# A Positive Solution to a Conjecture of A. Katok for Diffeomorphism Case

Xiongping Dai  Department of Mathematics
Nanjing University
Nanjing, 210093, P. R. CHINA<EMAIL_ADDRESS>
(Date: June 9, 2008)

###### Abstract.

A. Katok has conjectured that a $C^{1+\alpha}$ map $g\colon M^{n}\rightarrow
M^{n},n\geq 2$, which is Hölder conjugated to an Anosov diffeomorphism is also
an Anosov diffeomorphism. Using Pesin stable manifold theorem and Liao
spectrum theorem, we show that under the hypothesis of such a conjecture, $g$
is an Axiom A diffeomorphism having no cycles. Particularly, if $g$ is Hölder
conjugated to a hyperbolic toral automorphism, then $g$ is Anosov.

###### Key words and phrases:

Anosov diffeomorphism, Pesin stable manifold, Liao theory, shadowing property.

###### 2000 Mathematics Subject Classification:

37D05, 37D20, 37D25

This project was supported by NSFC (No. 10671088) and 973 project (No.
2006CB805903)

##  1\. Introduction

Let $M^{n}$ be a connected, compact, smooth, and closed Riemannian manifold of
dimension $n\geq 2$. A. Katok has conjectured that if
$g\in\mathrm{Diff}^{1+}(M)$ is Hölder conjugated to an Anosov
$f\in\mathrm{Diff}^{1}(M)$; i.e., there is a Hölder-homeomorphism $h$ of $M$
such that $f=h\circ g\circ h^{-1}$, then $g$ is also an Anosov diffeomorphism.
Here “Hölder-homeomorphism $h$” means that $h$ and its inverse $h^{-1}$ both
are Hölder continuous. And $\mathrm{Diff}^{1+}(M)$ is the set of all $C^{1}$
diffeomorphisms with $\alpha$-Hölder derivatives for some Hölder exponent
$\alpha$ with $0<\alpha\leq 1$.

For convenience, if $g\in\textrm{Diff}^{1+}(M)$ is Hölder conjugated to an
Anosov diffeomorphism, then $g$ is temporally said to be _Katok_. Under the
hypothesis of such a conjecture, in [7] the authors proved the following.

######  Theorem A ([7]).

If $g$ is Katok, then all periodic points of $g$ have only non-zero Lyapunov
exponents, and such exponents are uniformly bounded away from zero.

In this paper, using Pesin stable manifold theorem, Liao spectrum theorem and
Liao reordering theorem, and shadowing property, based on Theorem A above we
obtain a positive solution to Katok’s conjecture as follows.

######  Theorem B.

If $g\in\mathrm{Diff}^{1+}(M)$ is Katok, then $g$ is an Axiom A diffeomorphism
having no cycles.

Consequently, if $g$ is Hölder conjugated to an Anosov diffeomorphism $f$ that
satisfies $\Omega(f)=M$ such as a hyperbolic toral automorphism, then $g$ is
also Anosov. In addition, if $g$ is volume-preserving, then $g$ is Anosov too.
Theorem B shows that Anosov diffeomorphisms have strong rigidity.

This paper is organized as follows. In $\S\ref{sec2}$ we will introduce the
Liao spectrum theorem and reordering theorem for $C^{1}$ differential systems
on Euclidean spaces. Then we will prove a semi-uniform ergodic theorem which
provides us with a criterion from nonuniform hyperbolicity to uniform
hyperbolicity. In $\S\ref{sec3}$ we will prove an approximation theorem of
ergodic measure by periodic measures. In $\S\ref{sec4}$ we will first prove
that a Katok diffeomorphism is nonuniformly hyperbolic for any invariant
measures and then show that it is uniformly hyperbolic. We will consider a
volume-preserving Katok diffeomorphism in the last section.

##  2\. Liao spectrum and reordering theorem

In this section, we will introduce the Liao spectrum theorem and Liao
reordering theorem, which are basic in Liao theory. Then, applying the Liao
spectrum theorem, we will provide with a criterion of uniform contraction.

For simplicity, let us consider throughout this section a nonsingular
autonomous system of $C^{1}$-differential equations in an $(n+1)$-dimensional
Euclidean $w$-space $\mathbb{E}^{n+1},n\geq 2$

$\dot{w}=S(w)\quad w\in\mathbb{E}^{n+1},\
S(w)\in\mathbb{R}^{n+1}-\\{\textbf{0}\\},$

where we write $T_{w}\mathbb{E}^{n+1}=\mathbb{R}^{n+1}$ for all $w$ to
distinguish the $w$-state-space $\mathbb{E}^{n+1}$ from its tangent $x$-space
$\mathbb{R}^{n+1}$, which then naturally gives rise to a $C^{1}$-flow on the
state-space $\mathbb{E}^{n+1}$

$\phi\colon\mathbb{R}\times\mathbb{E}^{n+1}\rightarrow\mathbb{E}^{n+1};\
(t,w)\mapsto t_{\cdot}w.$

It further induces, on the tangent bundle
$T\mathbb{E}^{n+1}=\mathbb{E}^{n+1}\times\mathbb{R}^{n+1}$, a smooth linear
skew-product flow

$\displaystyle\varPhi\colon\mathbb{R}\times\mathbb{E}^{n+1}\times\mathbb{R}^{n+1}\rightarrow\mathbb{E}^{n+1}\times\mathbb{R}^{n+1};\
(t,(w,x))\mapsto(t_{\cdot}w,{\varPhi_{t,w}}{x})$ where
$\varPhi_{t,w}\colon\mathbb{R}^{n+1}\rightarrow\mathbb{R}^{n+1};\
x\mapsto\frac{\partial\phi(t,w)}{\partial w}x$, corresponding to the extended
system $\displaystyle\dot{w}=S(w),\quad\dot{x}=S^{\prime}(w)x$

on the extended $(w,x)$-phase-space $\mathbb{E}^{n+1}\times\mathbb{R}^{n+1}$.

### 2.1.

Let $\mathbb{T}=\bigcup_{w\in\mathbb{E}^{n+1}}\mathbb{T}_{w}$, where
$\mathbb{T}_{w}=\left\\{x\in\mathbb{R}^{n+1}\,|\,\langle
S(w),x\rangle=0\right\\}$, denote the subbundle of the tangent bundle
$\mathbb{E}^{n+1}\times\mathbb{R}^{n+1}$ transversal to $S$ over
$\mathbb{E}^{n+1}$, called the _transversal tangent bundle to $S$ over
$\mathbb{E}^{n+1}$_. Then there is another naturally induced smooth linear
skew-product flow

$\varPsi\colon\mathbb{R}\times\mathbb{T}\rightarrow\mathbb{T};\
(t,(w,x))\mapsto(t_{\cdot}w,\varPsi_{t,w}{x})$

where along the fiber direction,
$\varPsi_{t,w}\colon\mathbb{T}_{w}\rightarrow\mathbb{T}_{t_{\cdot}w}$ is
defined as the component of $\varPhi_{t,w}x$ transversal to $S(t_{\cdot}w)$
for any $(w,x)\in\mathbb{T}$; that is,
$\varPhi_{t,w}x=rS(t_{\cdot}w)+\varPsi_{t,w}{x}$,
$\varPsi_{t,w}x\in\mathbb{T}_{t_{\cdot}w}$, for some $r\in\mathbb{R}$.
Particularly, let

$\mathscr{F}_{1}^{*\sharp}=\left\\{(w,x)\in\mathbb{T}\colon\|x\|=1\right\\}$

be the unit transversal tangent bundle to $S$ over $\mathbb{E}^{n+1}$. Then,
there is a natural skew-product flow

$\varPsi^{\sharp}\colon\mathbb{R}\times\mathscr{F}_{1}^{*\sharp}\rightarrow\mathscr{F}_{1}^{*\sharp};\
(t,(w,x))\mapsto(t_{\cdot}w,\varPsi_{t,w}^{\sharp}{x})$

where

$\varPsi_{t,w}^{\sharp}{x}=\varPsi_{t,w}{x}/\|\varPsi_{t,w}^{\sharp}{x}\|.$

### 2.2.

By the _bundle of transversal orthogonal $n$-frames_ over $\mathbb{E}^{n+1}$,
it means $\mathscr{F}_{n}^{*}$, where the fiber at $w$ is given by

$\mathscr{F}_{n,w}^{*}=\left\\{\gamma\in\stackrel{{\scriptstyle
n\textrm{-time}}}{{\overbrace{\mathbb{T}_{w}\setminus\\{\textbf{0}\\}\times\cdots\times\mathbb{T}_{w}\setminus\\{\textbf{0}\\}}}}\,|\,\langle\textrm{col}_{i}{\gamma},\textrm{col}_{j}{\gamma}\rangle=0,\
1\leq i\neq j\leq n\right\\}.$

Here and in the future, for $1\leq i\leq n$

$\textrm{col}_{i}\colon(v_{1},\ldots,v_{n})\mapsto v_{i}.$

Using the well-known Gram-Schmidt orthogonalizing process, based on
$(\mathbb{E}^{n+1},\phi)$, we can obtain from $\varPsi$ the well-defined skew-
product flow on $\mathscr{F}_{n}^{*}$ as follows

$\chi^{*}\colon\mathbb{R}\times\mathscr{F}_{n}^{*}\rightarrow\mathscr{F}_{n}^{*};\
(t,(w,\gamma))\mapsto(t_{\cdot}w,\chi_{t,w}^{*}\gamma).$

The bundle of transversal orthonormal $n$-frames over $\mathbb{E}^{n+1}$ is
written as $\mathscr{F}_{n}^{*\sharp}$, where the fiber at $w$ is defined as

$\mathscr{F}_{n,w}^{*\sharp}=\left\\{{\gamma}\in\mathscr{F}_{n}^{*}\colon\|\textrm{col}_{j}{\gamma}\|=1,\
j=1,\ldots,n\right\\}.$

Furthermore, there is a natural skew-product flow based on
$(\mathbb{E}^{n+1},\phi)$

$\chi^{*\sharp}\colon\mathbb{R}\times\mathscr{F}_{n}^{*\sharp}\rightarrow\mathscr{F}_{n}^{*\sharp};\
(t,(w,\gamma))\mapsto(t_{\cdot}w,\chi_{t,w}^{*\sharp}\gamma)$

called the _Liao transversal orthonormal $n$-frame flow of $S$_. For
convenience, let

$\pi\colon\mathscr{F}_{n}^{*\sharp}\rightarrow\mathbb{E}^{n+1};\
(w,{\gamma})\mapsto w$

be the bundle projection. Then, the following commutativity holds:

$t_{\cdot}w=\phi(t,\pi(w,{\gamma}))=\pi(\chi^{*\sharp}(t,(w,\gamma)))\quad\forall\,(t,(w,\gamma))\in\mathbb{R}\times\mathscr{F}_{n}^{*\sharp}.$

### 2.3.

Now, the so-called _Liao qualitative functions_ of $S$ are the following

$\displaystyle\omega_{i}^{*}\colon\mathscr{F}_{n}^{*\sharp}\rightarrow{\mathbb{R}}$
for $i=1,\ldots,n$, given by
$\displaystyle\omega_{i}^{*}(w,{\gamma})=\left.\frac{d}{dt}\right|_{t=0}\|\textrm{col}_{i}\circ{\chi_{t,w}^{*}}{\gamma}\|$

where
$\chi_{t,w}^{*}\cdot\colon\mathscr{F}_{n,w}^{*\sharp}\rightarrow\mathscr{F}_{n,t_{\cdot}w}^{*}$
is as in $\S\ref{sec2.2}$ for any $(t,w)\in\mathbb{R}\times\mathbb{E}^{n+1}$.
Particularly, let

(2.1) $\omega^{*}\colon\mathscr{F}_{1}^{*\sharp}\rightarrow{\mathbb{R}};\
(w,x)\mapsto\left.\frac{d}{dt}\right|_{t=0}\|\varPsi_{t,w}x\|.$

Since $S$ is of class $C^{1}$, these functions
$\omega^{*},\omega_{i}^{*},1\leq i\leq n$, all are well defined and
continuous; see [14, 10].

### 2.4.

Let ${\mathcal{M}}_{\textit{inv}}$ and ${\mathcal{M}}_{\textit{erg}}$ denote
the set of all invariant Borel probability measures and ergodic Borel
probability measures of a dynamical system, respectively. The following
theorem equivalently describes the Lyapunov characteristic spectrum of
$(\mathbb{E}^{n+1},S)$.

######  Theorem 2.1 (Liao spectrum theorem [14, 10]).

Let $S(w),S^{\prime}(w)$ be bounded on $\mathbb{E}^{n+1}$ and assume that
there is an ergodic $\phi$-invariant Borel probability measure $\mu$ on
$\mathbb{E}^{n+1}$. Then, there exists a $\phi$-invariant Borel subset
$L(\mu)$ of $\mathbb{E}^{n+1}$ such that:

  1. (1)

$\mu(L(\mu))=1$;

  2. (2)

every point $w$ in $L(\mu)$ is Oseledets regular for $\varPsi$ based on
$(\mathbb{E}^{n+1},\phi)$;

  3. (3)

given any
$P\in{\mathcal{M}}_{\textit{erg}}({\mathscr{F}}_{n}^{*\sharp},\chi^{*\sharp})$
with marginal $\mu$; i.e., $\mu=P\circ\pi^{-1}$,

${\rm\textsf{Sp}}_{\textsl{Lia}}^{*}(S,\mu):=\left\\{\vartheta_{i}^{*}(P)\,|\,i=1,\ldots,n\right\\}$

is just the Lyapunov spectrum of $\varPsi$ based on $(\phi,\mu)$, counting
with multiplicity and ignoring the order, which is called the “spectrum of
transversal Lyapunov exponents” of $(S,\mu)$ and is independent of the choices
of $P$, where

$\vartheta_{i}^{*}(P):=\int_{{\mathscr{F}}_{n}^{*\sharp}}\omega_{i}^{*}(w,{\gamma})\,dP(w,{\gamma})\quad\textrm{for
}i=1,\ldots,n.$

Notice that under the hypotheses of Theorem 2.1 above, there
$P\in{\mathcal{M}}_{\textit{erg}}({\mathscr{F}}_{n}^{*\sharp},\chi^{*\sharp})$
with marginal $\mu$ is always existent from [10].

