp\O_L$.
  \item $\oplus_{\fP\mid \fp} \O_{L_\fP} \subset \E_\fp$ where the sum
    is over all primes $\fP \mid \fp\O_L$.
  \end{compactenum}
\end{nncor}



Since every maximal order in $M_n(k)$ can be identified with a vertex
in the affine building for $SL_n(k)$, we are able to leverage both
algebraic and geometric properties of buildings to give a very
explicit characterization of these embeddings
(Theorem~\ref{thm:OLembedslocally}):

\begin{nnthm}
  Let $B$ be a central simple algebra over a number field $K$ of
  dimension $n^2$ and $L$ a degree $n$ field extension of $K$ which
  embeds into $B$. Let $\fp$ be a prime of $K$ which splits in $B$ and
  is unramified in $L$, and write $\fp \O_L = \fP_1\fP_2\cdots \fP_g$,
  where $f_i = f(\fP_i | \fp)$ are the associated inertia degrees.
  Then $\O_L$ is contained in the maximal orders of $B_\fp\cong
  M_n(K_\fp)$ represented by the homothety class $[\mathcal L] =[a_1,
  \dots, a_n] \in \Z^n/\Z(1,\dots,1)$ if and only if there are $\ell_i
  \in \Z$ such that $[\mathcal L] = [\underbrace{\ell_1,
    \dots,\ell_1}_{f_1},\underbrace{\ell_2, \dots,\ell_2}_{f_2},\dots,
  \underbrace{\ell_g, \dots,\ell_g}_{f_g}]$.
\end{nnthm}




To apply these local results to the problem of selectivity, we next
construct a set of representatives of the isomorphism classes of
maximal orders in $B$, i.e., representatives of the genus of $\R$,
using a local-global correspondence.  We associate to $\R$ a class
field, $K(\R)$, which for maximal orders $\R$ is contained in the
Hilbert class field of $K$.  The class field corresponds to a class
group derived from the normalizer of $\R$.  We show that key to
determining whether $\O_L$ embeds into a maximal order $\E$ in $B$ is
the arithmetic of $L_0 = L \cap K(\R)$.  More precisely, using the
local characterization of maximal orders via affine buildings, we
define an idelic analog, $\delta(\R, \E)$, of the distance ideal (see
\cite{Chinburg-Friedman}) between the fixed maximal order $\R$ and any
other maximal order $\E$.  We prove (Theorem~\ref{thm:EmbedviaArtin})
that

\begin{nnthm}
  $\E$ admits an embedding of $\O_L$ if and only if the idelic Artin
  symbol $(\delta(\R, \E),L_0/K)$ is trivial in $Gal(L_0/K)$.
\end{nnthm}

From this we deduce a selectivity result,
Corollary~\ref{cor:selectivity}:


\begin{nncor}The ratio of the number of isomorphism classes of maximal
  orders in $B$ which admit an embedding of  $\O_L$ to the total number
  of isomorphism classes of maximal orders is $[L_0: K]^{-1}$ where
  $L_0 = K(\R)\cap L$.
\end{nncor}


This corollary has the same shape as many previous results giving the
proportion (in terms of an index of fields) of the number of
isomorphism classes of maximal orders in $B$ which admit an embedding
of $\O_L$, but differs in the choice of class field which is leveraged
to produce the proportion: not the Hilbert class field of Chevalley,
nor the spinor class field of Arenas-Carmona. On the other hand, it is
true that for maximal orders $\R$, the class field $K(\R)$ is a
subfield of the Hilbert class field of $K$ having exponent $n$. The
above corollary implies Chevalley's theorem (in the case that
$B=M_n(K)$) and should be viewed as a generalization of Chevalley's
result to arbitrary central simple algebras of finite degree.  But the
concreteness of our construction allows us to sharpen the result
further, by computing the index in specific cases:
Corollary~\ref{cor:fullyramified} shows that if $B_\nu$ is a division
algebra at any finite prime, then there is no selectivity (i.e., the index
is one).  In particular if $\Omega \subseteq \O_L$ is any
$\O_K$-order, then $\Omega$ is never selective; that is, every maximal
order in $B$ admits an embedding of $\Omega$.  We conclude with
Example~\ref{ex:partialramification} produced using Magma
\cite{Magma}, which demonstrates that selectivity in an algebra with 
partial ramification is still possible.







\section{Local embeddings via buildings}\label{sec:localembeddings}


Let $L/K$ be a degree $n$ extension of number fields, $B/K$ a central
simple algebra of dimension $n^2$ which contains $L$, and $\fp$ a
prime of $K$ which splits in $B$ and is unramified in $L$.  In this
section we characterize explicitly all the maximal orders of $B_\fp$
which admit an embedding of $\O_L$.

Since $\O_L$ has rank $n$ over $K$, any embedding $\varphi$ of $\O_L$
into a maximal order $\E \subset B_\fp$ extends to an inner
automorphism of $B_\fp$ by Skolem-Noether, thus $\E$ contains a
conjugate of $\O_L$, or equivalently, $\O_L$ is contained in a
conjugate of $\E$.  We therefore begin by characterizing those maximal
orders which contain $\O_L$.

Fix a maximal order $\R$ of $B$ containing $\O_L$.  The prime $\fp$
splits in $B$, so that $B_\fp \cong M_n(K_\fp)$ hence all maximal
orders are conjugate ((17.3), (17.4) of \cite{Reiner-book}).  By a
choice of basis we may assume that $B_\fp$ is identified with
$M_n(K_\fp)$ so that $\R_\fp = M_n(\O_\fp)$.

Write $\fp \O_L = \fP_1\fP_2\cdots \fP_g$ as a product of primes in
$L$, and let $f_i = f(\fP_i | \fp)$ be the associated inertia degrees.
Of course we have $f_1 + \cdots + f_g = n$, and
%
\begin{equation}\label{eq:block_structure}
  L\otimes_K K_\fp \cong \oplus_{i=1}^g L_{\fP_i} \hookrightarrow
  \oplus_{i=1}^g M_{f_i}(K_\fp) \hookrightarrow M_n(K_\fp).
\end{equation}

In particular, since all the maximal orders of $M_{f_i}(K_\fp)$ are
conjugate, by a change of basis we adjust the embeddings $L_{\fP_i}
\hookrightarrow M_{f_i}(K_\fp)$ so that $\O_{\fP_i} \hookrightarrow
M_{f_i}(\O_\fp)$.  Thus

\begin{equation} \label{eq:integralblockstructure} \O_L \subset
  \O_L\otimes_{\O_K}\O_\fp \hookrightarrow \oplus_{i=1}^g \O_{\fP_i}
  \hookrightarrow \oplus_{i=1}^g M_{f_i}(\O_\fp) \hookrightarrow
  M_n(\O_\fp).
\end{equation}

Fix a uniformizing parameter $\pi$ of $\O_\fp$, and let $D_k^\ell =
\diag(\underbrace{\pi^\ell,\dots,\pi^\ell}_k,1,\dots,1)\in
M_n(K_\fp)$.  Thus
\begin{equation}\label{eq:blockstr}
  \R(k,\ell) := D_{k}^\ell M_n(\O_\fp) D_k^{-\ell} = 
  \begin{pmatrix}
    M_k(\O_\fp)& \fp^{\ell}M_{k\times n-k}(\O_\fp)\\
    \fp^{-\ell}M_{n-k\times k}(\O_\fp)& M_{n-k}(\O_\fp)\\
  \end{pmatrix}\subset M_n(K_\fp).
\end{equation}



