#  Local-global principle for hermitian spaces over semi global fields

Jayanth Guhan

###### Abstract.

Let $K$ be a complete discrete valued field with residue field $k$ and $F$ the
function field of a curve over $K$. Let $A\in{}_{2}Br(F)$ be a central simple
algebra with an involution $\sigma$ of any kind and $F_{0}=F^{\sigma}$. Let
$h$ be an hermitian space over $(A,\sigma)$ and $G=SU(A,\sigma,h)$ if $\sigma$
is of first kind and $G=U(A,\sigma,h)$ if $\sigma$ is of second kind. Suppose
that char$(k)\neq 2$ and ind$(A)\leq 4$. Then we prove that projective
homogeneous spaces under $G$ over $F_{0}$ satisfy a local-global principle for
rational points with respect to discrete valuations of $F$.

Let $K$ be a complete discrete valued field and $F$ the function field of a
curve over $K$. Let $\Omega$ be a set of discrete valuations of $F$. For
$\nu\in F$, let $F_{\nu}$ denote the completion of $F$ at $\nu$. Let $G$ be a
connected linear algebraic group over $F$ and $X$ a projective homogeneous
variety under $G$ over $F$. We say that a local-global principle holds for $X$
with respect to $\Omega$ if $X(F_{\nu})\neq\emptyset$ for all $\nu\in\Omega$
implies $X(F)\neq\emptyset$.

If $K$ is a $p$-adic field, $F$ the function field of a curve over $K$ and $G$
a connected linear algebraic group which of classical type with characteristic
of the residue field $k$ a ’good’ prime for $G$, then local-global principle
holds for projective homogeneous spaces under $G$ over $F$ with respect to the
set $\Omega_{F}$ of all ’divisorial’ discrete valuations ([5], [20], [25],
[18]). The aim of this paper is to obtain analogues results for certain
unitary groups if $K$ is a complete discrete valued field with any residue
field.

Suppose $K$ is any complete discrete valued field with residue field $k$. Let
$q$ be a quadratic form over $F$ of rank at least 3 and $G=SO(q)$. If
char$(k)\neq 2$, then Colliot-Thélène-Parimala-Suresh proved that local-global
principle holds for projective homogeneous spaces under $G$ over $F$ with
respect to $\Omega_{F}$ ([5, Theorem 3.1]). Let $A$ be a central simple
algebra over $F$ of degree $n$ and $G=PGL_{1}(A)$. If $n$ is coprime to
char$(k)$ and $k$ contains a primitive $n^{\rm th}$ root of unity, then Reddy-
Suresh proved that local-global principle holds for projective homogeneous
spaces under $G$ over $F$ with respect to $\Omega_{F}$ ([20, Theorem 2.6]).

Let $A\in{}_{2}Br(F)$ be a central simple algebra over $F$ with an involution
$\sigma$ and $h$ a hermitian form over $(A,\sigma)$ and $G=SU(A,\sigma,h)$ if
$\sigma$ is of first kind and $G=U(A,\sigma,h)$ if $\sigma$ is of second kind.
Suppose that char$(k)\neq 2$. A theorem of Wu ([25, Theorem 1.2]) asserts that
if ind$(A)\leq 2$ or for every finite extension $\ell/k$,
$\mid\ell^{*}/\ell^{*2}\mid\leq 2$, then local-global principle holds for
projective homogeneous spaces under $G$ over $F$ with respect to $\Omega_{F}$.
Since for every finite field ${\mathbb{F}}$ of characteristic not 2,
$\mid{\mathbb{F}}^{*}/{\mathbb{F}}^{*2}\mid=2$, if $K$ is a $p$-adic field
with $p\neq 2$, then local-global principle holds for projective homogeneous
spaces under $G$ over $F$ with respect $\Omega_{F}$.

These results are in the direction of a Hasse principle for projective
homogeneous spaces over general semiglobal fields. We however point out that a
Hasse principle for principal or projective homogeneous spaces over function
fields of $p$-adic curves is a conjecture in ([5]) and proved in many cases
(cf. [18]). There are examples to show that a Hasse principle may fail for
principal homogeneous space for arbitrary semiglobal field ([6], [7]).

In this paper we prove the following (cf. 7.1, 7.2).

######  Theorem 0.1.

Let $K$ be a complete discretely valued field with valuation ring $T$ and
residue field $k$. Suppose that char$(k)\neq 2$. Let $F$ be the function field
of a smooth projective geometrically integral curve over $K$. Let
$A\in{}_{2}Br(F)$ be a central simple algebra over $F$ with an involution
$\sigma$ of any kind, $F_{0}=F^{\sigma}$ and $h$ a hermitian form over
$(A,\sigma)$. Let $G=SU(A,\sigma,h)$ if $\sigma$ is first kind or
$U(A,\sigma,h)$ if $\sigma$ is of second kind. Suppose that one of the
following holds:
i) ind$(A)\leq 4$
ii) for every finite extension $\ell/k$, every element $D\in{}_{2}Br(\ell)$
has index at most 2. Let $X$ be a projective homogeneous variety under $G$
over $F_{0}$. If $X(F_{0\nu})\neq\emptyset$ for all divisorial discrete
valuations $\nu$ of $F$, then $X(F_{0})\neq\emptyset$.

As a consequence we get the following (7.4).

######  Corollary 0.2.

Let $K$ be a complete discretely valued field with valuation ring $T$ and
residue field $k$. Suppose that $k$ is a local field with the characteristic
of the residue field not equal 2. Let $F$ be the function field of a smooth
projective geometrically integral curve over $K$. Let $A\in{}_{2}Br(F)$ be a
central simple algebra over $F$ with an involution $\sigma$ and $h$ a
hermitian form over $(A,\sigma)$ and $F_{0}=F^{\sigma}$. If $h\otimes L$ is
isotropic for some odd degree extension $L/F_{0}$, then $h$ is isotropic.

The main technique in the proof of the main theorem is the patching technique
of Harbater, Hartman and Krashen ([8, Theorem 3.7]) and the existence of
maximal orders over two dimensional complete regular local rings for some
suitable division algebras. We now give a brief description of the structure
of paper. In §(1), we recall the patching set up and some known results. Let
$R$ be a 2-dimensional complete regular local ring with residue field $k$,
maximal ideal $(\pi,\delta)$ and field of fractions $F$. Suppose that
char$(k)\neq 2$. Let $D$ be a division algebra over $F$ which is unramified on
$R$ except possibly at $(\pi)$ and $(\delta)$. If $D$ has an involution of any
kind, in §2 and §3, following Saltman ([22, Theorem 2.1] & [23, Theorem 1.2]),
we give a description of $D$ if ind$(D)\leq 4$ or if for every finite
extension $\ell$ of $k$, every element of ${}_{2}Br(\ell)$ has index at most
2. In §4, we produce some suitable maximal $R$-orders for such algebras. In
§5, under some assumptions on the existence of maximal $R$-orders in $D$, we
show that hermitian spaces satisfy a certain local-global principle. In §6, we
study the behavior of the maximal $R$-orders constructed in §4 under blowups.
Finally using all these we prove the main results in §7.

##  1\. Preliminaries

In this section we recall some basic facts about hermitian forms ([24, Chapter
7 & 8]), projective homogeneous spaces under unitary groups ([15], [16]) and
patching techniques ([8]).

Let $F$ be a field and $A$ a central simple algebra with an involution
$\sigma$. Let $F_{0}=F^{\sigma}=\\{a\in F\mid\sigma(a)=a\\}$. Then
$[F:F_{0}]\leq 2$. If $F=F_{0}$, then $\sigma$ is called an involution of the
first kind if $[F:F_{0}]=2$, then $\sigma$ is called an involution of the
second kind. We also say that the involution $\sigma$ is an
$F/F_{0}$-involution. Let $h$ be an hermitian form over $(A,\sigma)$. Let
$G(A,\sigma,h)=SU(A,\sigma,h)$ if $F=F_{0}$ and $G(A,\sigma,h)=U(A,\sigma,h)$
if $[F:F_{0}]=2$. Then $G(A,\sigma,h)$ is a connected linear algebraic group
over $F_{0}$ which is rational ([4, Lemma 5]).

By a theorem of Wedderburn ([24, Chapter 8, Corollary 1.6 & Theorem 1.9]), we
have $A\simeq M_{n}(D)$ for a central division algebra over $D$ over $F$.
Since $A$ has a $F/F_{0}$-involution, $D$ has a $F/F_{0}$-involution ([24,
Chapter 8, Corollary 8.3]). Let $\sigma_{0}$ be a $F/F_{0}$-involution on $D$.
Then, by Morita equivalence ([12, Chapter 1, 9.3.5]), $h$ corresponds to a
hermitian $h_{0}$ over $(D,\sigma_{0})$.

Let $X$ be a projective homogeneous space under $G(A,\sigma,h)$ over $F_{0}$.
Then $X$ corresponds to a projective homogeneous space $X_{0}$ under
$G(D,\sigma_{0},h_{0})$ over $F_{0}$ such that for any extension $L/F_{0}$,
$X(L)\neq\emptyset$ if and only if $X_{0}(L)\neq\emptyset$ ([15, Section 5],
[16, Section 9], [11, Theorem 16.10]). Hence the study of rational points on
projective homogeneous space under $G(A,\sigma,h)$ reduces to the study of
rational points on projective homogeneous space under $G(D,\sigma_{0},h_{0})$.
Suppose $\sigma^{\prime}$ is an involution on $A$ of the same kind as
$\sigma$. Then $\sigma^{\prime}=$ int$(u)\sigma$ for some $u\in A^{*}$ and
$h^{\prime}=uh$ is an hermitian form over $(A,\sigma^{\prime})$. Further $X$
corresponds to a projective homogeneous space $X^{\prime}$ under
$G(A,\sigma^{\prime},h^{\prime})$ over $F_{0}$ such that for any extension
$L/F_{0}$, $X(L)\neq\emptyset$ if and only if $X^{\prime}(L)\neq\emptyset$.
For more details on the structure of projective homogeneous space under
$G(A,\sigma,h)$, we refer to ([15], [16]).

Let $\mathscr{X}$ be a regular integral scheme with function field $F$. For a
point $P$ of $\mathscr{X}$, let $\mathcal{O}_{P}$ denote the local ring at $P$
on $\mathscr{X}$, $\hat{\mathcal{O}}_{P}$ the completion of the local ring
$\mathcal{O}_{P}$. Let $A$ be a central simple algebra over $F$. Let
$P\in\mathscr{X}$. We say that $A$ is unramified at $P$, if there exists an
Azumaya algebra $\mathscr{A}$ over the local ring $\mathcal{O}_{P}$ at $P$
such that $\mathscr{A}\otimes F\simeq A$. If $A$ is not unramified at $P$, we
say that $A$ is ramified  at $P$.

Let $T$ be a complete discrete valuation ring with residue field $k$ and field
of fractions $K$. Let $F$ be the function field of a smooth projective
geometrically integral curve over $K$. A two dimensional projective scheme
$\mathscr{X}\to$ Spec$(T)$ with function field $F$ is called a model of $F$.
The fibre $X_{0}$ over the closed point of Spec$(T)$ is called the closed
fibre of $\mathscr{X}$.

Let $\mathscr{X}$ be a model of $F$ with $X_{0}$ its closed fibre. Every
codimension one point $x$ of $\mathscr{X}$ gives a discrete valuation
$\nu_{x}$ on $F$. A discrete valuation of $F$ is called a divisorial discrete
valuation of $F$, if it is given by a codimension one point of a model of $F$.
For any discrete valuation $\nu$ of $F$, we denote the completion of $F$ at
$\nu$ by $F_{\nu}$.

Let $A$ be a central simple algebra over $F$. The ramification divisor
ram${}_{\mathscr{X}}(A)$ of $A$ on $\mathscr{X}$ is the union of codimension
one points of $\mathscr{X}$ where $A$ is ramified. Note that there are only
finitely many codimension one point of $\mathscr{X}$ where $A$ ramified.

For any point $P\in\mathscr{X}$, let $\hat{\mathcal{O}}_{P}$ be the completion
of the local ring $\mathcal{O}_{P}$ at $P$ on $\mathscr{X}$ and $F_{P}$ the
field of fractions of $\hat{\mathcal{O}}_{P}$. Let $U\subset X_{0}$ be a
nonempty open subset of the nonsingular points of $X_{0}$ which is properly
contained in an irreducible component of $X_{0}$. Let $R_{U}$ be the set of
$f\in F$ which are regular at every point of $U$. Then $T\subset R_{U}$. Let
$t\in T$ be a parameter and $\hat{R}_{U}$ be the $(t)$-adic completion of
$R_{U}$. Let $F_{U}$ be the field of fractions of $\hat{R}_{U}$. For any
nonempty subset $V$ of $U$, we have $F_{U}\subset F_{V}$. For any point $P\in
U$, we have $F_{U}\subset F_{P}$. For more details about $F_{U}$ and $F_{P}$,
we refer to [8].

We record the following theorem from ([9, Proposition 5.8]).

######  Theorem 1.1.

Let $T$ be a complete discrete valuation ring with residue field $k$ and field
of fractions $K$. Let $F$ be the function field of a smooth projective
geometrically integral curve over $K$ and $\mathscr{X}$ a model of $F$ with
$X_{0}$ its closed fibre. Let $Y$ be a variety over $F$. Suppose that
$Y(F_{\nu})\neq\emptyset$ for all divisorial discrete valuations $\nu$ of $F$.
For every irreducible component $C$ of $X_{0}$, there exists a nonempty proper
open subset $U$ of $C$ such that $Y(F_{U})\neq\emptyset$. In particular there
exists a finite subset $\mathscr{P}$ of closed points of $X_{0}$ such that
$Y(F_{P})\neq\emptyset$ for all $P\in X_{0}\setminus\mathscr{P}$.