######  Theorem 2.2 (Liao reordering theorem [14, 10]).

Under the hypotheses of Theorem 2.1, let
${\rm\textsf{Sp}}_{\textsl{Lia}}^{*}(S,\mu)=\left\\{\lambda_{i}^{*}(\mu)\,|\,i=1,\ldots,n\right\\}$
be the spectrum of transversal Lyapunov exponents of $(S,\mu)$. If
$i\mapsto\varrho(i)$ is any given permutation of $\\{1,\ldots,n\\}$, then
there is some
$P_{\varrho}\in{\mathcal{M}}_{\textit{erg}}({\mathscr{F}}_{n}^{*\sharp},\chi^{*\sharp})$
with marginal $\mu$ such that

$\vartheta_{i}^{*}(P_{\varrho})=\lambda_{\varrho(i)}^{*}(\mu)\quad\textrm{for
}i=1,\ldots,n.$

The above spectrum theorem 2.1 and reordering theorem 2.2 will play an
important role for the proof of Theorem B stated in $\S\ref{sec1}$.

### 2.5.

Based on Theorem 2.1, for any
$P\in{\mathcal{M}}_{\textit{erg}}({\mathscr{F}}_{1}^{*\sharp},\varPsi^{\sharp})$
with marginal $\mu$ we have

$\vartheta^{*}(P):=\int_{{\mathscr{F}}_{1}^{*\sharp}}\omega^{*}(w,x)\,dP(w,x)\in{\rm\textsf{Sp}}_{\textsl{Lia}}^{*}(S,\mu).$

Now, the following semi-uniform ergodic theorem will play an important role
for the proof of our main result.

######  Theorem 2.3.

Let $\Lambda$ be an $\phi$-invariant compact subset of $\mathbb{E}^{n+1}$ and
$\Delta$ an $\phi$-invariant Borel subset of $\Lambda$ with total measure $1$;
that is to say, $\mu(\Delta)=1$ for all
$\mu\in\mathcal{M}_{\textit{inv}}(\Lambda,\phi|\Lambda)$. Let

$\mathbb{D}\colon\Delta\ni w\mapsto D(w)\subset\mathbb{T}_{w}$

be an $\mathfrak{i}$-dimensional $\varPsi$-invariant measurable distribution
for some integer $1\leq\mathfrak{i}<n$. If $\varPsi|\mathbb{D}$ has only
negative Lyapunov exponents at almost every $w\in\Delta$ and $\mathbb{D}$ is
such that $\lim_{\ell\to\infty}D(w_{\ell})=D(w)$ provided, of course, that
this limit exists for $w_{\ell}\to w$ in $\Delta$, then $\varPsi|\mathbb{D}$
is uniformly contracting.

###### Proof.

Without any loss of generality, assume $\Lambda=\overline{\Delta}$. Let

$\mathscr{F}_{1}^{*\sharp}(\Delta)=\left\\{(w,x)\in\mathscr{F}_{1}^{*\sharp}\,|\,w\in\Delta\textrm{
and }x\in D(w)\right\\},$

which is $\varPsi^{\sharp}$-invariant. Then, it is easily seen that
$Y:=\overline{\mathscr{F}_{1}^{*\sharp}(\Delta)}$ is an
$\varPsi^{\sharp}$-invariant compact subbundle of $\mathscr{F}_{1}^{*\sharp}$
over $\Lambda$, such that

$P(Y-\mathscr{F}_{1}^{*\sharp}(\Delta))=0\quad\textrm{for all
}P\in\mathcal{M}_{\textit{inv}}(Y,\varPsi^{\sharp}|Y).$

Moreover, according to Theorem 2.1 we obtain

$\int_{Y}\omega^{*}\,dP<0\quad\forall\,P\in\mathcal{M}_{\textit{erg}}(Y,\varPsi^{\sharp}|Y),$

since $\varPsi|\mathbb{D}$ is nonuniformly contracting for any
$\mu\in\mathcal{M}_{\textit{erg}}(\Lambda,\phi|\Lambda)$. From the continuous-
time version of a semi-uniform theorem of [19] ([9, Lemma 3.1]), it follows
that there exist constants $\sigma>0$ and $T_{0}>0$ such that

$\displaystyle\int_{Y}\omega^{*}\,dP\leq-\sigma\quad\forall\,P\in\mathcal{M}_{\textit{erg}}(Y,\varPsi^{\sharp}|Y)$
and uniformly, for all $T\geq T_{0}$
$\displaystyle\frac{1}{T}\int_{0}^{T}\omega^{*}(\varPsi^{\sharp}(t+s,y))\,dt\leq-\sigma/2$

for all $y\in Y$ and for any $s\in\mathbb{R}$. Next, by the identity

$\frac{1}{T}\log\|\varPsi_{t,w}x\|=\frac{1}{T}\int_{0}^{T}\omega^{*}(\varPsi^{\sharp}(t,(w,x)))\,dt$

for all $(w,x)\in\mathscr{F}_{1}^{*\sharp}$ and for any $T\not=0$, we can
easily obtain that $\varPsi|\mathbb{D}$ is uniformly contracting.

This proves the theorem. ∎

Notice here that if the distribution $\mathbb{D}$ is continuous; that is,
$D(w_{\ell})\to D(w)$ provided that $w_{\ell}\to w$ in $\Lambda$, then from
[5] one can directly obtain the uniform contraction of $\varPsi|\mathbb{D}$.

##  3\. Shadowing property and approximation of ergodic measures

If $g\in\mathrm{Diff}^{1}(M)$ is conjugated to an Anosov diffeomorphism, then
$\overline{\textrm{Per}(g)}=\Omega(g)$ and $g$ has the shadowing property (see
[17, Proposition 8.5]). Naturally, we ask if every ergodic measure of $g$ can
be approximated by periodic measures or not. Under our context, this is the
case.

Let $S$ be a $C^{1}$-differential system on $\mathbb{E}^{n+1}$ as in
$\S\ref{sec2}$ and $\Lambda$ an $\phi$-invariant nonempty compact subset of
$\mathbb{E}^{n+1}$. We say that $(\phi,\Lambda)$ has the _shadowing by
periodic points property_ provided that to any $\epsilon>0$, there corresponds
to some $\alpha>0$, such that for any orbit arc
$\phi([0,\tau],w)\subset\Lambda,\tau\geq 2$ with $\|w-\tau_{\cdot}w\|<\alpha$,
there exists a periodic point $p\in\Lambda$ with period $\tau$ satisfying
$\|t_{\cdot}w-t_{\cdot}p\|<\epsilon$ for all $t\in[0,\tau]$. In this case, we
say that the orbit $\phi(t,p)$ $\epsilon$-shadows the orbit arc
$\phi([0,\tau],w)$. Notice here that we require $p\in\Lambda$.

An $\phi$-invariant Borel probability measure $\mu$ on $\mathbb{E}^{n+1}$ is
called a _periodic measure_ if it is supported on a periodic orbit of $\phi$;
that is, $\textrm{supp}(\mu)=\overline{\phi(\mathbb{R},p)}$ for some periodic
point $p$. The following result shows that periodic measures are dense in
$\mathcal{M}_{\textit{erg}}(\Lambda,\phi)$ under the weak $*$-topology.

######  Theorem 3.1.

If the compact subsystem $(\Lambda,\phi)$ has the shadowing by periodic points
property, then periodic measures are dense in
$\mathcal{M}_{\textit{erg}}(\Lambda,\phi)$; that is, for any $\mu$ in
$\mathcal{M}_{\textit{erg}}(\Lambda,\phi)$ there is a sequence of periodic
measures $(\mu_{k})$ on $\Lambda$ such that $\mu_{k}\to\mu$ as $k$ tends to
$\infty$.

###### Proof.

Let $\mu\in\mathcal{M}_{\textit{erg}}(\Lambda,\phi)$ be non-periodic, and let
$Q_{\mu}(\Lambda,\phi)$ be the quasi-regular point set of
$(\Lambda,\mu,\phi)$; that is, $w\in Q_{\mu}(\Lambda,\phi)$ if and only if

$\lim_{T\to\infty}T^{-1}\int_{0}^{T}\varphi(t_{\cdot}w)\,dt=\int_{\Lambda}\varphi\,d\mu$

for all $\varphi\in C(\Lambda)$.

For any $w\in\Lambda$ and $T>0$, using the Riesz representation theorem, we
define the empirical measure $\mu_{w,T}$ on $\Lambda$ by

$\mu_{w,T}(\varphi)=\frac{1}{T}\int_{0}^{T}\varphi(t_{\cdot}w)\,dt\quad\forall\,\varphi\in
C(\Lambda).$

Then, given any Poisson stable (recurrent) point $\hat{w}\in
Q_{\mu}(\Lambda,\phi)$ we have $\mu_{\hat{w},T}\to\mu$ as $T\to\infty$ in the
sense of weak $*$-topology; that is, $\mu_{\hat{w},T}(\varphi)\to\mu(\varphi)$
for all $\varphi\in C(\Lambda)$.

For any $\varphi\in C(\Lambda)$, let

$\|\varphi\|_{\infty}={\sup}_{x\in\Lambda}\|\varphi(x)\|$

and

$\|\varphi\|_{\textrm{L}}=\sup_{x,y\in\Lambda,x\not=y}\frac{\|\varphi(x)-\varphi(y)\|}{\|x-y\|}.$

Then $\textrm{BL}(\Lambda)=\\{\varphi\in
C(\Lambda);\,\|\varphi\|_{\infty}+\|\varphi\|_{\textrm{L}}<\infty\\}$ is dense
in $(C(\Lambda),\|\cdot\|_{\infty})$; see [12, Theorem 11.2.4].

Now, by the shadowing by periodic points property and the recurrence of the
motion $\phi(t,\hat{w})$, we can choose a sequence of periodic points $p_{i}$
with period $T_{i}\to\infty$ such that for any $i$,
$\|t_{\cdot}\hat{w}-t_{\cdot}p_{i}\|<1/i$ for all $t\in[0,T_{i}]$. Then, it is
easily seen that $\mu_{p_{i}}$, defined by

$\mu_{p_{i}}(\varphi)=\frac{1}{T_{i}}\int_{0}^{T_{i}}\varphi(t_{\cdot}p_{i})\,dt\quad\forall\,\varphi\in
C(\Lambda),$

is an ergodic periodic measure of the subsystem $(\Lambda,\phi)$.

Next, for any $\varphi\in\textrm{BL}(\Lambda)$ we have

$\displaystyle\lim_{i\to\infty}|\int\varphi\,d\mu-\int\varphi\,d\mu_{p_{i}}|$
$\displaystyle\leq\lim_{i\to\infty}|\int\varphi\,d\mu-\int\varphi\,d\mu_{\hat{w},T_{i}}|$
$\displaystyle{\quad}\quad+\limsup_{i\to\infty}|\int\varphi\,d\mu_{\hat{w},T_{i}}-\int\varphi\,d\mu_{p_{i}}|$
$\displaystyle\leq\limsup_{i\to\infty}\frac{1}{T_{i}}\int_{0}^{T_{i}}|\varphi(t_{\cdot}\hat{w})-\varphi(t_{\cdot}p_{i})|\,dt$
$\displaystyle\leq\limsup_{i\to\infty}\frac{1}{T_{i}}\int_{0}^{T_{i}}\|\varphi\|_{\textrm{L}}\|t_{\cdot}\hat{w}-t_{\cdot}p_{i}\|\,dt$
$\displaystyle\leq\limsup_{i\to\infty}\frac{\|\varphi\|_{\textrm{L}}}{i}$
$\displaystyle=0,$

which implies that $\mu_{p_{i}}\to\mu$ by the density of
$\textrm{BL}(\Lambda)$ in $(C(\Lambda),\|\cdot\|_{\infty})$, as required.

This proves the theorem. ∎

##  4\. Hyperbolicity of Katok maps

In this section, we will finish the proof of our main result Theorem B stated
in the Introduction, using the theorems introduced in $\S\S~{}\ref{sec2}$ and
3.

### 4.1.

Let $S$ be a $C^{1}$-differential system on $\mathbb{E}^{n+1}$ as in
$\S\ref{sec2}$ and $\Lambda$ an $\phi$-invariant nonempty compact subset of
$\mathbb{E}^{n+1}$.

######  Theorem 4.1.

Assume that the subsystem $(\phi,\Lambda)$ has the shadowing by periodic
points property, and, for each periodic point $p$ in $\Lambda$, let
$\lambda_{1}^{*}(p)\leq\cdots\leq\lambda_{n}^{*}(p)$ be the spectrum of
transversal Lyapunov exponents of $S$ at $p$, counting with multiplicity. If
there are some $\sigma<\varsigma$ such that $\lambda_{1}^{*}(p)\leq\sigma$ and
$\lambda_{n}^{*}(p)\geq\varsigma$ for all $p\in\mathrm{Per}(\Lambda,\phi)$,
then for all $\mu\in\mathcal{M}_{\textit{erg}}(\Lambda,\phi)$, $(S,\mu)$ has
at least two transversal Lyapunov exponents $\lambda_{-}^{*}(\mu)\leq\sigma$
and $\lambda_{+}^{*}(\mu)\geq\varsigma$.

###### Proof.

Let $\mu\in\mathcal{M}_{\textit{erg}}(\Lambda,\phi)$ be non-periodic. To prove
the statement, it is enough to show that $(S,\mu)$ has at least two
transversal Lyapunov exponents $\lambda_{-}^{*}(\mu)$ and
$\lambda_{+}^{*}(\mu)$ such that $\lambda_{-}^{*}(\mu)\leq\sigma$ and
$\lambda_{+}^{*}(\mu)\geq\varsigma$.

