\begin{thm}\label{thm:main} Let $F$ be a totally real field,
and $F'$ be a real quadratic extension of $F$. Assume that $p$ is
unramified in both $F$ and $F'$. Let $f$ be a Hilbert newform of
parallel even weight $k$ over $F$ with trivial central character and
level $\mathfrak{n}_f$. Let $f'$ be the base change of $f$ to $F'$.
Let $\mathfrak{n}_{f'}$ be the level of $f'$.

Let $p$ be a prime number satisfying $p\geq\mathrm{max}(k+2,7)$. We
assume that $\mathfrak{n}_f$ is prime to $p$ and $f$ is ordinary at
each prime above $p$. When $[F:\mathbb{Q}]$ is odd, we assume that
there exists at least one prime $\mathfrak{q}$ such that
$\mathfrak{q}|| \mathfrak{n}_f$, and $\mathfrak{q}$ is split in
$F'$.

Suppose the following conditions hold.
\begin{enumerate}
\item The restrictions of $\bar{\rho}_f$ to $G_{F'(\xi_p)}$ and $G_{F'(
\sqrt{ p^*})}$ are absolutely irreducible, where
$p^*=(-1)^{\frac{p-1}{2}}p$.
\item In the case of $k=2$, for each place $v$ of $F$ above $p$ we
have  $a_v^2(f)\ {\backslash\hskip -10pt \equiv } 1 \ (\mathrm{mod}
\ p)$; similarly, for each place $v'$ of $F'$ above $p$ we have
$a_{v'}^2(f')\ {\backslash\hskip -10pt \equiv } 1 \ (\mathrm{mod} \
p)$.
\item For each $\mathfrak{l}|\mathfrak{n}$, if
$\bar{\rho}_f|_{I_{F_\mathfrak{l}}}$ is absolutely irreducible, then
$\mathrm{N}(\mathfrak{l})\equiv  {\hskip -10pt /}-1 \ (\mathrm{mod
}\ p)$. Similarly, for each $\mathfrak{l}'|\mathfrak{n}'$, if
$\bar{\rho}_{f'}|_{I_{F'_{\mathfrak{l}'}}}$ is absolutely
irreducible, then $\mathrm{N}(\mathfrak{l}')\equiv  {\hskip -10pt
/}-1 \ (\mathrm{mod }\ p)$.
\item $\rho_f$ is a minimal modular lifting of $\bar{\rho}_f$.
\end{enumerate} Then $$
\Omega_{f'}^\mathrm{can}=(\Omega_f^\mathrm{can})^2 $$ up to a
$p$-adic unit.
\end{thm}

\subsection{Special value formula, Gross period and Hida canonical period}

In \cite{Gro87} Gross provided a formula expressing the special
values of $L$-functions via Heegner points. To describe his result,
let $f$ be a newform on $\Gamma_0(N)$  of even weight $k$, and let $K$ be an imaginary
quadratic field of discriminant $D$ such that $D$ is prime to $N$.

Assume $k=2$ and $N$ is a prime inert in $K$. Let $B$ be the definite quaternion algebra that is ramified exactly at $N$. Fix a prime number $p$. Let $H_n$ be the ring class field of $K$ of conductor $p^n$. When $\chi$ is a character of the Galois group $\mathrm{Gal}(H_n/K)$, one forms the Rankin-Selberg $L$-function $L(f,\chi,s)$.
Gross \cite{Gro87} showed that there exists a period $\Omega_{f,K}$
depending on $f$ and $K$ but independent of $n$ and $\chi$, called Gross
period, such that $\frac{L(f,\chi,1)}{\Omega_{f,K}}$ are algebraic
and they satisfy
$$\frac{L(f,\chi,1)}{\Omega_{f,K}}\cdot\sqrt{D}p^{n}=|\sum\limits_{\sigma\in\mathrm{Gal}(H_n/K)}\chi(\sigma)\psi(P^{\sigma})|^{2},$$
where $P$ is the Heegner point on the Shimura set attached to $B$.

Gross formula is generalized by Shouwu Zhang \cite[Theorem 7.1]{Zha04} for $k=2$ and general $N$, and by Haiping Yuan \cite{Yua05} for $k>2$.

In \cite{BD07}  Bertolini and Darmon used Gross' special value
formula to construct anticyclotomic $p$-adic $L$-functions, i.e.
$p$-adic $L$-functions for anticyclotomic $\BZ_p$-extension instead
of the cyclotomic $\BZ_p$-extension. In the same paper they proved
one divisibility for Iwasawa main conjecture for elliptic curves in
the setting of anticyclotomic $\BZ_p$-extension (\cite[Theorem
1]{BD07}), and also gave new evidence for Birch and Swinnerton-Dyer
conjecture (\cite[Corollary 4]{BD07}).

Gross period serves as a bridge between the complex $L$-function and
the $p$-adic $L$-function. But it depends on the choice of an imaginary
quadratic extension $K$.

Hida \cite{Hid81, Hid81.2} introduced a period that is independent
of $K$ and called the {\it canonical period}. Attached to $f$ there
exists a homomorphism $\lambda_f:T\longrightarrow \mathcal{O}_f$
from the Hecke algebra $T$ of level $N$ to a discrete valuation ring
$\mathcal{O}_f$ that is finite over $\BZ_p$. Let $\eta $ be the
congruence number for $\lambda_f$ defined by
$$\eta =\lambda_f(\mathrm{Ann}(\mathrm{Ker}(\lambda_f))).$$ Then the
canonical period is defined by
$$\Omega_{f}^{\mathrm{can}}=\frac{\langle f, f
\rangle_{\mathrm{Pet}}}{\eta }.$$ The reader may consult
\cite[Section 2]{Vat03} about more knowledge on Gross period and the
canonical period.

Vatsal \cite{Vat03} showed that
$\frac{\Omega_{f,K}}{\Omega_{f}^{\mathrm{can}}}$ lies in
$\mathcal{O}_f$. Pollack and Weston \cite{PW11} pointed out that
$\frac{\Omega_{f,K}}{\Omega_{f}^{\mathrm{can}}}\in\mathcal{O}_f^{\times}$
is equivalent to the freeness of spaces of modular forms on $B$ over
the associated Hecke algebra and the vanishing of certain local
Tamagawa components. In \cite{CH18} Chida and Hsieh actually proved
that $\frac{\Omega_{f,K}}{\Omega_{f}^{\mathrm{can}}}$ is a $p$-adic
unit, i.e.
$\frac{\Omega_{f,K}}{\Omega_{f}^{\mathrm{can}}}\in\mathcal{O}_f^{\times}$
under a condition ($\mathrm{CR^{+}}$). Wang \cite{Wang} generalized Chida and Hsieh's result to the setting of Hilbert
modular forms under similar hypothesis.
There are two normalizations of anticyclotomic $p$-adic
$L$-functions depending on which period one uses. However, in the
present paper we assume those hypothesis in \cite{Wang} in order
that there is essentially no difference between these two
normalizations.

\subsection{Comparison of periods under base change}

The goal of the present paper is to compare the canonical period of
a Hilbert modular form and that of its base change to a (totally)
real quadratic extension.

To make it precise, let $F$ be a totally real number field, and  $f$
be a Hilbert newform of parallel even weight over $F$. Let $F'$ be a
totally real quadratic extension of $F$, and $f'$ be the base change
of $f$ to $F'$. One may expect the following holds.

\begin{conj} We have
$$ \Omega_{f'}^\mathrm{can}=(\Omega_f^\mathrm{can})^2 $$ up to a
$p$-adic unit.
\end{conj}

The main result of this paper is the following theorem. Let $\rho_f$ be $p$-adic Galois representation attached to $f$ (see \cite{Wil88,Tay}), and let $\bar{\rho}_f$ denote the residue representation of $\rho_f$.

Assume $k=2$ and $N$ is a prime inert in $K$. Let $B$ be the definite quaternion algebra that is ramified exactly at $N$. Fix a prime number $p$. Let $H_n$ be the ring class field of $K$ of conductor $p^n$. When $\chi$ is a character of the Galois group $\mathrm{Gal}(H_n/K)$, one forms the Rankin-Selberg $L$-function $L(f,\chi,s)$.
Gross \cite{Gro87} showed that there exists a period $\Omega_{f,K}$
depending on $f$ and $K$ but independent of $n$ and $\chi$, called Gross
period, such that $\frac{L(f,\chi,1)}{\Omega_{f,K}}$ are algebraic
and they satisfy
$$\frac{L(f,\chi,1)}{\Omega_{f,K}}\cdot\sqrt{D}p^{n}=|\sum\limits_{\sigma\in\mathrm{Gal}(H_n/K)}\chi(\sigma)\psi(P^{\sigma})|^{2},$$
where $P$ is the Heegner point on the Shimura set attached to $B$.

Hida \cite{Hid81, Hid81.2} introduced a period that is independent
of $K$ and called the {\it canonical period}. Attached to $f$ there
exists a homomorphism $\lambda_f:T\longrightarrow \mathcal{O}_f$
from the Hecke algebra $T$ of level $N$ to a discrete valuation ring
$\mathcal{O}_f$ that is finite over $\BZ_p$. Let $\eta $ be the
congruence number for $\lambda_f$ defined by
$$\eta =\lambda_f(\mathrm{Ann}(\mathrm{Ker}(\lambda_f))).$$ Then the
canonical period is defined by
$$\Omega_{f}^{\mathrm{can}}=\frac{\langle f, f
\rangle_{\mathrm{Pet}}}{\eta }.$$ The reader may consult
\cite[Section 2]{Vat03} about more knowledge on Gross period and the
canonical period.

Vatsal \cite{Vat03} showed that
$\frac{\Omega_{f,K}}{\Omega_{f}^{\mathrm{can}}}$ lies in
$\mathcal{O}_f$. Pollack and Weston \cite{PW11} pointed out that
$\frac{\Omega_{f,K}}{\Omega_{f}^{\mathrm{can}}}\in\mathcal{O}_f^{\times}$
is equivalent to the freeness of spaces of modular forms on $B$ over
the associated Hecke algebra and the vanishing of certain local
Tamagawa components. In \cite{CH18} Chida and Hsieh actually proved
that $\frac{\Omega_{f,K}}{\Omega_{f}^{\mathrm{can}}}$ is a $p$-adic
unit, i.e.
$\frac{\Omega_{f,K}}{\Omega_{f}^{\mathrm{can}}}\in\mathcal{O}_f^{\times}$
under a condition ($\mathrm{CR^{+}}$). Wang \cite{Wang} generalized Chida and Hsieh's result to the setting of Hilbert
modular forms under similar hypothesis.
There are two normalizations of anticyclotomic $p$-adic
$L$-functions depending on which period one uses. However, in the
present paper we assume those hypothesis in \cite{Wang} in order
that there is essentially no difference between these two
normalizations.

To make it precise, let $F$ be a totally real number field, and $f$
be a Hilbert newform of parallel even weight over $F$. Let $F'$ be a
totally real quadratic extension of $F$, and $f'$ be the base change
of $f$ to $F'$. One may expect the following holds.

\begin{conj} We have
$$ \Omega_{f'}^\mathrm{can}=(\Omega_f^\mathrm{can})^2 $$ up to a
$p$-adic unit.
\end{conj}

The main result of this paper is the following theorem. Let $\rho_f$ be $p$-adic Galois representation attached to $f$ (see \cite{Wil88,Tay}), and let $\bar{\rho}_f$ denote the residue representation of $\rho_f$.

The main result of this paper is the following theorem. Let $\rho_f$ be $p$-adic Galois representation attached to $f$ (see \cite{Wil88,Tay}), and let $\bar{\rho}_f$ denote the residue representation of $\rho_f$.