###### Proof.

Let $C$ be an irreducible component of $X_{0}$. Then for the generic point
$\eta$ of $C$, $F_{\eta}$ is the completion of $F$ at the discrete valuation
given by $\eta$ and hence $Y(F_{\eta})\neq\emptyset$. Then, by ([9,
Proposition 5.8]), there exists a nonempty proper open subset $U_{\eta}$ of
$C$ such that $Y(F_{U_{\eta}})\neq\emptyset$. Let
$\mathscr{P}=X_{0}\setminus\cup_{\eta}U_{\eta}$. Since $U_{\eta}$ is a
nonempty open subset of the irreducible curve $C$, $C-U_{\eta}$ is a finite
set. Since there are only finitely many irreducible components of $X_{0}$,
$\mathscr{P}$ is a finite set. Let $P\in X_{0}\setminus\mathscr{P}$. Then
$P\in U_{\eta}$ for some $\eta$. Since $F_{U_{\eta}}\subset F_{P}$ and
$Y(F_{U_{\eta}})\neq\emptyset$, $Y(F_{P})\neq\emptyset$. ∎

We record the following from ([20, Proposition 2.4]).

######  Proposition 1.2.

Let $(R,\mathfrak{m})$ be a 2-dimensional complete regular ring with maximal
ideal $\mathfrak{m}=(\pi,\delta)$, field of fractions $F$ and residue field
$k$. Let $D\in{}_{d}\text{Br}(F)$ be a division algebra over $F$ which is
unramified on $R$ except possibly at $\langle\pi\rangle$ or
$\langle\delta\rangle$. Suppose that $d$ is coprime to $\text{char}(k)$ and
$F$ contains a primitive $d^{\rm th}$ root of unity. Then ind$(A)=$
ind$(A\otimes F_{\pi})$.

###### Proof.

Let $n=$ deg$(A)$. This result is proved in ([20, Proposition 2.4]) under the
assumption that $F$ contains a primitive $n^{\rm th}$ root of unity. However
we note that the same proof goes through if $F$ contains a primitive $d^{\rm
th}$ root of unity. ∎

We end this section with the following.

######  Theorem 1.3.

Let $T$ be a complete discrete valuation ring with residue field $k$ and field
of fractions $K$. Suppose that char$(k)\neq 2$. Let $F$ be the function field
of a smooth projective geometrically integral curve over $K$ and $\mathscr{X}$
a model of $F$. Let $A$ be a central simple algebra over $F$ of period at most
2. Let $L$ be the field $F$ or $F_{P}$ for some closed point $P$ of
$\mathscr{X}$. Let $Z$ be a projective homogeneous space under $PGL(A)$ over
$L$. If $Z(L_{\nu})\neq\emptyset$ for all divisorial discrete valuations of
$L$, then $Z(L)\neq\emptyset$.

###### Proof.

This follows from the proof of ([18, Theorem 10.1]) and (1.2). ∎

######  Corollary 1.4.

Let $L$ and $A$ be as in (1.3). Let $Z$ be a projective homogenous space under
$GL(A)$ over $L$. If $Z(L_{\nu})\neq\emptyset$ for all divisorial discrete
valuations of $L$, then $Z(L)\neq\emptyset$.

###### Proof.

This follows from the proof of ([18, Corollary 10.3]) and (1.3). ∎

##  2\. Division algebras with involution of first kind over two dimensional
local fields

Let $(R,\mathfrak{m})$ be a 2-dimensional complete regular ring with maximal
ideal $(\pi,\delta)$, field of fractions $F$ and residue field $k$. Suppose
that $\text{char}(k)\neq 2$. Let $D\in{}_{2}\text{Br}(F)$ be a division
algebra over $F$ which is unramified on $R$ except possibly at
$\langle\pi\rangle$ or $\langle\delta\rangle$. In this section we show that if
ind$(D)=4$, then $D$ is a tensor product of two quaternion algebras with some
properties. Suppose that for any central simple algebra
$A\in{}_{2}\text{Br}(k)$, ind$(A)\leq 2$. Then we show that ind$(D)\leq 8$.
Further we show that if ind$(D)=8$, then $D$ is isomorphic to a tensor product
of three quaternion algebras with some properties.

We begin with the following well know lemma

######  Lemma 2.1.

Let $A\in{}_{2}\text{Br}(k)$ be a central division algebra over $k$.
1) If $A\otimes_{k}k(\sqrt{a})$ has index at most 2 for some $a\in k^{*}$,
then $A=(a,c)\otimes(b,d)\in Br(k)$ for some $b,c,d,\in k^{*}$.
2) If $A\otimes_{k}k(\sqrt{a},\sqrt{b})$ is a matrix algebra for some $a,b\in
k^{*}$, then $A=(a,c)\otimes(b,d)\in Br(k)$ for some $c,d,\in k^{*}$.

###### Proof.

If deg$(A)=1$ or $2$, then 1) and 2) are immediate. Suppose that deg$(A)\geq
4$.

Suppose $A\otimes_{k}k(\sqrt{a})$ has index at most 2 for some $a\in k^{*}$.
Then deg$(A)=$ ind$(A)=4$ and $K=k(\sqrt{a})$ is a subfield of $A$. Let
$A_{1}$ be the commutant of $K$ in $A$. Then $A_{1}$ is a quaternion algebra
over $K$ ([24, Theorem 5.4]). Since $A\in{}_{2}\text{Br}(k)$, $A$ admits an
involution ([24, Chapter 8, Theorem 8.4]) and the non trivial automorphism of
$K/k$ can be extended to an involution $\sigma$ on $A$ ([24, Chapter 8,
Theorem 10.1]). Since $\sigma(K)=K$, the restriction of $\sigma$ to $A_{1}$ is
an involution of second kind. Thus, by a theorem of Albert ([2, Chapter 10,
Theorem 21]), $A_{1}=K\otimes Q_{1}$ for some quaternion algebra $Q_{1}$. Let
$Q_{2}$ be the commutant of $Q_{1}$ in $A$. Then $K\subset Q_{2}$ and
$A=Q_{2}\otimes Q_{1}$. Since $K\subset Q_{2}$, $Q_{2}=(a,c)$ for some $c\in
k^{*}$. Since $Q_{1}$ is a quaternion algebra, $Q_{1}=(b,d)$ for some $b,d\in
k^{*}$. Hence $A=(a,c)\otimes(b,d)$.

Suppose $A\otimes_{k}k(\sqrt{a},\sqrt{b})$ is a matrix algebra for some
$a,b\in k^{*}$. Then deg$(A)=$ ind$(A)=4$ and $K=k(\sqrt{a},\sqrt{b})$ is a
maximal subfield of $A$. Then as above we have $A=Q_{2}\otimes Q_{1}$ with
$K=k(\sqrt{a})\subset Q_{2}$. Since $k(\sqrt{a},\sqrt{b})$ is a maximal
subfield of $A$, it follows that $k(\sqrt{b})\subset Q_{1}$. Hence
$Q_{2}=(a,c)$ and $Q_{1}=(b,d)$ for some $c,d\in k^{*}$. Thus
$A=(a,c)\otimes(b,d)$. ∎

######  Lemma 2.2.

Let $R$ be a complete regular local ring with residue field $k$ and field of
fractions $F$. Suppose that char$(k)\neq 2$. Let $\Gamma_{0}$ be an Azumaya
algebra over $R$ and $D_{0}=\Gamma_{0}\otimes_{R}F\in{}_{2}\text{Br}(F)$. Let
$u\in R$ be a unit. If ind$(D_{0}\otimes_{F}(F(\sqrt{u})))\leq 2$. Then there
exists $v,w,t\in R^{*}$ such that $D_{0}=(u,v)\otimes(w,t)\in Br(F)$.

###### Proof.

Suppose that ind$(D_{0}\otimes_{F}(F(\sqrt{u})))\leq 2$. Let
$\mathfrak{D}_{0}=\Gamma_{0}\otimes_{R}k$. Since ind$(D_{0}\otimes
F(\sqrt{u})\leq 2$, $\mathfrak{D}_{0}\otimes k(\sqrt{\overline{u}})\leq 2$
([20, Lemma 1.1]), where $\overline{u}$ is the image of $u$ in $k$. Thus there
exist $b,c,d\in k^{*}$ such that
$\mathfrak{D}_{0}=(\overline{u},c)\otimes(b,d)$ (cf. 2.1). Let $v,w,t\in
R^{*}$ be the lifts of $b,c,d\in k^{*}$. Since $R$ is a complete regular local
ring, $\text{Br}(R)\cong\text{Br}(k)$ ([3], 6.5). Hence
$D_{0}=(u,v)\otimes(w,t)\in Br(F)$. ∎

######  Lemma 2.3.

Let $R$ be a complete regular local ring with residue field $k$ and field of
fractions $F$. Suppose that char$(k)\neq 2$. Let $\Gamma_{0}$ be an Azumaya
algebra over $R$ with $D_{0}=\Gamma_{0}\otimes_{R}F\in{}_{2}\text{Br}(F)$. Let
$u,v\in R$ be units. If $D_{0}\otimes_{F}(F(\sqrt{u},\sqrt{v}))$ is a matrix
algebra, then there exists $w,t\in R$ units such that
$D_{0}=(u,w)\otimes(v,t)\in\text{Br}(F)$.

###### Proof.

Let $\mathfrak{D}_{0}=\Gamma_{0}\otimes_{R}k$. Since
$D_{0}\otimes_{F}(F(\sqrt{u},\sqrt{v}))$ is a matrix algebra,
$\mathfrak{D}_{0}\otimes k(\sqrt{\overline{u}},\sqrt{\overline{v}})$ is a
matrix algebra. Hence
$\mathfrak{D}_{0}=(\overline{u},c)\otimes(\overline{v},d)\in Br(k)$ (cf. 2.1).
Let $w,t\in R^{*}$ be lifts of $c,d\in k^{*}$. Since $R$ is a complete regular
local ring, $D_{0}=(u,w)\otimes(v,t)\in\text{Br}(F)$. ∎

######  Lemma 2.4.

Let $R$ be a two dimensional complete regular local ring with residue field
$k$, maximal ideal $(\pi,\delta)$ and field of fractions $F$. Suppose that
char$(k)\neq 2$. Let $\Gamma_{0}$ be an Azumaya algebra on $R$ and
$D_{0}=\Gamma_{0}\otimes F$. Suppose that $D_{0}\in{}_{2}\mathrm{Br}(F)$. Let
$u,v\in R$ units.
i) If $D=D_{0}\otimes(u,\pi)\otimes(v,\delta)$, then ind$(D)=$
ind$(D_{0}\otimes F(\sqrt{u},\sqrt{v}))[F(\sqrt{u},\sqrt{v}):F]$.
ii) If $D=D_{0}\otimes(u\pi,v\delta)$, then ind$(D)=$ 2 ind$(D_{0})$.

###### Proof.

Let $\kappa(\pi)$ be the residue field at $\pi$. Then $\kappa(\pi)$ is the
field of fractions of the complete discrete valuation ring $R/(\pi)$ and the
the image $\overline{\delta}$ of $\delta$ in $\kappa(\pi)$ is a parameter.

Suppose $D=D_{0}\otimes(u,\pi)\otimes(v,\delta)$. Since
$D_{0}\otimes(v,\delta)$ is unramified at $\pi$, by ([20, Lemma 2.1]), we have
ind$(D\otimes F_{\pi})=$ ind$((D_{0}\otimes(v,\delta)\otimes
F_{\pi}(\sqrt{u}))[F_{\pi}(\sqrt{u}]:F_{\pi}]$. Since
$\Gamma_{0}\otimes\kappa(\pi)$ is unramified on $R/(\delta)$, by ([20, Lemma
2.1]), we have
ind$(\Gamma_{0}\otimes(\overline{v},\overline{\delta})\otimes\kappa(\pi)(\sqrt{\overline{u}}))=$
ind$(\Gamma_{0}\otimes\kappa(\pi)(\sqrt{\overline{u}},\sqrt{\overline{v}}))[\kappa(\pi)(\sqrt{\overline{u}},\sqrt{\overline{v}}):\kappa(\pi)(\sqrt{\overline{u}})]$.
Since $F_{\pi}$ is complete, we have

${\text{i}nd}((D_{0}\otimes(v,\delta)\otimes
F_{\pi}(\sqrt{u}))={\text{i}nd}(D_{0}\otimes
F_{\pi}(\sqrt{u},\sqrt{v}))[F_{\pi}(\sqrt{u},\sqrt{v}):F_{\pi}(\sqrt{u})].$

Hence ind$(D\otimes F_{\pi})=$ ind$((D_{0}\otimes
F_{\pi}(\sqrt{u},\sqrt{v}))[F_{\pi}(\sqrt{u},\sqrt{v}]:F_{\pi}]$. By (1.2), we
have ind$(D)=$ ind$(D\otimes F_{\pi})$, ind$(D_{0}\otimes
F_{\pi}(\sqrt{u},\sqrt{v}))=$ ind$((D_{0}\otimes F_{\pi}(\sqrt{u},\sqrt{v})))$
and $[F_{\pi}(\sqrt{u},\sqrt{v}):F_{\pi}]=[F(\sqrt{u},\sqrt{v}):F]$. Thus
ind$(D)=$ ind$(D_{0}\otimes F(\sqrt{u},\sqrt{v}))[F(\sqrt{u},\sqrt{v}):F]$.

Suppose $D=D_{0}\otimes(u\pi,v\delta)$. Then as above, we have ind$(D)=$
ind$(D_{0}\otimes F(\sqrt{v\delta}))[F(\sqrt{v\delta}):F]$. Since $D_{0}$ is
unramified at $\delta$, we have ind$(D_{0}\otimes
F_{\delta}(\sqrt{v\delta}))=$ ind$(D_{0}\otimes F_{\delta})$. Once again, by
([20]), we have ind$(D_{0}\otimes F(\sqrt{v\delta}))=$ ind$(D_{0}\otimes
F_{\delta}(\sqrt{v\delta}))=$ ind$(D_{0}\otimes F_{\delta})$ = ind$(D_{0})$.
Since $[F(\sqrt{v\delta}):F]=2$, we have ind$(D)=$ 2 ind$(D_{0})$. ∎

We recall the following.