Let
$\mathscr{F}_{n}^{*\sharp}(\Lambda)=\left\\{(w,\gamma)\in\mathscr{F}_{n}^{*\sharp}\,|\,w\in\Lambda\right\\}$.
From Theorem 3.1, we can take a sequence of periodic measures, say
$\\{\mu_{p_{i}}\\}$, in $\mathcal{M}_{\textit{erg}}(\Lambda,\phi)$ with
$\mu_{p_{i}}\to\mu$. By using Theorems 2.1 and 2.2, we can choose some

$P_{i}\in\mathcal{M}_{\textit{erg}}(\mathscr{F}_{n}^{*\sharp}(\Lambda),\chi^{*\sharp})\quad\textrm{with
marginal }\mu_{p_{i}}$

for all $i$, such that

$\lambda_{1}^{*}(p_{i})=\vartheta_{1}^{*}(P_{i})\leq\cdots\leq\vartheta_{n}^{*}(P_{i})=\lambda_{n}^{*}(p_{i}).$

Since
$\mathcal{M}_{\textit{inv}}(\mathscr{F}_{n}^{*\sharp}(\Lambda),\chi^{*\sharp})$
is compact under the weak $*$-topology, there is no loss of generality in
assuming that $P_{i}\to P$ for some
$P\in\mathcal{M}_{\textit{inv}}(\mathscr{F}_{n}^{*\sharp}(\Lambda),\chi^{*\sharp})$
with marginal $\mu$; i.e., $P\circ\pi^{-1}=\mu$. Thus,

$\displaystyle\lim_{i\to\infty}\int_{\mathscr{F}_{n}^{*\sharp}(\Lambda)}\omega_{1}^{*}\,dP_{i}=\int_{\mathscr{F}_{n}^{*\sharp}(\Lambda)}\omega_{1}^{*}\,dP\leq\sigma$
and
$\displaystyle\lim_{i\to\infty}\int_{\mathscr{F}_{n}^{*\sharp}(\Lambda)}\omega_{n}^{*}\,dP_{i}=\int_{\mathscr{F}_{n}^{*\sharp}(\Lambda)}\omega_{n}^{*}\,dP\geq\varsigma$

for $\omega_{1}^{*}$ and $\omega_{n}^{*}$ both are continuous on
$\mathscr{F}_{n}^{*\sharp}(\Lambda)$. Then, by the classical ergodic
decomposition theorem we can choose at least two $P_{-}$ and $P_{+}$ in
$\mathcal{M}_{\textit{erg}}(\mathscr{F}_{n}^{*\sharp}(\Lambda),\chi^{*\sharp})$
with marginal $\mu$ such that

$\lambda_{-}^{*}(\mu):=\int_{\mathscr{F}_{n}^{*\sharp}(\Lambda)}\omega_{1}^{*}\,dP_{-}\leq\sigma\quad\textrm{and}\quad\lambda_{+}^{*}(\mu):=\int_{\mathscr{F}_{n}^{*\sharp}(\Lambda)}\omega_{n}^{*}\,dP_{+}\geq\varsigma.$

By Theorem 2.1 again, $\lambda_{-}^{*}(\mu)$ and $\lambda_{+}^{*}(\mu)$ both
lie in ${\rm\textsf{Sp}}_{\textsl{Lia}}^{*}(S,\mu)$, as required.

This proves the theorem. ∎

### 4.2.

To prove Theorem B, we need a further remark on Theorem A stated in
$\S\ref{sec1}$. Let $g\in\mathrm{Diff}^{1+}(M)$ be Hölder conjugated to an
Anosov $f\in\mathrm{Diff}^{1}(M)$. Theorem A asserts that all periodic points
of $g$ have only non-zero Lyapunov exponents. However, there is no information
that if there are any contracting or expanding periodic orbits. The following
lemma shows that a Katok map has no any contracting and expanding periodic
orbits.

Recall $f\in\textrm{Diff}^{1}(M)$ is said to be _Anosov_ if there is a
continuous splitting of $T_{x}M=E^{s}(x)\oplus E^{u}(x)$ for every $x\in M$
and constants $C>0,\lambda>1$ such that

$\displaystyle D_{x}f(E^{s}(x))$ $\displaystyle=E^{s}(f(x))\textrm{ and
}D_{x}f(E^{u}(x))=E^{u}(f(x)),$ $\displaystyle\|(D_{x}f^{n})\vec{v}\|$
$\displaystyle\leq C\lambda^{-n}\|\vec{v}\|\quad\forall\,\vec{v}\in E^{s}(x),\
n\in\mathbb{N},$ $\displaystyle\|(D_{x}f^{-n})\vec{u}\|$ $\displaystyle\leq
C\lambda^{-n}\|\vec{u}\|\quad\forall\,\vec{u}\in E^{u}(x),\ n\in\mathbb{N}.$

Then, the nonnegative integer

$\textrm{Ind}(x):=\dim E^{s}(x)$

for all $x\in M$ is called the _index_ of $f$ at $x$.

For any $f\in\textrm{Diff}^{1}(M)$ and $\delta>0$, as usual, for $x\in M$ let

$\displaystyle W^{s}(x)$ $\displaystyle=\left\\{y\in
M\,|\,\textrm{dist}(f^{k}x,f^{k}y)\to 0\textrm{ as }k\to\infty\right\\}$ and
$\displaystyle W_{\delta}^{s}(x)$ $\displaystyle=\left\\{y\in
M\,|\,\textrm{dist}(f^{k}x,f^{k}y)\leq\delta\textrm{ and
}\lim_{k\to\infty}\textrm{dist}(f^{k}x,f^{k}y)=0\right\\}$

be the stable set and the local stable set of $f$ at $x$, respectively. If $f$
is partially hyperbolic or $f$ is a nonuniformly partially hyperbolic
$C^{1+\alpha}$ diffeomorphism, then $W^{s}(x)$ has local smooth manifold
structure with $T_{x}W^{s}(x)=E^{s}(x)$ a.e. ([13, 16]).

Similarly, one can define the unstable manifolds $W^{u}(x)$ and
$W_{\delta}^{u}(x)$.

######  Lemma 4.2.

If $g\in\mathrm{Diff}^{1+}(M)$ is Hölder conjugated to an Anosov
$f\in\mathrm{Diff}^{1}(M)$, then all periodic points of $g$ have only non-zero
Lyapunov exponents, and such exponents are uniformly bounded away from zero,
and

$\mathrm{Ind}(p)\equiv\mathfrak{i}\quad\forall\,p\in\mathrm{Per}(g)$

for some integer $\mathfrak{i}$ with $1\leq\mathfrak{i}<n$.

###### Proof.

We first assert that for an Anosov $f\colon M^{n}\rightarrow M^{n}$, there is
an integer $\mathfrak{i}$ with $1\leq\mathfrak{i}<n$ such that

$\textrm{Ind}(\hat{x},f)=\mathfrak{i}\quad\textrm{for all }\hat{x}\in M.$

In fact, let $\Lambda_{i}=\\{\hat{x}\in M\colon\textrm{Ind}(\hat{x},f)=i\\}$
for $i=0,1,\ldots,n$. Since the splitting $T_{\hat{x}}M=E^{s}(\hat{x},f)\oplus
E^{u}(\hat{x},f)$ is continuous with respect to $\hat{x}\in M$, $\Lambda_{i}$
is closed and further open in $M$ for all $i=0,1,\ldots,n$. Thus, every
$\Lambda_{i}$ is either equal to $\varnothing$ or to $M$. Clearly,
$\Lambda_{0}=\Lambda_{n}=\varnothing$. This shows the assertion.

Let $h\colon M\rightarrow M$ be a Hölder conjugacy from $g$ to $f$. Given any
$p\in\mathrm{Per}(g)$ and let $\hat{p}=h(p)$. From Theorem A it follows that
$h(W^{s}(p;g))\subset W^{s}(\hat{p};f)$ and $h(W^{u}(p;g))\subset
W^{u}(\hat{p};f)$. Since $h$ is Hölder homeomorphic, we have $\dim
W_{\textit{loc}}^{s}(p;g)\leq\dim W_{\textit{loc}}^{s}(\hat{p};f)$ and $\dim
W_{\textit{loc}}^{u}(p;g)\leq\dim W_{\textit{loc}}^{u}(\hat{p};f)$. Then, we
can obtain that $\dim W_{\textit{loc}}^{s}(p;g)=\dim
W_{\textit{loc}}^{s}(\hat{p};f)$ and so $\mathrm{Ind}(p)=\mathfrak{i}$
constant with $1\leq\mathrm{Ind}(p)<n$.

This proves the lemma. ∎

### 4.3.

Before proving Theorem B, we first prove that a Katok map is non-uniformly
hyperbolic.

######  Theorem 4.3.

Let $g\in\mathrm{Diff}^{1+}(M)$ be Katok. Then

  1. (1)

there is a $\sigma>0$ such that to any
$\mu\in\mathcal{M}_{\textit{erg}}(M,g)$, $(g,\mu)$ is non-uniformly hyperbolic
and has at least two Lyapunov exponents, say $\lambda_{-}(\mu)$ and
$\lambda_{+}(\mu)$, with $\lambda_{-}(\mu)\leq-\sigma$ and
$\lambda_{+}(\mu)\geq\sigma$;

  2. (2)

there is an invariant subset $\varGamma$ and a measurable function
$\delta\colon\varGamma\rightarrow(0,\infty)$ such that $\mu\varGamma=1$ for
all $\mu\in\mathcal{M}_{\textit{erg}}(M,g)$ and $x\mapsto
W_{\delta(x)}^{s}(x),\ x\mapsto W_{\delta(x)}^{u}(x)$ both are well defined
and continuous for $x$ in $\varGamma$.

Here $W_{\delta}^{s}(x)$ and $W_{\delta}^{u}(x)$ mean the local stable and
unstable manifolds of $g$ at $x$, respectively.

###### Proof.

Let $g\in\mathrm{Diff}^{1+}(M)$ be Hölder conjugated to an Anosov
diffeomorphism $f\colon M\rightarrow M$. Given any
$\mu\in\mathcal{M}_{\textit{erg}}(M,g)$.

By the so-called suspension technique, from Lemma 4.2 and Theorem 4.1 it
follows immediately that $(g,\mu)$ has at least two Lyapunov exponents, say
$\lambda_{-}(\mu)$ and $\lambda_{+}(\mu)$, such that
$\lambda_{-}(\mu)\leq-\sigma$ and $\lambda_{+}(\mu)\geq\sigma$, where $\sigma$
is some positive constant which is independent of $\mu$.

We next proceed to prove that $(g,\mu)$ is non-uniformly hyperbolic. Let

$T_{x}M=E^{s}(x,g)\oplus E^{c}(x,g)\oplus E^{u}(x,g)$

for $\mu$-a.e. $x\in M$, where $E^{s}(x,g),E^{c}(x,g)$, and $E^{u}(x,g)$ stand
for the stable direction, central direction and unstable direction,
respectively, associated to the Oseledets splitting of $Dg$ at $x$ [15, 14].
We have $\dim E^{s}(x,g)\geq 1$ and $\dim E^{u}(x,g)\geq 1$ for a.e. $x$.
Since $g$ is of class $C^{1+\alpha}$ for some Hölder exponent $0<\alpha<1$,
according to Pesin theory [16] there are local stable manifold
$W_{\textit{loc}}^{s}(x;g)$ and local unstable manifold
$W_{\textit{loc}}^{u}(x;g)$ with $\dim W_{\textit{loc}}^{s}(x;g)=\dim
E^{s}(x,g)$ and $\dim W_{\textit{loc}}^{u}(x;g)=\dim E^{u}(x,g)$ for
$\mu$-a.e. $x\in M$. On the other hand, by the $C^{\alpha}$-conjugation
$h^{-1}\colon M\rightarrow M$ from the Anosov diffeomorphism $f$ to $g$, we
obtain that for $\hat{x}=h(x)$

$\dim W_{\textit{loc}}^{s}(x;g)\geq\dim
W_{\textit{loc}}^{s}(\hat{x};f)\quad\textrm{and}\quad\dim
W_{\textit{loc}}^{u}(x;g)\geq\dim W_{\textit{loc}}^{u}(\hat{x};f),$

which implies $\dim W_{\textit{loc}}^{s}(x;g)+\dim
W_{\textit{loc}}^{u}(x;g)=n$ and so $E^{c}(x,g)=\textbf{0}$ for $\mu$-a.e.
$x\in M$. Thus, $(g,\mu)$ is non-uniformly hyperbolic. This proves the
statement (1).

Next, we are going to prove the statement (2). Since $f$ is Anosov, there is
some constant $\hat{\delta}>0$ such that the local stable foliation
$\mathscr{W}^{s}=(W_{\hat{\delta}}^{s}(\hat{x};f))_{\hat{x}\in M}$ is
continuous in $\hat{x}\in M$. Let $\varGamma$ be the non-uniformly hyperbolic
Pesin regular set of $g$ [3]. Noticing that $h,h^{-1}$ both are Hölder and
$h^{-1}(W_{\hat{\delta}}^{s}(\hat{x};f))\subset W^{s}(x;g)$ where
$h(x)=\hat{x}$ for all $x\in\varGamma$, we can easily find some measurable
function $\delta\colon\varGamma\rightarrow(0,\infty)$, which satisfies the
requirements of the statement (2). This proves the statement (2).

Thus, Theorem 4.3 is proved. ∎

### 4.4.

In [6], the authors exhibit an example of a non-hyperbolic horseshoe such that
all Lyapunov exponents are non-zero and uniformly bounded away from zero for
all invariant measures. This phenomenon is named “completely nonuniformly
hyperbolic.” Theorem 4.3 implies that a Katok map $g$ of $M^{2}$ has just two
Lyapunov exponents $\lambda_{-}(\mu)<0<\lambda_{+}(\mu)$, uniformly bounded
away from zero, for all $\mu$ in $\mathcal{M}_{\textit{erg}}(M^{2},g)$.