Note that $\R(0,\ell) = \R(n,\ell) =\R(k,0) =\R_\fp= M_n(\O_\fp)$.
From equations~(\ref{eq:integralblockstructure}),(\ref{eq:blockstr})
above, it is evident that for all $\ell_1, \dots, \ell_g \in \Z$,
\begin{equation}
  \O_L \subset \R(f_1,\ell_1) \ \cap \R(f_1+f_2,\ell_2) 
  \cap \cdots \cap \R(f_1+\cdots+f_g,\ell_g), 
\end{equation}
that is,
\begin{equation}\label{eq:orderscontainingomega}
  \O_L \subset \bigcap_{\ell_i \in \Z}\big[\R(f_1,\ell_1) \ \cap 
  \R(f_1+f_2,\ell_2) \cap \cdots \cap \R(f_1+\cdots+f_g,\ell_g)\big] = 
  \bigoplus_{i=1}^g M_{f_i}(\O_\fp).
\end{equation}

We now translate this to the language of affine buildings.  We have
picked a basis $\{\omega_1, \dots, \omega_n\}$ of $K_\fp^n$ (and hence
in particular fixed an apartment of the building associated to
$SL_n(K_\fp)$), so that with respect to this basis, $\R_\fp =
M_n(\O_\fp) = \End(\Lambda)$, where $\Lambda = \oplus_{i=1}^n \O_\fp
\omega_i$. Also we have $\R(k,\ell) = \End(\M(k,\ell))$ where
$\M(k,\ell) = \oplus_{i=1}^{k}\O_\fp \pi^\ell \omega_i \oplus
\oplus_{i={k+1}}^n\O_\fp\omega_i$ and $\pi$ is our fixed uniformizer
in $\O_\fp$.  As usual, this maximal order in $B_\fp$ can be
represented by the homothety class of the lattice $\M(k,\ell)$,
$[\M(k,\ell)] := [\underbrace{\ell,\dots,\ell}_k,0,\dots,0] \in
\Z^n/\Z(1,\dots,1)$.  Observe that $\R(k,\ell)$ has type $k\ell\pmod
n$ (see \cite{Brown}).

With the notation fixed as above, we characterize precisely which
maximal orders in this apartment contain $\O_L$. 


\begin{thm}\label{thm:OLembedslocally}
  Let $B$ be a central simple algebra over a number field $K$ of
  dimension $n^2$ and $L$ a degree $n$ field extension of $K$ which
  embeds into $B$. Let $\fp$ be a prime of $K$ which splits in $B$ and
  is unramified in $L$, and write $\fp \O_L = \fP_1\fP_2\cdots \fP_g$,
  where $f_i = f(\fP_i | \fp)$ are the associated inertia degrees.
  Then $\O_L$ is contained in the maximal orders of $B_\fp$
  represented by the homothety class $[\mathcal L] =[a_1, \dots, a_n]
  \in \Z^n/\Z(1,\dots,1)$ if and only if there are $\ell_i \in \Z$
  such that $[\mathcal L] = [\underbrace{\ell_1,
    \dots,\ell_1}_{f_1},\underbrace{\ell_2, \dots,\ell_2}_{f_2},\dots,
  \underbrace{\ell_g, \dots,\ell_g}_{f_g}]$.
  
\end{thm}


The following corollary is immediate.

\begin{cor}\label{cor:verticesinfundamentalchamber} $\O_L$ is contained in maximal orders in $B_\fp$ of types
  $td$ where $d = \gcd(f_1, \dots, f_g)\mid n$ and $t = 1, 2, \dots,
  n/d$.  In particular in a standard fundamental chamber containing
  $\R_\fp \leftrightarrow [0,\dots,0]$, $\O_L$ is contained in the
  maximal orders corresponding to the vertices $v_0=[0,\dots,0]$, $v_1
  = [\underbrace{1, \dots,1}_{f_1},0, \dots, 0]$, $v_2 =
  [\underbrace{1, \dots,1}_{f_1+f_2},0,\dots, 0]$, \dots,
  $v_{g-1}=[\underbrace{1,\dots, 1}_{f_1+\cdots+f_{g-1}}, 0, \dots,0]$.
\end{cor}

\begin{remark}
Theorem~\ref{thm:OLembedslocally} answers the question of which
maximal orders contain $\O_L$. The answer to which orders 
admit an embedding of $\O_L$ follows quite easily:  If an order $\E$
admits an embedding of $\O_L$, then a conjugate of $\E$ contains
$\O_L$.  But conjugation just amounts to a change of basis and hence
a change of apartment in the building, so we can simply translate the
results for containment to those of embedding by
conjugation. 
\end{remark}


To proceed, we need a technical lemma.

\begin{lemma}\label{lemma:IntegralClosure}
  Let $R$ be the valuation ring of $\fp$ in $K$, and $S$ its integral
  closure in $L$.  Suppose that $\E_\fp$ is a maximal order in $B_\fp
  = M_n(K_\fp)$ and that $\O_L \subset \E_\fp$.  Then $S \subset \E_\fp$.
\end{lemma}


\begin{proof}
  By Corollary 5.22 of \cite{Atiyah-Macdonald}, $S$ is the
  intersection of all valuation rings of $L$ which contain $R$. The
  valuation ring $R$ is equal to the localization of $\O_K$ at the
  prime ideal $\fp$, and the valuation rings of $L$ which contain $R$
  are precisely the localizations of $\O_L$ at the primes $\fP_1,
  \dots, \fP_g$ of $L$ which lie above $\fp$. By p43 of
  \cite{Reiner-book}, the intersection of these localizations, $S$, is
  equal to the localization $T^{-1}\O_L$, where $T = \O_L \setminus
  (\fP_1\cup \cdots \cup \fP_g)$.  To show $S \subset \E_\fp$ we
  choose $\alpha/\beta \in S$ with $\alpha, \beta \in \O_L$ and $\beta
  \notin \fP_1\cup \cdots \cup \fP_g$.  We are assuming $\O_L \subset
  \E_\fp = \gamma_\fp M_n(\O_\fp) \gamma_\fp^{-1}$ for some
  $\gamma_\fp \in B_\fp^\times$, so write $\alpha = \gamma_\fp A
  \gamma_\fp^{-1}$ and $\beta = \gamma_\fp B \gamma_\fp^{-1}$ for $A,
  B \in M_n(\O_\fp)$. Finally recall that $\O_L \subset
  \O_L\otimes_{\O_K}\O_\fp \hookrightarrow \oplus_{i=1}^g \O_{\fP_i}
  \hookrightarrow \oplus_{i=1}^g M_{f_i}(\O_\fp) \hookrightarrow
  M_n(\O_\fp),$ so that $\det(\beta) =
  \prod_{i=1}^gN_{L_{\fP_i}/K_\fp}(\beta) = N_{L/K}(\beta)$. As $\beta
  \in \O_{\fP_i}^\times$ for each $i$, its local norm is in
  $\O_\fp^\times$.  Thus $\beta$ is a unit in $\E_\fp$ from which the
  lemma follows.
\end{proof}

\begin{proof}[Proof of Theorem]
  Consider equation~(\ref{eq:orderscontainingomega}).  We know that
  $\O_L$ is contained in $\R(f_1,\ell_1)\cap \R(f_1+f_2, \ell_2) \cap
  \cdots \cap \R(f_1+\cdots+f_g,\ell_g)$ for any choice of $\ell_i\in
  \Z$.  These orders correspond to homothety classes of lattices
  $[\M(f_1+\cdots+f_i,\ell_i)]=\ell_i [\M(f_1+\cdots+f_i,1)]=\ell_i
  [\underbrace{1,\dots,1}_{f_1+\cdots+f_i},0,\dots,0]$ as an element
  of $\Z^n/\Z(1,\dots,1)$.  In \cite{Ballantine-Rhodes-Shemanske}, it
  is shown that walks in an apartment are consistent with the natural
  group action on $\Z^n/\Z(1,\dots,1)$, and since by
  \cite{Shemanske-Split} the intersection of any finite number of
  maximal orders (containing $\O_\fp^n$) in an apartment is the same
  as the intersection of all the maximal orders in the convex hull they
  determine, we deduce that $\O_L$ is contained in maximal orders
  corresponding to
  \begin{align*}
    [\M(f_1,\ell_1) &+ \M(f_1+f_2, \ell_2)
    + \cdots + \M(f_1 + \cdots + f_g,\ell_g)] = \\
    & [\underbrace{\ell_1+\cdots+\ell_g,
      \dots,\ell_1+\cdots+\ell_g}_{f_1},\underbrace{\ell_2+\cdots+\ell_g,
      \dots,\ell_2+\cdots+\ell_g}_{f_2},\dots, \underbrace{\ell_g,
      \dots,\ell_g}_{f_g}].
  \end{align*}

  Since the $\ell_i\in \Z$ are arbitrary, a simple change of variable
  ($\ell_k + \cdots+ \ell_g \mapsto \ell_k$) shows that $\O_L$ is
  contained in the maximal orders specified in the proposition.  We
  now show these are the only maximal orders in the apartment which
  contain $\O_L$.