######  Lemma 2.5.

([25, Lemma 3.6]) Let $R$ be a two dimensional complete regular local ring
with residue field $k$, maximal ideal $(\pi,\delta)$ and field of fractions
$F$. Suppose that char$(k)\neq 2$. Let $D$ be quaternion division algebra over
$F$ which is unramified on $R$ except possibly at $(\pi)$ and $(\delta)$. Then
$D$ is isomorphic to one of the following:
i) $(u,w)$
ii) $(u,v\pi)$ or $(u,v\delta)$
iii) $(u,v\pi\delta)$
iv) $(u\pi,v\delta)$
for some units $u,v,w,\in R$.

######  Proposition 2.6.

Let $R$ be a two dimensional complete regular local ring with residue field
$k$, maximal ideal $(\pi,\delta)$ and field of fractions $F$. Suppose that
char$(k)\neq 2$. Let $D\in{}_{2}\text{Br}(F)$ be a division algebra over $F$
which is unramified on $R$ except possibly at $(\pi)$ and $(\delta)$. Suppose
that ind$(D)=4$. Then $D$ is isomorphic to one of the following:
i) $(u,w)\otimes(v,t)$
ii) $(u,w)\otimes(v,t\pi)$ or $(u,w)\otimes(v,t\delta)$
iii) $(u,v)\otimes(w,t\pi\delta)$
iv) $(u,w\pi)\otimes(v,t\delta)$
v) $(u,v)\otimes(w\pi,t\delta)$
for some units $u,v,w,t\in R$.

###### Proof.

Suppose that $D$ is unramified on $R$. Let $\Gamma$ be an Azumaya algebra on
$R$ with $\Gamma\otimes F\simeq D$ ([3, Theorem 7.4]). Since ind$(D)=4$, we
have ind$(\Gamma\otimes k)=4$. Hence $\Gamma\otimes k=(a,b)\otimes(c,d)$ for
some $a,b,c,d\in k^{*}$. Let $u,w,v,t\in R$ be lifts of $a,b,c,d$. Since $R$
is complete, we have $\Gamma=(u,w)\otimes(v,t)\in\text{Br}(R)$ and hence
$D=(u,w)\otimes(v,t)\in\text{Br}(F)$. Since deg$(D)=4$,
$D\simeq(u,w)\otimes(v,t)$.

Suppose that $D$ is ramified only at $\langle\pi\rangle$. Then, by Saltman’s
classification ([25], Proposition 3.5), we have $D=D_{0}\otimes(u,\pi)$, where
$D_{0}$ is unramified on $R$ and $u\in R$ a unit which is not a square. Since
$D=D_{0}\otimes(u,\pi)\otimes(1,\delta)$, by (2.4(i)), we have ind$(D)=2$
ind$(D_{0}\otimes F(\sqrt{u}))$. Since ind$(D)=4$, we have ind$(D_{0}\otimes
F(\sqrt{u}))=2$. Hence, by (2.2), we have $D_{0}=(u,w)\otimes(v,t)$ for some
$u,v,w,t\in R$ units. We have
$D=D_{0}\otimes(u,\pi)=(u,w)\otimes(v,t)\otimes(u,\pi)=(u,w\pi)\otimes(v,t)$.
Since ind$(D)=4$, $D\simeq(u,w\pi)\otimes(v,t)$. Similarly if $D$ is ramified
only at $\langle\delta\rangle$, then $D\simeq(u,w\delta)\otimes(v,t)$.

Suppose that $D$ is ramified both at $\langle\pi\rangle$ and
$\langle\delta\rangle$. Then by ([22, Theorem 2.1] & [23, Theorem 1.2]), we
have $D=D_{0}\otimes(u,\pi)\otimes(v,\delta)$ or
$D=D_{0}\otimes(w\pi,t\delta)$ for some $u,v\in R-R^{2}$ units, $w,t\in R$
units and $D_{0}$ unramified on $R$.

Suppose $D=D_{0}\otimes(u,\pi)\otimes(v,\delta)$. Suppose that $uv\in R$ is a
square. Then $D=D_{0}\otimes(u,\pi\delta)$. By (2.4(i)), we have ind$(D)=$ 2
ind$(D_{0}\otimes F(\sqrt{u}))$ and hence ind$(D_{0}\otimes F(\sqrt{u}))=2$.
Thus, by (2.2), we have $D_{0}=(u,w)\otimes(v,t)$ for some $u,v,w,t\in R$
units. In particular $D\simeq(v,t)\otimes(u,w\pi\delta)$.

Suppose that $uv$ not a square in $R$. By (2.4(i)), we have ind$(D)=$ 4
ind$(D_{0}\otimes F(\sqrt{u}))$ and hence ind$(D_{0}\otimes F(\sqrt{u}))=1$.
Hence $D_{0}\otimes F(\sqrt{u},\sqrt{v})$ is a matrix algebra. Thus, by (2.3),
we have $D_{0}=(u,w)\otimes(v,t)$ for some units $w,t\in R$. In particular
$D\simeq(u,w\pi)\otimes(v,t\delta)$.

Suppose $D=D_{0}+(w\pi,t\delta)$. By (2.4(ii)), we have ind$(D)=$ 2
ind$(D_{0})$ and hence ind$(D_{0})=2$. Thus $D_{0}=(u,v)$ and
$D\simeq(u,v)\otimes(w\pi,t\delta)$. ∎

######  Proposition 2.7.

Let $R$ be a two dimensional complete regular local ring with residue field
$k$, maximal ideal $(\pi,\delta)$ and field of fractions $F$. Suppose that
char$(k)\neq 2$ and every central simple algebra in ${}_{2}\text{Br}(k)$ has
index at most 2. Let $D\in{}_{2}\text{Br}(F)$ be a division algebra over $F$
which is unramified on $R$ except possibly at $(\pi)$ and $(\delta)$. Then
ind$(D)\leq 8$. Further if ind$(D)=8$, then
$D\simeq(w,t)\otimes(u,\pi)\otimes(v,\delta)$ for some units $u,v,w,t\in R$.

###### Proof.

Suppose that $D$ is unramified on $R$. Let $\Gamma$ be an Azumaya algebra on
$R$ with $\Gamma\otimes F\simeq D$ ([3, Theorem 7.4]). By the assumption on
$k$, ind$(\Gamma\otimes k)\leq 2$. Since $R$ is complete, we have ind$(D)\leq
2$ ([3, Theorem 6.5]).

Suppose that $D$ is ramified only at $\langle\pi\rangle$. Then, by Saltman’s
classification ([25], Proposition 3.5), we have $D=D_{0}\otimes(u,\pi)$, where
$D_{0}$ is unramified on $R$ and $u\in R$ a unit which is not a square. Since
$D_{0}$ is unramified on $R$, by the assumption on $k$, ind$(D_{0})\leq 2$ and
hence ind$(D)\leq 4$. Similarly if $D$ is ramified only at
$\langle\delta\rangle$, then ind$(D)\leq 4$.

Suppose that $D$ is ramified both at $\langle\pi\rangle$ and
$\langle\delta\rangle$. Then by ([22, Theorem 2.1] & [23, Theorem 1.2]), we
have $D=D_{0}\otimes(u,\pi)\otimes(v,\delta)$ or
$D=D_{0}\otimes(w\pi,t\delta)$ for some $u,v\in R-R^{2}$ units, $w,t\in R$
units and $D_{0}$ unramified on $R$. Since $D_{0}$ is unramified on $R$, by
the assumption on $k$, ind$(D_{0})\leq 2$. In particular ind$(D)\leq 8$.

Suppose ind$(D)=8$. Then $D=D_{0}\otimes(u,\pi)\otimes(v,\delta)$ for some
units $u,v\in R$ units and $D_{0}$ unramified on $R$. Once again by the
assumption on $k$, we have $D_{0}=(w,t)$ for some units $w,t\in R$. Hence
$D\simeq(w,t)\otimes(u,\pi)\otimes(v,\delta)$. ∎

##  3\. Two torsion division algebras with involution of second kind over two
dimensional local fields

Let $R_{0}$ be a 2-dimensional complete regular local ring with maximal ideal
$\mathfrak{m}_{0}=(\pi_{0},\delta_{0})$ and residue field $k_{0}$. Suppose
that char$(k_{0})\neq 2$. Let $F_{0}$ be the field of fractions of $R_{0}$ and
let $F=F_{0}(\sqrt{\lambda})$ be an extension of degree 2, with $\lambda$ a
unit in $R_{0}$ or $\lambda=w\pi_{0}$ for some unit $w\in R_{0}$. Let
$D\in{}_{2}\text{Br}(F)$ be a division algebra with $F/F_{0}$-involution
$\sigma$. In this section we show that if ind$(D)=4$, then $D$ is a tensor
product of two quaternion algebras with some properties. Suppose that for any
central simple algebra $A\in{}_{2}\text{Br}(k)$, ind$(A)\leq 2$. Then we show
that ind$(D)\leq 8$. Further we show that if ind$(D)=8$, then $D$ is
isomorphic to a tensor product of three quaternion algebras with some
properties.

Let $R$ be the integral closure of $R_{0}$ in $F$. By the assumption on
$\lambda$, $R$ is a 2-dimensional regular local ring with maximal ideal
$\mathfrak{m}=(\pi,\delta)$ ([19, Theorem 3.1, 3.2]), where; if $\lambda$ is a
unit in $R_{0}$, then $\pi=\pi_{0}$ and $\delta=\delta_{0}$ and if
$\lambda=w\pi_{0}$, then $\pi=\sqrt{\lambda}$ and $\delta=\delta_{0}$.

######  Proposition 3.1.

Let $R$ and $F$ be as above. Let $D\in{}_{2}\text{Br}(F)$ with an $F/F_{0}$
involution. Suppose that $D$ is unramified on $R$ except possibly at $(\pi)$
and $(\delta)$. If $\lambda=w\pi_{0}$ for some unit $w\in R_{0}$, then
$D=(D_{0}\otimes(v_{0},\delta_{0}))\otimes_{F_{0}}F$ for some
$D_{0}\in{}_{2}\text{Br}(F_{0})$ which is unramified on $R_{0}$ and $v_{0}\in
R_{0}$ a unit.

###### Proof.

Since $F/F_{0}$ is ramified at $\pi_{0}$, by ([18, Lemma 6.3]), $D$ is
unramified at $\pi$. Hence, by([25, Proposition 3.5]),
$D=D^{\prime}\otimes(v,\delta)$ for some unit $v\in R$ and $D^{\prime}$
unramified on $R$. Let $\Gamma^{\prime}$ be an Azumaya $R$-algebra with
$\Gamma^{\prime}\otimes F\simeq D^{\prime}$. Since
$R/\mathfrak{m}=R_{0}/\mathfrak{m}_{0}$ and $R$ is complete, there exists an
Azumaya $R_{0}$-algebra $\Gamma_{0}$ with
$\Gamma^{\prime}\simeq\Gamma_{0}\otimes R$. Let $D_{0}=\Gamma_{0}\otimes
F_{0}$. Since $R/\mathfrak{m}=R_{0}/\mathfrak{m}_{0}$, we have
$v=v_{0}v_{1}^{2}$ for some $v_{1}\in R_{0}$ a unit. Since
$\delta=\delta_{0}$, we have $D=(D_{0}\otimes(v_{0},\delta_{0}))\otimes F$ as
required. ∎

For the rest of the section, we assume that $\lambda\in R_{0}$ is a unit. In
particular $\pi=\pi_{0}$ and $\delta=\delta_{0}$ and $F/F_{0}$ is unramified
on $R_{0}$. Let $\tau$ denote the non trivial automorphism of $F/F_{0}$.

######  Proposition 3.2.

Let $\Gamma^{\prime}$ be an Azumaya $R$-algebra and
$D^{\prime}=\Gamma^{\prime}\otimes F$. Suppose that
$D^{\prime}\in{}_{2}\mathrm{Br}(F)$. Let
$D=D^{\prime}\otimes(u,\pi)\otimes(v,\delta)$ (resp.
$D=D^{\prime}\otimes(u\pi,v\delta))$. If $D$ has a $F/F_{0}$-involution, then
$D^{\prime}$, $(u,\pi)$ and $(v,\delta)$ (resp. $D^{\prime}$ and
$(u\pi,v\delta)$) have $F/F_{0}$-involutions.

###### Proof.

Suppose $D=D^{\prime}\otimes(u,\pi)\otimes(v,\delta)$ has a
$F/F_{0}$-involution. Since cores${}_{F/F_{0}}(D)=0$, by ([18, Lemma 6.4]),
cores${}_{F_{\delta}/F_{0\delta_{0}}}(D^{\prime}\otimes(u,\pi))=0$ and
cores${}_{F_{\delta}/F_{0\delta_{0}}}(v,\delta)=0$. Since $D^{\prime}$ is
unramified on $R$, cores${}_{F/F_{0}}(D^{\prime})$ is unramified on $R_{0}$.
Since $\pi=\pi_{0}\in R_{0}$ and $\delta=\delta_{0}\in R_{0}$,
cores${}_{F/F_{0}}(u,\pi)=(N_{F/F_{0}}(u),\pi_{0})\otimes F_{0}$ and
cores${}_{F/F_{0}}(v,\delta)=(N_{F/F_{0}}(v),\delta_{0})\otimes F_{0}$. In
particular cores${}_{F/F_{0}}(D^{\prime}\otimes(u,\pi))$ and
cores${}_{F/F_{0}}(v,\delta)$ are unramified on $R_{0}$ except possibly at
$(\pi_{0})$. Hence, by (1.2), cores${}_{F/F_{0}}(D^{\prime}\otimes(u,\pi))=0$
and cores${}_{F/F_{0}}(v,\delta)=0$. The same argument implies that
cores${}_{F/F_{0}}(D^{\prime})=0$ and cores${}_{F/F_{0}}(u,\pi)=0$. Hence
$D^{\prime}$, $(u,\pi)$ and $(v,\delta)$ have $F/F_{0}$-involutions.