Nevertheless, there is still an essential gap from Theorem 4.3 to Katok’s
conjecture even though in the $2$-dimensional case, the continuity of the
foliation $(W_{\delta}^{s}(x))_{x\in\varGamma}$ guaranteed by Theorem 4.3(2),
can avoid the occurrence of the completely nonuniformly hyperbolic phenomenon.

Now we can finish the proof of Theorem B using the semi-uniform ergodic
theorem Theorem 2.3.

###### Proof of Theorem B.

Let $\varGamma$ be defined by Theorem 4.3(2) and let

$T_{x}M=E^{s}(x)\oplus E^{u}(x)$

for all $x\in\varGamma$ be the Oseledets splitting of $g$ according to the
multiplicative ergodic theorem. Let $\mathfrak{i}$ be the index of $g$ and
$\mathscr{G}_{\mathfrak{i}}(T_{\varGamma}M)$ the Grassmannian manifold of
$\mathfrak{i}$-dimensional linear subspaces of $T_{\varGamma}M$. Then

$\mathbb{D}\colon\varGamma\rightarrow\mathscr{G}_{\mathfrak{i}}(T_{\varGamma}M);\
x\mapsto E^{s}(x)$

is a $Dg$-invariant measurable distribution over $\varGamma$ such that
$T_{x}W_{\delta(x)}^{s}(x)=E^{s}(x)$ and $\dim E^{s}(x)=\mathfrak{i}$ for all
$x\in\varGamma$.

Let $x_{\ell}\to x$ in $\varGamma$ and
$\lim_{\ell\to\infty}E^{s}(x_{\ell})=E(x)$ for some $\mathfrak{i}$-dimensional
linear subspace $E(x)\subset T_{x}M$. As
$W_{\delta(x_{\ell})}^{s}(x_{\ell})\to W_{\delta(x)}^{s}(x)$ as
$\ell\to\infty$ by Theorem 4.3, it follows from
$T_{x_{\ell}}W_{\delta(x_{\ell})}^{s}(x_{\ell})=E^{s}(x_{\ell})$ that
$T_{x}W_{\delta(x)}^{s}(x)=E(x)$. Thus, $E(x)=E^{s}(x)$. Then, by the
discretization of Theorem 2.3 using the terms introduced in [8], we obtain
that
$Dg\colon\bigcup_{x\in\varGamma}E^{s}(x)\rightarrow\bigcup_{x\in\varGamma}E^{s}(x)$
is uniformly contracting.

Similarly, we can show that
$Dg\colon\bigcup_{x\in\varGamma}E^{u}(x)\rightarrow\bigcup_{x\in\varGamma}E^{u}(x)$
is uniformly expanding.

Therefore, $g$ is uniformly hyperbolic on $\overline{\varGamma}$. Since
$\Omega(g)=\overline{\textrm{Per}(g)}$, we have that
$\overline{\varGamma}=\Omega(g)$. Thus, $g$ is of Axiom A. Clearly, $g$ has no
cycles.

This proves the theorem. ∎

##  5\. Volume-preserving Katok maps

Let Leb denote the standard volume measure of $M^{n}$. A Borel probability
measure $\mu$ on $M$ is called a _smooth probability measure_ if $\mu$ is
absolutely continuous with respect to Leb such that

$C\leq d\mu/d\textrm{Leb}\leq K$

for some constants $C,K>0$. If $g$ is a $C^{1+\alpha}$ volume-preserving
Anosov diffeomorphism, $g$ is ergodic due to Anosov [1, 2]. For other proof of
Anosov’s ergodicity theorem, see [20]. However, we are going to prove that if
a Katok map preserves a smooth probability measure, then it is Anosov and thus
also ergodic.

Using different approaches, it was proved independently by Bochi $\&$ Viana
[4] and Xia [20] that the uniformly hyperbolic closed sets of every
$C^{1+\alpha}$ volume-preserving diffeomorphisms have zero Lebesgue measure,
unless they coincide with the whole ambient compact manifold (Anosov case).
This result was generalized by using Pesin theory as follows:

######  Lemma 5.1 ([11]).

Let $f$ be a diffeomorphism preserving a smooth probability measure $\mu$ on a
compact, connected, and closed Riemannian manifold $M$. Let $\Lambda\subset M$
be a uniformly hyperbolic invariant Borel set (not necessarily closed). If
$\mu(\Lambda)>0$, then $f$ is Anosov and $\Lambda=M\ (\textrm{mod }0)$.

Now we can prove the ergodicity of a Katok map.

######  Corollary 5.2.

Let $g$ be a $C^{1+\alpha}$ Katok diffeomorphism of $M$ which preserves a
smooth probability measure $\mu$. Then $g$ is Anosov and ergodic with
$\Omega(g)=M$ .

###### Proof.

Theorem B and Lemma 5.1 imply that $g$ is Anosov with $\Omega(g)=M$.

This proves Corollary 5.2. ∎

######  Remark 5.3.

We noted that it was recently announced by Zhihong Xia [18] that an Anosov
diffeomorphism must be topologically transitive. Then, our Theorem B implies
that every Katok diffeomorphism must be Anosov.

### Acknowledgments

The author is grateful to Professors Huyi Hu and Yunping Jiang for discussions
on this paper.

## References

  * [1] D. V. Anosov, Ergodic properties of geodesic flows on closed Riemannian manifolds of negative curvature. Soviet Math. Dokl., 4 (1963), 1153–1156.
  * [2] D. V. Anosov, Geodesic flows on closed Riemannian manifolds of negative curvature. Proc. Steklov Math. Inst., 90 (1967), 1–235.
  * [3] L. Barreira and Ya. Pesin, Lectures on Lyapunov Exponents and Smooth Ergodic Theory. Appendix A by M. Brin and Appendix B by D. Dolgopyat, H. Hu and Pesin. Proc. Sympos. Pure Math., 69, Smooth ergodic theory and its applications (Seattle, WA, 1999), 3–106, Amer. Math. Soc., Providence, RI, 2001.
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  * [14] S.-T. Liao, On characteristic exponents construction of a new Borel set for the multiplicative ergodic theorem for vector fields. Acta Sci. Natur. Univ. Pekinensis, 29 (1993), 277–302.
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# On the sum of two integral squares in quadratic fields
${\mathbb{Q}}(\sqrt{\pm p})$

Dasheng Wei  Academy of Mathematics and System Science, CAS, Beijing 100190,
P.R.China<EMAIL_ADDRESS>

## Abstract

We propose a method for determining which integers can be written as a sum of
two integral squares for quadratic fields ${\mathbb{Q}}(\sqrt{\pm p})$, where
$p$ is a prime.

MSC classification : 11D09; 11E12

Keywords : integral points, ring class field, reciprocity law.

## Introduction

Gauss first determined which integers can be written as a sum of two integral
squares. And Niven determined which integers can be written as a sum of two
integral squares for the imaginary quadratic field ${\mathbb{Q}}({\sqrt{-1}})$
in [Niv]. Nagell further studied the question for the twenty quadratic fields
${\mathbb{Q}}(\sqrt{d})$ in [Nag1] and [Nag2], where

$d=\pm 2,\pm 3,\pm 5,\pm 7,\pm 11,\pm 13,\pm 19,\pm 43,\pm 67,\pm 163.$

His method essentially depends on the fact that the class number of
${\mathbb{Q}}(\sqrt{d},\sqrt{-d})$ is 1 when $d$ is one of the above integers.
However, this method can not apply for general quadratic fields. Recently,
Harari showed that the Brauer-Manin obstruction is the only obstruction for
existence of the integral points of a scheme over a ring of integers of a
number field whose generic fiber is a principal homogeneous space of tori in
[Ha08]. However, the Brauer-Manin obstruction of tori given in [Ha08] is not
constructive and one can not use that result to determine the existence of
integral points for the scheme. Fei Xu and the author gave another proof of
the result in [WX] and [WX2] which is constructive. In this paper we apply the
method in [WX] for the question for quadratic fields ${\mathbb{Q}}(\sqrt{\pm
p})$, where $p$ is a prime.

Notation and terminology are standard if not explained. Let $F$ be a number
field, $\mathfrak{o}_{F}$ the ring of integers of $F$, $\Omega_{F}$ the set of
all primes in $F$ and $\infty$ the set of all infinite primes in $F$. For
simplicity, we write $\mathfrak{p}<\infty$ for
$\mathfrak{p}\in\Omega_{F}\setminus\infty$. Let $F_{\mathfrak{p}}$ be the
completion of $F$ at $\mathfrak{p}$ and $\mathfrak{o}_{F_{\mathfrak{p}}}$ be
the local completion of $\mathfrak{o}_{F}$ at $\mathfrak{p}$ for each
$\mathfrak{p}\in\Omega_{F}$. Write
$\mathfrak{o}_{F_{\mathfrak{p}}}=F_{\mathfrak{p}}$ for
$\mathfrak{p}\in\infty$. We also denote the adele ring (resp. the idele ring)
of $F$ by $\mathbb{A}_{F}$ (resp. $\mathbb{I}_{F}$) and

$F_{\infty}=\prod_{\mathfrak{p}\in\infty}F_{\mathfrak{p}}.$

Let $E=F(\sqrt{-1})$ and let $T$ be the torus
$R^{1}_{E/F}(\mathbb{G}_{m})={\rm
Ker}[R_{E/F}(\mathbb{G}_{m,E})\rightarrow\mathbb{G}_{m,F}]$, here $R$ denotes
the Weil’s restriction (see [Milne98]). Denote $\lambda$ to be the embedding
from $T$ to $R_{E/F}(\mathbb{G}_{m,E})$. Obviously $\lambda$ induces a natural
group homomorphism

$\lambda_{E}:\ \ \ T(\mathbb{A}_{F})\rightarrow\mathbb{I}_{E}.$

Let $\mathbb{X}_{\alpha}$ be the affine scheme over $\mathfrak{o}_{F}$ defined
by the equation $x^{2}+y^{2}=\alpha$ for a non-zero integer
$\alpha\in\mathfrak{o}_{F}$. The generic fiber of $\mathbb{X}_{\alpha}$ is a
principle homogenous space of the torus $T$. The equation $x^{2}+y^{2}=\alpha$
is solvable over $\mathfrak{o}_{F}$ if and only if
$\mathbb{X}_{\alpha}(\mathfrak{o}_{F})\neq\emptyset$.

######  Definition 0.1.

Let $K/E$ be a finite abelian extension. Let
$\psi_{K/E}:\mathbb{I}_{E}\rightarrow Gal(K/E)$ be the Artin map. We say that
$\alpha$ satisfies the Artin condition of $K$ if there is

$\prod_{\mathfrak{p}\leq\infty}(x_{\mathfrak{p}},y_{\mathfrak{p}})\in\prod_{\mathfrak{p}\leq\infty}\mathbb{X}_{\alpha}(\mathfrak{o}_{F_{\mathfrak{p}}})\text{
such that
}\psi_{K/E}(f_{E}[\prod_{\mathfrak{p}\leq\infty}(x_{\mathfrak{p}},y_{\mathfrak{p}})])=1$

where $1$ is the identity element of $Gal(K/E)$ and
$f_{E}:\prod_{\mathfrak{p}\leq\infty}\mathbb{X}_{\alpha}(\mathfrak{o}_{F_{\mathfrak{p}}})\rightarrow\mathbb{I}_{E}$
is defined by

$f_{E}[(x_{\mathfrak{p}},y_{\mathfrak{p}})]=\begin{cases}(x_{\mathfrak{p}}+y_{\mathfrak{p}}\sqrt{-1},x_{\mathfrak{p}}-y_{\mathfrak{p}}\sqrt{-1})\
\ \ &\text{if $\mathfrak{p}$ splits in $E/F$}\\\
x_{\mathfrak{p}}+y_{\mathfrak{p}}\sqrt{-1}\ \ \ &\text{otherwise.}\end{cases}$

By the class field theory, it is a necessary condition for
$\mathbb{X}_{\alpha}(\mathfrak{o}_{F})\neq\emptyset$ that $\alpha$ satisfies
the Artin condition of $K$. In fact there is a finite abelian extension $K/E$
that is independent on $\alpha$, such that the Artin condition of $K$ is also
sufficient for $\mathbb{X}_{\alpha}(\mathfrak{o}_{F})\neq\emptyset$ (see
[WX]).

Let $\mathbb{T}$ be the group scheme over $\mathfrak{o}_{F}$ defined by
$x^{2}+y^{2}=1$, which is an integral model of $T$. Since $\mathbb{T}$ is
separated over $\mathfrak{o}_{F}$, we can view
$\mathbb{T}(\mathfrak{o}_{F_{\mathfrak{p}}})$ as a subgroup of
$T(F_{\mathfrak{p}})$. Furthermore, the following result is proved in [WX].

######  Proposition 0.2.

Let $K/E$ be a finite abelian extension such that the group homomorphism
$\widetilde{\lambda}_{E}$ induced by $\lambda_{E}$

$\widetilde{\lambda}_{E}:\nicefrac{{T(\mathbb{A}_{F})}}{{T(F)\prod_{\mathfrak{p}\leq\infty}\mathbb{T}(\mathfrak{o}_{F_{\mathfrak{p}}})}}\longrightarrow\nicefrac{{\mathbb{I}_{E}}}{{E^{*}N_{K/E}(\mathbb{I}_{K})}}$

is well-defined and injective, where well-defined means

$\lambda_{E}(T(F)\prod_{\mathfrak{p}\leq\infty}\mathbb{T}(\mathfrak{o}_{F_{\mathfrak{p}}}))\subset
E^{*}N_{K/E}(\mathbb{I}_{K}).$

Then $\mathbb{X}_{\alpha}(\mathfrak{o}_{F})\neq\emptyset$ if and only if
$\alpha$ satisfies the Artin condition of $K$.

In this paper, we mainly prove the following result.