We choose an imaginary quadratic extension $K$ of $F$ that is
inertia at $\mathfrak{q}$ when $[F:\mathbb{Q}]$ is odd. Denote $KF'$
by $K'$. Fix a place $\mathfrak{p}$ of $F$ above $p$. Let $K_\infty$ be the $\mathfrak{p}$-anticyclotomic
$\mathbb{Z}_p$-extension of $K$.

\begin{lem}\label{lem:fitt} Let $S$ be the set of primes dividing $\mathfrak{n}^-$. Then
$$ \mathrm{Fitt}(\prod_{\mathfrak{l}|
\mathfrak{n}^- }H^1((\mathcal{K}_\infty)_\mathfrak{l},
F^-_\mathfrak{l}A_\rho)^\vee) =\left(\frac{L^S_p (\mathcal{K}_\infty,
f)}{L_p (\mathcal{K}_\infty, f)}\right).
$$
\end{lem} This should be well known.
However, the authors do not have a reference about it.
\begin{proof}
By definition (see Section \ref{ss:selmer}) $H^{1}((\mathcal{K}_{\infty})_{\mathfrak{l}},F_{\mathfrak{l}}^{-}A_{\rho})$ is the inductive limit $\lim\limits_{\overrightarrow{\;\;L\;\;}} H^{1}(L_{\mathfrak{l}},F_{\mathfrak{l}}^{-}A_{\rho}),$ where $L$ runs through all finite extensions of $K$ contained in $\mathcal{K}_\infty$. For each $L$, $H^{1}(L_{\mathfrak{l}},F_{\mathfrak{l}}^{-}A_{\rho})$ is isomorphic to
$\bigoplus_{\mathfrak{L}|\mathfrak{l}} H^1(G_{L_\mathfrak{L}}, F_{\mathfrak{l}}^{-}A_{\rho})$ where $\mathfrak{L}$ runs over all places of $L$ above $\mathfrak{l}$,
and sits in the following exact sequence
{\small $$ \bigoplus_{\mathfrak{L}|\mathfrak{l}} H^1(G_{L_\mathfrak{L}}/I_{L_\mathfrak{L}}, (F_{\mathfrak{l}}^{-}A_{\rho})^{I_{L_\mathfrak{L}}})\rightarrowtail \bigoplus_{\mathfrak{L}|\mathfrak{l}} H^1(G_{L_\mathfrak{L}}, F_{\mathfrak{l}}^{-}A_{\rho})\twoheadrightarrow \bigoplus_{\mathfrak{L}|\mathfrak{l}} H^1(I_{L_\mathfrak{L}}, F_{\mathfrak{l}}^{-}A_{\rho})^{G_{L_\mathfrak{L}}/I_{L_\mathfrak{L}}} . $$}
As $G_{L_{\mathfrak{L}}}$ acts trivially on $F_\mathfrak{l}^-A_\rho$, we have
$$ H^1(G_{L_\mathfrak{L}}/I_{L_\mathfrak{L}}, (F_{\mathfrak{l}}^{-}A_{\rho})^{I_{L_\mathfrak{L}}}) \cong F_{\mathfrak{l}}^{-}A_{\rho}. $$ If $L'$ is a finite extension of $L$ contained in $\mathcal{K}_\infty$, and $\mathfrak{L}'$ is a place of $L'$ above $\mathfrak{L}$, we have the following commutative diagram
\[ \xymatrix{ H^1(G_{L_\mathfrak{L}}/I_{L_\mathfrak{L}}, (F_{\mathfrak{l}}^{-}A_{\rho})^{I_{L_\mathfrak{L}}})\ar[rr]^{\hskip45pt \simeq}\ar[d] && F_{\mathfrak{l}}^{-}A_{\rho} \ar[d]^{\times[G_{L_\mathfrak{L}}/I_{L_\mathfrak{L}}:G_{L'_{\mathfrak{L}'}}/I_{L'_{\mathfrak{L}'}}]} \\ H^1(G_{L'_{\mathfrak{L}'}}/I_{L'_{\mathfrak{L}'}}, (F_{\mathfrak{l}}^{-}A_{\rho})^{I_{L'_{\mathfrak{L}'}}})\ar[rr]^{\hskip45pt\simeq} && F_{\mathfrak{l}}^{-}A_{\rho}, } \] where the left vertical arrow is the restriction map. Since both $L_{\mathfrak{L}}$ and $L'_{\mathfrak{L}'}$ are unramified over $K_\mathfrak{l}$, $$[G_{L_\mathfrak{L}}/I_{L_\mathfrak{L}}:G_{L'_{\mathfrak{L}'}}/I_{L'_{\mathfrak{L}'}}]
=|\mathrm{Gal}(L'_{\mathfrak{L}'}/L_{\mathfrak{L}})|=[L'_{\mathfrak{L}'}:L_{\mathfrak{L}}]$$ is a power of $p$. Thus taking limit we have $$\lim\limits_{\overrightarrow{\;\;L\;\;}}\bigoplus_{\mathfrak{L}: \text{places of } L \text{ above }\mathfrak{l}} H^1(G_{L_\mathfrak{L}}/I_{L_\mathfrak{L}}, (F_{\mathfrak{l}}^{-}A_{\rho})^{I_{L_\mathfrak{L}}})=0.$$

\begin{proof} The representation $\bar{\rho}_f$ factors through
$\mathrm{Gal}(F_1/F)$ for a finite Galois extension $F_1$ of $F$.
Take a finite place $\mathfrak{q}\nmid p\mathfrak{n}$. For each
$\mathfrak{l}|p\mathfrak{n}$ take a unit $u_\mathfrak{l}$ of
$\mathcal{O}_{F_{\mathfrak{l}}}$ such that the image
$u_\mathfrak{l}$ in $k_{\mathfrak{l}}$ by the natural homomorphism
$$ \mathcal{O}_{F_\mathfrak{l}}\rightarrow
\mathcal{O}_{F_\mathfrak{l}}/\mathfrak{l}\cong k_\mathfrak{l} $$
generates the multiplicative group $k_\mathfrak{l}^\times$. Take a
uniformizing element $\omega_\mathfrak{q}$ of
$\mathcal{O}_{F_{\mathfrak{q}}}$. By the weak approximation theorem
there exists a totally positive element $a$ of $F$ such that
$a_{\mathfrak{l}}-u_{\mathfrak{l}} \in
\mathfrak{l}\mathcal{O}_{F_\mathfrak{l}}$ for each
$\mathfrak{l}|p\mathfrak{n}$, and
$a_\mathfrak{q}-\omega_\mathfrak{q}\in
\mathfrak{q}^2\mathcal{O}_{F_\mathfrak{q}}$. Put $F'=F(\sqrt{a})$.
Then each place dividing $p\mathfrak{n}$ splits in $F'$. As $a$ is
totally positive, $F'$ is totally real. Let $f'$ be the base change
of $f$ to $F'$.

\begin{thm}\label{thm:Iw-main} Assume $f$ satisfies
$(\mathrm{CR}^+)$, $(\mathfrak{n}^+\text{-}\mathrm{DT})$,
$(\mathrm{PO})$ and $(\mathrm{Fuji}1$-$4)$.
\begin{enumerate}
\item\label{it:Iw-main-a}
We have \begin{equation}\label{eq:char} \mathrm{char}\ \mathrm{Sel}
(K_\infty, A_{\rho})^\vee = (L_p(K_\infty, f)). \end{equation}
\item\label{it:Iw-main-b} Let $\mathcal{L}$ be $\mathcal{K}_\infty,
\mathcal{K}^-_\infty$ or $K^-_J$. If $[F:\BQ]$ is even, then
$$ \mathrm{char}\ \mathrm{Sel}(\mathcal{L}, A_\rho)^\vee = (L_p(\mathcal{L}, f))
$$ in $\mathcal{O}_\mathfrak{P}[[\mathrm{Gal}(\mathcal{L}/K)]]_E$.
\end{enumerate}
\end{thm}
\begin{proof} Let $F'$ be a totally real quadratic extension of $F$, as in Lemma \ref{lem:quad-ext}. Then $[F':\BQ]$ is even.
Let $\chi_{F'/F}$ be the quadratic character corresponding to the
extension $F'/F$. Put $K'=F'K$. Let $\mathcal{K}'^-_\infty$ be the
maximal abelian anticyclotomic $\BZ_p$-extension of $K'$ unramified
outside $p$. Put $\mathcal{K}'^+_\infty=F_\infty K'$,
$\mathcal{K}'_\infty=\mathcal{K}'^+_\infty\mathcal{K}'^-_\infty$ and
$\Gamma_{K'}=\mathrm{Gal}(\mathcal{K}'_\infty/K')$. Then
$K'\mathcal{K}_\infty=F'\mathcal{K}_\infty$ is contained in
$\mathcal{K}'_\infty$.

\begin{thm}\label{thm:main-stronger} Let $F$ be a totally real field,
and $F'$ be a real quadratic extension of $F$. Assume that $p$ is
unramified in both $F$ and $F'$. Let $f$ be a Hilbert newform of
parallel even weight $k$ over $F$ with trivial central character and
level $\mathfrak{n}_f$. Let $f'$ be the base change of $f$ to $F'$.
Let $\mathfrak{n}_{f'}$ be the level of $f'$.

We assume that $\mathfrak{n}_f$ is prime to $p$ and $f$ is ordinary
at each place above $p$. When $[F:\mathbb{Q}]$ is odd, we assume
that there exists at least one prime $\mathfrak{q}$ such that
$\mathfrak{q}|| \mathfrak{n}_f$, $\mathfrak{q}$ is split in $F'$,
and $\bar{\rho}_{f}$ is ramified at $\mathfrak{q}$. Suppose that, if
$\mathfrak{l}'||\mathfrak{n}_{f'}$ and
$\mathrm{N}(\mathfrak{l}')\equiv 1 \ (\mathrm{mod} \ p)$, then
$\bar{\rho}_{f'}$ is ramified at $\mathfrak{l}'$.

\begin{conj}\label{conj-mu} Let $F$ be a totally real number field, $K$ a quadratic imaginary
extension of $F$. Let $F'$ be a finite totally real extension of
$F$, and $f_{F'}$ be a Hilbert modular form over $F'$ that is
ordinary at each place of $F'$ above $p$. Then
$$\mu(L_p(F'K_\infty, f_{F'}))=0.$$
\end{conj}
Corollary \ref{thm:mu} says that Conjecture \ref{conj-mu} holds when
$F'$ is a quadratic real extension of $F$ and $f_{F'}$ is the base
change of a Hilbert modular form over $F$ (that satisfies some
conditions). It seems impossible to use the method in \cite{CH18,
Hung} to prove Conjecture \ref{conj-mu}, though it works when
$F'=F$.

We choose an imaginary quadratic extension $K$ of $F$ that is
inertia at $\mathfrak{q}$ when $[F:\mathbb{Q}]$ is odd. Denote $KF'$
by $K'$. Fix a place $\mathfrak{p}$ of $F$  above $p$. Let $K_\infty$ be the $\mathfrak{p}$-anticyclotomic
$\mathbb{Z}_p$-extension of $K$.

Using the canonical periods one attaches to $f$,
$f\otimes\chi_{F'/F}$ and $f'$ the $p$-adic $L$-functions
$$L_{p}(K_{\infty} ,f), \
L_{p}(K_{\infty} ,f\otimes\chi_{F'/F})  \ \text{ and }
L_{p}(K'K_{\infty} ,f').$$ All of them can be considered as elements
of $\mathcal{O}_{\mathfrak{P}}[[\mathrm{Gal}(K_{\infty} /K)]]$. The
ring $\mathcal{O}_{\mathfrak{P}}\supset \mathcal{O}_f$ is some
coefficient ring that is clearly defined in our context. To prove
Theorem \ref{thm:main} we only need to show
\begin{equation}\label{eq:main}
(L_{p}(K'K_{\infty} ,f'))=(L_{p}(K_{\infty} ,f))\cdot
(L_{p}(K_{\infty} ,f\otimes\chi_{F'/F}))  \end{equation} in
$\mathcal{O}_{\mathfrak{P}}[[\mathrm{Gal}(K_{\infty} /K)]].$

Our strategy is to use Iwasawa main conjecture that provides an
equality between a quantity measuring Selmer groups and $p$-adic
$L$-functions. Actually what we need is Iwasawa main conjecture for
Hilbert modular forms in the anticyclotomic setting.

Its proof is divided into two parts, one part proving one
divisibility by Ribet's method, and the other proving the converse
divisibility by Euler systems. The former divisibility was proved by
Skinner and Urban \cite{S-U} for elliptic modular forms, and was
proved by Wan \cite{Wan} for Hilbert modular forms. When
$[F:\mathbb{Q}]$ is odd, Wan needs the condition that Ihara Lemma
for Shimura curves holds. The latter divisibility was proved by
Bertolini and Darmon \cite{BD07} for elliptic curves, and by Chida
and Hsieh \cite{CH15} for elliptic modular forms. It was proved by
Longo \cite{Longo} and Wang \cite{Wang} for Hilbert modular forms
assuming Ihara Lemma for Shimura curves. This condition was removed
by the second author \cite{Xie}.

Combining results in \cite{Wan} and \cite{Xie} we obtain the
following theorem. Remark that we do not need Ihara Lemma for
Shimura curves even when $[F:\mathbb{Q}]$ is odd, though such a
condition is needed in \cite{Wan}.

\begin{thm}\label{thm:Iw-main-int}
Assume that $f$ satisfies conditions $(\mathrm{CR}^+)$,
$(\mathfrak{n}^+\text{-}\mathrm{DT})$, $(\mathrm{PO})$ and $(\mathrm{Fuji}1$-$4)$.
Then we have $$(L_p(K_\infty, f))=\mathrm{char}\ \mathrm{Sel}
(K_\infty, A_{\rho})^\vee .$$
\end{thm} See our context for Conditions $(\mathrm{CR}^+)$,
$(\mathfrak{n}^+\text{-}\mathrm{DT})$, $(\mathrm{PO})$ and
$(\mathrm{Fuji}1$-$4)$.