The case $D=D^{\prime}\otimes(u\pi,v\delta)$ is similar. ∎

######  Lemma 3.3.

Let $D_{1}=(u,\pi)$ (resp. $(u\pi,v\delta)$, $(u,v)$) for some units $u,v\in
R$. If $D_{1}$ has a $F/F_{0}$ involution, then $D_{1}=(u_{0},\pi)$ (resp.
$(u_{0}\pi,v_{0}\delta)$, $(u_{0},v_{0})$) for some $u_{0},v_{0}\in R_{0}$
units.

###### Proof.

Suppose $D_{1}=(u,\pi)$ for some $u\in R$ unit. Since $\pi=\pi_{0}\in R_{0}$,
we have cores${}_{F/F_{0}}(u,\pi)=(N_{F/F_{0}}(u),\pi_{0})$. Since $D_{1}$ has
a $F/F_{0}$-involution, we have $(N_{F/F_{0}}(u),\pi_{0})=0\in Br(F_{0})$.
Since the residue of $(N_{F/F_{0}}(u),\pi_{0})$ at $\pi_{0}$ is the image of
$N_{F/F_{0}}(u)$ in $\kappa(\pi_{0})^{*}/\kappa(\pi_{0})^{*2}$, the image of
$N_{F/F_{0}}(u)$ is a square in $\kappa(\pi_{0})$. Since $R_{0}$ is a complete
local ring with $\pi_{0}$ a regular prime, $N_{F/F_{0}}(u)$ is a square in
$R_{0}$. Let $w_{0}\in R_{0}$ be such that $N_{F/F_{0}}(u)=w_{0}^{2}$. Then
$N_{F/F_{0}}(w_{0}^{-1}u)=1$ and hence $w_{0}^{-1}u=\theta\tau(\theta)^{-1}$
for some $\theta\in R$. We have
$u\tau(\theta)^{2}=w_{0}\theta\tau(\theta)=u_{0}\in R_{0}$ and
$(u,\pi)=(u_{0},\pi)$.

Suppose $D_{1}=(u\pi,v\delta)$. As above, by taking the residues at $\pi$ and
$\delta$, we see that $N_{F/F_{0}}(u)$ and $N_{F/F_{0}}(v)$ are squares. Hence
as above, we can replace $u$ and $v$ by $u_{0}$ and $v_{0}$ for some
$u_{0},v_{0}\in R_{0}$ units.

Suppose that $D_{1}=(u,v)$ for some $u,v\in R$. Since $D_{1}$ has an
$F/F_{0}$-involution and $D_{1}$ is unramified on $R$, $D_{1}=D_{0}\otimes F$
for some quaternion algebra $D_{0}$ over $F_{0}$ which is unramified on
$R_{0}$ ([3, Theorem 7.4]). In particular $D_{0}=(u_{0},v_{0})$ for some units
$u_{0},v_{0}\in R_{0}$ ∎

######  Corollary 3.4.

Let $D\in{}_{2}Br(F)$ with an $F/F_{0}$-involution. Suppose that $D$ is
unramified on $R$ except possibly at $(\pi)$ and $(\delta)$ and ind$(D)=2$.
Then one of the following holds
i) $D$ is unramified on $R$
ii) $D\simeq(u_{0},u_{1}\pi_{0})$ or $D\simeq(v_{0},v_{1}\delta_{0})$
iii) $D\simeq(u_{0},u_{1}\pi_{0}\delta_{0})$
iv) $D\simeq(u_{0}\pi_{0},v_{0}\delta_{0})$
for some units $w_{i},u_{i},v_{i}\in R_{0}$

###### Proof.

Follows from (2.5), (3.2) and (3.3). ∎

######  Corollary 3.5.

Let $D\in{}_{2}Br(F)$ with an $F/F_{0}$-involution. Suppose that $D$ is
unramified on $R$ except possibly at $(\pi)$ and $(\delta)$ and ind$(D)=4$.
Then one of the following holds
i) $D$ is unramified on $R$
ii) $D\simeq(w_{0},w_{1})\otimes(u_{0},u_{1}\pi_{0})$ or
$D\simeq(w_{0},w_{1})\otimes(v_{0},v_{1}\delta_{0})$
iii) $D\simeq(w_{0},w_{1})\otimes(u_{0},u_{1}\pi_{0}\delta_{0})$
iv) $D\simeq(u_{0},u_{1}\pi_{0})\otimes(v_{0},v_{1}\delta_{0})$
v) $D\simeq(w_{0},w_{1})\otimes(u_{0}\pi_{0},v_{0}\delta_{0})$
for some units $w_{i},u_{i},v_{i}\in R_{0}$

###### Proof.

Follows from (2.6), (3.2) and (3.3). ∎

######  Corollary 3.6.

Let $D\in{}_{2}Br(F)$ with an $F/F_{0}$-involution. Suppose that $D$ is
unramified on $R$ except possibly at $(\pi)$ and $(\delta)$ and every element
of ${}_{2}Br(k)$ has index at most 2. If ind$(D)=8$, then Then
$D\simeq(w_{0},w_{1})\otimes(u_{0},\pi_{0})\otimes(v_{0},\delta_{0})$ for some
units $w_{0},w_{1},u_{0},v_{0}\in R_{0}$.

###### Proof.

By the assumptions on $k$ and $D$, by (2.7),
$D\simeq(w_{0},w_{1})\otimes(u_{0},\pi)\otimes(v_{0},\delta)$ for some units
$w_{0},w_{1},u_{0},v_{0}\in R$. Since $D$ has a $F/F_{0}$-involution, by
(3.2), $(w_{0},w_{1})$, $(u_{0},\pi)$ and $(v_{0},\delta)$ have
$F/F_{0}$-involutions. As in the proof of (3.5), we can assume
$w_{0},w_{1},u_{0},v_{0}\in R_{0}$. ∎

##  4\. maximal orders

Let $R$ be a complete regular local ring with residue field $k$,
$(\pi,\delta)$ maximal ideal and field of fractions $F$. Suppose that
char$(k)\neq 2$. Let $D\in{}_{2}\text{Br}(F)$ be a division algebra which is
unramified on $R$ except possibly at $(\pi)$ and $(\delta)$. By (([25],
Proposition 3.5)), we know that $D=D_{0}\otimes D_{1}$ for some
$D_{0}\in{}_{2}\text{Br}(F)$ which is unramified on $R$ and $D_{1}$ is
$(u,v\pi)$ or $(u,v\delta)$ or $(u,w\pi)\otimes(v,t\delta)$ or
$(u,v\pi\delta)$ or $(u\pi,v\delta)$ for some units $u,v\in R$. If $D\simeq
D_{0}\otimes D_{1}$, then in this section we show that there is a maximal
$R$-order with some properties.

For an integral domain $R$ and $a,b\in R$ non zero elements, let $R(a,b)$ be
the $R$-algebra generated by $i,j$ with $i^{2}=a$, $j^{2}=b$ and $ij=-ji$.
Suppose that $2\in R$ is a unit. Then $R(a,b)$ is a $R$-order in the
quaternion algebra $(a,b)$ over $F$. Further note that if $a,b\in R$ are
units, then $R(a,b)$ is an Azumaya $R$-algebra.

######  Proposition 4.1.

Let $R$ be a complete discrete valuation ring with residue field $k$ and field
of fractions $F$. Suppose that char$(k)\neq 2$. Let $\Gamma_{0}$ be an Azumaya
algebra over $R$ and $D_{0}=\Gamma_{0}\otimes_{R}K$. Let $u\in R$ be a unit
and $\pi\in R$ a parameter. If $D=D_{0}\otimes(u,\pi)$ is a division algebra,
then $\Gamma=\Gamma_{0}\otimes_{R}R(u,\pi)$ is the maximal $R$-order of $D$.

###### Proof.

Suppose that $D$ is division. Let $d=$ deg$(D)$ and $d_{0}=$ deg$(D_{0})$.
Then $d=2d_{0}$.

There is a discrete valuation on $D$ given by $\nu_{D}(z)=\nu(Nrd_{D}(z))$
([21, 139]). Furthermore $\Gamma^{\prime}=\\{z\in D^{*}\mid\nu_{D}(z)\geq
0\\}\cup\\{0\\}=\\{z\in D\mid z~{}\text{is ~{}integral~{} over}~{}R\\}$ is the
unique maximal $R$-order of $D$ ([21, Theorem 12.8]).

Since $\Gamma_{0}$ and $R(u,\pi)$ are finitely generated $R$-modules, $\Gamma$
is a finitely generated $R$-module and hence every element of $\Gamma$ is
integral over $R$. Hence $\Gamma\subseteq\Gamma^{\prime}$. We now show that
$\Gamma^{\prime}\subseteq\Gamma$.

Let $i,j\in(u,\pi)$ be the standard generators with $i^{2}=u$, $j^{2}=\pi$ and
$ij=-ji$. Let $D_{1}=D_{0}\otimes F(i)\subset D$ and
$\Gamma_{1}=\Gamma_{0}\otimes R[i]$. Then $D=D_{1}+D_{1}j$ and
$\Gamma=\Gamma_{1}\oplus\Gamma_{1}j$. Since $D_{0}$ is unramified, $D_{1}$ is
unramified. Since $D$ is a division algebra, $D_{1}$ is a division algebra.
Since $\Gamma_{0}$ is an Azumaya algebra over $R$, $\Gamma_{1}$ is an Azumaya
algebra over $R[i]$. In particular $\Gamma_{1}$ is the maximal $R[i]$-order in
$D_{1}$. Since $F(i)/F$ is an unramified extension and $D_{0}$ is unramified
on $R$, $\pi$ is a parameter in $D_{1}$. Therefore, $\nu_{D}(z)$ is a multiple
of $2d_{0}$ for all $z\in D_{1}$ ([21, 139]). Since $j^{2}=\pi$ and
$Nrd_{D}(j)=\pi^{d_{0}}$, we have $\nu_{D}(j)=d_{0}$.

Let $z\in\Gamma^{\prime}$. Then $z=z_{1}+z_{2}j$ for some $z_{1},z_{2}\in
D_{1}$. Suppose that $\nu_{D}(z_{1})=\nu_{D}(z_{2}j)$. Then
$\nu_{D}(z_{1})-\nu_{D}(z_{2})=\nu_{D}(j)=d_{0}$. This is a contradiction,
since $\nu_{D}(z_{1})$ and $\nu_{D}(z_{2})$ are multiple of $2d_{0}$. Hence
$\nu_{D}(z_{1})\neq\nu_{D}(z_{2}j)$. Then
$\nu_{D}(z)=\text{min}\\{\nu_{D}(z_{1}),\nu_{D}(z_{2}j)\\}\geq 0$. In
particular $\nu_{D}(z_{1})\geq 0$ and hence $z_{1}\in\Gamma_{1}$. Since
$\nu_{D}(z_{2}j)=\nu_{D}(z_{2})+\nu_{D}(j)=\nu_{D}(z_{2})+d_{0}$, we have
$\nu_{D}(z_{2})\geq-d_{0}$. Since $\nu_{D}(z_{2})$ is a multiple of $2d_{0}$,
it follows that $\nu_{D}(z_{2})\geq 0$ and hence $z_{2}\in\Gamma_{1}$. Thus
$z\in\Gamma$. ∎

######  Proposition 4.2.

Let $R$ be a discrete valuation ring with residue field $k$, field of
fractions $F$ and $\hat{F}$ the completion of $F$. Suppose that char$(k)\neq
2$. Let $\Gamma_{0}$ be an Azumaya algebra over $R$ and
$D_{0}=\Gamma_{0}\otimes_{R}F$. Let $u\in R$ be a unit and $\pi\in R$ a
parameter. If $(D_{0}\otimes_{F}(u,\pi))\otimes_{F}\hat{F}$ is a divison
algebra, then $\Gamma_{0}\otimes_{R}R(u,\pi)$ is the maximal $R$-order of
$D_{0}\otimes_{F}(u,\pi)$.

###### Proof.

Let $\hat{R}$ be the completion of $R$. Let
$\hat{\Gamma}_{0}=\Gamma_{0}\otimes\hat{R}$. Then $\hat{\Gamma}_{0}$ is an
Azumaya algebra over $\hat{R}$. Since
$(D_{0}\otimes_{F}(u,\pi))\otimes_{F}\hat{F}$ is a divison algebra, by (4.1),
$\hat{\Gamma}_{0}\otimes\hat{R}(u,\pi)$ is a maximal $\hat{R}$-order of
$(D_{0}\otimes_{F}(u,\pi))\otimes_{F}\hat{F}$. Thus, by ([21], Theorem 11.5),
$\Gamma_{0}\otimes_{R}R(u,\pi)$ is the maximal $R$-order of
$D_{0}\otimes_{F}(u,\pi)$. ∎

######  Corollary 4.3.

Let $R$ be a two dimensional complete regular local ring with residue field
$k$, field of fractions $F$ and maximal ideal $\mathfrak{m}=(\pi,\delta)$. For
units $u,v\in R$, let $D_{1}$ and $\Gamma_{1}$ be one of the following:
i) $D_{1}=(u,v)$, $\Gamma_{1}=R(u,v)$
ii) $D_{1}=(u,\pi)$, $\Gamma_{1}=R(u,\pi)$
iii) $D_{1}=(\pi,\delta)$, $\Gamma_{1}=R(\pi,\delta)$
iv) $D_{1}=(u,\pi\delta)$, $\Gamma_{1}=R(u,\pi\delta)$
v) $D_{1}=(u,\pi)\otimes(v,\delta)$, $\Gamma_{1}=R(u,\pi)\otimes R(v,\delta)$.
Let $\Gamma_{0}$ be an Azumaya algebra over $R$ and
$D_{0}=\Gamma_{0}\otimes_{R}F$. If $D_{0}\otimes_{F}D_{1}$ is a division
algebra, then $\Gamma=\Gamma_{0}\otimes_{R}\Gamma_{1}$ is a maximal $R$-order
of $D_{0}\otimes_{F}D_{1}$.

###### Proof.

An order of a Noetherian integrally closed domain is maximal if and only if it
is reflexive and its localization at all height one prime ideals are maximal
orders ([21], Theorem 11.4). Since $\Gamma$ is a finitely generated free
module, it is reflexive. Furthermore, $R$ is a regular local ring, hence it is
Noetherian and integrally closed. We only need to show that $\Gamma_{P}$ is a
maximal $R_{P}$-order for all height one prime ideals $P$.