######  Theorem 0.3.

Let $p$ be a prime number and $F$ the quadratic field ${\mathbb{Q}}(\sqrt{p})$
or ${\mathbb{Q}}(\sqrt{-p})$. Then the diophantine equation
$x^{2}+y^{2}=\alpha$ is solvable over $\mathfrak{o}_{F}$ if and only if
$\alpha$ satisfies the Artin condition of $H_{L}$, where $H_{L}$ is the ring
class field corresponding to the order
$L=\mathfrak{o}_{F}+\mathfrak{o}_{F}\sqrt{-1}$.

##  1\. The sum of two squares in imaginary quadratic fields

Let $d$ be a square-free positive integer here $d\geq 2$. Let
$F={\mathbb{Q}}(\sqrt{-d})$, $\mathfrak{o}_{F}$ be the integral ring of $F$
and $E=F(\sqrt{-1})$. One takes the order
$L=\mathfrak{o}_{F}+\mathfrak{o}_{F}\sqrt{-1}$ inside $E$. Let $H_{L}$ be the
ring class field corresponding to the order $L$.

######  Proposition 1.1.

Suppose one of the following conditions holds:

(1) The equation $x^{2}+y^{2}=-1$ has an integer solution in
$\mathfrak{o}_{F}$.

(2) The equation $x^{2}+y^{2}=-1$ has no local integral solutions at a place
of $F$.

Then the diophantine equation $x^{2}+y^{2}=\alpha$ is solvable over
$\mathfrak{o}_{F}$ if and only if $\alpha$ satisfies the Artin condition of
$H_{L}$.

###### Proof.

(1) First we assume $d\neq 3$. Let $\mathfrak{p}$ be a place of $F$ and
$L_{\mathfrak{p}}$ be the $\mathfrak{p}$-adic completion of $L$ inside
$E_{\mathfrak{p}}=E\otimes_{F}F_{\mathfrak{p}}$. Recall
$T=R^{1}_{E/F}({\mathbb{G}}_{m,F})$ and $\mathbb{T}$ is the scheme defined by
the equation $x^{2}+y^{2}=1$, we have

$T(F)=\\{\beta\in E^{*}:\ N_{E/F}(\beta)=1\\}$

and

$\mathbb{T}(\mathfrak{o}_{F_{\mathfrak{p}}})=\\{\beta\in
L_{\mathfrak{p}}^{\times}:\
N_{E_{\mathfrak{p}}/F_{\mathfrak{p}}}(\beta)=1\\}.$

Since the ring class field $H_{L}$ of the order $L$ corresponds to the open
subgroup $E^{*}(\prod_{\mathfrak{p}\leq\infty}L_{\mathfrak{p}}^{\times})$ of
$\mathbb{I}_{E}$ by the class field theory, the natural group homomorphism

$\widetilde{\lambda}_{E}:\ \ \
\nicefrac{{T(\mathbb{A}_{F})}}{{T(F)\prod_{\mathfrak{p}\leq\infty}\mathbb{T}(\mathfrak{o}_{F_{\mathfrak{p}}})}}\longrightarrow\nicefrac{{\mathbb{I}_{E}}}{{E^{*}\prod_{\mathfrak{p}\leq\infty}L_{p}^{\times}}}$

is well-defined. By Proposition 0.2, we only need to show
$\widetilde{\lambda}_{E}$ is injective.

Suppose there are

$\beta\in E^{*}\ \ \ \text{ and}\ \ \ i\in
E_{\infty}^{*}\prod_{\mathfrak{p}<\infty}L_{\mathfrak{p}}^{\times}$

such that $\beta\cdot i\in T(\mathbb{A}_{E})$. Then

$N_{E/F}(\beta i)=N_{E/F}(\beta)N_{E/F}(i)=1$

and

$N_{E/F}(\beta)\in
F^{*}\cap\prod_{\mathfrak{p}<\infty}\mathfrak{o}_{F_{\mathfrak{p}}}^{\times}=\\{\pm
1\\}.$

If $N_{E/F}(\beta)=1$, one concludes that

$N_{E/F}(\beta)=N_{E/F}(i)=1\ \ \ \Rightarrow\ \ \ \beta\in T(E)\ \ \
\text{and}\ \ \
i\in\prod_{\mathfrak{p}\leq\infty}\mathbb{T}(\mathfrak{o}_{F_{\mathfrak{p}}}).$

So $\beta i\in
T(E)\prod_{\mathfrak{p}\leq\infty}\mathbb{T}(\mathfrak{o}_{F_{\mathfrak{p}}})$.

If $N_{E/F}(\beta)\neq 1$, then $N_{E/F}(\beta)=N_{E/F}(i)=-1$. That is to say
that $x^{2}+y^{2}=-1$ has local integral solutions at every local place of
$F$. By the assumption, we have $x^{2}+y^{2}=-1$ has an integral solution
$(x_{0},y_{0})$. Let

$\zeta=x_{0}+y_{0}\sqrt{-1}\ \text{ and }\ \gamma=\beta\zeta,j=i/\zeta.$

Then $\beta i=\gamma j$ and

$N_{E/F}(\gamma)=N_{E/F}(j)=1\ \ \ \Rightarrow\ \ \ \gamma\in T(F)\ \ \
\text{and}\ \ \
j\in\prod_{\mathfrak{p}}\mathbb{T}(\mathfrak{o}_{F_{\mathfrak{p}}}).$

So $\beta i=\gamma j\in
T(F)\prod_{\mathfrak{p}\leq\infty}\mathbb{T}(\mathfrak{o}_{F_{\mathfrak{p}}})$.
Therefore $\widetilde{\lambda}_{E}$ is injective.

(2) If $d=3$, then $\mathfrak{o}_{F}^{\times}=<\pm 1,\zeta_{3}>$, where
$\zeta_{3}$ is a primitive 3-rd root of unity. Since $\zeta_{3}=\zeta_{3}^{4}$
is a square, we can give a proof for this case with similar arguments as
above. ∎

In the rest of this section we consider the case that $d$ is a prime. First we
need the following result that can be found in [Yo].

######  Proposition 1.2.

Let $p$ be a prime. Then

(1) If $p\equiv 1\mod 4$, then $x^{2}-py^{2}=-1$ is solvable over
${\mathbb{Z}}$.

(2) If $p\equiv-1\mod 8$, then $x^{2}-py^{2}=2$ is solvable over
${\mathbb{Z}}$.

(2) If $p\equiv 3\mod 8$, then $x^{2}-py^{2}=-2$ is solvable over
${\mathbb{Z}}$.

Now we can prove the following lemma.

######  Lemma 1.3.

Let $p$ be a prime and $F={\mathbb{Q}}(\sqrt{-p})$. Then

(1) If $p\equiv-1\mod 8$, then $x^{2}+y^{2}=-1$ is not solvable over
$\mathfrak{o}_{F_{\mathfrak{p}}}$, where $\mathfrak{p}\mid 2$.

(2) If $p\not\equiv-1\mod 8$, then $x^{2}+y^{2}=-1$ is solvable over
$\mathfrak{o}_{F}$.

###### Proof.

(1) If $p\equiv-1\mod 8$, then $2$ splits into $\mathfrak{p}_{1}$ and
$\mathfrak{p}_{2}$ in the field $F/{\mathbb{Q}}$. So the Hilbert symbol

$(-1,-1)_{\mathfrak{p}_{1}}=(-1,-1)_{\mathfrak{p}_{2}}=-1.$

Therefore the equation $x^{2}+y^{2}=-1$ is not solvable over
$\mathfrak{o}_{F_{\mathfrak{p}_{1}}}$ and
$\mathfrak{o}_{F_{\mathfrak{p}_{2}}}$.

If $p\equiv 1\mod 4$ or $p=2$, then $x^{2}-py^{2}=-1$ has an integral solution
in ${\mathbb{Z}}$ by Proposition 1.2. Choose one solution $(x_{0},y_{0})$, we
have $x_{0}^{2}+(y_{0}\sqrt{-p})^{2}=-1$.

If $p\equiv 3\mod 8$, then $x^{2}-py^{2}=-2$ has an integral solution in
${\mathbb{Z}}$ by Proposition 1.2. We can choose one solution $(x_{0},y_{0})$
and it’s easy to see $x_{0},y_{0}\equiv 1\mod 2$. So

$\frac{x_{0}\pm y_{0}\sqrt{-p}}{2}\in\mathfrak{o}_{F}\text{ and
}(\frac{x_{0}+y_{0}\sqrt{-p}}{2})^{2}+(\frac{x_{0}-y_{0}\sqrt{-p}}{2})^{2}=-1.$

∎

By Proposition 1.1 and Lemma 1.3, we can now prove the following result.

######  Theorem 1.4.

Let $p$ be a prime number and $F={\mathbb{Q}}(\sqrt{-p})$. Let $H_{L}$ be the
ring class field corresponding to the order
$L=\mathfrak{o}_{F}+\mathfrak{o}_{F}\sqrt{-1}$. Then the diophantine equation
$x^{2}+y^{2}=\alpha$ is solvable over $\mathfrak{o}_{F}$ if and only if
$\alpha$ satisfies the Artin condition of $H_{L}$.

######  Remark 1.5.

It is possible that the family of equations
$x^{2}+y^{2}=\alpha,\alpha\in\mathfrak{o}_{F}\text{ and }\alpha\neq 0$
satisfies Hasse principle even if the ring class field $H_{L}$ is not trivial.
For example, the equation $x^{2}+y^{2}=\alpha$ satisfies Hasse principle and
$H_{L}$ is not trivial for $F={\mathbb{Q}}(\sqrt{-p})$ with
$p=23,31,47,59,71$. The reason is $H_{L}=EH$ for the above $p$, where
$E=F(\sqrt{-1})$ and $H$ is the Hilbert class field of $F$. If
$x^{2}+y^{2}=\alpha$ has local solutions for every place, then $\alpha$
automatically satisfies the Artin condition of $H_{L}$ by the class field
theory.

Now we use Theorem 1.4 to give an explicit example. Let
$F={\mathbb{Q}}(\sqrt{-79})$. We write
$N_{F/{\mathbb{Q}}}(\alpha)=2^{s_{1}}79^{s_{2}}p_{1}^{e_{1}}\cdots
p_{g}^{e_{g}}$ for any $\alpha\in\mathfrak{o}_{F}$. Let
$D(n)=\\{p_{1},\cdots,p_{g}\\}$ and $h(x)=x^{3}-307x+1772$. Denote

$\displaystyle D_{1}=\\{p\in D(n):(\frac{79}{p})=(\frac{-1}{p})=1\text{ and
}h(x)\equiv 0\mod p\text{ is not solvable}\\}$ $\displaystyle D_{2}=\\{p\in
D(n):(\frac{79}{p})=-(\frac{-1}{p})=1\text{ and }h(x)\equiv 0\mod p\text{ is
not solvable}\\}.$

It’s easy to see that $e_{i}$ is even when $p_{i}\in D_{2}$.

######  Example 1.6.

Let $F={\mathbb{Q}}(\sqrt{-79})$ and let $\alpha$ be an integer in $F$ with
the above notation. Then $x^{2}+y^{2}=\alpha$ is solvable over
$\mathfrak{o}_{F}$ if and only if

(1) The equation $x^{2}+y^{2}=\alpha$ has integral solutions at every place of
$F$.

(2) The sum

$\sum_{p_{i}\in D_{1}}e_{i}+\sum_{p_{i}\in D_{2}}e_{i}/2\neq 1.$

##  2\. The sum of two squares in real quadratic fields

Let $d$ be a square-free positive integer and $F={\mathbb{Q}}(\sqrt{d})$. Let
$\mathfrak{o}_{F}$ be the ring of integers of $F$, $\varepsilon$ the
fundamental unit of $\mathfrak{o}_{F}$ and $\varepsilon=a+b\sqrt{d}\text{ with
}a,b>0$. Let $E=F(\sqrt{-1})$. One takes the order
$L=\mathfrak{o}_{F}+\mathfrak{o}_{F}\sqrt{-1}$ inside $E$. Let $H_{L}$ be the
ring class field corresponding to the order $L$.

######  Proposition 2.1.

Suppose one of the following conditions holds:

(1) The equation $x^{2}+y^{2}=\varepsilon$ has an integer solution in
$\mathfrak{o}_{F}$.

(2) The equation $x^{2}+y^{2}=\varepsilon$ has no local integral solutions at
a place of $F$.

Then the diophantine equation $x^{2}+y^{2}=\alpha$ is solvable over
$\mathfrak{o}_{F}$ if and only if $\alpha$ satisfies the Artin condition of
$H_{L}$.

###### Proof.

Let $\mathfrak{p}$ be a place of $F$ and $L_{\mathfrak{p}}$ be the
$\mathfrak{p}$-adic completion of $L$ inside
$E_{\mathfrak{p}}=E\otimes_{F}F_{\mathfrak{p}}$. Since the ring class field
$K_{L}$ of the order $L$ corresponds to the open subgroup
$E^{*}(\prod_{\mathfrak{p}\leq\infty}L_{\mathfrak{p}}^{\times})$ of
$\mathbb{I}_{E}$ by the class field theory, the natural group homomorphism

$\widetilde{\lambda}_{E}:\ \ \
\nicefrac{{T(\mathbb{A}_{F})}}{{T(F)\prod_{\mathfrak{p}\leq\infty}\mathbb{T}(\mathfrak{o}_{F_{\mathfrak{p}}})}}\longrightarrow\nicefrac{{\mathbb{I}_{E}}}{{E^{*}\prod_{\mathfrak{p}\leq\infty}L_{p}^{\times}}}$

is well-defined. By Proposition 0.2, we only need to show
$\widetilde{\lambda}_{E}$ is injective.

Suppose there are

$\beta\in E^{*}\ \ \ \text{ and}\ \ \ i\in
E_{\infty}^{*}\prod_{\mathfrak{p}<\infty}L_{\mathfrak{p}}^{\times}$

such that $\beta\cdot i\in T(\mathbb{A}_{E})$. Then

$N_{E/F}(\beta i)=N_{E/F}(\beta)N_{E/F}(i)=1$

and

$N_{E/F}(\beta)\in
F^{*}\cap\prod_{\mathfrak{p}<\infty}\mathfrak{o}_{F_{\mathfrak{p}}}^{\times}=\\{\pm\varepsilon^{n}\\}.$

Since $N_{E/F}(\beta)$ is totally positive, we have
$N_{E/F}(\beta)=\varepsilon^{n}$.