By Theorem \ref{thm:Iw-main-int} we have $$
(L_{p}(K_{\infty},f))=\mathrm{char}\
\mathrm{Sel}(K_{\infty},A_{\rho})^{\vee} $$ and
    $$  (L_{p}(K_{\infty},f\otimes\chi_{F'/F}))=\mathrm{char}\ \mathrm{Sel}(K_{\infty},A_{\rho\otimes\chi_{F'/F}})^{\vee}    .$$
Using the relation
$$\mathrm{Sel}(K'K_{\infty},A_{\rho})=\mathrm{Sel}(K_{\infty},A_{\rho})\bigoplus\mathrm{Sel}(K_{\infty},A_{\rho\otimes\chi_{F'/F}})$$
we obtain \begin{equation}\label{eq:Lp-eq} (L_{p}(K_{\infty},f)\cdot
L_{p}(K_{\infty},f\otimes\chi_{F'/F}))= \mathrm{char}\
\mathrm{Sel}(K'K_{\infty},A_{\rho})^{\vee}. \end{equation}

Let $\mathfrak{p}'$ be a place of $F'$ above $\mathfrak{p}$. Let
$K'_\infty$ be the $\mathfrak{p}'$-anticyclotomic
$\mathbb{Z}_p$-extension of $K'$.    With $K'_\infty$ instead of
$K_\infty$ we also have $$ (L_{p}(K_{\infty}',f'))=\mathrm{char}\
\mathrm{Sel}(K_{\infty}',A_{\rho})^{\vee},$$ from which we deduce
that \begin{equation}\label{eq:con}(L_{p}(K'K_{\infty},f'))\supseteq
\mathrm{char}\ \mathrm{Sel}(K'K_{\infty},A_{\rho})^{\vee}.
\end{equation} Combining (\ref{eq:Lp-eq}) and (\ref{eq:con}) we
obtain
\begin{equation}\label{eq:contain} (L_{p}(K'K_{\infty},f'))\supseteq (L_{p}(K_{\infty},f)\cdot
L_{p}(K_{\infty},f\otimes\chi_{F'/F})) . \end{equation} As
(\ref{eq:main}) holds in
$\mathcal{O}_{\mathfrak{P}}[[\mathrm{Gal}(K_{\infty}
/K)]][\frac{1}{p}]$, and as the $\mu$-invariants of
$L_{p}(K_{\infty} ,f)$ and $L_{p}(K_{\infty} ,f\otimes\chi_{F'/F})$
are zero, (\ref{eq:contain}) implies (\ref{eq:main}).

The paper is organized as follows. In Section \ref{sec:a} we collect
some basic facts about automorphic forms on definite quaternion
algebras. In Section \ref{sec:b} we recall the construction of
anticyclotomic $p$-adic $L$-functions via Theta elements. In Section
\ref{sec:d}, we combine Wan's result in \cite{Wan} with the result
in \cite{Xie} to prove Iwasawa main conjecture for Hilbert modular
form in the anticyclotomic setting (under certain conditions).
Finally we prove Theorem \ref{thm:main} in Section \ref{sec:e}.

In Sections \ref{sec:a}-\ref{sec:d}, the imaginary quadratic
extension $K$ of $F$ is fixed. In Section \ref{sec:e}, we will
choose a suitable $K$.

The authors thank B.H. Gross for his helpful advice.

\subsection*{Notations}

We fix a totally real number field \(F\) over \(\mathbb{Q}\). Let
\(\Sigma_{F}\) be the set of all real embeddings of \(F\) and let
\(\Sigma_{p}\) be the set of all places of \(F\) above \(p\).

For a number field \(L\), and each place \(v\) of \(L\) above a
prime number \(l\), let \(|\cdot|_{v}\) or \(|\cdot|_{L_{v}}\) be
the absolute value on \(L_{v}\) defined by
\(|x|_{v}=|N_{L_{v}/\mathbb{Q}_{l}}(x)|_{l}\), and  let
\(\varpi_{v}\) be a uniformizer of \(\mathcal{O}_{L_{v}}\).

Let $E$ be a finite extension of $\mathbb{Q}$, $\mathcal{O}$ be the ring of
integers in $E$ and $\mathfrak{P}$ be a prime  of $\mathcal{O}$
above $p$ such that $\mathcal{O}_\mathfrak{P}$ contains
$\mathcal{O}_f$. Let $\omega$ be a uniformizer of
$\mathcal{O}_\mathfrak{P}$. For each positive integer $n$ we put
$\mathcal{O}_{n}=\mathcal{O}_\mathfrak{P}/\omega^n$. We consider
$E_\mathfrak{P}$, $\mathcal{O}_\mathfrak{P}$ and $\mathcal{O}_n$ as
coefficient rings, and let $G_F=\mathrm{Gal}(\overline{F}/F)$ act trivially on them.

Let $\epsilon$ be the $p$-adic cyclotomic character of $G_F$.

Choose an imaginary quadratic extension $K/F$ (inert at $\mathfrak{q}$ when $[F:\mathbb{Q}]$ is odd) and put $K'=KF'$. Fix $\mathfrak{p}\mid p$ of $F$ and let $K_\infty$ be the $\mathfrak{p}$-anticyclotomic $\mathbb{Z}_p$-extension of $K$. Using the canonical periods, attach $p$-adic $L$-functions $L_p(K_\infty,f)$, $L_p(K_\infty,f\otimes\chi_{F'/F})$ and $L_p(K'K_\infty,f')$ in $\mathcal{O}_{\mathfrak{P}}[[\mathrm{Gal}(K_\infty/K)]]$. To prove Theorem~\ref{thm:main} it suffices to show the ideal identity
\[\tag{\ref{eq:main}} (L_{p}(K'K_{\infty} ,f'))=(L_{p}(K_{\infty} ,f))\cdot (L_{p}(K_{\infty} ,f\otimes\chi_{F'/F}))\]
inside $\mathcal{O}_{\mathfrak{P}}[[\mathrm{Gal}(K_\infty/K)]]$.

The strategy is to use the anticyclotomic Iwasawa main conjecture, i.e. an equality between $p$-adic $L$-functions and characteristic ideals of Selmer groups. By Theorem~\ref{thm:Iw-main-int},
\[(L_p(K_\infty,f))=\mathrm{char}\,\mathrm{Sel}(K_\infty,A_{\rho})^\vee,\qquad (L_p(K_\infty,f\otimes\chi_{F'/F}))=\mathrm{char}\,\mathrm{Sel}(K_\infty,A_{\rho\otimes\chi_{F'/F}})^\vee.
\]
Using the Selmer decomposition
\[\mathrm{Sel}(K'K_{\infty},A_{\rho})=\mathrm{Sel}(K_{\infty},A_{\rho})\bigoplus\mathrm{Sel}(K_{\infty},A_{\rho\otimes\chi_{F'/F}}),\]
one gets
\[\tag{\ref{eq:Lp-eq}} (L_p(K_\infty,f)\cdot L_p(K_\infty,f\otimes\chi_{F'/F}))=\mathrm{char}\,\mathrm{Sel}(K'K_\infty,A_{\rho})^\vee.
\]
Then, letting $\mathfrak{p}'\mid\mathfrak{p}$ in $F'$ and $K'_\infty$ be the $\mathfrak{p}'$-anticyclotomic $\mathbb{Z}_p$-extension of $K'$, one also has $(L_p(K'_\infty,f'))=\mathrm{char}\,\mathrm{Sel}(K'_\infty,A_{\rho})^\vee$, which implies the containment
\[\tag{\ref{eq:con}} (L_p(K'K_\infty,f'))\supseteq \mathrm{char}\,\mathrm{Sel}(K'K_\infty,A_{\rho})^\vee.
\]
Combining \eqref{eq:Lp-eq} and \eqref{eq:con} yields
\[\tag{\ref{eq:contain}} (L_{p}(K'K_{\infty},f'))\supseteq (L_{p}(K_{\infty},f)\cdot L_{p}(K_{\infty},f\otimes\chi_{F'/F})).
\]
Finally, since \eqref{eq:main} holds after inverting $p$ and the $\mu$-invariants of $L_p(K_\infty,f)$ and $L_p(K_\infty,f\otimes\chi_{F'/F})$ are zero, the containment \eqref{eq:contain} implies the equality \eqref{eq:main}, hence Theorem~\ref{thm:main}.

Choose an imaginary quadratic extension $K/F$ (inert at $\mathfrak{q}$ when $[F:\mathbb{Q}]$ is odd) and put $K'=KF'$. Fix $\mathfrak{p}\mid p$ of $F$ and let $K_\infty$ be the $\mathfrak{p}$-anticyclotomic $\mathbb{Z}_p$-extension of $K$. Using the canonical periods, attach $p$-adic $L$-functions $L_p(K_\infty,f)$, $L_p(K_\infty,f\otimes\chi_{F'/F})$ and $L_p(K'K_\infty,f')$ in $\mathcal{O}_{\mathfrak{P}}[[\mathrm{Gal}(K_\infty/K)]]$. To prove the main theorem, it suffices to show the ideal identity
\begin{equation}\label{eq:main}
(L_{p}(K'K_{\infty} ,f'))=(L_{p}(K_{\infty} ,f))\cdot
(L_{p}(K_{\infty} ,f\otimes\chi_{F'/F}))  \end{equation}
inside $\mathcal{O}_{\mathfrak{P}}[[\mathrm{Gal}(K_\infty/K)]]$.

The strategy is to use the anticyclotomic Iwasawa main conjecture, i.e. an equality between $p$-adic $L$-functions and characteristic ideals of Selmer groups. We first use the following theorem.
\begin{thm}\label{thm:Iw-main-int}
Assume that $f$ satisfies conditions $(\mathrm{CR}^+)$,
$(\mathfrak{n}^+\text{-}\mathrm{DT})$, $(\mathrm{PO})$ and $(\mathrm{Fuji}1$-$4)$.
Then we have $$(L_p(K_\infty, f))=\mathrm{char}\ \mathrm{Sel}
(K_\infty, A_{\rho})^\vee .$$
\end{thm}
Thus
\[(L_p(K_\infty,f))=\mathrm{char}\,\mathrm{Sel}(K_\infty,A_{\rho})^\vee,\qquad (L_p(K_\infty,f\otimes\chi_{F'/F}))=\mathrm{char}\,\mathrm{Sel}(K_\infty,A_{\rho\otimes\chi_{F'/F}})^\vee.
\]
Using the Selmer decomposition
\[\mathrm{Sel}(K'K_{\infty},A_{\rho})=\mathrm{Sel}(K_{\infty},A_{\rho})\bigoplus\mathrm{Sel}(K_{\infty},A_{\rho\otimes\chi_{F'/F}}),\]
one gets
\begin{equation}\label{eq:Lp-eq} (L_{p}(K_{\infty},f)\cdot
L_{p}(K_{\infty},f\otimes\chi_{F'/F}))= \mathrm{char}\
\mathrm{Sel}(K'K_{\infty},A_{\rho})^{\vee}. \end{equation}
Then, letting $\mathfrak{p}'\mid\mathfrak{p}$ in $F'$ and $K'_\infty$ be the $\mathfrak{p}'$-anticyclotomic $\mathbb{Z}_p$-extension of $K'$, one also has $(L_p(K'_\infty,f'))=\mathrm{char}\,\mathrm{Sel}(K'_\infty,A_{\rho})^\vee$, which implies the containment
\begin{equation}\label{eq:con}(L_{p}(K'K_{\infty},f'))\supseteq
\mathrm{char}\ \mathrm{Sel}(K'K_{\infty},A_{\rho})^{\vee}.
\end{equation}
Combining the equation above and the containment above yields
\begin{equation}\label{eq:contain} (L_{p}(K'K_{\infty},f'))\supseteq (L_{p}(K_{\infty},f)\cdot
L_{p}(K_{\infty},f\otimes\chi_{F'/F})) . \end{equation}
Finally, since the displayed ideal identity above holds after inverting $p$ and the $\mu$-invariants of $L_p(K_\infty,f)$ and $L_p(K_\infty,f\otimes\chi_{F'/F})$ are zero, the containment above implies the ideal identity
\begin{equation}\label{eq:main}
(L_{p}(K'K_{\infty} ,f'))=(L_{p}(K_{\infty} ,f))\cdot
(L_{p}(K_{\infty} ,f\otimes\chi_{F'/F}))  \end{equation}
and this completes the proof of the main theorem.