Suppose that $D=D_{0}\otimes_{F}D_{1}$ is a division algebra. Let $P$ be a
height one prime ideal of $R$. Suppose
$P\neq\langle\pi\rangle,\langle\delta\rangle$. Since $u,v\in R$ are units,
$u,v,\pi,\delta$ are units in $R_{P}$ and hence $\Gamma_{1}\otimes R_{P}$ is
an Azumaya $R_{P}$-algebra. In particular $\Gamma\otimes R_{P}$ is an Azumaya
$R_{P}$-algebra. Hence $\Gamma_{P}$ is a maximal $R_{P}$-order of $D$. Suppose
that $P=\langle\pi\rangle,\langle\delta\rangle$.

  1. i)

Since $u,v\in R$ are units, ${(\Gamma_{1})}_{P}$ is an Azumaya algebra over
$R_{P}$. Hence $\Gamma_{P}$ is a maximal $R_{P}$-order on $D$.

  2. ii)

If $P\neq\langle\pi\rangle)$, then $\Gamma_{1}$ is an Azumaya $R_{P}$-algebra
and hence $\Gamma_{P}$ is a maximal $R_{P}$-order on $D$. If
$P=\langle\pi\rangle$, then $\Gamma_{P}$ is a maximal $R_{P}$-order on $D$ by
(4.2).

  3. iii), iv)

If $P=\langle\pi\rangle$ or $P=\langle\delta\rangle$, then $\Gamma_{P}$ is a
maximal $R_{P}$-order on $D$ by (4.2).

  4. v)

Suppose $P=\langle\pi\rangle$. Let $\Gamma_{0}^{\prime}=\Gamma_{0}\otimes
R_{P}(v,\delta)$. Since $v,\delta$ are units in $R_{P}$, $R_{P}(v,\delta)$ is
an Azumaya $R_{P}$-algebra. Since $D=(D_{0}\otimes(v,\delta))\otimes(u,\pi)$
and $\Gamma=(\Gamma_{0}\otimes R_{P}(v,\delta))\otimes R_{P}(u,\pi)$, by
(4.2), $\Gamma_{P}$ is a maximal $R_{P}$-order on $D$. If
$P=\langle\delta\rangle$, a similar argument holds.

∎

##  5\. A local global principle for hermitian forms over two dimensional
local fields

Let $R_{0}$ be a 2-dimensional complete regular local ring with maximal ideal
$\mathfrak{m}_{0}=(\pi_{0},\delta_{0})$ and residue field $k_{0}$. Suppose
that char$(k_{0})\neq 2$. Let $F_{0}$ be the field of fractions of $R_{0}$ and
let $F=F_{0}(\sqrt{\lambda})$ be an extension of degree at most 2, with
$\lambda$ a unit in $R_{0}$ or a unit times $\pi_{0}$. Let
$D\in{}_{2}\text{Br}(F)$ be a division algebra with $F/F_{0}$-involution
$\sigma$ and $h$ an hermitian form over $(D,\sigma)$. In this section, under
some assumptions on $D$, $F_{0}$ and $h$, we prove that if $h\otimes F_{0\pi}$
or $h\otimes F_{0\delta}$ is isotropic, then $h$ is isotropic.

Let $R$ be the integral closure of $R_{0}$ in $F$. By the assumption on
$\lambda$, $R$ is a 2-dimensional regular local ring with maximal ideal
$(\pi,\delta)$ ([19, 3.1, 3.2]), where; if $\lambda$ is a unit in $R_{0}$,
then $\pi=\pi_{0}$ and $\delta=\delta_{0}$ and if $\lambda$ is a unit times
$\pi_{0}$, then $\pi=\sqrt{\lambda}$ and $\delta=\delta_{0}$.

Let $G(D,\sigma,h)=SU(D,\sigma,h)$ if $F=F_{0}$ and
$G(D,\sigma,h)=U(A,\sigma,h)$ if $[F:F_{0}]=2$.

We begin with the following, which is proved by Wu ([25, Corollary 3.12]) for
$D$ a quaternion algebra.

######  Proposition 5.1.

Let $F_{0}$ and $F$ be as above. Let $D\in{}_{2}\text{Br}(F)$ and $\sigma$ an
$F/F_{0}$\- involution. Suppose that $D$ is a division algebra which
unramified on $R$ except possibly at $(\pi)$ and $(\delta)$. Let $d=$deg$(D)$,
$e_{0}$ the ramification index of $D$ at $\pi$ and $e_{1}$ the ramification
index of $D$ at $\delta$. Suppose that there exists a maximal $R$-order
$\Gamma$ of $D$ and $\pi_{D},\delta_{D}\in\Gamma$ such that
$\sigma(\pi_{D})=\pm\pi_{D}$, $\sigma(\delta_{D})=\pm\pi_{D}$ and
$\pi_{D}\delta_{D}=\pm\delta_{D}\pi_{D}$ and
Nrd$(\pi_{D})=\theta_{0}\pi^{\frac{d}{e_{0}}}$ and
Nrd$(\delta_{D})=\theta_{1}\delta^{\frac{d}{e_{1}}}$ for some units
$\theta_{0},\theta_{1}\in R$. Let $h=\langle a_{1},\cdots,a_{n}\rangle$ be an
hermitian form over $(D,\sigma)$. Suppose that for $1\leq i\leq n$,
$a_{i}\in\Gamma$ and Nrd$(a_{i})$ is a product of a unit in $R$, a power of
$\pi$ and a power of $\delta$. If $h\otimes F_{\pi}$ or $h\otimes F_{\delta}$
is isotropic, then $h$ is isotropic.

###### Proof.

Follows from ([25, Corollary 3.3]). ∎

As a consequence we have the following (cf. [25, Corollary 3.12])

######  Proposition 5.2.

Let $F_{0}$ and $F$ be as above. Let $D\in{}_{2}\text{Br}(F)$ and $\sigma$ an
$F/F_{0}$\- involution. Suppose that $D$ is a division algebra which is
unramified on $R$ except possibly at $(\pi)$ and $(\delta)$. Let $d=$deg$(D)$,
$e_{0}$ the ramification index of $D$ at $\pi$ and $e_{1}$ the ramification
index of $D$ at $\delta$. Suppose that there exists a maximal $R$-order
$\Gamma$ of $D$ and $\pi_{D},\delta_{D}\in\Gamma$ such that
$\sigma(\pi_{D})=\pm\pi_{D}$, $\sigma(\delta_{D})=\pm\pi_{D}$,
$\pi_{D}\delta_{D}=\pm\delta_{D}\pi_{D}$,
Nrd$(\pi_{D})=\theta_{0}\pi^{\frac{d}{e_{0}}}$ and
Nrd$(\delta_{D})=\theta_{1}\delta^{\frac{d}{e_{1}}}$ for some units
$\theta_{0},\theta_{1}\in R$. Let $h=\langle a_{1},\cdots,a_{n}\rangle$ be an
hermitian form over $(D,\sigma)$. Suppose that for $1\leq i\leq n$,
$a_{i}\in\Gamma$ and Nrd$(a_{i})$ is a product of a unit in $R$, a power of
$\pi$ and a power of $\delta$. Let $X$ be a projective homogeneous space under
$G(D,\sigma,h)$. If $X(F_{0\pi})\neq\emptyset$ or
$X(F_{0\delta})\neq\emptyset$, then $X(F_{0})\neq\emptyset$.

###### Proof.

Since the period of $D$ divides 2 by our assumption on $D$, we have ind$(D)=$
ind$(D\otimes F_{\pi})$ (1.2). Suppose that $X(F_{0\pi})\neq\emptyset$ or
$X(F_{0\delta})\neq\emptyset$. Using, (5.1), the rest of the proof of ([25,
Corollary 3.12]) can be applied here to show that $X(F_{0})\neq\emptyset$. ∎

We fix the following.

######  Notation 5.3.

Let $R_{0}$ be a 2-dimensional complete regular local ring with maximal ideal
$\mathfrak{m}_{0}=(\pi_{0},\delta_{0})$ and residue field $k_{0}$. Suppose
that char$(k_{0})\neq 2$. Let $F_{0}$ be the field of fractions of $R_{0}$ and
let $F=F_{0}(\sqrt{\lambda})$ be an extension of degree at most 2, with
$\lambda$ a unit in $R_{0}$ or a unit times $\pi_{0}$. Let $R$ be the integral
closure of $R_{0}$ in $F$. By the assumption on $\lambda$, $R$ is a
2-dimensional regular local ring with maximal ideal $(\pi,\delta)$ ([19, 3.1,
3.2]), where; if $\lambda$ is a unit in $R_{0}$, then $\pi=\pi_{0}$ and
$\delta=\delta_{0}$ and if $\lambda$ is a unit times $\pi_{0}$, then
$\pi=\sqrt{\lambda}$ and $\delta=\delta_{0}$. Let $u_{i},v_{i}\in R_{0}$ be
units and $D_{1}$, $\Gamma_{1}$ and $\sigma_{1}$ denote one of the following:
i) $D_{1}=F_{0}$, $\Gamma_{1}=R$, $\sigma_{1}=id$
ii) $D_{1}=(u_{0},u_{1}\pi_{0})$, $\Gamma_{1}=R(u_{0},u_{1}\pi_{0})$,
$\sigma_{1}$ the canonical involution
iii) $D_{1}=(u_{0}\pi_{0},v_{0}\delta_{0})$,
$\Gamma_{1}=R(u_{0}\pi_{0},v_{0}\delta_{0})$, $\sigma_{1}$ the canonical
involution
iv) $D_{1}=(u_{0},u_{1}\pi_{0})\otimes(v_{0},v_{1}\delta_{0})$,
$\Gamma_{1}=R(u_{0},u_{1}\pi_{0})\otimes R(v_{0},v_{1}\delta_{0})$,
$\sigma_{1}$ the tensor product of the canonical involutions
Let $\Gamma_{0}$ be an Azumaya algebra over $R$ with an $R/R_{0}$\- involution
$\tilde{\sigma}_{1}$. Let $D_{0}\simeq\Gamma_{0}\otimes F$ and
$\sigma_{0}=\tilde{\sigma_{0}}\otimes 1$. Let $D\simeq D_{0}\otimes D_{1}$,
$\Gamma=\Gamma_{0}\otimes\Gamma_{1}$ and $\sigma=\sigma_{0}\otimes\sigma_{1}$.
Then $\sigma$ is a $F/F_{0}$-involution on $D$ and $\sigma(\Gamma)=\Gamma$.
Let $d_{i}$ denote the degree of $D_{i}$. The following table gives a choice
of $\pi_{D},\delta_{D}\in\Gamma$ and some of their properties.

D | $\pi_{D}$ | $\delta_{D}$ | $\text{Nrd}(\pi_{D})$ | $\text{Nrd}(\delta_{D})$ | $\sigma(\pi_{D})$ | $\sigma(\delta_{D})$ | $\sigma(\pi_{D}\delta_{D})$
---|---|---|---|---|---|---|---
$D_{0}$ | $\pi_{0}$ | $\delta_{0}$ | $\pi_{0}^{d_{0}}$ | $\delta_{0}^{d_{0}}$ | $\pi_{D}$ | $\delta_{D}$ | $\pi_{D}\delta_{D}$
$D_{0}\otimes(u_{0},u_{1}\pi_{0})$ | $1\otimes j$ | $1\otimes\delta$ | $(u_{1}\pi_{0})^{d_{0}}$ | $\delta_{0}^{2d_{0}}$ | $-\pi_{D}$ | $\delta_{D}$ | $-\pi D\delta_{D}$
$D_{0}\otimes(u_{0}\pi_{0},v_{0}\delta_{0})$ | $1\otimes i$ | $1\otimes j$ | $(u_{0}\pi_{0})^{d_{0}}$ | $(v_{0}\delta_{0})^{d_{0}}$ | $-\pi_{D}$ | $-\delta_{D}$ | $-\pi_{D}\delta_{D}$
$D_{0}\otimes(u_{0},u_{1}\pi_{0})\otimes(v_{0},v_{1}\delta_{0})$ | $1\otimes j_{1}\otimes 1$ | $1\otimes 1\otimes j_{2}$ | $(u_{1}\pi_{0})^{2d_{0}}$ | $(v_{1}\delta_{0})^{2d_{0}}$ | $-\pi_{D}$ | $-\delta_{D}$ | $\pi_{D}\delta_{D}$

######  Corollary 5.4.

Let $F$, $D$, $\sigma$ and $\Gamma$ be as in (5.3). Suppose that $D$ is a
division algebra. Let $h=\langle a_{1},\cdots,a_{n}\rangle$ be a hermitian
form over $(D,\sigma)$ with $a_{i}\in\Gamma$. Suppose Nrd$(a_{i})$ is a unit
times a power of $\pi$ and a power of $\delta$. Let $X$ be a projective
homogeneous space under $G(D,\sigma,h)$ over $F_{0}$. If
$X(F_{0\pi_{0}})\neq\emptyset$ or $X(F_{0\delta})\neq\emptyset$, then
$X(F_{0})\neq\emptyset$.