When $n$ is even, let $\gamma=\beta\varepsilon^{n/2},j=i\varepsilon^{-n/2}$.
Then $\beta i=\gamma j$ and

$N_{E/F}(\gamma)=N_{E/F}(j)=1\ \ \ \Rightarrow\ \ \ \gamma\in T(F)\ \ \
\text{and}\ \ \
j\in\prod_{\mathfrak{p}}\mathbb{T}(\mathfrak{o}_{F_{\mathfrak{p}}}).$

So $\beta i=\gamma j\in
T(F)\prod_{\mathfrak{p}\leq\infty}\mathbb{T}(\mathfrak{o}_{F_{\mathfrak{p}}})$.

When $n$ is odd, we have $N_{E/F}(i)=\varepsilon^{-n}$. That is to say that
$x^{2}+y^{2}=\varepsilon^{-n}$ has integral solutions at every local place of
$F$. Since $n$ is odd and $\varepsilon\in\mathfrak{o}_{F}^{\times}$, we have
$x^{2}+y^{2}=\varepsilon$ has integral solutions at every place of $F$. By the
assumption, we have $x^{2}+y^{2}=\varepsilon$ has an integer solution
$(x_{0},y_{0})$. Let $\zeta=x_{0}+y_{0}\sqrt{-1}$ and
$\gamma=\beta\varepsilon^{(n-1)/2}\zeta,j=i\varepsilon^{(1-n)/2}\zeta^{-1}$.
Then $\beta i=\gamma j.$ And

$N_{E/F}(\gamma)=N_{E/F}(j)=1\ \ \ \Rightarrow\ \ \ \gamma\in T(F)\ \ \
\text{and}\ \ \
j\in\prod_{\mathfrak{p}\leq\infty}\mathbb{T}(\mathfrak{o}_{F_{\mathfrak{p}}}).$

So $\beta i=\gamma j\in
T(F)\prod_{\mathfrak{p}\leq\infty}\mathbb{T}(\mathfrak{o}_{F_{\mathfrak{p}}})$.
Therefore $\widetilde{\lambda}_{E}$ is injective. ∎

In the following, we consider the case that $d$ is a prime number.

######  Lemma 2.2.

Let $p$ be a prime and $F={\mathbb{Q}}(\sqrt{p})$. Let $\varepsilon$ be the
fundamental unit of $\mathfrak{o}_{F}$ and $\varepsilon=a+b\sqrt{p}\text{ with
}a,b>0$. Then there is a place $\mathfrak{p}$ of $F$, such that the equation
$x^{2}+y^{2}=\varepsilon$ is not solvable over
$\mathfrak{o}_{F_{\mathfrak{p}}}$.

###### Proof.

If $p\equiv 1\mod 4$ or $p=2$, then $x^{2}-py^{2}=-1$ has integral solutions
by Proposition 1.2. Therefore $N_{F/{\mathbb{Q}}}(\varepsilon)=-1$. There
exists a real place $\mathfrak{p}$ of $F$ such that
$|\varepsilon|_{\mathfrak{p}}<0$. So the equation $x^{2}+y^{2}=\varepsilon$ is
not solvable at the real place $\mathfrak{p}$.

If $p\equiv 3\mod 4$, then $x^{2}-py^{2}=-1$ is not solvable over
${\mathbb{Z}}$ by Proposition 1.2. Therefore $N_{F/Q}(\varepsilon)=1$ and
$\varepsilon$ is totally positive. And we know one of the equations
$x^{2}-py^{2}=\pm 2$ has an integral solution in ${\mathbb{Z}}$ by Proposition
1.2. We can choose one solution $(x_{0},y_{0})$ and it is easy to see that
$x_{0}$ and $y_{0}$ are odd. Let

$A=(x_{0}^{2}+py_{0}^{2})/2\text{ and }B=x_{0}y_{0}.$

Since $x_{0},y_{0}$ are odd, we can see $A,B$ are integers and $B$ is odd. And

$A^{2}-pB^{2}=(x_{0}^{2}-py_{0}^{2})^{2}/4=1.$

Let $\varepsilon_{1}=A+B\sqrt{p}$. Obviously $\varepsilon_{1}$ is totally
positive and $\varepsilon_{1}=\varepsilon^{m},m\in{\mathbb{Z}}$.

Let $\mathfrak{p}$ be the unique place of $F$ over $2$. We assume the equation
$x^{2}+y^{2}=\varepsilon$ is solvable over $\mathfrak{o}_{F_{\mathfrak{p}}}$.
Since $\varepsilon_{1}=\varepsilon^{m}$, the equation
$x^{2}+y^{2}=\varepsilon_{1}$ is also solvable over
$\mathfrak{o}_{F_{\mathfrak{p}}}$. For any solution
$(x_{1},y_{1})=(a_{1}+b_{1}\sqrt{p},a_{2}+b_{2}\sqrt{p})$, we have

$(a_{1}+b_{1}\sqrt{p})^{2}+(a_{2}+b_{2}\sqrt{p})^{2}=A+B\sqrt{p}.$

Then

$2a_{1}b_{1}+2a_{2}b_{2}=B.$

However, we know $B$ is odd, a contradiction is derived. ∎

By Proposition 2.1 and Lemma 2.2, we can now prove the following result.

######  Theorem 2.3.

Let $p$ be a prime number and $F={\mathbb{Q}}(\sqrt{p})$. Let $H_{L}$ be the
ring class field corresponding to the order
$L=\mathfrak{o}_{F}+\mathfrak{o}_{F}\sqrt{-1}$. Then the diophantine equation
$x^{2}+y^{2}=\alpha$ is solvable over $\mathfrak{o}_{F}$ if and only if
$\alpha$ satisfies the Artin condition of $H_{L}$.

Now we use Theorem 2.3 to give an explicit example. Let
$F={\mathbb{Q}}(\sqrt{17})$. We write
$N_{F/{\mathbb{Q}}}(\alpha)=2^{s_{1}}17^{s_{2}}p_{1}^{e_{1}}\cdots
p_{g}^{e_{g}}$ for any $\alpha\in\mathfrak{o}_{F}$. Let
$D(n)=\\{p_{1},\cdots,p_{g}\\}$ and $h(x)=x^{4}-2x^{2}+17$. Denote

$\displaystyle D_{1}=\\{p\in D(n):(\frac{-17}{p})=-(\frac{-1}{p})=1\\}$
$\displaystyle D_{2}=\\{p\in D(n):(\frac{-17}{p})=(\frac{-1}{p})=1\text{ and
}h(x)\equiv 0\mod p\text{ is not solvable}\\}.$

We can see $e_{i}$ is even if $p_{i}\in D_{1}$.

######  Example 2.4.

Let $F={\mathbb{Q}}(\sqrt{17})$ and let $\alpha$ be an integer in $F$ with the
above notation. Then $x^{2}+y^{2}=\alpha$ is solvable over $\mathfrak{o}_{F}$
if and only if

(1) The equation $x^{2}+y^{2}=\alpha$ has integral solutions at every place of
$F$.

(2) The sum

$s_{1}+\sum_{p_{i}\in D_{1}}e_{i}/2+\sum_{p_{i}\in D_{2}}e_{i}\equiv 0\mod 2.$

Acknowledgment  The author would like to thank Fei Xu and Chungang Ji for some
helpful discussions. The work is supported by the Morningside Center of
Mathematics and NSFC, grant # 10901150 and # 10671104.

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# IRAM 30 m CO-line Observation toward PeVatron Candidate G106.3+2.7: Direct
Interaction between the Shock and the Molecular Cloud Remains Uncertain

Qian-Cheng Liu1, Ping Zhou1,2, Yang Chen  Department of Astronomy, Nanjing
University, 163 Xianlin Avenue, Nanjing 210023, People’s Republic of China;
<EMAIL_ADDRESS><EMAIL_ADDRESS>Key Laboratory of Modern Astronomy and
Astrophysics, Nanjing University, Ministry of Education, Nanjing 210093,
People’s Republic of China

###### Abstract

Supernova remnant (SNR) G106.3+2.7 was recently found to be one of the few
potential Galactic hadronic PeVatrons. Aiming to test how solid the SNR is
associated with the molecular clouds (MCs) that are thought to be responsible
for hadronic interaction, we performed a new CO observation with the IRAM 30 m
telescope toward its “belly” region, which is coincident with the centroid of
the $\gamma$-ray emission. There is a filament structure in the local-
standard-of-rest velocity interval $-8$ to $-5\,{\rm km}\,{\rm s}^{-1}$ that
nicely follows the northern radio boundary of the SNR. We have seen asymmetric
broad profiles of 12CO lines, with widths of a few $\,{\rm km}\,{\rm s}^{-1}$,
along the northern boundary and in the “belly” region of G106.3+2.7, but
similar 12CO line profiles are also found outside the SNR boundary. Further,
the low 12CO $J$= 2–1/$J$= 1–0 line ratios suggest the MCs are cool.
Therefore, it is still uncertain whether the MCs are directly disturbed by the
SNR shocks, but we do find some clues that the MCs are nearby and thus can
still be illuminated by the escaped protons from the SNR. Notably, we find an
expanding molecular structure with a velocity of $\sim 3.5\,{\rm km}\,{\rm
s}^{-1}$ and a velocity gradient of the MCs across the SNR from $\sim-3$ to
$-7\,{\rm km}\,{\rm s}^{-1}$, which could be explained as the effect of the
wind blown by the SNR’s progenitor star.

ISM: individual objects (G106.3+2.7) – ISM: supernova remnants – ISM:
molecular clouds

##  1 Introduction

Supernova remnants (SNRs) are suggested to be one of the major factories to
produce cosmic rays (CRs) up to $\sim 10^{15}$ eV. High-energy GeV/TeV
$\gamma$-ray photons can be produced via either the interaction between the CR
protons and the gas material (hadronic origin) or inverse-Compton scattering
of relativistic electrons (leptonic origin). The former scenario requires a
dense environmental medium, such as molecular clouds (MCs), while the latter
origin is not subject to the nearby density. Therefore, an investigation of
the molecular environment of an $\gamma$-ray-emitting SNR is vital to
distinguish the origin of the high-energy emission.

Figure 1: 12CO $J$= 1–0 intensity map integrated in velocity range $-10$ to
$-2\,{\rm km}\,{\rm s}^{-1}$, overlaid by 1.4 GHz radio contours from the CGPS
(Taylor et al., 2003) in levels 6.9, 7.3, and 8 K. The magenta circle and
black ellipse represent the gamma-ray emission detected by the Tibet
AS$\gamma$ (Tibet AS$\gamma$ Collaboration, 2021) and VERITAS (Acciari et al.,
2009), respectively. The red and orange crosses show the best-fit positions of
the LHAASO (Cao et al., 2021) and HAWC (Albert et al., 2020) sources. The
green arcs crudely separate the “head”, “belly”, and “tail” regions (referring
to the upper, middle, and lower parts, respectively) of the SNR described in
this paper. The upper arc is adopted from Ge et al. (2021), and the lower arc
follows the southwestern edge of the field of view of our IRAM observation.
Figure 2:  12CO $J$= 1–0 intensity map integrated at $1\,{\rm km}\,{\rm
s}^{-1}$ in the velocity range $-13$ to $-2\,{\rm km}\,{\rm s}^{-1}$, overlaid
by radio contours the same as those in Figure 1. The region enclosed with the
green polygon in the top middle panel delineates the region we observed with
the IRAM 30 m telescope, and the data within this region for all the panels
are from IRAM 30 m telescope. The data outside this region are from the
archived PMOD. The data from the IRAM 30 m telescope have been resampled and
re-grid to match the velocity resolution and pixel size of the PMOD data, and
the data from PMOD has been multiplied by a factor of 2 for better visibility.
The top left panel shows the averaged CO spectra of the polygon region in the
velocity range $-30$ to $20\,{\rm km}\,{\rm s}^{-1}$ from the IRAM
observation. The black line is for 12CO $J$= 1–0, the red line is for 12CO
$J$= 2–1, the blue line is for 13CO $J$= 1–0, and the dashed line is for the 0
K main-beam temperature. The 13CO $J$= 1–0 line has been multiplied by a
factor of three for better visibility. The lines used to make the position-
velocity diagrams in Figure 3 are indicated in the bottom right panel in dark
blue.

SNR G106.3+2.7 has attracted substantial attention recently, as it was found
to be one of the few potential Galactic hadronic PeVatrons (Albert et al.,
2020; Tibet AS$\gamma$ Collaboration, 2021; Cao et al., 2021). Extended
$\gamma$-ray emission has been detected with Fermi-LAT (Xin et al., 2019),
Milagro (Abdo et al., 2007), VERITAS (Acciari et al., 2009), and the Tibet
AS$\gamma$ experiment (Tibet AS$\gamma$ Collaboration, 2021) from G106.3+2.7
(see Figure 1). Recently, nonthermal X-ray emission was also discovered based
on the analysis of _Chandra_ , _XMM-Newton_ and _Suzaku_ data (Ge et al.,
2021; Fujita et al., 2021). There is a bright X-ray point source toward the
radio peak of G106.3+2.7 (Halpern et al., 2001b), and it was later confirmed
to be a pulsar in the radio, X-rays (Halpern et al., 2001a), and the
$\gamma$-ray (Abdo et al., 2009) bands. The existence of a pulsar establishes
the core-collapse origin of G106.3+2.7.