Question:
Let $F$ be a totally real field and let $F'/F$ be a real quadratic extension. Let $f$ be a Hilbert newform over $F$ of parallel even weight $k$, trivial central character, and level $\mathfrak{n}_f$, and let $f'$ be the base change of $f$ to $F'$ with level $\mathfrak{n}_{f'}$. Let $p$ be a prime such that $p\ge \max(k+2,7)$ and $p$ is unramified in both $F$ and $F'$. Assume that $\mathfrak{n}_f$ is prime to $p$ and that $f$ is ordinary at every prime of $F$ above $p$. If $[F:\mathbb{Q}]$ is odd, assume there exists a prime $\mathfrak{q}$ such that $\mathfrak{q}\parallel \mathfrak{n}_f$ (that is, $\mathfrak{q}$ divides $\mathfrak{n}_f$ exactly once) and $\mathfrak{q}$ is split in $F'$. Let $\rho_f$ be the $p$-adic Galois representation attached to $f$, and let $\bar\rho_f$ and $\bar\rho_{f'}$ denote the residual Galois representations attached to $f$ and $f'$, respectively. Suppose moreover that:

1. the restrictions of $\bar\rho_f$ to $G_{F'(\xi_p)}$ and to $G_{F'(\sqrt{p^*})}$ are absolutely irreducible, where $p^*=(-1)^{\frac{p-1}{2}}p$;
2. if $k=2$, then for every place $v$ of $F$ above $p$, $a_v(f)^2\not\equiv 1\pmod p$, and for every place $v'$ of $F'$ above $p$, $a_{v'}(f')^2\not\equiv 1\pmod p$;
3. for every prime $\mathfrak{l}\mid \mathfrak{n}_f$, if $\bar\rho_f|_{I_{F_{\mathfrak{l}}}}$ is absolutely irreducible, then $\mathrm N(\mathfrak{l})\not\equiv -1\pmod p$, and for every prime $\mathfrak{l}'\mid \mathfrak{n}_{f'}$, if $\bar\rho_{f'}|_{I_{F'_{\mathfrak{l}'}}}$ is absolutely irreducible, then $\mathrm N(\mathfrak{l}')\not\equiv -1\pmod p$;
4. $\rho_f$ is a minimal modular lifting of $\bar\rho_f$.

For a modular form $g$, write its canonical period as $\Omega_g^{\mathrm{can}}=\langle g,g\rangle_{\mathrm{Pet}}/\eta_g$, where $\langle g,g\rangle_{\mathrm{Pet}}$ is the Petersson inner product and $\eta_g$ is the associated congruence number. Under these hypotheses, which statement about the canonical periods of $f$ and $f'$ is valid?

The canonical periods satisfy
$$\Omega_{f'}^{\mathrm{can}}=(\Omega_f^{\mathrm{can}})^2$$
up to a $p$-adic unit; equivalently, the ratio $\Omega_{f'}^{\mathrm{can}}/(\Omega_f^{\mathrm{can}})^2$ is a $p$-adic unit.

Choices:
B.
The canonical periods satisfy
$$\Omega_{f'}^{\mathrm{can}}=(\Omega_f^{\mathrm{can}})^2$$
exactly in the coefficient ring, i.e. the ratio $\Omega_{f'}^{\mathrm{can}}/(\Omega_f^{\mathrm{can}})^2$ is equal to $1$.
C.
The canonical periods satisfy
$$\Omega_{f'}^{\mathrm{can}}=u\, (\Omega_f^{\mathrm{can}})^2$$
for some nonzero element $u$ in the coefficient field; in particular, $\Omega_{f'}^{\mathrm{can}}$ and $(\Omega_f^{\mathrm{can}})^2$ agree up to multiplication by a scalar.
D.
The canonical periods satisfy
$$\Omega_{f'}^{\mathrm{can}}=(\Omega_f^{\mathrm{can}})^2$$
up to a $p$-adic unit whenever the restrictions of $\bar\rho_f$ to $G_{F(\xi_p)}$ and to $G_{F(\sqrt{p^*})}$ are absolutely irreducible; no hypothesis involving the corresponding restrictions over $F'$ is needed.
E.
The canonical periods satisfy
$$\Omega_{f'}^{\mathrm{can}}=(\Omega_f^{\mathrm{can}})^2$$
up to multiplication by an arbitrary $p$-power; equivalently, the ratio $\Omega_{f'}^{\mathrm{can}}/(\Omega_f^{\mathrm{can}})^2$ is merely $p$-adically integral.

unit-ambiguity from ideal identity and period normalization

dropped the requirement that the scalar be a p-adic unit

base-change irreducibility hypotheses over F'

mu=0 used to upgrade containment to equality up to unit

The stem gives hypotheses but does not reveal the conclusion. The correct choice is not explicitly stated, and the reader must distinguish it from closely related alternatives.

This is essentially a theorem-recall question: under a long list of hypotheses, the task is to identify the exact conclusion. It is not a pure verbatim restatement because the options vary in strength and precision, but it remains close to theorem matching.

Some reasoning is required to separate 'up to a p-adic unit' from stronger or weaker formulations, but the main task is recognizing the precise theorem statement rather than generating a new argument or synthesis.

The distractors are mathematically plausible and target common failure modes: overstating to exact equality, weakening to mere scalar commensurability, allowing extra p-power ambiguity, or adding unnecessary hypotheses. They are distinct and well-designed.

A solid theorem-discrimination MCQ with strong distractors and no answer leakage, but it mainly tests precise recall of a known conclusion rather than deeper generative reasoning.


\label{main}
    Let $C\subset\mathbb CP^2$ be an algebraic plane curve of degree at most five. Then the fundamental group $\pi_1(\mathbb CP^2\setminus C)$ is linear and virtually polyfree. In particular, $\pi_1(\mathbb CP^2\setminus C)$ is residually finite.

Let $C$ be an algebraic curve in the projective plane $\mathbb CP^2$. We consider the following question.

\begin{ques}
\label{q1}
 Is the fundamental group $\pi_1(\mathbb CP^2\setminus C)$ of an algebraic plane curve residually finite?
\end{ques}

This question has been open for a long time. Zariski had already posed the question of the existence of non-residually finite groups, cf. § 1 and Appendix 1) of Chapter VIII of \cite{Zar} (see also Libgober \cite[Problem 3.1] {Lib}).
Even in the low-degree cases (degree up to five), such a question was still unknown (see the survey article of Artal Bartolo-Cogolludo Agustın-Tokunaga \cite{bartolo2008survey}, page 3). Toledo \cite{Toledo} constructed a smooth projective variety with a non-residually finite fundamental group, answering in the negative a question of Serre. However, it is not clear whether one can choose the example as the complement of an algebraic curve with a non-residually finite fundamental group. Koberda and Suciu \cite[Problem 1.9]{KS} asked Question \ref{q1} for affine line arrangements.

The purpose of this article is to answer Question \ref{q1} positively for algebraic curves of degree at most five. Recall that a group $G$ is linear if it can be embedded into a general linear group $\mathrm{GL}(n,F)$ for some field $F$ and positive integer $n$. A linear group is residually finite. A group $G$ is called poly-free if there is a finite
subnormal sequence\begin{equation*}
1<G_{1}\trianglelefteq G_{2}\trianglelefteq \cdots \trianglelefteq G_{n}=G
\end{equation*}such that the quotient $G_{i}/G_{i+1}$ is free for each $i$. Polyfree groups have many nice properties, like left-orderable, having Tits alternative, and so on. We say a group $G$ virtually has property P, if there is a finite-index subgroup $H<G$ that has the property P. The following is our main result.

\begin{ques}
\label{q1}
    Is the fundamental group $\pi_1(\mathbb CP^2\setminus C)$ of an algebraic plane curve residually finite?
\end{ques}

\begin{abstract}
We prove that for any algebraic plane curve $C$ of degree at most $5$, the fundamental group $\pi_1(\mathbb CP^2\setminus C)$ is linear and virtually polyfree. As a consequence, we answer positively the open question on the residual finiteness of these groups for all plane curves of degree at most $5$.
\end{abstract}

\begin{ques}
\label{q1}
 Is the fundamental group $\pi_1(\mathbb CP^2\setminus C)$ of an algebraic plane curve residually finite?
\end{ques}

The purpose of this article is to answer Question \ref{q1} positively for algebraic curves of degree at most five. Recall that a group $G$ is linear if it can be embedded into a general linear group $\mathrm{GL}(n,F)$ for some field $F$ and positive integer $n$. A linear group is residually finite. A group $G$ is called poly-free if there is a finite
subnormal sequence\begin{equation*}
1<G_{1}\trianglelefteq G_{2}\trianglelefteq \cdots \trianglelefteq G_{n}=G
\end{equation*}such that the quotient $G_{i}/G_{i+1}$ is free for each $i$. Polyfree groups have many nice properties, like left-orderable, having Tits alternative, and so on. We say a group $G$ virtually has property P, if there is a finite-index subgroup $H<G$ that has the property P. The following is our main result.

It is well-known (cf.\cite[2.3.2]{cogolludo2011braid}) that for any plane curve $C\subset\mathbb CP^2$ of degree $\le3$, the fundamental group $\pi_1(\mathbb CP^2\setminus C)$ is either finitely generated abelian or a free group $\mathbb Z\ast\mathbb Z$. Thus $\pi_1(\mathbb CP^2\setminus C)$ is linear. For degree-$4$ curves, we applied the results due to Nori \cite{nori1983zariski} and Cogolludo-Agustin and Elduque \cite{cogolludo2025quasi} to conclude that
that $\pi_1(\mathbb{C}P^2\setminus C)$ must be one of the following groups: a (virtually) finitely generated abelian group, a finite group, the braid group on the sphere $B_3(S^2)$, the free group $F_3$, the braid group $B_3$, the free product $\mathbb{Z} \ast \mathbb{Z}/2\mathbb{Z}$, or the semi-direct product $F_2 \rtimes \mathbb{Z}$, (see Subsection \ref{degree4} for details), which are all virtually poly-free and linear. For degree-$5$ curves, Degtyarev \cite{degtyarev1987topology} obtained presentations of all the $\pi_1(\mathbb{C}P^2\setminus C)$ when $\deg C=5$. In Section \ref{degree5}, we will review the presentations of these groups from Degtyarev and prove the virtual polyfreeness and linearity of these fundamental groups. Our key observation is that most of the non-abelian groups are virtual surface groups or (triangle or right-angled) Artin groups.

\subsection{Curves of degree $\le3$}
\noindent\\
It is well-known (cf.\cite[2.3.2]{cogolludo2011braid}) that for any plane curve $C\subset\mathbb CP^2$ of degree $\le3$, $\pi_1(\mathbb CP^2\setminus C)$ is either finitely generated abelian or a free group $\mathbb Z\ast\mathbb Z$. Thus $\pi_1(\mathbb CP^2\setminus C)$ is linear and virtually polyfree.

\subsection{Quartic curves}\label{degree4}
\noindent\\
If $C\subset \mathbb CP^2$ is an irreducible quartic curve that is not a 3-cuspidal quartic, then the fundamental group $\pi_1(\mathbb CP^2\setminus C)$ is abelian (cf. \cite[p.130, 4.3]{dimca2012singularities}). When $C$ is a 3-cuspidal quartic $\pi_1(\mathbb CP^2\setminus C) \cong B_3(S^2)$, the sphere braid group (cf. \cite{zariski1936poincare}), a finite group of order 12 and thus linear (see also\cite[4.8, page 131]{dimca2012singularities}). Since any finitely generated abelian group is linear, we know that for any irreducible quartic curve $C$, $\pi_1(\mathbb CP^2\setminus C)$ is linear. Also, these groups are known to be virtually polyfree.

\begin{cor}
 If $C$ is a plane curve of degree $\le 4$, then $\pi_1(\mathbb CP^2\setminus C)$ is linear and virtually polyfree.
\end{cor}

\section{Quintic}\label{degree5}
The presentation of fundamental groups of $\mathbb{C}P^2 \setminus\ C$ for degree-5 algebraic curves $C$ has been classified by Degtyarev \cite{Deg}. Based on this classification, we will prove Theorem \ref{main} for these curves by studying the presentations.
\subsection{Presentation of fundamental groups}
In this subsection, we will review the results proved by Degtyarev \cite{Deg}. Before introducing the list of Degtyarev, we need some notations.
\begin{enumerate}
\item $F_n$ is the free group of rank $n$, and $B_n$ is the braid group with $n$-strands.
\item $T_{p,q}$ is the fundamental group of a toric link of type $(p, q)$. In particular, if $p = 2$, then
$T_{2,2r}=\langle a,b| (ab)^r=(ba)^r\rangle$; if $\text{gcd}(p,q)=1$ then $T_{p,q}=\langle a,b| a^p=b^q\rangle$.
\item $G(T)$ and $G_p(T)$, where $T\in\mathbb Z[t]$ is an integral polynomial, are the extensions
$$1\to \mathbb Z[t]/T\to G(T)\to \mathbb Z\to 1$$ and
$$1\to (\mathbb Z/{p\mathbb Z})[t]/T\to G(T)\to \mathbb Z\to 1,$$ where the conjugation action of the generator of the quotient $\mathbb Z$ on the kernel is the multiplication by $t$.
\item $\text{Gr}\langle p,q,r \rangle :=\langle a,b,c|a^p=b^q=c^r=abc\rangle$.
\end{enumerate}

\begin{ques}
\label{q1}
    Is the fundamental group $\pi_1(\mathbb CP^2\setminus C)$ of an algebraic plane curve residually finite?
\end{ques}