  Suppose that $\O_L$ is contained in a maximal order $\Lambda(a_1,
  \dots, a_n)$ where
  \begin{align*}
    \Lambda(a_1, \dots, a_n) &= \diag(\pi^{a_1}, \dots,
  \pi^{a_n})M_n(\O_\fp)\diag(\pi^{a_1}, \dots, \pi^{a_n})^{-1} =\\
  &\begin{pmatrix}
    \O_\fp&\fp^{a_1 - a_2}&\fp^{a_1-a_3}& \dots&\fp^{a_1-a_n}\\
    \fp^{a_2-a_1}&\O_\fp&\fp^{a_2-a_3}&\dots&\fp^{a_2 - a_n}\\
    \fp^{a_3-a_1}&\fp^{a_3-a_2}&\ddots&\dots&\fp^{a_3 - a_n}\\
    \vdots&\vdots&&\O_\fp&\vdots\\
    \fp^{a_n - a_1}&\dots&&\fp^{a_{n} - a_{n-1}}&\O_\fp
  \end{pmatrix},
  \end{align*}
  that is $\Lambda(a_1, \dots, a_n)$ corresponds to the homothety
  class of the lattice $[a_1, \dots, a_n]$ relative to our fixed basis
  $\{\omega_1, \dots, \omega_n\}$ of $K_\fp^n$.  By
  equation~(\ref{eq:orderscontainingomega}), we can reorder subsets of
  the basis $\{\omega_1,\dots, \omega_{f_1}\}$, $\{\omega_{f_1+1},
  \dots, \omega_{f_1+f_2}\}$, \dots, $\{\omega_{f_1+\dots+f_{g-1}+1},
  \dots, \omega_n\}$ so that equation~(\ref{eq:orderscontainingomega})
  remains valid and $a_1 \le \cdots\le a_{f_1}$, $a_{f_1+1} \le \cdots
  \le a_{f_1+f_2}$, \dots, $a_{f_1+\cdots+f_{g-1}+1} \le \cdots \le
  a_n$.

  Now we assume that $[a_1, \dots, a_n]$ is not of the form
  $[\underbrace{\ell_1, \dots,\ell_1}_{f_1},\underbrace{\ell_2,
    \dots,\ell_2}_{f_2},\dots, \underbrace{\ell_g,
    \dots,\ell_g}_{f_g}]$ for $\ell_i \in \Z$.  Since we can permute
  the order in which we list the primes $\fP_i$ of $L$ lying above
  $\fp$, we may assume that there is an $r_0$ with $1 < r_0 \le f_1$
  so that $a_1 = \cdots = a_{r_0-1}< a_{r_0} \le \cdots \le a_{f_1}$.
  We already know that $\O_L \hookrightarrow \oplus_{i=1}^g
  M_{f_i}(\O_\fp) \hookrightarrow M_n(\O_\fp)$, so we focus on the
  upper $f_1\times f_1$ block of $\oplus_{i=1}^g M_{f_i}(\O_\fp) \cap
  \Lambda(a_1, \dots, a_n)$.  That intersection is contained in
  $\Gamma_\fp := \begin{pmatrix}
    M_{r_0-1}(\O_\fp)& \pi^{a_1-a_{f_1}} M_{r_0-1\times f_1-r_0+1}(\O_\fp)\\
    \pi M_{f_1-r_0+1\times r_0-1}(\O_\fp)& M_{f_1-r_0+1}(\O_\fp)\\
  \end{pmatrix} \cap  M_{f_1}(\O_\fp)$.

  Since $\O_L \subset \Lambda(a_1, \dots, a_n)$,
  Lemma~\ref{lemma:IntegralClosure} gives us that $S$, the integral
  closure of $R = (\O_\fp\cap K)$ in $L$, is contained in
  $\Lambda(a_1, \dots, a_n)$.  Thus $S\otimes_R \O_\fp \subset
  \Lambda(a_1, \dots, a_n)$.  By Proposition II.4 of
  \cite{Serre-LocalFields}, $S\otimes_R \O_\fp \cong \oplus_{i=1}^g
  \O_{\fP_i}$, so $ \oplus_{i=1}^g \O_{\fP_i} \subset \Lambda(a_1,
  \dots, a_n)$.

  As $ \oplus_{i=1}^g \O_{\fP_i} \subset \oplus_{i=1}^g
  M_{f_i}(\O_\fp)$, we may assume that $\O_{\fP_1} \subset
  \Gamma_\fp$, from which we shall derive a contradiction.  Since
  $L_{\fP_1}/K_\fp$ is unramified, we can write $\O_{\fP_1} =
  \O_\fp[\alpha]$ for some $\alpha$ whose reduction modulo $\fp$
  generates the residue field extension (see Theorem 5.8 of
  \cite{Reiner-book}).  In particular if $h$ is the minimal polynomial
  of $\alpha$ over $K_\fp$, then $h \in \O_\fp[x]$ is irreducible of
  degree $f_1$ and the reduction of $h$ modulo $\fp$ is
  irreducible.  Now viewing $\alpha$ as an element of $\Gamma_\fp$, we
  consider its characteristic polynomial $\chi_\alpha \in \O_\fp[x]$,
  also of degree $f_1$. Since $h$ is irreducible, we have $h \mid
  \chi_\alpha$ and so by degree arguments, $h = \chi_\alpha$.  On the
  other hand $\Gamma_\fp$, viewed modulo $\fp\O_\fp$, is upper
  triangular, so the reduction of $h = \chi_\fp$ modulo $\fp$ is
  reducible in $\O_\fp/\fp\O_\fp[x]$, a contradiction.
\end{proof}
 
We derive a few interesting corollaries.

\begin{cor}\label{cor:embeddingisalocalquestion} Let $B$ be a central simple algebra over a number field
  $K$ of dimension $n^2$, and $L$ a degree $n$ field extension of $K$
  which embeds into $B$. Let $\fp$ be a prime of $K$ which splits in
  $B$ and is unramified in $L$, and let $\E_\fp$ a maximal order in
  $B_\fp$.  Then the following are equivalent:
  \begin{compactenum}
  \item $\O_L \subset \E_\fp$
  \item $\O_{L_\fP} \subset \E_\fp$ for some prime $\fP$ of $L$
    dividing $\fp\O_L$.
  \item  $\oplus_{\fP\mid \fp} \O_{L_\fP} \subset \E_\fp$ where the
    sum is over all primes $\fP \mid \fp\O_L$.
  \end{compactenum}
  \end{cor}

  \begin{proof}
    The implications (3) implies (2) implies (1) are
    obvious.  That (1) implies (3) is exactly as in the proof of
    Theorem~\ref{thm:OLembedslocally} using
    Lemma~\ref{lemma:IntegralClosure}: If $S$ is the integral closure
    in $L$ of $(\O_K)_\fp$ (the localization of $\O_K$ at the prime
    $\fp$), then Lemma~\ref{lemma:IntegralClosure} shows that
    $\O_L\subset \E_\fp$ implies that $S\subset \E_\fp$.  Thus $S
    \otimes_{(\O_k)_\fp} \O_\fp \subset \E_\fp$.   But by Proposition II.4 of
  \cite{Serre-LocalFields}, $S\otimes_{(\O_k)_\fp} \O_\fp \cong \oplus_{\fP\mid \fp}
  \O_{L_\fP}$, so $\oplus_{\fP\mid \fp} \O_{L_\fP} \subset \E_\fp$.
  \end{proof}