G106.3+2.7 has a comet-shaped morphology in the radio band, with a bright head
in the northeast and a tail toward the southwest (Pineault & Joncas, 2000). It
was suggested to be located at the edge of a large HI bubble and associated
with the MCs at a systemic local-standard-of-rest (LSR) velocity of about
$-6\,{\rm km}\,{\rm s}^{-1}$ based mainly on the spatial correspondence
(Kothes et al., 2001). This HI bubble was suggested to be created by stellar
winds and SNe in the previous generation. Furthermore, the head of the SNR is
found to be located in a small HI bubble (Kothes et al., 2001). Spatial
coincidence has also been used to indicate a possible association between the
$\gamma$-ray emission, the SNR, and the MCs (e.g., Acciari et al., 2009; Xin
et al., 2019; Tibet AS$\gamma$ Collaboration, 2021; Bao & Chen, 2021).
However, the nature of the $\gamma$-ray emission (hadronic or leptonic origin)
is still under debate (e.g., Acciari et al., 2009; Xin et al., 2019; Albert et
al., 2020; Liu et al., 2020b), and there is little dedicated investigation
that aims at kinematically clarifying if the MCs are directly disturbed by the
SNR shock (e.g., asymmetric broad-line profiles of 12CO lines, high 12CO $J$=
2–1/$J$= 1–0 line ratios, etc.; Jiang et al. 2010; Chen et al. 2014). A new,
independent CO observation with higher spatial and velocity resolution is
therefore needed, which helps explore the astrophysical environment toward the
direction of G106.3+2.7.

Here, we present a new CO-line observation using the IRAM 30 m telescope
toward G106.3+2.7 and search for kinematic and physical signatures of the SNR-
MC interaction. The observation and the data reduction process are described
in Section 2, and the results are presented in Section 3; the discussion of
the main results and the summary are presented in Section 4 and 5,
respectively.

Figure 3: Position-velocity diagrams of 12CO $J$= 1–0 emission for two lines
that go from R.A.=22h28m07s.58,
Decl.=$+60$∘33${}^{\prime}16.92^{\prime\prime}$ to R.A.=22h23m54s.65,
Decl.=$+61$∘12${}^{\prime}39.6^{\prime\prime}$ with a $52.4^{\circ}$
inclination (line1, the top panel), and from R.A.= 22h29m31s.56,
Decl.=$+$60∘40${}^{\prime}46.92^{\prime\prime}$ to R.A.=22h25m45s.84,
Decl.=$+61$∘09${}^{\prime}54.00^{\prime\prime}$ with a $46.7^{\circ}$
inclination (line2, the bottom panel), respectively (see the bottom right
panel of Figure 2). The red lines in each diagram delineate the edge of
G106.3+2.7 (see the red small rectangles in the bottom right panel of Figure
2), and the ellipse in the top panel indicates the expanding molecular
structure in the position-velocity diagram.

##  2 Observation and data reduction

Our observations of molecular lines toward G106.3+2.7 was made simultaneously
in 12CO $J$= 1–0, 12CO $J$= 2–1, and 13CO $J$= 1–0 with the IRAM 30 m
telescope during 2020 December, 22 – 30, for a total of 40 hr. The
observations covered an irregular polygon region toward the “belly” region of
G106.3+2.7 with an area of about 680 arcmin2 (see the top middle panel of
Figure 2). The mapping was conducted with the on-the-fly position-switching
mode, using the Eight Mixer Receiver in E0 and E1 band and the fast Fourier
transform spectrometers (FTS). The backend FTS provided a bandwidth of 16 GHz
and a spectral resolution of 200 kHz. The velocity resolution is thus $\sim
0.5\,{\rm km}\,{\rm s}^{-1}$ for 12CO $J$= 1–0 and 13CO $J$= 1–0, and $\sim
0.25\,{\rm km}\,{\rm s}^{-1}$ for 12CO $J$= 2–1. The half-power beamwidth
(HPBW) of the telescope is about $21^{\prime\prime}$ at 115 GHz and about
$10^{\prime\prime}.7$ at 230 GHz, and the main-beam efficiency is about 0.78
at 115 GHz and about 0.59 at 230 GHz. To better study the three CO
transitions, we have convolved the angular resolution to $22^{\prime\prime}$
and resampled the velocity resolution to $0.5\,{\rm km}\,{\rm s}^{-1}$ for all
line data. The rms noise levels of the main-beam temperature ($T_{\rm mb}$) of
the convolved and resampled data are $\sim 0.07$ – 0.2 K for 13CO $J$= 1–0
lines, $\sim 0.2$ – 0.8 K for 12CO $J$= 1–0 lines, and $\sim 0.06$ – 0.2 K for
12CO $J$= 2–1 lines. All the IRAM 30 m data were reduced with the GILDAS/CLASS
package developed by the IRAM observatory 111http://www.iram.fr/IRAMFR/GILDAS.

To study the larger-scale environment of the SNR, we have used the archived
12CO $J$= 1–0 data obtained with the 13.7 m millimeter-wavelength telescope of
the Purple Mountain Observatory at Delingha (PMOD). The HPBW of the data is
about $50^{\prime\prime}$, and the velocity resolution of the data is about
$0.16\,{\rm km}\,{\rm s}^{-1}$. The rms noise level of the data varies in a
range of $\sim 0.1$ – 0.2 K. We also used 1.4 GHz radio continuum emission
data from the Canadian Galactic Plane Survey (CGPS; Taylor et al., 2003).

Figure 4:  Averaged spectra of regions “1” to “8” in the velocity range $-20$
to $5\,{\rm km}\,{\rm s}^{-1}$. The eight regions are defined in Figure B.4.
The black lines denote the 12CO $J$= 1–0 spectra, the red lines denote the
12CO $J$= 2–1 spectra, the blue lines denote the 13CO $J$= 1–0 spectra, and
the dashed lines represent the 0 K main-beam temperature. The 13CO $J$= 1–0
spectra have been multiplied by a factor of three for better visibility.

##  3 Results

###  3.1 Spatial Distribution of the Molecular Clouds

The top left panel of Figure 2 shows the 12CO $J$= 1–0, 12CO $J$= 2–1, and
13CO $J$= 1–0 spectra averaged from the polygon region that was observed with
the IRAM 30 m telescope. There are two prominent CO emission peaks, at around
$-11\,{\rm km}\,{\rm s}^{-1}$ and $-4.5\,{\rm km}\,{\rm s}^{-1}$. To examine
the spatial distribution of these MCs, we have made 12CO $J$= 1–0 emission
channel maps in the velocity range $-13\,{\rm km}\,{\rm s}^{-1}$ to $-2\,{\rm
km}\,{\rm s}^{-1}$ using the highest spatial resolution CO data to date
(Figure 2, where we have inserted the data from the IRAM 30 m telescope into
the data from the PMOD). The maps cover the main components of the two
emission peaks.

There are a few noticeable structures in this velocity range. (1) In velocity
interval $-8$ to $-5\,{\rm km}\,{\rm s}^{-1}$, except for the envelope around
the head region of the SNR as noticed by Kothes et al. (2001), there is a
filament-like structure nicely aligning the northwestern edge of the SNR, with
an orientation from the northeast to the southwest. (2) In velocity interval
$-6$ to $-3\,{\rm km}\,{\rm s}^{-1}$, there is a ring-like structure and some
diffuse 12CO emission toward the “belly” region (see Figure 1) of the SNR.
Notably, the spatial distribution of the diffuse 12CO emission happens to
overlap with the $\gamma$-ray emissions detected by Fermi-LAT (Xin et al.,
2019) and the Tibet AS$\gamma$ experiment (Tibet AS$\gamma$ Collaboration,
2021), while the head/PWN part of the SNR seems faint in 12CO line. Actually,
the LSR velocity of the MCs toward the “belly” region is very similar to that
of environmental HI gas (Kothes et al., 2001).

###  3.2 Position-Velocity Diagram of the Molecular Clouds

We note some molecular structures with unusual velocity features toward the
“belly” region of the SNR. Figure 3 shows the position-velocity (P-V) diagrams
of slices taken along the two lines, labeled in the bottom right panel of
Figure 2. The P-V diagram along line 1 shows an expanding molecular structure,
with an angular extent larger than that of the SNR. This expanding structure
is centered at $\sim-7.5\,{\rm km}\,{\rm s}^{-1}$ with an expansion velocity
of $\sim 3.5\,{\rm km}\,{\rm s}^{-1}$ and an angular size of $\sim
26^{\prime}$. Also, we find a velocity gradient that changes from
$\sim-3\,{\rm km}\,{\rm s}^{-1}$ at angular offset $10^{\prime}$ to
$\sim-7\,{\rm km}\,{\rm s}^{-1}$ at angular offset $30^{\prime}$ along line 2.
The measured velocity variation is about $4\,{\rm km}\,{\rm s}^{-1}$, which is
similar to the expansion velocity of the expanding structure. Because there
lacks conclusive kinematic evidence of disturbation of the MCs by the shock of
the SNR (see § 3.3 and 3.4), the P-V structures could be a result of other
processes, such as stellar wind(s), instead of the shock of SNR G106.3+2.7.

Figure 5:  Ratio maps of IRAM 12CO $J$= 2–1/$J$= 1–0 for LSR velocity ranges
$-15$ to $-9\,{\rm km}\,{\rm s}^{-1}$ (left panel) and $-9$ to $-1\,{\rm
km}\,{\rm s}^{-1}$. The pixels toward which the value are smaller than
$5\sigma$ for either 12CO $J$= 2–1 or 12CO $J$= 1–0 have been omitted. The
regions delineated by white rectangles are used to extract CO spectra (Figure
4), and the region delineated by the cyan rectangle is used to estimate the
properties of the MC toward the “belly” region of G106.3+2.7 in velocity range
$-9$ to $-1\,{\rm km}\,{\rm s}^{-1}$ (Section 4). The radio contours are the
same as those in Figure 1.

###  3.3 Molecular Line Profiles

The physical properties of an MC can be strongly affected if it is shocked by
the SNR blast wave. Observationally, the 12CO line profiles can be one/two-
sided broadened if the MC is located at the edge of or within the SNR, and
such kinematic signatures are present in many SNR-MC interacting systems like
IC 443 (e.g., White et al., 1987), W28 (e.g., Arikawa et al., 1999), Kes 75
(Su et al., 2009), 3C 397 (Jiang et al., 2010), Kes 78 (Zhou & Chen, 2011),
G357.7+0.3 (Rho et al., 2017), and CTB 87 (Liu et al., 2018).

To examine whether SNR G106.3+2.7 is interacting with the MCs, we have
inspected the CO line profiles toward the region observed by the IRAM 30 m
telescope (see Appendix A). Figure 4 shows the averaged emission line profiles
delineated by the white rectangles in Figure B.4, where regions 1 to 3 are at
the edge of the SNR, region 4 is outside the SNR boundary, and regions 5 to 8
are in the “belly” region, projectively. There are separated 12CO components
at $\sim-11$ and $\sim-6\,{\rm km}\,{\rm s}^{-1}$ in both regions 1 and 2. The
12CO components at $\sim-6\,{\rm km}\,{\rm s}^{-1}$ are weaker and show weaker
13CO line emission, which is either the red-wing broadened part of the main
components at $\sim-11\,{\rm km}\,{\rm s}^{-1}$ or a separated, irrelevant
velocity component. The 12CO line profiles in regions 3 to 6 are similar to
those in regions 1 and 2, except that the main components are at $\sim-8\,{\rm
km}\,{\rm s}^{-1}$ in regions 3 to 5 and at $\sim-6\,{\rm km}\,{\rm s}^{-1}$
in region 6. Further, there is a weak peak at $\sim-11\,{\rm km}\,{\rm
s}^{-1}$ in region 5. There are main 12CO components at $\sim-5\,{\rm
km}\,{\rm s}^{-1}$ in both regions 7 and 8, while there seem to be blue-wing
broadened parts at $\sim-11$ to $-7\,{\rm km}\,{\rm s}^{-1}$. We note that the
profiles of the 12CO lines are systematically asymmetric in the “belly” region
of the SNR (regions 1–3 and 5–8). However, there are similar 12CO line
profiles in the region outside the SNR boundary (region 4). Furthermore, the
widths of the asymmetric wings are only a few $\,{\rm km}\,{\rm s}^{-1}$,
indicating that the disturbance in the molecular gas is not strong. Therefore,
it is uncertain whether the broadened profiles in the region projected inside
the SNR are the results of interaction between the SNR shock and the MC.
Alternatively, they could be a signature of other perturbed processes, such as
perturbation by the stellar wind(s).

###  3.4 The 12CO $J$= 2–1/$J$= 1–0 Line Ratio

The 12CO $J$= 2–1/$J$= 1–0 line ratios ($R_{21/10}$) at the CO line wings
could be enhanced ($\gtrsim 1$) if the molecular gas is disturbed and heated
by the SNR shock (e.g., Seta et al., 1998; Jiang et al., 2010; Sano et al.,
2019, 2020). By inspecting the spectral properties of the CO emissions, we
have not found enhanced 12CO $J$= 2–1/$J$= 1–0 line ratios at the CO line
wings toward the region observed with IRAM 30 m telescope (see Figures 4, A.1,
A.2, and A.3).

Figure 5 shows the spatial distribution of the ratio $R_{21/10}$ of the
molecular gas in velocity intervals $-15$ to $-9\,{\rm km}\,{\rm s}^{-1}$ and
$-9$ to $-1\,{\rm km}\,{\rm s}^{-1}$. We do not show the pixels where the
values are smaller than $5\sigma$ for either 12CO $J$= 2–1 or 12CO $J$= 1–0
emission. Although the 12CO $J$= 2–1/$J$= 1–0 line ratios in some pixels just
outside the northern edge of the SNR could reach unity in velocity interval
$-15$ to $-9\,{\rm km}\,{\rm s}^{-1}$, they mainly arise from the CO line
centers instead of the line wings (see Figure A.1). This means that this MC
with high $R_{21/10}$ has a relatively higher kinetic temperature (see
Appendix B) but is not shocked and is a quiescent MC outside the SNR. On the
other hand, the line ratios in all other pixels are clearly smaller than
unity.