It is well-known (cf.\cite[2.3.2]{cogolludo2011braid}) that for any plane curve $C\subset\mathbb CP^2$ of degree $\le3$, the fundamental group $\pi_1(\mathbb CP^2\setminus C)$ is either finitely generated abelian or a free group $\mathbb Z\ast\mathbb Z$. Thus $\pi_1(\mathbb CP^2\setminus C)$ is linear. For degree-$4$ curves, we applied the results due to Nori \cite{nori1983zariski} and Cogolludo-Agustin and Elduque  \cite{cogolludo2025quasi} to conclude that
that $\pi_1(\mathbb{C}P^2\setminus C)$ must be one of the following groups: a (virtually) finitely generated abelian group, a finite group, the braid group on the sphere $B_3(S^2)$, the free group $F_3$, the braid group $B_3$, the free product $\mathbb{Z} \ast \mathbb{Z}/2\mathbb{Z}$, or the semi-direct product $F_2 \rtimes \mathbb{Z}$, (see Subsection \ref{degree4} for details), which are all virtually poly-free and linear. For degree-$5$ curves, Degtyarev \cite{degtyarev1987topology} obtained presentations of all the $\pi_1(\mathbb{C}P^2\setminus C)$ when $\deg C=5$. In Section \ref{degree5}, we will review the presentations of these groups from Degtyarev and prove the virtual polyfreeness and linearity of these fundamental groups. Our key observation is that most of the non-abelian groups are virtual surface groups or (triangle or right-angled) Artin groups.

\subsection*{Acknowledgments}
We would like to thank Anatoly Libgober, Jose Ignacio Cogolludo, Corey Bregman for useful discussions and Alex Degtyarev for pointing us the results from his thesis and book.

To prove Theorem~\ref{main}, the argument is by degree.

- For $\deg C\le 3$: it is "well-known" that $\pi_1(\mathbb CP^2\setminus C)$ is "either finitely generated abelian or a free group $\mathbb Z\ast\mathbb Z$"; hence it is linear.

- For $\deg C=4$: using results of Nori and Cogolludo-Agustin--Elduque, one concludes that $\pi_1(\mathbb CP^2\setminus C)$ is one of an explicit list ("a (virtually) finitely generated abelian group, a finite group, the braid group on the sphere $B_3(S^2)$, the free group $F_3$, the braid group $B_3$, the free product $\mathbb{Z} \ast \mathbb{Z}/2\mathbb{Z}$, or the semi-direct product $F_2 \rtimes \mathbb{Z}$"), and these groups are "all virtually poly-free and linear."

- For $\deg C=5$: Degtyarev "obtained presentations" of all groups $\pi_1(\mathbb{C}P^2\setminus C)$ when $\deg C=5$; the plan is to "review the presentations" and "prove the virtual polyfreeness and linearity" of these groups, with the "key observation" that "most of the non-abelian groups are virtual surface groups or (triangle or right-angled) Artin groups."

To prove the main theorem, the argument is by degree.

- For $\deg C\le 3$: it is "well-known" that $\pi_1(\mathbb CP^2\setminus C)$ is "either finitely generated abelian or a free group $\mathbb Z\ast\mathbb Z$"; hence it is linear.

- For $\deg C=4$: using results of Nori and Cogolludo-Agustin--Elduque, one concludes that $\pi_1(\mathbb CP^2\setminus C)$ is one of an explicit list ("a (virtually) finitely generated abelian group, a finite group, the braid group on the sphere $B_3(S^2)$, the free group $F_3$, the braid group $B_3$, the free product $\mathbb{Z} \ast \mathbb{Z}/2\mathbb{Z}$, or the semi-direct product $F_2 \rtimes \mathbb{Z}$"), and these groups are "all virtually poly-free and linear."

- For $\deg C=5$: Degtyarev "obtained presentations" of all groups $\pi_1(\mathbb{C}P^2\setminus C)$ when $\deg C=5$; the plan is to "review the presentations" and "prove the virtual polyfreeness and linearity" of these groups, with the "key observation" that "most of the non-abelian groups are virtual surface groups or (triangle or right-angled) Artin groups."

Question: Let \(C\subset \mathbb{CP}^2\) be an algebraic plane curve of degree at most \(5\). Recall that a group is called linear if it embeds into \(\mathrm{GL}(n,F)\) for some field \(F\) and some positive integer \(n\), and it is called virtually polyfree if it has a finite-index subgroup admitting a finite subnormal series whose successive quotients are free groups. Which statement holds for every such curve \(C\)?

The fundamental group \(\pi_1(\mathbb{CP}^2\setminus C)\) is linear and virtually polyfree; in particular, it is residually finite.

Choices:
B. The fundamental group \(\pi_1(\mathbb{CP}^2\setminus C)\) is residually finite and virtually polyfree; in fact, for every such curve it admits a finite-index subgroup which is a free group.
C. The fundamental group \(\pi_1(\mathbb{CP}^2\setminus C)\) is residually finite.
D. The fundamental group \(\pi_1(\mathbb{CP}^2\setminus C)\) is linear; moreover, there exists a field \(F\) and a positive integer \(n\), depending only on the degree bound \(5\), such that every such group embeds into \(\mathrm{GL}(n,F)\).
E. The fundamental group \(\pi_1(\mathbb{CP}^2\setminus C)\) is either finitely generated abelian or a free group, and hence is linear and virtually polyfree.

drops linear and virtually polyfree, keeping only the stated consequence

non-uniform linearity strengthened to uniform embedding dimension/field

extends the degree \(\le 3\) classification to all degrees \(\le 5\)

The stem does not state the theorem or uniquely point to the correct option. The recalled definitions of 'linear' and 'virtually polyfree' are needed to read several choices, not just the correct one.

The correct choice is essentially a direct statement of the target result for degree <= 5 curves, with only an immediate consequence appended. This makes the item close to theorem recall rather than a non-tautological inference task.

There is some reasoning required to reject stronger or overextended variants (virtually free, uniform linearity, low-degree classification extended to degree 5) and to notice that one option is merely weaker. But if the theorem is known, the answer is mostly straightforward.

The distractors are plausible and mathematically meaningful: one is weaker but true, others are stronger false claims or overgeneralizations that reflect common failure modes in theorem recall and logical strength.

A solid theorem-recall MCQ with good distractors and little answer leakage, but it scores low on tautology avoidance because the correct option essentially restates the theorem rather than forcing substantial generative reasoning.


For any $d \geqslant 1$ and $j \in [0, d]$, we have $c_{d, 0}(j) = |s(d + 1, j + 1)|$.

\begin{equation} \label{eq:polynomial def}
  \sum_{n \geq 0} P_{d,k}(n)\, x^n = \frac{(1+x)^k}{(1-x)^{d+1}}
\end{equation}

\begin{remark}\label{ehrbij}
    The fact that the objects described above are counted by $d!P_{d, k}(n)$ also follows by the interpretation of $P_{d, k}(n)$ as an Ehrhart polynomial.
    More specifically, $P_{d, k}(n)$ counts points $\bm x \in \ZZ^d$, such that
    \begin{itemize}
        \item $|x_1| + \ldots +|x_k| + x_{k + 1} + \ldots + x_d \leq n$, and
        \item $x_{k + 1}, \ldots, x_d \geq 0$.
    \end{itemize}
    One way to represent a sequence of length $d$ of integers with sum of absolute values at most $n$ is as a sequence consisting of $n$ balls and $d$ dots.
    For example, with $n = 6$ and $d = 3$, the sequence $(2, 0, 3)$ (with only nonnegative terms) is represented as
    \begin{equation*}
        \emptycircle[0pt]\, \emptycircle[0pt]\, {\bullet}\, {\bullet}\, \emptycircle[0pt]\, \emptycircle[0pt]\, \emptycircle[0pt]\, {\bullet}\, \emptycircle[0pt].
    \end{equation*}
    The total number of balls is $n$, the first entry is the number of balls before the first dot, and for any $i \geq 2$, the $i$-th entry is the number of balls between the $(i - 1)$-th and $i$-th dots.
    If we want the $i$-th entry to be negative, we move the $i$-th dot into the ball immediately before it.
    For example, the sequence $(-1, 3, -2)$ is represented as
    \begin{equation*}
        \circled[0pt]{\bullet}\, \emptycircle[0pt]\, \emptycircle[0pt]\, \emptycircle[0pt]\, {\bullet}\, \emptycircle[0pt]\, \circled[0pt]{\bullet}.
    \end{equation*}
    Hence, $P_{d, k}(n)$ counts such configurations with $n$ balls and $d$ dots where only the first $k$ dots can be in balls.
    This interpretation can be used to show $d!P_{d, k}(n) = |W_{d, k}(n)|$.
\end{remark}

\label{sec:intro}

The Ehrhart polynomial \(\ehr_{\mathcal P}(n)\) of a lattice polytope \(\mathcal P\) counts the number of integer lattice points in the dilate \(n \mathcal P\). It is known that the Ehrhart polynomial of the cross-polytope
\begin{equation*}
  \mathcal P_d \coloneqq \left\{\bm x \in \mathbb{R}^d : |x_1| + |x_2| + \ldots + |x_d| \leqslant 1\right\}
\end{equation*}
is determined by the generating function
\begin{equation*}
  \sum_{n \geqslant 0} \ehr_{\mathcal P_d}(n)\, x^n = \frac{(1+x)^d}{(1-x)^{d+1}}.
\end{equation*}
More generally, if \(k \in [0,d]\), the polynomial \(P_{d,k}(n)\) defined by
\begin{equation} \label{eq:polynomial def}
  \sum_{n \geqslant 0} P_{d,k}(n)\, x^n = \frac{(1+x)^k}{(1-x)^{d+1}}
\end{equation}
is the Ehrhart polynomial of the \((d-k)\)-fold pyramid over \(\mathcal P_k\). The relevant background can be found for example in \cite{ccd}. The figure below illustrates \(\mathcal P_3\) (left), the pyramid over \(\mathcal P_2\) (middle) and \(\mathcal P_2\) itself (right).

\bigskip
\begin{center}
  \begin{tikzpicture}[scale=1.7]
    \coordinate (1) at (1,0,0);
    \coordinate (-1) at (-1,0,0);
    \coordinate (2) at (0,1,0);
    \coordinate (-2) at (0,-1,0);
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  \end{tikzpicture}
\end{center}

\noindent
The fact that \(P_{d,k}(n)\) is a polynomial is not immediately clear from \eqref{eq:polynomial def}, but one can check that
\begin{equation*}
  P_{d,k}(n) = \sum_{i=0}^k \binom{k}{i} \binom{n + d - i}{d}.
\end{equation*}

The coefficients of Ehrhart polynomials have been studied extensively, and they are usually hard to understand -- it is uncommon to be able to fully describe the coefficients for a given polytope. It is difficult even to determine if the coefficients are nonnegative, see \cite{ehrpos}. In this paper, we will examine these questions for (pyramids over) cross-polytopes. We define \(c_{d,k}(j)\) by
\begin{equation} \label{eqn}
  d! P_{d,k}(n) = \sum_{j=0}^d c_{d,k}(j) n^j,
\end{equation}
where the factor \(d!\) ensures that every coefficient is integral. Our main results are a combinatorial interpretation of the coefficients \(c_{d,k}(j)\), and a bijective proof of equation \eqref{eqn}. In particular, we prove combinatorially that the Ehrhart coefficients for (pyramids over) cross-polytopes are positive, which answers a question posed by Stanley \cite[Chapter 4, Exercise 1(b)]{ecnewex}.

The outline of the paper is as follows.
In Section \ref{sec:sri}, we construct a class of objects that are counted by the coefficients \(c_{d, k}(j)\), in particular showing that these coefficients are positive.
In Section \ref{sec:bij}, we give a combinatorial argument that proves \eqref{eqn} by constructing objects that are counted by both sides of the equation and exhibiting a bijection.
In Section \ref{sec:prop}, we use the combinatorial meaning we have for \(c_{d, k}(j)\) to obtain some properties of these coefficients.
Finally, in Section \ref{sec:poly}, we illustrate how the method we have used might be helpful for studying the Ehrhart coefficients of other polytopes.

\bigskip\noindent
Before proceeding with the general results, we will briefly describe a proof strategy for the simplest case \(k = 0\). The same ideas will serve as our guiding light for the rest of the paper.