  It is clear from Theorem~\ref{thm:OLembedslocally}, that when $\O_L
  \subset \R$ and a prime $\fp$ of $K$ splits completely in $L$, then
  $\O_L$ is contained in every maximal order in an apartment of the
  $SL_n(K_\fp)$ building which contains $\R_\fp$.  In particular
  $\O_L$ is contained in every vertex of a fundamental chamber
  containing $\R_\fp$ (corresponding to the homothety class
  $[0,\dots,0]$).  On the other hand, if $\fp$ is inert in $L$, then
  similarly, we see from
  Corollary~\ref{cor:verticesinfundamentalchamber} that $O_L$ is
  contained in precisely one vertex of the fundamental chamber;
  indeed it is contained in a unique vertex in the building.  This
  last observation can be cast in purely local terms.

  
\begin{cor}
  Let $F$ be a nondyadic nonarchimedean local field, and $E$ the
  degree $n \ge 3$ unramified extension of $F$.  Then there is a
  unique maximal order $\E$ of $M_n(F)$ containing $\O_E$.  
\end{cor}

\begin{proof}
  Let $K$ be a number field with a finite prime $\nu$ so that $F =
  K_\nu$, and let $B$ be a central simple algebra of dimension $n^2$
  over $K$ which is split at $\nu$, that is the local index $m_\nu =
  1$.  Let $m_{\fp_i}$, $i = 1, \dots , s$ be the nontrivial local
  indices associated to $B$.  By the Grunwald-Wang theorem
  \cite{Artin-Tate}, we may choose a cyclic extension $L/K$ of degree
  $n$ so that $m_{\fp_i} = [L_\fP:K_{\fp_i}]$ for all
  $\fP\mid \fp_i$, $i = 1, 2, \dots, s$, and $[L_\fP: k_\nu] = n$.
  Since $\nu \nmid 2$ we may choose the extension so that
  $L_\fP/K_\nu$ is unramified which means that $\nu$ is inert in $L$,
  so that $E = L_\fP$.  The conditions on the local indices guarantee
  that $L$ embeds in $B$ via the Albert-Brauer-Hasse-Noether theorem.
  So we assume $K \subset L \subset B$, and choose a maximal order
  $\R$ containing $\O_L$.  Then $\O_L \subset \R_\nu$.  By the
  previous corollary, $\O_{L_\fP} \subset \R_\nu$ for the unique prime
  $\fP$ lying above $\nu$.  We claim that this is the unique maximal
  order of $\B_\nu = M_n(K_\nu)$ containing $\O_{\L_\fP}$.  Indeed
  since $\nu$ is inert in $L$, Theorem~\ref{thm:OLembedslocally} characterizes the homothety
  classes of lattices which correspond to the maximal orders
  containing  $\O_{\L_\fP}$, as $[\ell, \dots, \ell ] = [0,\dots,0]$,
  the class corresponding only to $\R_\nu$.
\end{proof}


\section{Global Embeddings}

In this section we apply Theorem~\ref{thm:OLembedslocally} to the
problem of determining which isomorphism classes of maximal orders in
the global algebra $B$ admit an embedding of $\O_L$.



To begin we need to parametrize the isomorphism classes of maximal
orders in $B$.  Recall we have fixed a maximal order $\R$ which
contains $\O_L$, and now need to give a set of representatives of the
distinct isomorphism classes comprising the genus of maximal orders.

\subsection{Parametrizing the Genus}

We want to fix a convenient parametrization of the isomorphism classes
of maximal orders in $B$ which is compatible with the local embedding
theory of the previous section.  We begin by defining a class field
$K(\R)$ associated to the maximal order $\R$. We refer to
\cite{Linowitz-Shemanske-EmbeddingPrimeDegree} for most of the
details.

Given a maximal order $\R \subset B$, and a prime $\nu$ of $K$, we
define localizations $\R_\nu \subseteq B_\nu$ by:

\[
\R_\nu =
\begin{cases}
  \R \otimes_\O \O_\nu&\textrm{if $\nu$ is finite}\\
  \R \otimes_\O K_\nu = B_\nu&\textrm{if $\nu$ is infinite}\\
\end{cases}
\]



For every finite prime $\nu$, it is well-known that $\R_\nu$ is a
maximal order of $B_\nu$.  Let $\N(\R_\nu)$ denote the normalizer of
$\R_\nu$ in $B_\nu^\times$, and $nr(\N(\R_\nu))$ its reduced norm in
$K_\nu^\times$.  When $\nu$ is an infinite prime, $\N(\R_\nu) =
B_\nu^\times$.  If $\nu$ is
finite, it is convenient to distinguish three cases: If $\nu$ splits
in $B$, then $B_\nu \cong M_n(K_\nu)$ and every maximal order is
conjugate by an element of $B_\nu^\times$ to $M_n(\O_\nu)$, so every
normalizer is conjugate to $GL_n(\O_\nu) K_\nu^\times$ (37.26 of
\cite{Reiner-book}), while if $\nu$ totally ramified in $B$, $\R_\nu$
is the unique maximal order of the division algebra $B_\nu$
\cite{Reiner-book}, so $\N(\R_\nu) = B_\nu^\times$.  For $\nu$
partially ramified in $B$, $B_\nu \cong M_{\kappa_\nu}(D_\nu)$ with
$D_\nu$ a division algebra of degree $m_\nu$ over $K_\nu$, so $\R_\nu$
is conjugate to $M_{\kappa_\nu}(\Lambda_\nu)$ where $\Lambda_\nu$ is
the unique maximal order of $D_\nu$ (17.3 of \cite{Reiner-book}).
If we choose $u \in
\Lambda_\nu^\times$ then a conjugate of $U = \diag(u,1,\dots 1)\in
\N(R_\nu)$, and $nr(U) = nr(u) \in nr(\N(\R_\nu))$.  Finally, we recall (see
p145 of \cite{Reiner-book}) that $\Lambda_\nu$ contains the ring of
integers of an unramified cyclic extension of $K_\nu$.


It follows (p 153 of \cite{Reiner-book}) that for $\nu$ infinite or totally
ramified, $nr(\N(\R_\nu)) =
nr(B_\nu^\times) = K_\nu^\times,$ and in all other cases $\O_\nu^\times
(K_\nu^\times)^n \subseteq nr(\N(\R_\nu))$, with equality when $\nu$
is split.  Thus in all cases $\O_\nu^\times \subset nr(\N(\R_\nu))$
and for $\nu$ infinite or totally ramified in $B$, $K_\nu^\times =
nr(\N(\R_\nu))$, a fact that will be important in associating a class
field to $\R$.


The type number of $\R$ is the cardinality of the double coset space
$B^\times\backslash J_B/\fN(\R)$ where $J_B$ are the ideles of $B$.
To make use of class field theory, we need to realize this quotient in
terms of the arithmetic of $K$.  Let $J_K$ denote the idele group of $K$.

\begin{theorem}\label{thm:type_number_bijection}The reduced norm on
  $B$ induces a bijection 
\[
nr: B^\times\backslash J_B/\fN(\R) \to K^\times \backslash J_K/nr(\fN(\R)).
\]  
The group $K^\times \backslash J_K/nr(\fN(\R))$ is abelian with exponent~$n$.
\end{theorem}

\begin{proof}
  Here of course we also use $nr$ as the induced reduced norm map on
  ideles.  The proof is exactly as in Theorems 3.1 and 3.2 of
  \cite{Linowitz-Shemanske-EmbeddingPrimeDegree}.
\end{proof}


We have seen above that the distinct isomorphism classes of maximal
orders in $B$ are in one-to-one correspondence with the double
cosets in the group $G = K^\times \backslash J_K/nr(\fN(\R))$.  Put
$H_\R = K^\times nr(\fN(\R))$ and $G_\R = J_K/H_\R$.  Since $J_K$ is
abelian, $G$ and $G_\R$ are naturally isomorphic, and since $H_\R$
contains a neighborhood of the identity in $J_K$, it is an open
subgroup (Proposition II.6 of \cite{Higgins}).