###### Proof.

By (4.3), $\Gamma$ is a maximal $R$-order of $D$. Let $e_{0}$ be the
ramification index of $D$ at $\pi$ and $e_{1}$ be the ramification index of
$D$ at $\delta$. If $D_{1}$ as in (5.3(i)), then $e_{0}=e_{1}=1$. If $D_{1}$
is as in (5.3(ii)), then $e_{0}=2$ and $e_{1}=1$. If $D_{1}$ as in (5.3(iii)
or (iv)), then $e_{0}=e_{1}=2$. Let $\pi_{D}$ and $\delta_{D}$ be as in (5.3).
Then $\pi_{D}$ and $\delta_{D}$ satisfy the assumptions of (5.2). Hence, by
(5.2), $X(F_{0})\neq\emptyset$. ∎

##  6\. Behavior under blowups

Let $R_{0}$ be a 2-dimensional complete regular local ring with maximal ideal
$\mathfrak{m}_{0}=(\pi_{0},\delta_{0})$ and residue field $k_{0}$. Suppose
that char$(k_{0})\neq 2$. Let $F_{0}$ be the field of fractions of $R_{0}$ and
let $F=F_{0}(\sqrt{\lambda})$ be an extension of degree at most 2. Let
$D\in{}_{2}\text{Br}(F)$ be a division algebra with $F/F_{0}$-involution
$\sigma$ and $h$ an hermitian form over $(D,\sigma)$ and $G(A,\sigma,h)$ be as
in (§5). Let $X$ be a projective homogeneous variety under $G(A,\sigma,h)$
over $F_{0}$. Suppose that $X(F_{0\nu})\neq\emptyset$ for all divisorial
discrete valuations $\nu$ of $F_{0}$. Under some assumptions on $D$, in this
section we prove that there exists a sequence of blowups $\mathscr{Y}$ of
Spec$(R_{0})$ such that $X(F_{0P})\neq\emptyset$ for all closed points $P$ of
$\mathscr{Y}$.

Let $\mathscr{X}_{0}=Proj(R_{0}[x,y]/(\pi_{0}x-\delta_{0}y))$. Let $Q_{1}$ and
$Q_{2}$ be the closed points of $\mathscr{X}_{0}$ given by the homogeneous
ideals $(\pi_{0},\delta_{0},y)$ and $(\pi_{0},\delta_{0},x)$. Let $\tau$ be
the nontrivial automorphism of $F/F_{0}$ if $F\neq F_{0}$ and let $\tau$ be
the identity if $F=F_{0}$.

We begin with the following.

######  Lemma 6.1.

Suppose that $\lambda$ a unit in $R_{0}$ or a unit times $\pi_{0}$. Let
$a,b\in R_{0}$ be nonzero and square free. Suppose that the support of $a$ and
$b$ is at most $\pi_{0}$ and $\delta_{0}$ and have no common factors. Then for
any closed point $P\in\mathscr{X}_{0}$, there exist
$a^{\prime},b^{\prime},\pi^{\prime},\delta^{\prime}\in{\mathcal{O}}_{P}$ such
that the maximal ideal at $P$ is generated by $\pi^{\prime}$ and
$\delta^{\prime}$, $a^{\prime}$ and $b^{\prime}$ are square free, have no
common factors, the support is at most $(\pi^{\prime})$ or $(\delta^{\prime})$
and $(a,b)\otimes F_{0P}=(a^{\prime},b^{\prime})$ and
$R_{0}(a,b)\subset\hat{\mathcal{O}}_{P}(a^{\prime},b^{\prime})$.

###### Proof.

Suppose $a$ is a unit $R_{0}$. Then $b=v_{0}$ or $v_{0}\pi_{0}$ or
$v_{0}\delta_{0}$ or $v_{0}\pi_{0}\delta_{0}$ for some unit $v_{0}\in R_{0}$.
If $b=v_{0}$ or $v_{0}\pi_{0}$ or $v_{0}\delta_{0}$, then it is easy to see
that $a^{\prime}=a$ and $b^{\prime}=b$ have the required properties. Suppose
$b=v_{0}\pi_{0}\delta_{0}$. Suppose $P\neq Q_{1}$, $Q_{2}$. Then the maximal
ideal at $P$ is given by $(\pi_{0},\delta^{\prime})$ with
$\pi_{0}=w^{\prime}\delta_{0}$ for some unit $w^{\prime}$ in
$\mathcal{O}_{P}$. We have
$(a,b)=(a,v_{0}\pi_{0}\delta_{0})=(a,v_{0}w^{\prime})$. In this case it is
easy to see that $a^{\prime}=a$ and $b^{\prime}=v_{0}w^{\prime}$ have the
required property. Suppose $P=Q_{1}$. Then the maximal ideal at $P$ is given
by $(t,\delta_{0})$ with $\pi_{0}=t\delta_{0}$. We have
$(a,b)=(a,v_{0}t\delta_{0})$ and $a^{\prime}=a$, $b^{\prime}=v_{0}t\delta_{0}$
have the required properties. The case $P=Q_{2}$ is similar.

Suppose neither $a$ nor $b$ is a unit in $R_{0}$. Then by the assumption on
$a,b$, we have $\\{a,b\\}=\\{u_{0}\pi_{0},v_{0}\delta_{0}\\}$ for some units
$u_{0},v_{0}\in R_{0}$. Suppose $P=Q_{1}$. Since the maximal ideal of
$\hat{\mathcal{O}}_{Q_{1}}$ is given by $(t,\delta_{0})$ with
$\pi_{0}=t\delta_{0}$, we have $(a,b)\otimes
F_{0Q_{1}}=(u_{0}\pi_{0},v_{0}\delta_{0})\otimes
F_{0Q_{1}}=(u_{0}t\delta,v_{0}\delta)\otimes
F_{0Q_{1}}\simeq(-u_{0}v_{0}t,v_{0}\delta_{0})$. It is easy to see that
$a^{\prime}=-u_{0}v_{0}t$ and $b^{\prime}=v_{0}\delta_{0}$ have the required
properties. The case $P=Q_{2}$ is similar. Suppose $P\neq Q_{1},Q_{2}$. Since
the maximal ideal at $P$ is given by $(\pi_{0},\delta^{\prime})$ with
$\pi_{0}=w^{\prime}\delta_{0}$ for some unit $w^{\prime}$ in $\mathcal{O}_{P}$
and
$(a,b)=(u_{0}\pi_{0},v_{0}\delta_{0})=(u_{0}\pi_{0},v_{0}w^{\prime}\pi_{0})=(u_{0}\pi_{0},v_{0}w^{\prime}u_{0})$,
$a^{\prime}=u_{0}\pi_{0}$ and $b^{\prime}=v_{0}w^{\prime}u_{0}$ have the
required properties. ∎

######  Lemma 6.2.

Suppose that $\lambda$ a unit in $R_{0}$ or a unit times $\pi_{0}$. Suppose
$D=(u_{0},u_{1}\pi_{0})\otimes(v_{0},v_{1}\delta_{0})$ is a division algebra
for some units $u_{i},v_{i}\in R_{0}$. Let $\sigma$ be the tensor product of
canonical involutions on $(u_{0},u_{1}\pi_{0})$, $(v_{0},v_{1}\delta_{0})$.
Then for $P=Q_{1}$ or $Q_{2}$, there exist an isomorphism $\phi_{P}:D\otimes
F_{0P}\to(u_{0}^{\prime},u_{1}^{\prime}\pi^{\prime})\otimes(v_{0}^{\prime},v_{1}^{\prime}\delta^{\prime})$
for some $u_{i}^{\prime},v_{i}^{\prime}$ units in the local ring
${\mathcal{O}}_{P}$ at $P$ and the maximal ideal of ${\mathcal{O}}_{P}$ is
given by $(\pi^{\prime},\delta^{\prime})$ such that
$\phi_{P}(R_{0}(u_{0},u_{1}\pi)\otimes
R_{0}(v_{0},v_{1}\delta))\subset{\mathcal{O}}_{P}(u_{0}^{\prime},u_{1}^{\prime}\pi^{\prime})\otimes{\mathcal{O}}_{P}(v_{0}^{\prime},v_{1}^{\prime}\delta^{\prime})$.
Further if $\sigma^{\prime}$ is the tensor product of canonical involutions on
$(u_{0}^{\prime},u^{\prime}_{1}\pi^{\prime})$ and
$(v_{0}^{\prime},v_{1}^{\prime}\delta^{\prime})$, then there exists
$\phi_{P}\in{\mathcal{O}}_{P}(u_{0}^{\prime},u_{1}^{\prime}\pi^{\prime})\otimes{\mathcal{O}}_{P}(v_{0}^{\prime},v_{1}^{\prime}\delta^{\prime})$
such that int$(\theta_{P})\sigma^{\prime}=\phi_{P}\sigma\phi_{P}^{-1}$ and the
support of Nrd$(\theta_{P})$ at most $(\pi^{\prime})$ and $(\delta^{\prime})$.

###### Proof.

Since $Q_{1}$ is the closed point given by the homogeneous ideal
$(\pi_{0},\delta_{0},y)$, the maximal ideal of ${\mathcal{O}}_{Q_{1}}$ is
given by $(t,\delta_{0})$ with $\pi_{0}=t\delta_{0}$. Thus we have $D\otimes
F_{0P}=(u_{0},u_{1}\pi_{0})\otimes(v_{0},v_{1}\delta_{0})=(u_{0},u_{1}t\delta_{0})\otimes(v_{0},v_{1}\delta_{0})\simeq(u_{0},u_{1}v_{1}^{-1}t)\otimes(u_{0}v_{0},v_{1}\delta_{0})$.

Let $i_{1},j_{1}\in(u_{0},u_{1}\pi_{0})$,
$i_{2},j_{2}\in(v_{0},v_{1}\delta_{0})$,
$i_{3},j_{3}\in(u_{0},u_{1}v_{1}^{-1}t)$ and
$i_{4},j_{4}\in(u_{0}v_{0},v_{1}\delta_{0})$ be the standard generators. Then
we have an isomorphism
$\phi_{P}:(u_{0},u_{1}\pi_{0})\otimes(v_{0},v_{1}\delta_{0})\to(u_{0},u_{1}v_{1}^{-1}t)\otimes(u_{0}v_{0},v_{1}\delta_{0})$
given by $\phi(i_{1}\otimes 1)=i_{3}\otimes 1$, $\phi(j_{1}\otimes
1)=j_{3}\otimes j_{4}$, $\phi(1\otimes i_{2})=u_{0}^{-1}(i_{3}\otimes i_{4})$
and $\phi(1\otimes j_{2})=1\otimes j_{4}$. Since $u_{0}$ is a unit in $R_{0}$,
$\phi(R_{0}(u_{0},u_{1}\pi_{0})\otimes
R_{0}(v_{0},v_{1}\delta_{0}))\subset{\mathcal{O}}_{Q_{i}}(u_{0},u_{1}v_{1}^{-1}t)\otimes{\mathcal{O}}_{Q_{i}}(u_{0}v_{0},v_{1}\delta_{0})$.
Let $\theta_{Q_{1}}=i_{3}\otimes j_{4}$. Then it is easy to see that
$\theta_{Q_{1}}$ has the required properties. A similar computation gives the
required $\theta_{Q_{2}}$. ∎

######  Lemma 6.3.

Suppose that $\lambda$ a unit in $R_{0}$ or a unit times $\pi_{0}$. Suppose
$D=(u_{0},u_{1}\pi_{0})\otimes(v_{0},v_{1}\delta_{0})$ is a division algebra
for some units $u_{i},v_{i}\in R_{0}$. Let $\sigma$ be the tensor product of
canonical involutions on $(u_{0},u_{1}\pi_{0})$, $(v_{0},v_{1}\delta_{0})$.
Let $P\in\mathscr{X}_{0}$ be a closed point not equal to $Q_{1}$ or $Q_{2}$.
Then there exists an isomorphism $\phi_{P}:D\otimes
F_{0P}\simeq(u_{0}^{\prime},u_{1}^{\prime})\otimes(v_{0}^{\prime},v_{1}^{\prime}\pi^{\prime})$
for some $u_{0}^{\prime},v_{0}^{\prime},u_{1}^{\prime},v_{1}^{\prime}$ units
in ${\mathcal{O}}_{P}$ and $m_{P}=(\pi^{\prime},\delta^{\prime})$ such that
$\phi_{P}(R_{0}(u,\pi)\otimes
R_{0}(v,\delta))\subset{\mathcal{O}}_{P}(u_{0}^{\prime},u_{1}^{\prime})\otimes{\mathcal{O}}_{P}(v_{0}^{\prime},v_{1}^{\prime}\pi)$.
Further if $\sigma^{\prime}$ is the tensor product of canonical involutions on
$(u_{0}^{\prime},u^{\prime}_{1})$ and
$(v_{0}^{\prime},v_{1}^{\prime}\pi^{\prime})$, then there exists
$\theta_{P}\in{\mathcal{O}}_{P}(u_{0}^{\prime},u_{1}^{\prime})\otimes{\mathcal{O}}_{P}(v_{0}^{\prime},v_{1}^{\prime}\pi)$
such that int$(\theta_{P})\sigma^{\prime}=\phi_{P}\sigma\phi_{P}^{-1}$ and the
support of Nrd$(\theta_{P})$ at most $(\pi^{\prime})$ and $(\delta^{\prime})$.

###### Proof.

Since $P$ is a closed point not equal to $Q_{1}$ or $Q_{2}$, the maximal ideal
at $P$ is given by $(\pi_{0},\delta^{\prime})$ and
$\delta_{0}=w^{\prime}\pi_{0}$ for some unit $w^{\prime}$ in
$\mathcal{O}_{P}$. Thus we have $D\otimes
F_{0P}=(u_{0},u_{1}\pi_{0})\otimes(v_{0},v_{1}\delta_{0})=(u_{0},u_{1}\pi_{0})\otimes(v_{0},v_{1}w^{\prime}\pi_{0})\simeq(v_{0},v_{1}w^{\prime}u_{1}^{-1})\otimes(u_{0}v_{0},u_{1}\pi_{0})$.

Let $i_{1},j_{1}\in(u_{0},u_{1}\pi)$, $i_{2},j_{2}\in(v_{0},v_{1}\delta_{0})$,
$i_{3},j_{3}\in(v_{0},v_{1}w^{\prime}u_{1}^{-1})$ and
$i_{4},j_{4}\in(u_{0}v_{0},u_{1}\pi_{0})$ be the standard generators. Then we
have an isomorphism
$\phi_{P}:(u_{0},u_{1}\pi_{)})\otimes(v_{0},v_{1}\delta_{0})\to(v_{0},v_{1}w^{\prime}u_{1}^{-1})\otimes(u_{0}v_{0},u_{1}\pi_{0})$
given by $\phi(i_{1}\otimes 1)=v_{0}^{-1}(i_{3}\otimes i_{4})$,
$\phi(j_{1}\otimes 1)=1\otimes j_{4}$, $\phi(1\otimes i_{2})=(i_{3}\otimes 1)$
and $\phi(1\otimes j_{2})=j_{3}\otimes j_{4}$. Since $v_{0}\in R_{0}$ is a
unit, we have $\phi_{P}(R_{0}(u_{0},u_{1}\pi)\otimes
R_{0}(v_{0},v_{1}\delta))\subset{\mathcal{O}}_{P}(v_{0},v_{1}w^{\prime}u_{1}^{-1})\otimes{\mathcal{O}}_{P}(u_{0}v_{0},u_{1}\pi)$.
Let $\theta_{P}=i_{3}\otimes j_{4}$. Then $\theta_{P}$ has the required
properties. ∎

The following two results are extracted from ([25, §4]).