The numbers \(c_{d, 0}(j)\) are just unsigned Stirling numbers of the first kind.
Recall that the (signed) Stirling number of the first kind \(s(m, i)\) is defined by
\begin{equation*}
    s(m, i) \coloneqq (-1)^{m - i} \cdot |\{\sigma \in S_m : \sigma \text{ has } i \text{ cycles}\}|.
\end{equation*}

The Ehrhart polynomial \(\ehr_{\mathcal P}(n)\) of a lattice polytope \(\mathcal P\) counts the number of integer lattice points in the dilate \(n \mathcal P\). It is known that the Ehrhart polynomial of the cross-polytope
\begin{equation*}
 \mathcal P_d \coloneqq \left\{\bm x \in \mathbb{R}^d : |x_1| + |x_2| + \ldots + |x_d| \leqslant 1\right\}
\end{equation*}
is determined by the generating function
\begin{equation*}
 \sum_{n \geqslant 0} \ehr_{\mathcal P_d}(n)\, x^n = \frac{(1+x)^d}{(1-x)^{d+1}}.
\end{equation*}
More generally, if \(k \in [0,d]\), the polynomial \(P_{d,k}(n)\) defined by
\begin{equation} \label{eq:polynomial def}
 \sum_{n \geqslant 0} P_{d,k}(n)\, x^n = \frac{(1+x)^k}{(1-x)^{d+1}}
\end{equation}
is the Ehrhart polynomial of the \((d-k)\)-fold pyramid over \(\mathcal P_k\). The relevant background can be found for example in \cite{ccd}. The figure below illustrates \(\mathcal P_3\) (left), the pyramid over \(\mathcal P_2\) (middle) and \(\mathcal P_2\) itself (right).

\bigskip
\begin{center}
 \begin{tikzpicture}[scale=1.7]
 \coordinate (1) at (1,0,0);
 \coordinate (-1) at (-1,0,0);
 \coordinate (2) at (0,1,0);
 \coordinate (-2) at (0,-1,0);
 \coordinate (3) at (0,0,1);
 \coordinate (-3) at (0,0,-1);

The coefficients of Ehrhart polynomials have been studied extensively, and they are usually hard to understand -- it is uncommon to be able to fully describe the coefficients for a given polytope. It is difficult even to determine if the coefficients are nonnegative, see \cite{ehrpos}. In this paper, we will examine these questions for (pyramids over) cross-polytopes. We define \(c_{d,k}(j)\) by
\begin{equation} \label{eqn}
 d! P_{d,k}(n) = \sum_{j=0}^d c_{d,k}(j) n^j,
\end{equation}
where the factor \(d!\) ensures that every coefficient is integral. Our main results are a combinatorial interpretation of the coefficients \(c_{d,k}(j)\), and a bijective proof of equation \eqref{eqn}. In particular, we prove combinatorially that the Ehrhart coefficients for (pyramids over) cross-polytopes are positive, which answers a question posed by Stanley \cite[Chapter 4, Exercise 1(b)]{ecnewex}.

The outline of the paper is as follows.
In Section \ref{sec:sri}, we construct a class of objects that are counted by the coefficients \(c_{d, k}(j)\), in particular showing that these coefficients are positive.
In Section \ref{sec:bij}, we give a combinatorial argument that proves \eqref{eqn} by constructing objects that are counted by both sides of the equation and exhibiting a bijection.
In Section \ref{sec:prop}, we use the combinatorial meaning we have for \(c_{d, k}(j)\) to obtain some properties of these coefficients.
Finally, in Section \ref{sec:poly}, we illustrate how the method we have used might be helpful for studying the Ehrhart coefficients of other polytopes.

\bigskip\noindent
Before proceeding with the general results, we will briefly describe a proof strategy for the simplest case \(k = 0\). The same ideas will serve as our guiding light for the rest of the paper.

The numbers \(c_{d, 0}(j)\) are just unsigned Stirling numbers of the first kind.
Recall that the (signed) Stirling number of the first kind \(s(m, i)\) is defined by
\begin{equation*}
 s(m, i) \coloneqq (-1)^{m - i} \cdot |\{\sigma \in S_m : \sigma \text{ has } i \text{ cycles}\}|.
\end{equation*}

\begin{equation} \label{eqn}
  d! P_{d,k}(n) = \sum_{j=0}^d c_{d,k}(j) n^j,
\end{equation}

\renewcommand{\epsilon}{\varepsilon}
 \renewcommand{\phi}{\varphi}
 \renewcommand{\emptyset}{\varnothing}
 \renewcommand{\leq}{\leqslant}
 \renewcommand{\geq}{\geqslant}
 }
lib.bib0000664000000000000000000000366315116032500011003 0ustar rootroot@article {bona_permutations_2000,
 AUTHOR = {B\'ona, Mikl\'os and McLennan, Andrew and White, Dennis},
 TITLE = {Permutations with roots},
 JOURNAL = {Random Struct. Algorithms},
 VOLUME = {17},
 YEAR = {2000},
 NUMBER = {2},
 PAGES = {157--167},
 ISSN = {1042-9832,1098-2418},
 DOI = {10.1002/1098-2418(200009)17:2<157::AID-RSA4>3.0.CO;2-2}
}

The numbers \(c_{d, 0}(j)\) are just unsigned Stirling numbers of the first kind.
Recall that the (signed) Stirling number of the first kind \(s(m, i)\) is defined by
\begin{equation*}
 s(m, i) \coloneqq (-1)^{m - i} \cdot |\{\sigma \in S_m : \sigma \text{ has } i \text{ cycles}\}|.
\end{equation*}

\begin{proof}
 From the definition of $P_{d, k}$, we see that $P_{d, 0}(n) = \binom{n + d}{d}$.
 The result now follows from the fact that $s(m, i)$ is the coefficient of $x^i$ in $x(x - 1) \cdots (x - m + 1)$ (see, for example, \cite[Proposition 1.3.7]{ec1}).
\end{proof}

Although we could stop here and say we have given combinatorial meaning to the coefficients $c_{d, 0}(j)$, we would ideally like to give combinatorial meaning to the fact that for any $n \geq 0$,
\begin{equation*}
 d!P_{d, 0}(n) = \sum_{j = 0}^d c_{d, 0}(j)n^j.
\end{equation*}
Especially since the polynomials $P_{d, k}$ are Ehrhart polynomials, it makes more sense to interpret $c_{d, k}(j)$ in this context rather than doing so independently.
Also note that if we show that $|s(d + 1, j + 1)|$ satisfies the above equation for all $n \geq 0$, then this implies that $c_{d, 0}(j) = |s(d + 1, j + 1)|$.

\begin{lemma}\label{lem:str}
 For any $d \geq 1$ and $k, j \in [0, d]$, we have
 \begin{equation*}
 c_{d, k}(j) = \sum_{i = 0}^k \binom{k}{i} \sum_{\ell = 0}^{d - i} |s(d - i + 1, \ell + 1)|s(i, j - \ell).
 \end{equation*}
\end{lemma}

\begin{proof}
 Note that for any permutation $\sigma \in S_m$ with $i$ cycles, the parity of $m - i$ is the same as the number of even cycles in $\sigma$.
 We now interpret the expression from Lemma \ref{lem:str} for $c_{d, k}(j)$ as the signed count of the following objects:
 Permutations of size $d + 1$ with $j + 1$ cycles where
 \begin{itemize}
 \item any cycle with all elements in $[k]$ is colored either red or blue,
 \item any cycle containing an element not in $[k]$ is uncolored, and
 \item the sign associated to the permutation is the number of even cycles that are colored red.
 \end{itemize}

\begin{lemma}
 For any $d \geq 1$ and $k \in [0, d]$, we have $c_{d, k}(0) = d!$ and $c_{d, k}(d) = 2^k$.
\end{lemma}

\begin{lemma}
 For any $d \geq 1$ and $k, j \in [0, d]$, we have $c_{d, k}(j) \leq c_{d + 1, k}(j)$.
\end{lemma}

\begin{proof}
    From the definition of $P_{d, k}$, we see that $P_{d, 0}(n) = \binom{n + d}{d}$.
    The result now follows from the fact that $s(m, i)$ is the coefficient of $x^i$ in $x(x - 1) \cdots (x - m + 1)$ (see, for example, \cite[Proposition 1.3.7]{ec1}).
\end{proof}

Although we could stop here and say we have given combinatorial meaning to the coefficients $c_{d, 0}(j)$, we would ideally like to give combinatorial meaning to the fact that for any $n \geqslant 0$,
\begin{equation*}
    d!P_{d, 0}(n) = \sum_{j = 0}^d c_{d, 0}(j)n^j.
\end{equation*}
Especially since the polynomials $P_{d, k}$ are Ehrhart polynomials, it makes more sense to interpret $c_{d, k}(j)$ in this context rather than doing so independently.
Also note that if we show that $|s(d + 1, j + 1)|$ satisfies the above equation for all $n \geqslant 0$, then this implies that $c_{d, 0}(j) = |s(d + 1, j + 1)|$.

To do this, we construct objects counted by $d!P_{d, 0}(n) = d!\binom{n + d}{d}$.
One choice is objects that consist of a permutation of size $d$ with $n$ balls distributed among the $d + 1$ slots created by the terms of the permutation.
For example, for $d = 9$ and $n = 6$, such an object is
\begin{equation*}
    \emptycircle\, 281\, \emptycircle\, 9\, \emptycircle\, \emptycircle\, 354\, \emptycircle\, 67\, \emptycircle\,.
\end{equation*}
It can also be shown that these objects are counted by $d!P_{d, 0}(n)$ using the interpretation of $P_{d, 0}(n)$ as an Ehrhart polynomial (see Remark \ref{ehrbij}).

We interpret $|s(d + 1, j + 1)|n^j$ as choosing a permutation of size $d + 1$ with $j + 1$ cycles as well as assigning the labels $1, 2, \ldots, j$ to $n$ balls.
Such a pair for $d = 8$, $j = 3$, and $n = 4$ is
\begin{equation*}
    (2) (7\ 3\ 5\ 4) (8\ 1) (9\ 6)\quad \text{and}\quad \emptycircle\, \overset{1,3}{\emptycircle}\, \emptycircle\, \overset{2}{\emptycircle}\,.
\end{equation*}

We write cycles in a permutation in \emph{canonical cycle form}: each cycle has its largest element first, and the cycles are ordered in increasing order of their largest elements.
The labels on balls indicate which cycles must be placed before it.
The label $1, 3$ on the second ball in the example above indicates that the first and third cycle should be placed before the second ball.
We do this by placing the cycles as they are into that slot and then removing the brackets.
This is just an instance of the fundamental bijection on permutations (see \cite[Proposition 1.3.1]{ec1}).
The cycle containing $d + 1$ is placed after the last ball and then the brackets and $d + 1$ are deleted.
Hence, the object corresponding to the example above is
\begin{equation*}
    \emptycircle\, 281\, \emptycircle\, \emptycircle\, 7354\, \emptycircle\, 6.
\end{equation*}

It can be checked that this is a bijection and gives us a proof that for all $n \geqslant 0$,
\begin{equation*}
    d!P_{d, 0}(n) = \sum_{j = 0}^d |s(d + 1, j + 1)| n^j
\end{equation*}
and in particular $c_{d, 0}(j) = |s(d + 1, j + 1)|$.
We will generalize the interpretation we have for $c_{d, 0}(j)$ in Section \ref{sec:sri} and then in Section \ref{sec:bij}, we will present a bijection generalizing the one we have just seen.

From the definition of $P_{d,k}$, $P_{d,0}(n)=\binom{n+d}{d}$. The theorem then follows since $s(m,i)$ is the coefficient of $x^i$ in $x(x-1)\cdots(x-m+1)$.

Equivalently, to show $c_{d,0}(j)=|s(d+1,j+1)|$, it suffices to prove that for all $n\ge 0$,
\[
 d!P_{d,0}(n)=\sum_{j=0}^d |s(d+1,j+1)|\,n^j.
\]
A combinatorial proof is outlined by:
(1) constructing objects counted by $d!P_{d,0}(n)=d!\binom{n+d}{d}$, e.g. “a permutation of size $d$ with $n$ balls distributed among the $d+1$ slots created by the terms of the permutation”;
(2) interpreting $|s(d+1,j+1)|n^j$ as “choosing a permutation of size $d+1$ with $j+1$ cycles as well as assigning the labels $1,2,\ldots,j$ to $n$ balls”, with cycles written in canonical cycle form and ball labels indicating “which cycles must be placed before it”;
(3) producing the permutation-with-slots object by placing the cycles into the indicated slots and “removing the brackets” (an instance of “the fundamental bijection on permutations”), then placing the cycle containing $d+1$ after the last ball and deleting the brackets and $d+1$.
It is stated that “It can be checked that this is a bijection,” yielding the identity above and hence $c_{d,0}(j)=|s(d+1,j+1)|$.