Since $H_\R$ is an open subgroup of $J_K$ having finite index, there
is by class field theory \cite{Lang-ANT}, a class field, $K(\R)$,
associated to it.  The extension $K(\R)/K$ is an abelian extension
with $Gal(K(\R)/K) \cong G_\R = J_K/H_\R$ and with $H_\R = K^\times
N_{K(\R)/K}(J_{K(\R)})$.  Moreover, a prime $\nu$ of $K$ (possibly
infinite) is unramified in $K(\R)$ if and only if $\O_\nu^\times
\subset H_\R$, and splits completely if and only if $K_\nu^\times
\subset H_\R$.  Here if $\nu$ is archimedean, we take $\O_\nu^\times =
K_\nu^\times$.  From our computations just above, we saw that
$\O_\nu^\times$ is always contained in $H_\R$.  In particular $K(\R)/K$ is an everywhere unramified
abelian extension of $K$.


The Galois group $G = Gal(K(\R)/K)$ is a finite abelian
group of exponent $n$.  We wish to specify a set of generators for
the group as Artin symbols, $(\nu, K(\R)/K)$, in such a way that we can
control the splitting behavior of $\nu$ in the field $L$.  As $L$ is
an arbitrary extension of $K$ of degree $n$, this requires some care.

We have assumed that $L \subset
B$. Put $L_0 = K(\R) \cap L$ and $\wL_0 = \wL \cap
K(\R)$ where $\wL$ is the Galois closure of $L$. Then $L_0
\subset \wL_0$ and we define subgroups of $G$: $\wH =
Gal(K(\R)/\wL_0) \subseteq H = Gal(K(\R)/L_0)$.  We write the
finite abelian groups $\wH$, $H/\wH$, and $G/H$ as a direct product of
cyclic groups:
\begin{align}
  \label{eq:param1}
G/H = &\la\rho_1 H\ra \times \cdots \times \la \rho_r H\ra,\\
 \label{eq:param2}
H/\wH &= \la \sigma_1 \wH\ra \times \cdots \times \la \sigma_s \wH
\ra,\\
 \label{eq:param3}
\wH &= \la \tau_1\ra \times \cdots \times \la \tau_t\ra.
\end{align}

The following proposition is clear.
\begin{prop}\label{prop:Galoisrepn}
  Every element $\varphi \in G$ can be written uniquely as $\varphi =
  \rho_1^{a_1} \cdots \rho_r^{a_r}\sigma_1^{b_1} \cdots \sigma_s^{b_s}
  \tau_1^{c_1}\cdots \tau_t^{c_t}$ where $0 \le a_i < |\rho_i H|$, $0
  \le b_j < |\sigma_j\wH|$, and $0\le c_k < |\tau_k|$, with $|\cdot|$ the
  order of the element in the respective group.
\end{prop}

Next we characterize each of these generators in terms of Artin
symbols.  Since the vehicle to accomplish this is the Chebotarev
density theorem which provides an infinite number of
choices for primes, we may and do assume without loss
that the primes we choose to define the Artin symbols are unramified
in both $\wL$ and $B$.


First consider the elements $\tau_k \in \wH = Gal(K(\R)/\wL_0)$.  By
Lemma 7.14 of \cite{Narkiewicz-book}, there exist infinitely many
primes $\nu_k$ of $K$ so that $\tau_k = (\nu_k, K(\R)/K)$ and for
which there exists a prime $Q_k$ of $\wL$ with inertia degree
$f(Q_k\mid \nu_k) = 1$.  Since $\wL/K$ is Galois (and the prime
$\nu_k$ is unramified by assumption), this implies $\nu_k$ splits
completely in $\wL$, hence also in $L$.  



Next consider  $\sigma_j\wH$ with $\sigma_j \in H = Gal(K(\R)/L_0)$.  Again
by Lemma 7.14 of \cite{Narkiewicz-book}, there exist infinitely many
primes $\mu_j$ of $K$ so that $\sigma_j = (\mu_j, K(\R)/K)$ and for which there
exists a prime $Q_j$ of $L$ with inertia degree $f(Q_j\mid \mu_j) =
1$. 



Finally consider $\rho_kH$ with $\rho_k \in G =
Gal(K(\R)/K)$.  By Chebotarev, there exist infinitely many primes
$\lambda_i$ of $K$ so that $\rho_i = (\lambda_i, K(\R)/K)$.  For later
convenience, we note that by standard properties of the Artin symbol,
$\overline\rho_i = \rho_i\vert_{L_0} = (\lambda_i, L_0/K)$ whose order
in $Gal(L_0/K)$ is equal to the inertia degree $f(\lambda_i; L_0/K)$.

As we said above, we have assumed without loss that all the primes
$\lambda_i, \mu_j, \nu_k$ are unramified in $\wL$ and $B$.

\subsection{Fixing representatives of the isomorphism classes}

As above, $\R$ is a fixed maximal order of $B$ containing $\O_L$.
For a finite prime $\fp$ of $K$ which splits in $B$, we recall the
setup and notation of section ~\ref{sec:localembeddings} where we
fixed an apartment in the affine building for $SL_n(K_\fp)$ which
contains the vertex $\R_\fp \subset B_\fp \cong M_n(K_\fp)$.

We are interested in vertices of the form $\R(k,\ell)$ defined in
equation~(\ref{eq:blockstr}).  Because we shall vary the prime $\fp$
in the parametrization below, we will write $\R_\fp(k,\ell)$ for
$\R(k,\ell)$ to make the dependence on $\fp$ explicit. Recall that
$\R_\fp(k,\ell)$ corresponds to the homothety class
$[\underbrace{\ell, \dots, \ell}_k,0, \dots, 0] \in
\Z^n/\Z(1,\dots,1)$ which has type $k\ell\pmod n$.



Now for the primes $\lambda_i$, $\mu_j$, and $\nu_k$ we specified
above to parametrize $G = Gal(K(\R)/K)$, fix the following local orders
using the decomposition of $G$ into $G/H$, $H/\wH$, and $\wH$:

For each prime $\lambda_i$ ($i = 1, \dots, r$) whose Artin symbol
$(\lambda_i, K(\R)/K)) = \rho_i$ gives one of the generators $\rho_iH$
of $G/H$, we fix vertices $\R_{\lambda_i}(m,1)$, $m = 0, 1, \dots,
|\rho_i H| -1$ with associated homothety classes $[0, \dots, 0]$,
$[1,0,\dots, 0]$, $[1,1,0,\dots,0]$,\dots, $[\underbrace{1,\dots,1}_{
  |\rho_i H| -1}, 0,\dots, 0]$.

For each prime $\mu_j$ ($j = 1, \dots, s$) whose Artin symbol $(\mu_j,
K(\R)/K)) = \sigma_j$ gives one of the generators $\sigma_j\wH$ of
$H/\wH$, we fix vertices $\R_{\mu_j}(1,m)$, $m = 0, 1, \dots,
|\sigma_j \wH| -1$ with associated homothety classes $[0, \dots, 0]$,
$[1,0,\dots, 0]$, $[2,0,\dots,0]$, \dots, $[|\sigma_j\wH| -1,0,\dots,
0]$.  Recall that each $\mu_j$ factors into primes of $L$ with at
least one having inertia degree one over $\mu_j$. Since the
representation above depends on the order in which we list those
primes (see equation~(\ref{eq:integralblockstructure})), we assume the
first prime has degree one.