######  Lemma 6.4.

Let $F_{0}$ and $F$ be as above. Let $D\in Br(F)$ be a quaternion division
algebra over $F$ with a $F/F_{0}$-involution $\sigma$. Then there exists a
sequence of blowups $\mathscr{X}_{0}\to Spec{R_{0}}$ such that, the integral
closure $\mathscr{X}$ of $\mathscr{X}_{0}$ in $F$ is regular and
ram${}_{\mathscr{X}}(D)$ is a union of regular curves with normal crossings.
Further for every closed point $P$ of $\mathscr{X}_{0}$ with $D\otimes F_{0P}$
division, $D\otimes F_{0P}$ is as in (2.5 or 3.4) and not of the type
(2.5(iii) or 3.4(iii)).

###### Proof.

There exists a sequence of blowups $\mathscr{Y}_{0}\to Spec(R_{0})$ such that
the integral closure $\mathscr{Y}$ of $\mathscr{Y}_{0}$ is regular and
ram${}_{\mathscr{Y}}(D)$ is a union of regular curves with normal crossings
(cf. [18, Corollary 11.3]). Let $P$ be a closed point of $\mathscr{Y}_{0}$.
Since the integral closure of $\mathscr{Y}_{0}$ in $F$ is regular, the maximal
ideal at $P$ is generated by $(\pi_{P},\delta_{P})$ and
$F=F_{0}(\sqrt{\lambda_{P}})$ for some $\lambda_{P}=u$ or $u\pi_{P}$ for some
unit $u$ at $P$.

Suppose that $D\otimes F_{0P}$ is division. In particular $F\otimes F_{0P}$ is
a field. Let $R_{P}$ be the integral closure of $\mathcal{O}_{P}$ in $F$. Then
$R_{P}$ is a regular two dimensional local ring with maximal ideal
$(\pi^{\prime}_{P},\delta_{P}^{\prime})$ with
$\delta^{\prime}_{P}=\delta_{P}$, $\pi_{P}^{\prime}=\pi_{P}$ if $\lambda_{P}$
is a unit in $\mathcal{O}_{P}$ and $\pi_{P}^{\prime}=\sqrt{\lambda_{P}}$ if
$\lambda_{P}$ is not a unit in $\mathcal{O}_{P}$. Further $D$ is unramified on
$\mathcal{O}_{P}$ except possibly at $(\pi_{P}^{\prime})$ or
$(\delta_{P}^{\prime})$. In particular $D\otimes F_{0P}$ is as in (2.5 or
3.4).

Suppose $D\otimes F_{0P}$ as in (2.5(iii) or 3.4(iii)). Note that there are
only finitely many such closed points.

Suppose $F\otimes F_{0P}/F_{0P}$ is ramified. Then, by (3.1), we can assume
that $D\otimes F_{0P}$ is not of type (3.5(iii)).

Suppose that $F\otimes F_{0P}/F_{0P}$ is unramified. Let $\mathscr{X}_{P}\to
Spec(\mathcal{O}_{P})$ be the simple blow up. Then, it is easy to see that for
every closed point $Q$ of $\mathscr{X}_{P}$, $D\otimes F_{0Q}$ is not of type
(3.5(iii) or 3.4(iii)) (cf. [25, Lemma 4.1]). ∎

######  Proposition 6.5.

Let $D$ be a central simple algebra over $F$ with a $F/F_{0}$-involution
$\sigma$ and $h$ an hermitian form over $(D,\sigma)$. Let $X$ be a projective
homogeneous variety under $G(D,\sigma,h)$ over $F_{0}$. Suppose that
ind$(D)\leq 2$. If that $X(F_{0\nu})\neq\emptyset$ for all divisorial discrete
valuations $\nu$ of $F_{0}$, then there exists a sequence of blowups
$\mathscr{Y}\to Spec(R_{0})$ such that for every closed point $P$ of
$\mathscr{Y}$, $X(F_{0P})\neq\emptyset$.

###### Proof.

By Morita equivalence ([13, Theorem 3.1,3.11,3.20] & [12, Chapter 1, 9.3.5]),
we assume that $D$ is division. If $D=F$, then $X(F_{0})\neq\emptyset$ ([25,
Corollary 3.12]) and hence any blowup of Spec$(R)$ has the required property.

Suppose ind$(D)=2$. Then $D\simeq D_{0}\otimes_{F_{0}}F$ for some quaternion
division algebra over $F_{0}$. Without loss of generality we assume that
$\sigma$ is the canonical involution. Then using (6.4), we get a sequence of
blowups $\mathscr{Y}_{0}$ of Spec$(R_{0})$ such that for every closed point
$P$ of $\mathscr{Y}$ with $D\otimes F_{0P}$ division, $D\otimes F_{0P}$ is as
in (2.5 or 3.4) and not of the type (2.5(iii) or 3.4(iii)).

Suppose that $D\otimes F_{0P}$ is not division. Suppose $F\otimes F_{0P}$ is a
field. Since ind$(D)\leq 2$, $D\otimes F_{0P}$ is a matrix algebra and hence
by Morita equivalence, $h$ corresponds to a quadratic form over $F$. Thus
$X(F_{0P})\neq\emptyset$ ([5, Theorem 3.1]). Suppose $F\otimes F_{0P}$ is not
a field. Then $F\otimes F_{0P}\simeq F_{0P}\times F_{0P}$, $D\otimes
F_{0P}\simeq D_{0}\otimes F_{0P}\times D_{0}^{op}\otimes F_{0P}$ and
$G(D,\sigma,h)\otimes F_{0P}\simeq GL(M_{n}(D_{0}\otimes F_{0P}))$ ([13, p.
346]). Hence by (1.4), $X(F_{0P})\neq\emptyset$.

Suppose that $D\otimes F_{0P}$ is division. Then $D\otimes
F_{P}=(a_{p},b_{P})$ for some $a_{P},b_{P}$ as in (6.1). In particular
$\mathcal{O}_{P}(a_{P},b_{P})\otimes R$ is a maximal $\mathcal{O}_{P}$-order
of $D\otimes F_{0P}$. We have $h\otimes F_{0P}=\langle
a_{1P},\cdots,a_{nP}\rangle$ for some
$a_{iP}\in\hat{\mathcal{O}}_{P}(a,b)\otimes R$.

Let $Y_{0}$ be the special fibre of $\mathscr{Y}_{0}$. By (1.1), there exists
a finite subset $\mathscr{P}$ of $Y_{0}$ such that $X(F_{P})\neq\emptyset$ for
all $P\not\in\mathscr{P}_{0}$. Thus replacing $R$ by $\hat{\mathcal{O}}_{P}$,
we assume that $D=(a,b)$ for some $a,b$ as in (6.1) and $h=\langle
a_{1},\cdots,a_{n}\rangle$ for some $a_{i}\in R(a,b)$.

By ([25, Lemma 4.2]), there exists a sequence of blowups $\mathscr{Y}_{1}\to
Spec(R_{0})$ such that the support of Nrd$(a_{i})$ is a union of regular
curves with normal crossings and for every closed point $Q$ of
$\mathscr{Y}_{1}$ with $D\otimes F_{0Q}$ is division, $D\otimes F_{0Q}$ is not
of the form ( [25, Lemma 3.6(5)]).

Let $Q$ be a closed point of $\mathscr{Y}_{1}$. If $D\otimes F_{0Q}$ is a
matrix algebra, then by ([10, Corollary 4.7]) and Morita equivalence,
$X(F_{0Q})\neq\emptyset$. Suppose $D\otimes F_{0Q}$ is a division algebra.
Then, by (6.1), $R(a,b)$ is contained in the corresponding maximal order
$\hat{\mathcal{O}}_{Q}(a_{Q},b_{Q})$. Since $h=\langle
a_{1},\cdots,a_{n}\rangle$ with $a_{i}\in R(a,b)$ with support of Nrd$(a_{i})$
is a union of regular curves with normal crossings, by (5.4),
$X(F_{0Q})\neq\emptyset$. ∎

######  Lemma 6.6.

Let $D\in{}_{2}Br(F)$ be a central simple algebra over $F$ with a
$F/F_{0}$-involution $\sigma$ and $h$ an hermitian form over $(D,\sigma)$.
Suppose that ind$(D)=4$. Then there exists a sequence of blowups
$\mathscr{X}_{0}\to Spec{R_{0}}$ such that, the integral closure $\mathscr{X}$
of $\mathscr{X}_{0}$ in $F$ is regular and ram${}_{\mathscr{X}}(D)$ is a union
of regular curves with normal crossings. Further for every closed point $P$ of
$\mathscr{X}_{0}$ with ind$(D\otimes F_{0P})=4$, $D\otimes F_{0P}$ is as in
(2.6 or 3.5) and not of the type (2.6(iii) or 3.5(iii)).

###### Proof.

There exists a sequence of blowups $\mathscr{Y}_{0}\to Spec(R_{0})$ such that
the integral closure $\mathscr{Y}$ of $\mathscr{Y}_{0}$ is regular and
ram${}_{\mathscr{Y}}(D)$ is a union of regular curves with normal crossings
(cf. [18, Corollary 11.3]).

Let $P$ be a closed point of $\mathscr{Y}_{0}$. In particular if ind($D\otimes
F_{0P})=4$, then $D\otimes F_{0P}$ is as in (2.6 or 3.5). Suppose $D\otimes
F_{0P}=(u_{P},v_{P})\otimes(w_{P},w_{P}^{\prime}\pi_{P}\delta_{P})$ as in
(2.6(iii) or 3.5(iii)) for some units
$u_{P},v_{P},w_{P},w_{P}^{\prime}\in\mathcal{O}_{P}$. Note that there are only
finitely many such closed points.

Suppose $F\otimes F_{0P}/F_{0P}$ is ramified. Then, by (3.1), we can assume
that $D\otimes F_{0P}$ is not of type (3.5(iii)).

Suppose that $F\otimes F_{0P}/F_{0P}$ is unramified. Let $\mathscr{X}_{P}\to
Spec(\mathcal{O}_{P})$ be the simple blow up. Then, it is easy to see that for
every closed point $Q$ of $\mathscr{X}_{P}$, $D\otimes F_{0Q}$ is not of type
(2.6(iii) or 3.5(iii)). ∎

######  Proposition 6.7.

Let $D\in{}_{2}Br(F)$ be a central simple algebra over $F$ with a
$F/F_{0}$-involution $\sigma$ and $h$ an hermitian form over $(D,\sigma)$. Let
$X$ be a projective homogeneous variety under $G(D,\sigma,h)$ over $F_{0}$.
Suppose that ind$(D)\leq 4$. If $X(F_{0\nu})\neq\emptyset$ for all divisorial
discrete valuations $\nu$ of $F_{0}$, then there exists a sequence of blowups
$\mathscr{Y}\to Spec(R_{0})$ such that for every closed point $P$ of
$\mathscr{Y}$, $X(F_{0P})\neq\emptyset$.

###### Proof.

By (6.5), we assume that ind$(D)=4$. As in the proof of (6.5), we assume that
$D$ is division as in (2.6 or 3.5 ) and not of the type (2.6(iii) or
3.5(iii)). Let $\Gamma$ be the maximal $R$-order of $D$ as in (4.3) and write
$h=\langle a_{1},\cdots,a_{n}\rangle$ with $a_{i}\in\Gamma$.

Let $\mathscr{Y}_{1}\to Spec(R_{0})$ be a sequence of blowups such that the
support of Nrd$(a_{i})$ is a union of regular curves with normal crossings.
Further replacing $\mathscr{Y}$ by a sequence of blow ups (6.6), we assume
that for every closed point $P$ of $\mathscr{Y}$, $D\otimes F_{0P}$ is not of
the form (2.6(iii) or 3.5(iii)). Once again we have a finite set of closed
points $\mathscr{P}_{1}$ of $\mathscr{Y}_{1}$ such that
$X(F_{0P})\neq\emptyset$ for all $P\not\in\mathscr{P}$.

Let $P\in\mathscr{P}_{1}$ be a closed point. Suppose that $F\otimes F_{0P}$ is
not a field. Then as in the proof of (6.5), $X(F_{0P})\neq\emptyset$.

Suppose that $F\otimes F_{0P}$ is a field. If ind$(D\otimes F_{0P})\leq 2$,
then by (6.5), there exists a sequence of blowups $\mathscr{X}_{P}$ of
Spec$(\mathcal{O}_{P})$ such that for every closed point $Q$ of
$\mathscr{X}_{P}$, $X(F_{0Q})\neq\emptyset$.

Suppose ind$(D\otimes F_{P})=4$.

Suppose that $D$ is not of type (2.6(iv) or 3.5(iv)). Then
$D\simeq(u_{0},w_{0})\otimes(a,b)$ for some units $u,v,\in R_{0}$ and $a,b\in
R_{0}$ as in (6.1). Then, by the choice, we have $\Gamma=R(u_{0},v_{0})\otimes
R(a,b)$. By (6.1), $(a,b)\otimes F_{0P}\simeq(a_{P},b_{P})$ for some
$a_{P},b_{P}\in\mathcal{O}_{P}$ as in (6.1) and
$R_{0}(a,b)\subset\hat{\mathcal{O}}_{P}(a_{P},b_{P})$. In particular
$\Gamma\subset\hat{\mathcal{O}}_{P}(u_{0},v_{0})\otimes\hat{\mathcal{O}}_{P}(a_{P},b_{P})$.
Since $a_{i}\in\Gamma$ and $D\otimes F_{0P}$ is not of type (2.6(iii)), by
(5.4), $X(F_{0P})\neq\emptyset$.