##  4 Discussion

SNR G106.3+2.7 has been found to be a $\gamma$-ray emitting source in the band
from GeV to sub-PeV (e.g., Xin et al., 2019; Albert et al., 2020; Tibet
AS$\gamma$ Collaboration, 2021; Cao et al., 2021). The main part of the
$\gamma$-ray emission appears to be spatially correspondent with the CO
emission in LSR velocity interval $-6$ to $-3\,{\rm km}\,{\rm s}^{-1}$
projected in the “belly” region of the SNR (e.g., Xin et al., 2019; Tibet
AS$\gamma$ Collaboration, 2021). By parameterizing the distance to the MC as
$d_{0.8}=d/$(0.8 kpc) (Kothes et al., 2001), the mass of the molecular gas in
velocity range $-9$ to $-1\,{\rm km}\,{\rm s}^{-1}$ within the cyan rectangle
region delineated in Figure 5 is estimated to be about $330d_{0.8}M_{\odot}$
and $860d_{0.8}M_{\odot}$ from 13CO and 12CO emissions 222The mass of the MC
estimated using 12CO here could have large uncertainty, as the X-factor method
is more reliable for MC with a mass of order larger than $10^{4}M_{\odot}$
(Bolatto et al., 2013). respectively (see Appendix C). Assuming the length of
the cloud along the line of sight (LOS) is comparable to its projected size
($\sim 4.8\,{\rm pc}$), the molecular density of the cloud is estimated to be
about $30\,{\rm cm}^{-3}$ and $80\,{\rm cm}^{-3}$, respectively. Our
observation also shows that a filament follows the northern radio boundary of
the SNR in velocity interval $-8$ to $-5\,{\rm km}\,{\rm s}^{-1}$.
Asymmetrical 12CO line profiles are found in some locations along the northern
boundary and in the “belly” region of G106.3+2.7, but we could not conclude
that they represent the kinematic evidence of shock-MC interaction since we
also see similar 12CO line profiles outside the SNR (Figure 4). Physical
signature of interaction (such as elevated gas temperature or 12CO $J$=
2–1/$J$= 1–0 line ratio) has not been found either along the filament or in
the MC toward the “belly” of the SNR (Figures B.4 and 5). Nonetheless, we can
not exclude the possible proximity of the MC to the SNR along the LOS. In such
a scenario, the accelerated protons that escape from the SNR shock can
illuminate the MC via p-p hadronic interaction.

Except for SNR(s), other possible Galactic CR accelerators, such as cluster(s)
with massive stars (e.g., Aharonian et al., 2019; Morlino et al., 2021), are
also worth considering. We note that there are structural properties in the
MCs toward the gamma-ray emitting region in the velocity range of about $-11$
to $-3\,{\rm km}\,{\rm s}^{-1}$ (see § 3.1 and 3.2). Especially, an expanding
molecular structure and a velocity gradient across the “belly” region of
G106.3+2.7 in the P-V diagram (§ 3.2) might be the results of massive stellar
winds. The radius $R$ and the expansion velocity $v$ of the expanding
sturcture are about $12^{\prime}$ (corresponding to $\sim 2.9d_{0.8}$ pc) and
$3.5\,{\rm km}\,{\rm s}^{-1}$, respectively (§ 3.2). If it is blown by the
interior stellar wind(s), its age could be estimated to be (Weaver et al.,
1977) $t=3R/5v=5\times 10^{5}d_{0.8}(v/3.5\,{\rm km}\,{\rm s}^{-1})^{-1}\,{\rm
yr}$. Furthermore, the kinetic luminosity of the stellar wind(s) could be
(Weaver et al., 1977): $L_{w}\sim 1.2\times 10^{34}(n/100\,{\rm
cm}^{-3})(v/3.5\,{\rm km}\,{\rm s}^{-1})^{3}d_{0.8}^{2}\,{\rm erg}\,{\rm
s}^{-1}$, where $n$ is the atomic hydrogen number density of the molecular
gas. The estimated luminosity of the stellar wind(s) is typical for an O9V to
B0V type star (e.g., Elson et al., 1989; Chen et al., 2013). We have searched
for massive star(s) using the _Tycho_ -2 Spectral Type Catalog (Wright et al.,
2003) toward this region, yet no OB star cluster/association or massive main
sequence star has been found, and it is unlikely to be a result of
incompleteness of the catalog we use (see Appendix D). Therefore, we suggest
that the expanding structure and the velocity gradient of the molecular gas
could be due to the progenitor wind of G106.3+2.7. We note that similar
structures due to progenitor winds have also been found in a few other SNRs,
such as Tycho (Zhou et al., 2016; Chen et al., 2017), N132D (Sano et al.,
2020), and VRO 42.05.01 (Arias et al., 2019).

##  5 Summary

We have investigated the molecular environment of G106.3+2.7 using the CO-line
data obtained with the IRAM 30 m telescope toward the “belly” region of the
SNR, which is coincident with the centroid region of the $\gamma$-ray
emission. There is a filament that follows the north radio boundary of the SNR
in velocity interval $-8$ to $-5\,{\rm km}\,{\rm s}^{-1}$. Some asymmetric
broad profiles of 12CO lines are found not only toward the “belly” region of
the SNR but also outside the boundary. Further, the 12CO $J$= 2–1/$J$= 1–0
line ratios of the MCs are similar to the typical value for interstellar MCs.
Therefore, it is uncertain about whether the MCs are disturbed by the SNR
shock. Nonetheless, we can not exclude the possible proximity of the MC to the
SNR along the LOS, which facilitates illumination of the MC by the accelerated
protons that escape from the SNR shock via p-p hadronic interaction. Notably,
we find an expanding molecular structure and a velocity gradient toward the
“belly” region of the SNR, and we propose a possibility of their relations to
the progenitor wind of the SNR.

We are grateful to Xiao Zhang for the discussion about the gamma-ray emission
toward this SNR, Xin Zhou for the discussion about the methods to estimate the
properties of the MCs, Niu Liu and Yong Shao for a discussion about how to
search for OB stars in the field of view. We thank the staff of the IRAM 30 m
observatory for the help during the remote observation. This work is supported
by the National Key R&D Program of China under grants 2017YFA0402600, and the
NSFC under grants 11773014, 11633007, 11503008, and 11590781. Q.C.L.
acknowledges support from the program A for Outstanding PhD candidate of
Nanjing University.

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##  Appendix A Grid of the CO spectra observed by the IRAM 30 m telescope

We have inspected the CO line profiles toward the SNR G106.3+2.7 region
observed by the IRAM 30 m telescope. Because the pixel sizes of the original
datacubes are too small ($11^{\prime\prime}$), we have re-gridded the
datacubes to reach pixel sizes of $66^{\prime\prime}$ before making the grids
of 12CO $J$= 1–0, 12CO $J$= 2–1, and 13CO $J$= 1–0 spectra. Figures A.1, A.2,
and A.3 show the grids of CO spectra in velocity range $-15$ to $+5\,{\rm
km}\,{\rm s}^{-1}$ for the regions delineated by the white rectangles labeled
with “N”, “S”, and “W” in the right panel of Figure 5, respectively. There are
many pixels, either within, on, or outside the boundary of the SNR G106.3+2.7,
where the profiles of 12CO spectra seemingly to be asymmetric broadened. The
averaged CO line profiles of some of the pixels (regions “1” to “8” marked in
Figures B.4 and 5) are studied in detail in Section 3.3.

Figure A.1:  Grid of 12CO $J$= 1–0, 12CO $J$= 2–1, and 13CO $J$= 1–0 spectra
observed by the IRAM 30 m telescope in velocity range $-15$ to $+5\,{\rm
km}\,{\rm s}^{-1}$ for the region delineated by the white rectangle labeled
with “N” in the right panel of Figure B.4. The black lines denote the 12CO
$J$= 1–0 spectra, the red lines denote the 12CO $J$= 2–1 spectra, and the blue
lines denote the 13CO $J$= 1–0 spectra. The 13CO $J$= 1–0 spectra have been
multiplied by a factor of two for better visibility. The size of each pixel is
$66^{\prime\prime}\times 66^{\prime\prime}$. The contours are the same as
those in Figure 1.  Figure A.2:  The same as Figure A.1, but for the region
delineated by the white rectangle labeled with “S” in the right panel of
Figure B.4. The 13CO $J$= 1–0 spectra also have been multiplied by a factor of
two for better visibility.  Figure A.3:  The same as Figure A.1, but for the
region delineated by the white rectangle labeled with “W” in the right panel
of Figure B.4. The 13CO $J$= 1–0 spectra also have been multiplied by a factor
of two for better visibility.

##  Appendix B Temperature Distribution of the Molecular Clouds

The kinetic temperature of the molecular gas ($T_{\rm k}$), which is
approximately equal to the excitation temperature ($T_{\rm ex}$) under the
assumption of local thermodynamic equilibrium, can be estimated from the peak
of the main-beam temperature $T_{\rm mb}$ of the 12CO $J$= 1–0 line emission
with equation $T_{\rm k}=5.53/\\{\ln[1+5.53/(T_{\rm mb}+0.84)]\\}\,{\rm K}$,
where the beam filling factor is assumed to be equal to 1. Figure B.4 shows
the spatial distribution of the kinetic temperature of the molecular gas in
velocity intervals $-15$ to $-9\,{\rm km}\,{\rm s}^{-1}$ and $-9$ to $-1\,{\rm
km}\,{\rm s}^{-1}$, corresponding to the velocity ranges of the two components
shown in the top left panel of Figure 2. The kinetic temperature in each pixel
is estimated from the peak of the main-beam temperature in the given velocity
range. The kinetic temperatures are $\lesssim 10$ K for the most part of the
molecular gas in both velocity ranges, except for the clump at the northern
corner in velocity interval $-15$ to $-9\,{\rm km}\,{\rm s}^{-1}$. That is,
the distribution of the kinetic temperature is similar to that of the 12CO
$J$= 2–1/$J$= 1–0 line ratios. Actually, the higher line ratio toward the
northern corner in this velocity interval could be directly related to the
higher kinetic temperature there (see, e.g., Liu et al. 2020a).

Figure B.4:  Kinetic temperature ($T_{\rm k}$) maps of the MCs in velocity
intervals $-15$ to $-9\,{\rm km}\,{\rm s}^{-1}$ (left panel) and $-9$ to
$-1\,{\rm km}\,{\rm s}^{-1}$ (right panel) obtained from the IRAM 30 m
telescope observation, with the assumptions of no beam dilution ($f=1$) and
$\tau_{{}^{12}CO}\gg 1$. The contours are the same as in Figure 1. The three
regions delineated by large white rectangles and labeled with “N”, “S”, and
“W” are used to extract the CO grid spectra (Figure A.1, A.2, and A.3). The
regions delineated by small white rectangles are the same as those in Figure
5.

##  Appendix C The Mass of the Molecular Cloud toward the “belly” of
G106.3+2.7

The mass of the molecular gas in velocity range $-9$ to $-1\,{\rm km}\,{\rm
s}^{-1}$ toward the “belly” of G106.3+2.7 could be roughly estimated from the
column density of the MC in the given region. The angular size of the region
delineated by the cyan rectangle in the right panel of Figure 5 is $\sim
18^{\prime}.5\times 22^{\prime}.5$, which can be translated to a physical size
of $\sim 4.3d_{0.8}\,{\rm pc}\times 5.2d_{0.8}\,{\rm pc}$. The column density
of H2 within this region is estimated from either 13CO $J$= 1–0 or 12CO $J$=
1–0 line. When assuming the molecular gas to be in local thermodynamic
equilibrium and the 12CO $J$= 1–0 line to be optically thick, the column
density could be estimated using $N({\rm H}_{2})\approx 7\times
10^{5}N(^{13}{\rm CO})$ (Frerking et al., 1982). Here $N(^{13}{\rm
CO})=2.42\times 10^{14}\int T_{\rm mb}(^{13}{\rm CO})dv/(1-{\rm
exp}(-5.3/T_{\rm ex}))\,{\rm cm}^{-2}$ is the column density of 13CO. On the
other hand, it could also be estimated using the CO-to-H2 mass conversion
factor ($N({\rm H}_{2})/W(^{12}{\rm CO})$; the “X-factor”), which is assumed
to be $1.8\times 10^{20}\,{\rm cm}^{-2}{\rm K}^{-1}\,{\rm km}^{-1}\,{\rm s}$
(Dame et al., 2001). The mass of this MC is thus estimated to be about
$330d_{0.8}M_{\odot}$ and $860d_{0.8}M_{\odot}$ from 13CO and 12CO lines,
respectively.

##  Appendix D The Massive Stars Toward G106.3+2.7

We use the _Tycho_ -2 Spectral Type Catalog (Wright et al., 2003), which is
about 90% complete at an apparent visual magnitude of $\sim 11.5$ (Høg et al.,
2000), to search for massive stars that can potentially produce wind-blown
structure (O–B3V star; e.g., Chen et al. 2013) toward G106.3+2.7 in a circular
region centered at R.A.=22h27m12s.58,
decl.=+60∘46${}^{\prime}19.9^{\prime\prime}$, $30^{\prime}$ in radius.
However, no massive star has been found toward G106.3+2.7, which is unlikely
to be a result of optical extinction. The optical extinction can be estimated
from the empirical relation $N_{\rm H}=(2.87\pm 0.12)\times 10^{21}A_{V}\,{\rm
cm}^{-2}$ (Foight et al., 2016), where $N_{\rm H}=2N({\rm H}_{2})+N_{\rm HI}$
($\sim 9.5\times 10^{21}\,{\rm cm}^{-2}$, Ge et al. 2021) is the foreground
hydrogen (including atoms and molecules) column density. Therefore, the
optical extinction can be estimated to be $A_{V}=3.31\pm 0.15$. According to
the relation $V=M_{V}-5+5\log D+A_{V}$, where $V$ is the apparent visual
magnitude, $M_{V}$ is the absolute magnitude ($\leq-1.68$ for stars earlier
than B3V type; e.g., Wegner 2006), $D$ is the distance to the star in units of
pc, the apparent visual magnitude of a massive star at the distance of
G106.3+2.7 (about 800 pc; Kothes et al. 2001) $\leq 11.1$, which should have
been smaller than 11.5.