Finally, for each prime $\nu_k$ ($k = 1, \dots, t$) whose Artin symbol
$(\nu_k, K(\R)/K)) = \tau_k$ is one of the generators of $\wH$, we fix
vertices $\R_{\nu_k}(m,1)$, $m = 0, 1, \dots, |\tau_k| -1$ with
associated homothety classes $[0, \dots, 0]$, $[1,0,\dots, 0]$,
$[1,1,0,\dots,0]$, \dots, $[\underbrace{1,\dots,1}_{ |\tau_k| -1},
0,\dots, 0]$. Again recall that each prime $\nu_k$ splits completely
in $L$, so all the inertia degrees are one.


\begin{remark}\label{rem:embedsinab} 
  We note that using Theorem~\ref{thm:OLembedslocally} and the
  conventions on the degree one primes listed above, we see that
  $\O_L$ is a subset of $\R_{\mu_j}(1,m)$ for
  every value of $m$, and of $\R_{\nu_k}(m',1)$ for $0\le m'\le n$.
  \end{remark}




Now we use the local-global correspondence for orders to define global
orders from the above local factors. Fix the
following notation:

\begin{align*}
  \ba &= (a_i) \in \Z/|\rho_1 H|\Z \times \cdots \times \Z/|\rho_r
  H|\Z,\\
  \bb &= (b_j) \in \Z/|\sigma_1 \wH|\Z \times \cdots \times \Z/
  |\sigma_s \wH|\Z,\\
  \bc &= (c_k) \in \Z/|\tau_1|\Z \times \cdots \times \Z/|\tau_t|\Z.
\end{align*}
Here we assume the coordinates $a_i, b_j, c_k$ are integers which are the least
non-negative residues corresponding to the moduli.
Define maximal orders in $B$ via the local-global correspondence:
\[
\D_\fp^{\ba,\bb,\bc} =
\begin{cases}
  \R_\fp& \textrm{if }\fp \notin \{\lambda_i, \mu_j, \nu_k\},\\
  \R_{\lambda_i}(a_i,1)& \textrm{if } \fp = \lambda_i, i=1, \dots, r,\\
 \R_{\mu_j}(1,b_j)& \textrm{if } \fp = \mu_j, j=1, \dots, s,\\
 \R_{\nu_k}(c_k,1)& \textrm{if } \fp = \nu_k, k=1, \dots, t.\\
\end{cases}
\]

We claim that such a collection of maximal orders parametrizes the
isomorphism classes of maximal orders in $B$.  That is, given any
maximal order $\E$ in $B$, we show there are unique tuples $\ba, \bb,
\bc$ so that $\E \cong \D^{\ba,\bb,\bc}$.  To see this let $\fM$
denote the set of all maximal orders in $B$, and define a map (called
the $G_\R$-valued distance idele) $\delta: \fM \times \fM \to G_\R =
J_K/H_\R$ (where $H_\R = K^\times nr(\fN(\R))$) as follows: Let $\R_1,
\R_2 \in \fM$.  For a finite prime $\fp$ of $K$ which splits in $B$,
we have the notion of the type distance between their localizations:
$td_\fp(\R_{1\fp}, \R_{2\fp}) \in \Z/n\Z$ (see section 2 of
\cite{Linowitz-Shemanske-EmbeddingPrimeDegree}). If $\fp$ is either
archimedean or finite and ramified in $B$, define $td_\fp(R_{1\fp},
R_{2\fp})=0$.  Recall that since $\R_{1\fp} = \R_{2\fp}$ for almost
all $\fp$, $td_\fp(\R_{1\fp}, \R_{2\fp})=0$ for almost all primes
$\fp$. Let $\delta(\R_1, \R_2)$ be the image in $G_\R$ of the idele
$(\pi_\fp^{td_\fp(\R_{1\fp}, \R_{2\fp})})$, where $\pi_\fp$ is a fixed
uniformizing parameter in $K_\fp$.  Note that while the idele is not
well-defined, its image in $G_\R$ is since the local factor at the
finite split primes has the form $K_\fp^\times/\O_\fp^\times
(K_\fp^\times)^n$.

We now show that the orders $\{\D^{\ba, \bb, \bc}\}$  parametrize the
isomorphism classes of maximal orders in $B$.

\begin{prop}\label{prop:parametrize}
Let $\R$ be a fixed maximal order in $B$, and consider the collection
of maximal orders $\D^{\ba,\bb,\bc}$ defined above.

\begin{compactenum}
\item If $\E$ is a maximal order in $B$ and $\E \cong \R$, then
  $\delta(\R, \E)$ is trivial.
\item If $\E \cong \E'$ are maximal orders in $B$, then $\delta(\R, \E) = \delta(\R,\E')$.
\item $\D^{\ba,\bb,\bc} \cong \D^{\ba',\bb',\bc'}$ if and only if $\ba
  = \ba'$, $\bb = \bb'$, and $\bc = \bc'$.
\end{compactenum}
\end{prop}

\begin{proof}
  The first two statements are proven exactly as in Proposition 3.3 of
  \cite{Linowitz-Shemanske-EmbeddingPrimeDegree}. For the last, let
  $\fq$ be a finite prime of $K$ and $\pi_\fq$ the corresponding
  uniformizing parameter. Let $e_\fq$ denote the idele with
  $\pi_\fq$ in the $\fq$th place and 1's elsewhere.  Observe that
  Artin reciprocity identifies the image of $e_\fq$ in $G_R =
  J_K/H_\R$ with the Artin symbol $(\fq, K(\R)/K)$.  It follows that 
\[
\delta(\D^{\ba,\bb,\bc}, \D^{\ba',\bb',\bc'}) \leftrightarrow \prod_{i=1}^g
\rho_i^{a_i' - a_i} \prod_{j=1}^s \sigma_j^{b_j'-b_j} \prod_{k=1}^t
\tau_k^{c_k'-c_k} \in Gal(K(\R)/K),
\]
which is trivial if and only if $\ba=\ba'$, $\bb = \bb'$, and
$\bc=\bc'$ by Proposition~\ref{prop:Galoisrepn}.  The result is now
immediate.
\end{proof}

\section{Embedding $\O_L$ into maximal orders in $B$}

Pivotal to understanding which maximal orders in $B$ admit an
embedding of $\O_L$ is the following characterization.


\begin{thm}\label{thm:EmbedviaArtin}
  Assume that $\O_L \subset \R \subset B$.  Let $\E$ be another
  maximal order in $B$, and let $\delta(\R, \E)$ denote the distance
  idele defined above.  Then $\E$ admits an embedding of $\O_L$ if and
  only if the idelic Artin symbol $(\delta(\R, \E),L_0/K)$ is trivial
  in $Gal(L_0/K)$.
  \end{thm}

  \begin{proof} Assume that $\E$ admits an embedding of $\O_L$; that
    is, $\E$ contains a conjugate of $\O_L$. By
    Proposition~\ref{prop:parametrize}, we may replace $\E$ by a
    conjugate order without changing the distance idele, so we may
    assume without loss that $\O_L \subset \R \cap \E$. The distance
    idele $\delta(\R,\E)$ is the coset $\tilde\delta H_\R \in G_\R$
    where $\tilde\delta = \prod_\fp e_\fp^{td_\fp(\R_\fp,\E_\fp)}$
    with the product over all the finite primes of $K$ which split in
    $B$ and for which $\R_\fp \ne \E_\fp$, and with $e_\fp = (1,\dots,
    1, \pi_\fp, 1\dots, 1) \in J_K$. Write $\sigma = (\delta(\R,
    \E),L_0/K) \in Gal(L_0/K)$.  Then $\sigma = \prod_{\fp} (\fp,
    L_0/K)^{td_\fp(\R_\fp,\E_\fp)}$ where $(\fp, L_0/K)$ is the Artin
    symbol.  Since $\O_L \subset \E_\fp$, we know by
    Theorem~\ref{thm:OLembedslocally} that $\E_\fp$ corresponds to the
    homothety class $[\underbrace{\ell_1,
      \dots,\ell_1}_{f_1},\underbrace{\ell_2,
      \dots,\ell_2}_{f_2},\dots, \underbrace{\ell_g,
      \dots,\ell_g}_{f_g}]$ for integers $\ell_i$ and where the $f_i$
    are the inertia degrees of the primes of $L$ lying above $\fp$.
    Thus, $td_\fp(\R_\fp,\E_\fp) \equiv \sum_{i=1}^g \ell_if_i \pmod
    n$. Now each $f_i$ is divisible by $f_0 = f(\fp; L_0/K)$, the
    inertia $\fp$ gains up to $L_0$ (recall $L_0/K$ is Galois).  But
    $f_0$ is order of the Artin symbol $(\fp;L_0/K)$, so each term in
    the product is trivial, hence so is $\sigma$.