Question:
For integers \(d \geq 1\) and \(0 \leq j \leq d\), let the polynomial \(P_{d,0}(n)\) be defined by the generating function
\[
\sum_{n \geq 0} P_{d,0}(n)x^n=\frac{1}{(1-x)^{d+1}},
\]
and define integers \(c_{d,0}(j)\) by
\[
d!\,P_{d,0}(n)=\sum_{r=0}^d c_{d,0}(r)n^r.
\]
Also, for \(m\ge 0\) and \(0\le i\le m\), let the signed Stirling number of the first kind be
\[
s(m,i)=(-1)^{m-i}\,\bigl|\{\sigma\in S_m:\sigma\text{ has }i\text{ cycles}\}\bigr|.
\]
Which statement holds for every such pair \((d,j)\)?

Choices:
B. \(c_{d,0}(j)=|s(d,j)|\).
C. \(c_{d,0}(j)\ge 0\).
D. \(c_{d,0}(j)=|s(d+1,j)|\).
E. \(c_{d,0}(j)=(-1)^{d-j}|s(d+1,j+1)|\).

The stem gives definitions but does not explicitly state the Stirling-number identity or otherwise reveal choice A. There is no direct answer leakage.

The item is not a mere restatement of a theorem in the stem; it asks the solver to connect a generating-function/coefficient definition to a precise Stirling-number formula and distinguish among close variants.

Identifying A requires meaningful reasoning through binomial/rising-factorial expansions and the correct Stirling index shift. However, the presence of choice C, which is also true, lowers generative pressure because a weaker conclusion can be selected without deriving the exact identity.

The distractors are mathematically themed and some are plausible index/sign confusions, but choice C is a genuine weaker true statement. That makes the item ambiguous rather than cleanly discriminative, which is a serious distractor-quality flaw.

The question has no answer leakage and is substantively non-tautological, but it is flawed as an MCQ because it includes at least one additional true option (C). This ambiguity substantially weakens its ability to test precise generative reasoning.


Classical Ramsey theory typically asks which structured subsets must appear in any finite colouring of the natural numbers. Raimi proposed a complementary point of view: he asked which partitions of $\mathbb{N}$ cannot be avoided by any finite colouring of $\mathbb{N}$, even after allowing a shift.

More precisely, given a partition $\mathbb{N}=E_1\cup E_2$, we say that $(E_1,E_2)$ is \emph{unavoidable} if for every finite partition $\mathbb{N}=\bigcup\limits_{j=1}^t F_j$ there exist $j\in\{1,\dots,t\}$ and $k\in\mathbb{N}$ such that both $(F_j+k)\cap E_1$ and $(F_j+k)\cap E_2$ are infinite. This formalises Raimi’s viewpoint on unavoidable partitions under shifts.

A classical theorem of Raimi \cite{Raimi} shows that such partitions do exist: there is a partition $\mathbb{N}=E_1\cup E_2$ with the property that for every finite partition $\mathbb{N}=\bigcup\limits_{j=1}^t F_j$ with $t\in\mathbb{N}$, one can find $j\in\{1,\dots,t\}$ and $k\in\mathbb{N}$ such that both $(F_j+k)\cap E_1$ and $(F_j+k)\cap E_2$ are infinite. Raimi’s original proof used topological methods. Hindman later gave an elementary proof \cite[p.~180, Theorem~11.15]{HM79} and showed that one may take $E_1$ to be the set of natural numbers whose last non-zero digit in the ternary expansion is $1$, and $E_2=\mathbb{N}\setminus E_1$.

Strengthenings of this phenomenon, imposing density conditions on the partition sets or guaranteeing positive densities in the conclusion, were obtained by Hegyv\'ari \cite{NH} and by Bergelson and Weiss \cite{Bergelson2}. More recently, Hegyv\'ari, Pach, and Pham \cite{HPP25} introduced a powerful and flexible framework, combining tools from harmonic analysis, additive combinatorics, and group theory, which yields polynomial and finite-group extensions of Raimi’s theorem and makes its connection to Ramsey theory explicit. Their beautiful construction in the finite-group setting has since been extended successfully to the continuous setting for circles by Kang, Koh, and the author in \cite{KKT25}, which is stated as follows.

A classical theorem of Raimi \cite{Raimi} shows that such partitions do exist: there is a partition $\mathbb{N}=E_1\cup E_2$ with the property that for every finite partition $\mathbb{N}=\bigcup\limits_{j=1}^t F_j$ with $t\in\mathbb{N}$, one can find $j\in\{1,\dots,t\}$ and $k\in\mathbb{N}$ such that both $(F_j+k)\cap E_1$ and $(F_j+k)\cap E_2$ are infinite. Raimi’s original proof used topological methods. Hindman later gave an elementary proof \cite[p.~180, Theorem~11.15]{HM79} and showed that one may take $E_1$ to be the set of natural numbers whose last non-zero digit in the ternary expansion is $1$, and $E_2=\mathbb{N}\setminus E_1$.

Before stating the main results, we need to introduce some notation.

For every finite measurable cover
\[
\mathbb{S}^{n-1} \subset F_1\cup\cdots\cup F_t,
\]
there exist an index $m_0\in\{1,\dots,t\}$ and $\theta_0\in C$ such that
\[
\sigma_{n-1}\big(R_{\theta_0}(F_{m_0})
 \cap E_i^{\mathbb{S}^{n-1}}\big)>0
\qquad\text{for all }1\le i\le r.
\]
where $R_\theta$ denotes the rotation defined in \eqref{definition-rotation} as restricted to $\mathbb{S}^{n-1}$.
\end{theorem}

\begin{theorem}[Rotational power surfaces]\label{thm:power-surface-Raimi}
Let $n\ge 3$ and $k>0$. Define
\[
\mathcal{S}_{k,R}
 := \big\{(x',x_n)\in\mathbb{R}^{n-1}\times\mathbb{R}
 : x_n = |x'|^{\,k},\ 0<|x'|\le R\big\}.
\]
Let
$\sigma_{k,R}$ be the normalized surface measure on $\mathcal{S}_{k,R}$.
Then there exists a measurable partition
$\{E_i^{\mathcal{S}_{k,R}}\}_{i=1}^r$ of $\mathcal{S}_{k,R}$ with the following property:
For every finite measurable cover
\[
\mathcal{S}_{k,R} \subset F_1\cup\cdots\cup F_t
\]
there exist an index $m_0\in\{1,\dots,t\}$ and $\theta_0\in C$ such that
\[
\sigma_{k,R}\big(R_{\theta_0}(F_{m_0})
 \cap E_i^{\mathcal{S}_{k,R}}\big)>0
\qquad\text{for all }1\le i\le r.
\]
In particular, $k=1$ yields the cone, and $k=2$ yields the paraboloid.
\end{theorem}

For every finite measurable cover
\[
\mathcal{C}_{R,\Omega} \subset F_1\cup\cdots\cup F_t
\]
there exist an index $m_0\in\{1,\dots,t\}$ and $\theta_0\in C$ such that
\[
\mu_{\text{\tiny $R,\Omega$}}\big(R_{\theta_0}(F_{m_0})\cap E_i^{\mathcal{C}_{R,\Omega}}\big)>0
\qquad\text{for all }1\le i\le r,
\]
where $R_\theta$ denotes the rotation defined in \eqref{definition-rotation} as restricted to $\mathcal{C}_{R,\Omega}$.
\end{theorem}

We now present the following general theorem, of which the spherical, cylindrical, and rotational power surface cases are specific instances.
\begin{theorem}\label{thm:general-circle-bundle}
Let $(X,\mu)$ be a probability space equipped with a measurable, measure-preserving action
\[
\{R_\theta\}_{\theta\in C},\qquad C:=\mathbb{R}/\mathbb{Z},
\]
of the circle group $C$. Assume there exist
\begin{itemize}
 \item a probability space $(Y,\nu)$,
 \item a measurable set $N\subset X$ with $\mu(N)=0$,
 \item a measurable bijection
 \[
 \Phi : C\times Y \longrightarrow X\setminus N,
 \]
\end{itemize}
satisfying:
\begin{enumerate}
 \item[(i)] (\emph{Equivariance}) For all $\theta,\alpha\in C$ and $y\in Y$,
 \[
 \Phi(\theta+\alpha,y) = R_\alpha(\Phi(\theta,y)).
 \]
 \item[(ii)] (\emph{Product disintegration}) For every bounded measurable function $f:X\to\mathbb{R}$,
 \[
 \int_X f(x)\,d\mu(x)
 = \int_C \int_Y f(\Phi(\theta,y))\,d\nu(y)\,d\mu_1(\theta),
 \]
 where $\mu_1$ is the normalized Lebesgue measure on $C$.
\end{enumerate}
Let $\{E_i^C\}_{i=1}^r$ be a measurable partition of $C$ with the Raimi property from Theorem~\ref{thm:KKT-circle}. Define
\begin{equation*}
 \begin{cases}
 E_1^X := \Phi(E_1^C\times Y) \cup N,\\
 E_i^X:=\Phi(E_i^C\times Y), ~~~~~~~~2 \leq i\leq r.
 \end{cases}
\end{equation*}
Then $\{E_i^X\}_{i=1}^r$ is a measurable partition of $X$ with the following property:

Now define a measurable partition of $C$ by
\[
 C_m := \{\theta\in C : m(\theta)=m\},\qquad 1\le m\le t.
\]
By construction, for every $\theta\in C_m$ we have
\begin{equation}\label{eq:Am-lower-bound}
 \nu(A_m(\theta))\ge \frac1t.
\end{equation}
By applying Theorem~ \ref{thm:KKT-circle} to the partition $\{C_m\}_{m=1}^t$ of $C$, there exist an index $m_0\in\{1,\dots,t\}$ and a rotation
\[
 R_{\theta_0}: C\to C,\qquad R_{\theta_0}(\theta)=\theta+\theta_0,
\]
such that
\begin{equation}\label{eq:KKT-circle-hit}
 \mu_1\big(R_{\theta_0}(C_{m_0})\cap E_i^C\big)>0,
 \qquad\, \forall\, 1\le i\le r.
\end{equation}

To complete the proof of the theorem, it remains to use \eqref{eq:KKT-circle-hit} to show that
\begin{align}\label{ineq:intersection-rotation-E_i^X}
 \mu\bigl(R_{\theta_0}(F_{m_0}) \cap E_i^X\bigr) > 0
 \qquad \text{for all } 1 \le i \le r.
\end{align}
Let
\[
 R_{\theta_0}(\theta,y):=(\theta+\theta_0,y)
\]
be the induced rotation on $C\times Y$. By equivariance, we have
\[
 R_{\theta_0}(F_{m_0})\cap (X\setminus N)
 = R_{\theta_0}\big(\Phi(F_{m_0}^{\mathrm e})\big)
 = \Phi\big(R_{\theta_0}(F_{m_0}^{\mathrm e})\big),
\]
which, together with the definition of $E_i^X$, gives
\[
 R_{\theta_0}(F_{m_0})\cap E_i^X
 = \Phi\Big(R_{\theta_0}(F_{m_0}^{\mathrm e})\cap (E_i^C\times Y)\Big)
 \cup \Big(R_{\theta_0}(F_{m_0})\cap N\cap E_i^X\Big).
\]
By the assumption on $N$, the definition of definition of $F_{m_0}^{\mathrm e}$, and the rotation on $C\times Y$, we have
\begin{align*}
 \mu\big(R_{\theta_0}(F_{m_0})\cap E_i^X\big)
 =& (\mu_1\times\nu)\big(R_{\theta_0}(F_{m_0}^{\mathrm e})\cap (E_i^C\times Y)\big) \\
 = &\int_{C}\int_Y
 \mathbf{1}_{R_{\theta_0}(F_{m_0}^{\mathrm e})}(\theta,y)
 \mathbf{1}_{E_i^C}(\theta)\,d\nu(y)\,d\mu_1(\theta) \\
 =& \int_{C}\int_Y
 \mathbf{1}_{A_{m_0}(\theta-\theta_0)}(y)\mathbf{1}_{E_i^C}(\theta)
 \,d\nu(y)\,d\mu_1(\theta) \\
 =& \int_{C} \nu\big(A_{m_0}(\theta-\theta_0)\big)\,\mathbf{1}_{E_i^C}(\theta)\,d\mu_1(\theta).
\end{align*}
By restricting the outer integral to those $\theta$ satisfying
$\theta-\theta_0\in C_{m_0}$, i.e.\ $\theta\in R_{\theta_0}(C_{m_0})$, and using \eqref{eq:Am-lower-bound} together with \eqref{eq:KKT-circle-hit}, we obtain
\begin{align*}
 \mu\big(R_{\theta_0}(F_{m_0})\cap E_i^X\big)
 \geq & \int_{R_{\theta_0}(C_{m_0})\cap E_i^C} \nu\big(A_{m_0}(\theta-\theta_0)\big)\,d\mu_1(\theta) \\
 &\ge \frac1t\,\mu_1\big(R_{\theta_0}(C_{m_0})\cap E_i^C\big)>0
\end{align*}
This is precisely \eqref{ineq:intersection-rotation-E_i^X}. This completes the proof.
\end{proof}

\begin{align}\label{definition-rotation}
R_\theta(x_1,x_2,x'',x_n)
 :=\bigl(\,
        x_1\cos(2\pi\theta) - x_2\sin(2\pi\theta),\
        x_1\sin(2\pi\theta) + x_2\cos(2\pi\theta),\
        x'',\ x_n
      \,\bigr),
\end{align}

\begin{theorem}[Kang--Koh--Tran {\cite{KKT25}}]\label{thm:KKT-circle}
Let $r,t\in\mathbb{N}$ with $r,t\ge2$. There exists a measurable partition
\[
C = \bigcup_{i=1}^r E_i
\]
such that for every finite measurable cover
\[
C \subset F_1 \cup \cdots \cup F_t,
\]
there exist an index $m \in \{1,\dots,t\}$ and a rotation $R_{\theta}$ satisfying
\begin{align*}
    \mu_1\big(R_{\theta}(F_m) \cap E_i\big) > 0
\quad\text{for all } 1\le i\le r.
\end{align*}
\end{theorem}