Suppose that $D$ is of type (2.6(iv) or 3.5(iv)). Then
$D\simeq(u_{0},u_{1}\pi_{0})\otimes(v_{0},v_{1}\delta_{0})$ for some units
$u_{i},v_{i}\in R_{0}$.

Suppose $P$ is a nodal point of $\mathscr{Y}_{1}$. Then, by (6.2), there
exists an isomorphism $\phi_{P}:D\otimes
F_{0P}\to(u_{0}^{\prime},w_{1}^{\prime}\pi_{P})\otimes(v_{0}^{\prime},v_{1}^{\prime}\delta_{P})$
and
$\theta_{P}\in\hat{\mathcal{O}}_{P}(u_{0}^{\prime},w_{1}^{\prime}\pi_{P})\otimes\hat{\mathcal{O}}_{P}(v_{0}^{\prime},v_{1}^{\prime}\delta_{P})$
such that $\phi_{P}(R_{0}(u_{0},u_{1}\pi_{0})\otimes
R_{0}(v_{0},v_{1}\pi_{0}))\subset\hat{\mathcal{O}}_{P}(u_{0}^{\prime},w_{1}^{\prime}\pi_{P})\otimes\hat{\mathcal{O}}_{P}(v_{0}^{\prime},v_{1}^{\prime}\delta_{P})$
and $int(\theta_{P})\sigma=\phi_{P}^{-1}\sigma^{\prime}\phi_{P}$, where
$\sigma^{\prime}$ is the product of the canonical involutions on the right
hand side. Let $h^{\prime}$ be the hermitian form on
$((u_{0}^{\prime},w_{1}^{\prime}\pi_{P})\otimes(v_{0}^{\prime},v_{1}^{\prime}\delta_{P})),\phi_{P}\sigma\phi_{P}^{-1})$
which is the image of $h$ under $\phi_{P}$. Since $h=\langle
a_{1},\cdots,a_{n}\rangle$, we have
$h_{1}=\langle\phi_{P}(a_{1}),\cdots,\phi_{P}(a_{n})\rangle$. Let
$h^{\prime}=\theta_{P}h_{1}$. Then $h^{\prime}$ is an hermitian form with
respect $\sigma^{\prime}$. Let $X^{\prime}$ be the projective homogeneous
variety under
$G((u_{0}^{\prime},w_{1}^{\prime}\pi_{P})\otimes(v_{0}^{\prime},v_{1}^{\prime}\delta_{P}),\sigma^{\prime},h^{\prime})$
associated to $X$. Then $X(F_{0P})\neq\emptyset$ if and only if
$X^{\prime}(F_{0P})\neq\emptyset$. Since
$\phi_{P}(a_{i}),\theta_{P}\in\hat{\mathcal{O}}_{P}(u_{0}^{\prime},w_{1}^{\prime}\pi_{P})\otimes\hat{\mathcal{O}}_{P}(v_{0}^{\prime},v_{1}^{\prime}\delta_{P})$,
by (5.4), $X^{\prime}(F_{0P})\neq\emptyset$ and hence
$X(F_{0P})\neq\emptyset$.

If $P$ is a non-nodal point, then using (6.3), we get $X(F_{0P})\neq\emptyset$
as above. ∎

######  Proposition 6.8.

Let $k$, $F_{0}$ and $F$ be as above. Suppose that for finite extension
$\ell/k$, every element in ${}_{2}Br(\ell)$ has index at most 2. Let
$D\in{}_{2}Br(F)$ be a central simple algebra over $F$ with a
$F/F_{0}$-involution $\sigma$ and $h$ an hermitian form over $(D,\sigma)$. Let
$X$ be a projective homogeneous variety under $G(D,\sigma,h)$ over $F_{0}$. If
$X(F_{0\nu})\neq\emptyset$ for all divisorial discrete valuations $\nu$ of
$F_{0}$, then there exists a sequence of blowups $\mathscr{Y}\to Spec(R_{0})$
such that for every closed point $P$ of $\mathscr{Y}$,
$X(F_{0P})\neq\emptyset$.

###### Proof.

By (6.7), we assume that ind($D)>4$. As in the proof of (6.5), we assume that
$D$ is unramified on $R_{0}$ except possibly at $(\pi_{0})$ and
$(\delta_{0})$. Then, by (2.7, 3.6), ind$(D)=8$ and
$D\simeq(u_{0},u_{1})\otimes(v_{0},v_{1}\pi_{0})\otimes(w_{0},w_{1}\delta_{0})\otimes
F$ for some units $u_{,}v_{i},w_{i}\in R_{0}$. Without loss of generality we
assume that $\sigma$ is the tensor product of canonical involutions on
$(u_{0},u_{1})$, $(v_{0},v_{1}\pi)$ and $(w_{0},w_{1}\delta)$ and $\tau$. Let
$\Gamma=R(u_{0},u_{1})\otimes R(v_{0},v_{1}\pi)\otimes R(w_{0},w_{1}\delta)$.
Then $\Gamma$ is a maximal $R$-order of $D$. We have $h=\langle
a_{1},\cdots,a_{n}\rangle$ for some $a_{i}\in\Gamma$.

Let $\mathscr{X}_{0}\to Spec(R_{0})$ be a sequence of blowups such that the
integral closure of $\mathscr{Y}_{0}$ in $F$ is regular and the support of
$Nrd(a_{i})$ and ram${}_{\mathscr{Y}}(D)$ is a union of regular curves with
normal crossings. By (1.1), there exists a finite subset $\mathscr{P}$ of
$Y_{0}$ such that $X(F_{0P})\neq\emptyset$ for all $P\not\in\mathscr{P}_{0}$.

Let $P\in\mathscr{P}_{0}$. Suppose that $F\otimes F_{0P}$ is not a field. Then
as in the proof of (6.5), $X(F_{0P})\neq\emptyset$.

Suppose that $F\otimes F_{0P}$ is a field. If ind$(D\otimes F_{0P})\leq 4$,
then by (6.7), there exists a sequence of blowups $\mathscr{X}_{P}$ of
Spec$(\mathcal{O}_{P})$ such that for every closed point $Q$ of
$\mathscr{X}_{P}$, $X(F_{0Q})\neq\emptyset$.

Suppose ind$(D\otimes F_{0P})>4$. Then as above we have ind$(D\otimes
F_{0P})=8$. Arguing as in the proof of (6.7), we get that
$X(F_{0P})\neq\emptyset$. ∎

##  7\. Main theorem

In this section we prove the main theorems.

######  Theorem 7.1.

Let $K$ be a complete discretely valued field with valuation ring $T$ and
residue field $k$. Suppose that char$(k)\neq 2$. Let $F$ be the function field
of a smooth projective geometrically integral curve over $K$. Let
$A\in{}_{2}Br(F)$ be a central simple algebra over $F$ with an involution
$\sigma$ of any kind, $F_{0}=F^{\sigma}$ and $h$ a hermitian form over
$(A,\sigma)$. Suppose that ind$(A)\leq 4$. Let $G=SU(A,\sigma,h)$ if $\sigma$
is first kind or $U(A,\sigma,h)$ if $\sigma$ is of second kind. Let $X$ be a
projective homogeneous variety under $G$ over $F_{0}$. If
$X(F_{0\nu})\neq\emptyset$ for all divisorial discrete valuations $\nu$ of
$F$, then $X(F_{0})\neq\emptyset$.

###### Proof.

Since $F$ is the function field of a curve over $K$, $F_{0}$ is also the
function field of a curve over $K$. Let $\mathscr{X}_{0}\to Spec(R_{0})$ be a
regular proper model of $F_{0}$ with the closed fibre $X_{0}$ a union of
regular curves with normal crossings ([1]). Let $\eta\in X_{0}$ be a generic
point. Then $\eta$ gives a divisorial discrete valuation $\nu$ of $F$. Since
$X(F_{0\eta})\neq\emptyset$, by ([9, 5.8]), there exists a nonempty open set
$U_{\eta}$ of the closure of $\eta$ in $X_{0}$ such that
$X(F_{0U_{\eta}})\neq\emptyset$. By shrinking $U_{\eta}$, we assume that
$U_{\eta}$ does not contain any singular points of $X_{0}$.

Let $\mathscr{P}=X_{0}\setminus\cup_{\eta}U_{\eta}$. Then $\mathscr{P}$ is a
finite set of closed points of $X_{0}$ containing all the singular points of
$X_{0}$. Let $P$ be a closed point of $X_{0}$. Suppose $P\not\in\mathscr{P}$.
Then $P\in U_{\eta}$ for some $\eta$. Since $F_{0U_{\eta}}\subset F_{0P}$,
$X(F_{0P})\neq\emptyset$.

Let $P\in\mathscr{P}$. Since ind$(A)\leq 4$ and $A\in{}_{2}Br(F)$, by (6.7),
there exists a sequence of blowups $\mathscr{X}_{P}$ of
Spec$(\mathcal{O}_{P})$ such that $X(F_{0Q})\neq\emptyset$ for all closed
points of $\mathscr{X}_{P}$. Thus replacing $\mathscr{X}$ by these finitely
many sequences of blowups at all $P\in\mathscr{P}$, we assume that
$X(F_{0Q})\neq\emptyset$ for all closed points $Q$ of $X_{0}$. Since for any
generic point $\eta$ of $X_{0}$, $X(F_{0\eta})\neq\emptyset$, we have
$X(F_{0x})\neq\emptyset$ for all points $x\in X_{0}$. Since $G$ is a connected
rational group ([4, Lemma 5]), by ([8, Theorem 3.7]), we have
$X(F)\neq\emptyset$. ∎

######  Theorem 7.2.

Let $K$ be a complete discretely valued field with valuation ring $T$ and
residue field $k$. Suppose that char$(k)\neq 2$. Let $F$ be the function field
of a smooth projective geometrically integral curve over $K$. Let $A$ be a
central simple algebra over $F$ with an $\sigma$ of any kind,
$F_{0}=F^{\sigma}$ and $h$ a hermitian form over $(A,\sigma)$. Suppose that
for every finite extension $\ell/k$, every element in ${}_{2}Br(\ell)$ has
index at most 2. Let $G=SU(A,\sigma,h)$ if $\sigma$ is first kind or
$U(A,\sigma,h)$ if $\sigma$ is of second kind. Let $X$ be a projective
homogeneous variety under $G$ over $F_{0}$. If $X(F_{0\nu})\neq\emptyset$ for
all divisorial discrete valuations $\nu$ of $F_{0}$, then
$X(F_{0})\neq\emptyset$.

###### Proof.

Using (6.8), the proof is similar to the proof of (7.1). ∎

######  Corollary 7.3.

Let $K$ be a complete discretely valued field with valuation ring $T$ and
residue field $k$. Suppose that $k$ is a global field, local field or a
$C_{2}$-field with char$(k)\neq 2$. Let $F$ be the function field of a smooth
projective geometrically integral curve over $K$. Let $A\in{}_{2}Br(F)$ be a
central simple algebra over $F$ with an involution $\sigma$ of any kind and
$h$ a hermitian form over $(A,\sigma)$. Let $F_{0}=F^{\sigma}$,
$G=SU(A,\sigma,h)$ if $\sigma$ is first kind and $G=U(A,\sigma,h)$ if $\sigma$
is of second kind. Let $X$ be a projective homogeneous variety under $G$ over
$F_{0}$. If $X(F_{0\nu})\neq\emptyset$ for all divisorial discrete valuations
$\nu$ of $F_{0}$, then $X(F_{0})\neq\emptyset$.

###### Proof.

Suppose $k$ is a global field or a local field or a $C_{2}$-field, then any
finite extension $\ell$ of $k$ is also same type and hence every element in
${}_{2}Br(k)$ is of index at most 2 ([24, Chapter 10, 2.3(vi)], [24, Chapter
10, 2.2(i)], [14, Theorem 4.8]). Hence the corollary follows from (7.2). ∎

######  Corollary 7.4.

Let $K$ be a complete discretely valued field with valuation ring $T$ and
residue field $k$. Suppose that $k$ is a local field with the characteristic
of the residue field not equal 2. Let $F$ be the function field of a smooth
projective geometrically integral curve over $K$. Let $A\in{}_{2}Br(F)$ be a
central simple algebra over $F$ with an involution $\sigma$ and $h$ a
hermitian form over $(A,\sigma)$ and $F=F^{\sigma}$. If $h\otimes L$ is
isotropic for some odd degree extension $L/F_{0}$, then $h$ is isotropic.

###### Proof.

Let $\nu$ be a divisorial discrete valuation of $F_{0}$. Then the residue
field $\kappa(\nu)$ is either a finite extension of $K$ or a function field of
a curve over a finite extension of $k$ ([17, Theorem 8.1]).

Let $\kappa^{\prime}$ be an extension of $\kappa(\nu)$ with
$[\kappa^{\prime}:\kappa(\nu)]\leq 2$. Let $D$ be a central simple algebra
over $\kappa^{\prime}$ with a
$\kappa^{\prime}/\kappa(\nu^{\prime})$-involution $\theta$ and $f$ an
hermitian form over $(D,\theta)$. Let $\ell$ be an extension of $\kappa(\nu)$
of odd degree. Suppose that $f$ is isotropic over $\ell$.

Suppose $\kappa(\nu)$ is a finite extension of $K$. Since $K$ is a complete
discretely valued field with residue field $k$, $\kappa(\nu)$ is a compete
discretely valued field with residue field a finite extension of $k$. Then by
([25, Lemma 5.6] and [25, Lemma 5.1]), we get that $f$ is isotropic.

Suppose the residue field $\kappa(\nu)$ is a function field of a curve over a
finite extension of $k$. Since $k$ is local field, by ([25, Lemma 5.1]) and
([25, Theorem 4.4]), $f$ is isotropic.

Suppose that $h\otimes L$ is isotropic for some odd degree extension $L$ of
$F_{0}$. Then, as in the proof of ([25, Theorem 5.8]) and using (7.3), we get
that $h$ is isotropic.

∎

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