    For the converse, we suppose that $\sigma = (\delta(\R,
    \E),L_0/K)$ is the trivial element of $Gal(L_0/K)$ and show that
    $\E$ admits an embedding of $\O_L$.  In parametrizing the
    isomorphism classes of maximal orders of $B$, we wrote
    $Gal(L_0/K)$ as a product of cyclic groups: With $G= Gal(K(\R)/K)$
    and $H = Gal(K(\R)/L_0)$ we have $G/H \cong \la \rho_1H \ra \times
    \cdots \times \la \rho_rH \ra$, so $Gal(L_0/K) \cong \la\overline
    \rho_1\ra \times \cdots \times \la\overline\rho_r \ra$ where
    $\overline \rho_i$ is the restriction of $\rho_i$ to $L_0$.  Each
    $\rho_i$ is the Artin symbol $(\lambda_i, K(\R)/K)$.


    Our given $\E$ is isomorphic to a unique element,
    $\D^{\ba,\bb,\bc}$, of our parametrizing set of representatives of
    the isomorphism
    classes of maximal orders, and since $\delta(\R,\E) = \delta(\R, \D^{\ba,\bb,\bc})$ by
    Proposition~\ref{prop:parametrize}, it is sufficient to show that
    $\sigma = (\delta(\R,\D^{\ba,\bb,\bc}),L_0/K)$ trivial in $Gal(L_0/K)$  implies that
    $\O_L \subset \D^{\ba,\bb,\bc}$.

    We are assuming that $\O_L \subset \R$, and since $\R$ and
    $\D^{\ba, \bb, \bc}$ are equal at all primes outside the finite
    set $T = \{\lambda_1, \dots, \lambda_r, \mu_1, \dots, \mu_s,
    \nu_1,\dots, \nu_t\}$ of parametrizing primes, $\O_L \subset
    \D^{\ba, \bb, \bc}_\fp$ for all primes $\fp \notin T$.  For
    $\fp=\mu_j$ or $\nu_k$, we have $\O_L \subset \D^{\ba, \bb,
      \bc}_\fp$ by Remark~\ref{rem:embedsinab}.  So we need to deal
    only with the primes $\lambda_i$.



    Recall that for $\fp \notin T$ we have $td_\fp(R_\fp,\D_\fp^{\ba,
      \bb, \bc}) = 0$ so that $(\delta(\R,\D^{\ba, \bb, \bc}), L_0/K)
    = \prod_{\fp \in T} (\fp, L_0/K)^{td_{\fp}(\R_\fp, \D_\fp^{\ba,
        \bb, \bc})}$.  Furthermore the primes $\mu_j, \nu_k \in T$
    correspond to elements in $Gal(K(\R)/L_0)$ so their restriction to
    $L_0$ is trivial. Thus our condition reduces to
    $\prod_{i=1}^r(\lambda_i, L_0/K)^{td_{\lambda_i}(\R_{\lambda_i},
      \D^{\ba, \bb, \bc}_{\lambda_i})}=1$.  Since $Gal(L_0/K)$ is the
    direct product $\prod_{i=1}^r \la (\lambda_i, L_0/K) \ra$, we have
    that $\prod_{i=1}^r(\lambda_i,
    L_0/K)^{td_{\lambda_i}(\R_{\lambda_i}, \D^{\ba, \bb,
        \bc}_{\lambda_i})}=1$ implies each of the terms $(\lambda_i,
    L_0/K)^{td_{\lambda_i}(\R_{\lambda_i}, \D^{\ba, \bb,
        \bc}_{\lambda_i})}=1$, so $td_{\lambda_i}(\R_{\lambda_i},
    \D^{\ba, \bb, \bc}_{\lambda_i}) \equiv 0 \pmod
    {f(\lambda_i;L_0/K)}$ since $f(\lambda_i;L_0/K)$ is the order of
    the Artin symbol.  Since we have parametrized
    $\D_{\lambda_i}^{\ba, \bb, \bc}$ with maximal orders whose
    corresponding homothety classes have types $0, 1, \dots,
    f(\lambda_i;L_0/K)-1$, we have that $\D^{\ba, \bb,
      \bc}_{\lambda_i} = \R_{\lambda_i}$, so $\O_L \subset
    \D_{\lambda_i}^{\ba, \bb, \bc}$, and hence $\O_L \subset
    \D_\fp^{\ba, \bb, \bc}$ for all primes $\fp$ which completes the
    proof.
\end{proof}
 



\begin{cor}\label{cor:selectivity}The ratio of the number of
  isomorphism classes of maximal orders in $B$ which admit an
  embedding of $\O_L$ to the total number of isomorphism classes of maximal
  orders is $[L_0: K]^{-1}$ where $L_0 = K(\R)\cap L$.
\end{cor}



\begin{proof} Let $\{\D^{\ba,\bb,\bc}\}$ be the parametrization of the
  isomorphism classes of maximal orders of $B$ given in section 2.  We
  must show that exactly $[K(\R):L_0]$ of these orders admit an
  embedding of $\O_L$.  By the proof of
  Theorem~\ref{thm:EmbedviaArtin}, we have that $\sigma =
  (\delta(\R,\D^{\ba,\bb,\bc}),L_0/K) = \prod_{i=1}^r(\lambda_i,L_0/K)
  = 1$ in $Gal(L_0/K)$ if and only if  $\D_{\lambda_i}^{\ba, \bb,
    \bc}= \R_{\lambda_i}$ for $i = 1, \dots, r$; or equivalently, $\ba
  = 0$.  But the number of orders $\{\D^{\ba,\bb,\bc}\}$ with $\ba = 0$
  is $|H/\wH| |\wH| = |H| = [K(\R):L_0]$, which finishes the proof.
\end{proof}



\section{Examples and Applications}

Corollary~\ref{cor:selectivity} allows a significant refinement of the
contexts in which selectivity can occur.  We begin with the first
instance in which selectivity was noted, by recovering Chevalley's
\cite{Chevalley-book} elegant result when $B = M_n(K)$.

\begin{cor}[Chevalley]\label{cor:Chevalley}The
  ratio of the number of isomorphism classes of maximal orders in
  $B=M_n(K)$ into which $\O_L$ can be embedded to the total number of
  isomorphism classes of maximal orders is $[\widetilde K\cap L:
  K]^{-1}$ where $\widetilde K$ is the Hilbert class field of $K$.
\end{cor}

\begin{proof}We are assuming that $\O_L \subset \R$, where $\R$ our fixed
  maximal order in $B$.  We need only show that $\widetilde K\cap L =
  K(\R) \cap L$.

  When $B = M_n(K)$, we have that $H_\R = K^\times J_K^n
  J_{K,S_\infty}$ so that $G_\R = C_K/C_K^n$ where $C_K$ is the ideal
  class group of $K$. Since $G_\R$ is a quotient of the ideal class
  group (i.e., $Gal (\widetilde K/K)$), $K(\R) \subset \widetilde K$ is
  the subfield corresponding to $C_k^n$.  In particular $K(\R)\cap L =
  L_0 \subset \widetilde K \cap L$.

  Conversely, since $K(\R)$ corresponds to $C_K^n$, it is the maximal
  abelian unramified extension of $K$ with exponent $n$.  Since $[L:K]
  = n$, $\widetilde K \cap L$ is an unramified extension of $K$ having
  exponent $n$, so 
  by maximality $\widetilde K \cap L \subset K(\R)\cap L$.
\end{proof}

Examples of selective and non-selective orders in matrix algebras are
given in  \cite{Linowitz-Shemanske-EmbeddingPrimeDegree}.  Also in
that paper, we prove that orders in central simple division algebras
of (odd) prime degree are never selective.  We now show exactly how this
generalizes to arbitrary degree $n \ge 3$.


Recall that locally there is an isomorphism
$B_\nu \cong M_{\kappa_\nu}(D_\nu)$, where $D_\nu$ is a central simple
division algebra of dimensions $m_\nu^2$ over $K_\nu$, and we have $n =
\kappa_\nu m_\nu$. We recall the Albert-Brauer-Hasse-Noether theorem:
\begin{nnthm} (ABNH)
  Let the notation be as above, and suppose that $[L:K] = n$.  Then
  there is an embedding of $L/K$ into $B$ if and only if for each
  prime $\nu$ of $K$ and for all primes $\fP$ of $L$ lying above
  $\nu$, $m_\nu \mid [L_\fP:K_\nu]$.
\end{nnthm}


The first case we consider is a non-split algebra $B$ which is fully
ramified at some prime (necessarily a finite prime since $n > 2$). We show
that in this case there is never selectivity.


\begin{cor}\label{cor:fullyramified}
  Suppose there is a prime $\nu$ of $K$ so that $B_\nu$ is a division
  algebra, i.e., $m_\nu = n$ and $L \subset B$.   Let $\Omega
  \subseteq \O_L$ be any $\O_K$-order.  Then $\Omega$ is never
  selective; that is, every maximal order in $B$ admits an embedding of
  $\Omega$.  
\end{cor}


\begin{proof} It is enough to show that every maximal order in $B$
  admits an embedding of $\O_L$.  We suppose that $\R$ is our fixed maximal order
  containing $\O_L$.  That $B_\nu$ is a division algebra means
  $\R_\nu$ is the  unique maximal order in $B_\nu$, whose normalizer is all of
  $B_\nu^\times$ and so $K_\nu^\times \subset H_\R$.  As we have seen
  this implies that $\nu$ splits completely in the class field $K(\R)$,
  and hence also in $L_0 = K(\R)\cap L$.

  By the Albert-Brauer-Hasse-Noether theorem, $m_\nu = n \mid
  [L_\fP:K_\nu]$ for all $\fP \mid \nu \O_L$.  But this implies there
  is a unique prime $\fP$ of $L$ lying above $\nu$, and $n =
  [L_\fP:K_\nu] = e(\fP | \nu)f(\fP | \nu)$.  Now $\fP_0 = \fP \cap
  L_0$ is the unique prime of $L_0$ lying above $\nu$, so $[L_0:K] =
  e(\fP_0 | \nu)f(\fP_0 |\nu)$.  But $\nu$ splits completely in $L_0$
  which implies $[L_0:K] = 1$.  The result is now immediate from
  Corollary~\ref{cor:selectivity}.
\end{proof}


What remains to resolve is the question of how partial ramification affects
selectivity.  We demonstrate by means of an explicit example that
selectivity can occur in a partially ramified algebra.


\begin{example}\label{ex:partialramification}
  To compute our example, we employ Magma \cite{Magma}.  We construct
  a central simple algebra $B$ of degree 16 over $K =
  \Q(\sqrt{-14})$ which is ramified at two finite primes $K$ so that
  at each of those primes $\fP$, $B_\fP \cong M_2(D_\fp)$ with $D_\fP$
  a quaternion division algebra over $K_\fP$.  We construct a field
  $L$ which is contained in $B$ and for which $\O_L$ is selective.  By
  Corollary~\ref{cor:selectivity}, we need only show that $L_0 =
  K(\R)\cap L \ne K$.  In the example we construct, the class field
  $K(\R)$ is simply the Hilbert class field of $K$.

 The following calculations were produced by Magma \cite{Magma}.
 Let $K = \Q(\sqrt{-14})$.  The Hilbert class field of $K$ is
  $H_K = K(\alpha)$ where $\alpha^4 + 4\alpha^2 -28 = 0$, and the ideal
  class group of $K$, $C_K$, is isomorphic to  $\Z/4\Z$.  


  Let $L_0 = K(\beta)$ where $\beta ^2 + (4\sqrt{-14} +22)\beta +
  (44\sqrt{-14} + 33)=0$, and let $L = L_0(\sqrt{-5})$; then $L \cap
  H_K = L_0\ne K$.  Let $\fP_{137,1}$, $\fP_{137,2}$ be the two primes
  of $K$ lying above 137 in $\Z$.  One checks they are principal in
  $K$, so their classes $[\fP_{137,i}]$ are trivial in $C_K$, and each
  prime splits completely in $H_K$.

  Let $B$ be the central simple algebra over $K$ defined by the data:
  $\dim_KB = 16$; $Ram(B) = \{\fP_{137,1}, \fP_{137,2}\}$; and
  $m_{\fP_{137,i}} = 2$ for $i = 1,2$, that is $B_{\fP_{137,i}} \cong
  M_2(D_{\fP_{137,i}})$ with $D_{\fP_{137,i}}$ a quaternion division
  algebra over $K_{\fP_{137,i}}$, and is split at every other prime of
  $K$. The existence of $B$ is guaranteed by short exact sequence of
  Brauer groups induced by the Hasse invariants (see e.g., (32.13) of
  \cite{Reiner-book}).  To see that $L$ embeds in $B$, we need only
  verify the conditions of the Albert-Brauer-Hasse-Noether theorem
  above.  Since $m_\nu = 1$ for all primes except
  $\fP_{137,1},\fP_{137,2}$, we simply need to observe that
  $m_{\fP_{137,i}} = 2 \mid [L_Q:K_{\fP_{137,i}}]$ for every prime $Q$
  of $L$ lying above ${\fP_{137,i}}$ which is verified by Magma which
  shows (by factoring) that each prime is unramified and has inertia degree
  2 in $L$.

  Then for a maximal order $\R \subset B$, with $\O_L \subset \R$, the
  associated class field has Galois group $Gal(K(\R)/K) \cong
  I_K/(P_KI_K^4) \cong C_K$, since $[\fP_{137,i}]$ is trivial. So $G =
  Gal(K(\R)/K) = Gal(H_K/K) \cong C_K$.  Since $K(\R) \subseteq H_K$
  we have equality, and selectivity is demonstrated.

  We continue this example and give our parametrization as in
  Equations (\ref{eq:param1}) - (\ref{eq:param3}).  Put $H =
  Gal(K(\R)/L_0) = \wH$.  Then $Gal(L_0/K) \cong G/H = \la \sigma H\ra$
  where $\sigma$ is the Artin symbol $(\fP_3, H_K/K)$ with $\fP_3$ is
  the prime of $K$ with $\fP_3 \cap \Z = 3\Z$.  Finally choose $\fP_7$
  with $7\O_K = \fP_7^2$.  Then
\[ f(\fP_7; L_0/K) = 1; \qquad f(\fP_7; K(\R)/K) = 2;\qquad f(\fP_7;
L/K) = 1.
\]
So $H = Gal(K(\R)/L_0) = \la (\fP_7; K(\R)/K)\ra$.
\end{example}





%\newpage




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\end{document}

We note the inertia degrees which occur in our theorem refer to the
full extension $L/K$ which may not be Galois, and so are not the same
as in Theorem 1.1 of \cite{Arenas-Carmona-2011-Representation-Fields}
which focus on a subfield of the class field.