The theorem provides a partition of the circle with the property that every finite measurable cover admits a translate meeting each partition element in positive measure.  This paper shows that this phenomenon extends beyond compact abelian groups to a wide class of non-group geometric surfaces that still exhibit \textit{a hidden one-dimensional symmetry}. This answers a question raised in \cite{KKT25} concerning the extension
from the circle $C$ to the unit sphere $\mathbb{S}^{n-1}\subset \mathbb{R}^n$.

Before stating the main results, we need to introduce some notation.

For notational convenience, given a point
$x=(x_1,\dots,x_n)\in\mathbb{R}^n$, we write
\[
x' := (x_1,\dots,x_{n-1}) \in \mathbb{R}^{\,n-1},
\qquad
x'' := (x_3,\dots,x_{n-1}) \in \mathbb{R}^{\,n-3}.
\]
Throughout the paper, we denote by $C=\mathbb{R}/\mathbb{Z}$ the circle group. Let $\{E_i^C\}_{i=1}^r$ be the measurable partition of $C$ provided by Theorem~\ref{thm:KKT-circle}.

We also introduce the rotation
\begin{align}\label{definition-rotation}
R_\theta(x_1,x_2,x'',x_n)
 :=\bigl(\,
        x_1\cos(2\pi\theta) - x_2\sin(2\pi\theta),\
        x_1\sin(2\pi\theta) + x_2\cos(2\pi\theta),\
        x'',\ x_n
      \,\bigr),
\end{align}
that is, rotation by angle $2\pi\theta$ in the $(x_1,x_2)$–plane.

\begin{theorem}[Spheres]\label{thm:sphere-Raimi}
Let $n\ge 3$,
\[
\mathbb{S}^{n-1} := \{x\in\mathbb{R}^n : |x|=1\}
\]
be the unit sphere equipped with the normalized surface measure $\sigma_{n-1}$.
Then there exists a measurable partition $\{E_i^{\mathbb{S}^{n-1}}\}_{i=1}^r$ of $\mathbb{S}^{n-1}$ such that:

For every finite measurable cover
\[
\mathbb{S}^{n-1} \subset F_1\cup\cdots\cup F_t,
\]
there exist an index $m_0\in\{1,\dots,t\}$ and $\theta_0\in C$ such that
\[
\sigma_{n-1}\big(R_{\theta_0}(F_{m_0})
    \cap E_i^{\mathbb{S}^{n-1}}\big)>0
\qquad\text{for all }1\le i\le r.
\]
where $R_\theta$ denotes the rotation defined in \eqref{definition-rotation} as restricted to $\mathbb{S}^{n-1}$.
\end{theorem}

\begin{theorem}[Rotational power surfaces]\label{thm:power-surface-Raimi}
Let $n\ge 3$ and $k>0$. Define
\[
\mathcal{S}_{k,R}
 := \big\{(x',x_n)\in\mathbb{R}^{n-1}\times\mathbb{R}
        : x_n = |x'|^{\,k},\ 0<|x'|\le R\big\}.
\]
Let
$\sigma_{k,R}$ be the normalized surface measure on $\mathcal{S}_{k,R}$.
Then there exists a measurable partition
$\{E_i^{\mathcal{S}_{k,R}}\}_{i=1}^r$ of $\mathcal{S}_{k,R}$ with the following property:
For every finite measurable cover
\[
\mathcal{S}_{k,R} \subset F_1\cup\cdots\cup F_t
\]
there exist an index $m_0\in\{1,\dots,t\}$ and $\theta_0\in C$ such that
\[
\sigma_{k,R}\big(R_{\theta_0}(F_{m_0})
       \cap E_i^{\mathcal{S}_{k,R}}\big)>0
\qquad\text{for all }1\le i\le r.
\]
In particular, $k=1$ yields the cone, and $k=2$ yields the paraboloid.
\end{theorem}

\begin{theorem}[Cylindrical surface]\label{thm:cylinder-Raimi}
Let $n\ge 3$, $R>0$, and $\Omega\subset\mathbb{R}^{n-2}$ be a
bounded Borel set. Define the cylindrical surface
\[
\mathcal{C}_{R,\Omega}
:= \big\{(x_1,x_2, x'', x_n)\in\mathbb{R}^2\times\mathbb{R}^{n-2}
 : x_1^2+x_2^2=R^2,\ (x'', x_n)\in\Omega\big\}.
\]
Let $\mu_{\text{\tiny $R,\Omega$}}$ be the normalized surface measure on $\mathcal{C}_{R,\Omega}$.
Then there exists a measurable partition $\{E_i^{\mathcal{C}_{R,\Omega}}\}_{i=1}^r$ of $\mathcal{C}_{R,\Omega}$ with the following property:

For every finite measurable cover
\[
\mathcal{C}_{R,\Omega} \subset F_1\cup\cdots\cup F_t
\]
there exist an index $m_0\in\{1,\dots,t\}$ and $\theta_0\in C$ such that
\[
\mu_{\text{\tiny $R,\Omega$}}\big(R_{\theta_0}(F_{m_0})\cap E_i^{\mathcal{C}_{R,\Omega}}\big)>0
\qquad\text{for all }1\le i\le r,
\]
where $R_\theta$ denotes the rotation defined in \eqref{definition-rotation} as restricted to $\mathcal{C}_{R,\Omega}$.
\end{theorem}

\medskip
\paragraph{Sketch of proof.}
The essential observation is that spheres, rotational power surfaces, and cylinders all carry a natural measure-preserving action of the circle $C$ given by rotation in the $(x_1,x_2)$-plane.  Moreover, each of these surfaces admits a measurable, measure-preserving trivialization
\[
\Phi : C\times Y \longrightarrow X\setminus N,
\]
where $Y$ is a suitable parameter space and $N$ is a lower-dimensional set of measure zero.  This allows the surface measure on $X$ to disintegrate as
\[
d\mu_X(x)
 = d\mu_1(\theta)\, d\nu_Y(y),
\]
mirroring the product structure of $C\times Y$.
The equivariance identity
\[
\Phi(\theta+\alpha,y)=R_\alpha(\Phi(\theta,y))
\]
then enables the measurable Raimi partition on the base circle to be lifted directly to a partition of $X$.

The proofs of our three geometric theorems reduce to verifying the structural hypotheses of the general circle-bundle theorem introduced in Section~\ref{section-main-result}.  Once this framework is in place, the partition constructed in the previous work of Kang, Koh, and the author on the circle $C$ automatically induces the desired Raimi-type partitions on $\mathbb{S}^{n-1}$, on rotational power surfaces, and on circular cylindrical surfaces.

Our approach naturally leads to the following open question:
Let $M$ be a compact hyperbolic surface with its normalized area measure.
Does $M$ admit a measurable Raimi--type partition (in the sense of this paper), even though it has no circle action and therefore lies outside our circle--bundle framework?

{\bf The paper is organized as follows.}
In Section~\ref{section-main-result}, we establish the general circle-bundle theorem, which serves as the main structural tool of the paper.  Sections~\ref{section-sphere}, \ref{section-power-surface}, and \ref{section-cylinder} are devoted to the proofs of the three principal applications: the sphere, the rotational power surfaces, and the circular cylindrical surfaces, respectively.  Each result follows by verifying the hypotheses of the general theorem and applying the measurable Raimi partition on the base circle.

The post-theorem text does not sketch a proof of Theorem~\ref{thm:KKT-circle}; it only states that Theorem~\ref{thm:KKT-circle} provides a partition of the circle with a translate/rotation intersection property and then uses it as an input for later results. The only explicit \paragraph{Sketch of proof.} concerns the later geometric theorems (sphere/power surfaces/cylinders): it observes these surfaces have a measure-preserving circle action by rotation in the $(x_1,x_2)$-plane, admits a measurable measure-preserving trivialization \(\Phi: C\times Y\to X\setminus N\) leading to disintegration \(d\mu_X=d\mu_1(\theta)\,d\nu_Y(y)\), uses equivariance \(\Phi(\theta+\alpha,y)=R_\alpha(\Phi(\theta,y))\) to lift the partition from the base circle, and reduces the proofs to verifying hypotheses of a general circle-bundle theorem and then applying the partition from Kang--Koh--Tran on \(C\).

The post-theorem text does not sketch a proof of the main theorem; it only states that the main theorem provides a partition of the circle with a translate/rotation intersection property and then uses it as an input for later results. The only explicit \paragraph{Sketch of proof.} concerns the later geometric theorems (sphere/power surfaces/cylinders): it observes these surfaces have a measure-preserving circle action by rotation in the $(x_1,x_2)$-plane, admits a measurable measure-preserving trivialization \(\Phi: C\times Y\to X\setminus N\) leading to disintegration \(d\mu_X=d\mu_1(\theta)\,d\nu_Y(y)\), uses equivariance \(\Phi(\theta+\alpha,y)=R_\alpha(\Phi(\theta,y))\) to lift the partition from the base circle, and reduces the proofs to verifying hypotheses of a general circle-bundle theorem and then applying the partition from the main theorem on \(C\).

Question: Let $C:=\mathbb{R}/\mathbb{Z}$ be the circle group with normalized Haar measure $\mu_1$, and for $\theta\in C$ let the rotation $R_\theta:C\to C$ be given by $R_\theta(x)=x+\theta \pmod 1$. Fix integers $r,t\in\mathbb{N}$ with $r,t\ge 2$. Which statement holds about measurable partitions of $C$ and measurable covers of $C$?

There exists a measurable partition $C=\bigcup_{i=1}^r E_i$ such that for every measurable cover $C\subset F_1\cup\cdots\cup F_t$, there exist an index $m\in\{1,\dots,t\}$ and a rotation $R_\theta$ with \[\mu_1\bigl(R_\theta(F_m)\cap E_i\bigr)>0\qquad\text{for all }1\le i\le r.\]

Choices:
B. There exists a measurable partition $C=\bigcup_{i=1}^r E_i$ such that for every measurable cover $C\subset F_1\cup\cdots\cup F_t$, there exist an index $m\in\{1,\dots,t\}$ and a rotation $R_\theta$ with \[\mu_1\bigl(R_\theta(F_m)\cap E_i\bigr)=1/r\qquad\text{for all }1\le i\le r.\]
C. There exists a measurable partition $C=\bigcup_{i=1}^r E_i$ such that for every measurable cover $C\subset F_1\cup\cdots\cup F_t$, there exist an index $m\in\{1,\dots,t\}$, a rotation $R_\theta$, and an index $i_0\in\{1,\dots,r\}$ with \[\mu_1\bigl(R_\theta(F_m)\cap E_{i_0}\bigr)>0.\]
D. For every measurable partition $C=\bigcup_{i=1}^r E_i$ and every measurable cover $C\subset F_1\cup\cdots\cup F_t$, there exist an index $m\in\{1,\dots,t\}$ and a rotation $R_\theta$ with \[\mu_1\bigl(R_\theta(F_m)\cap E_i\bigr)>0\qquad\text{for all }1\le i\le r.\]
E. There exists a measurable partition $C=\bigcup_{i=1}^r E_i$ and a rotation $R_\theta$ such that for every measurable cover $C\subset F_1\cup\cdots\cup F_t$, there exists an index $m\in\{1,\dots,t\}$ with \[\mu_1\bigl(R_\theta(F_m)\cap E_i\bigr)>0\qquad\text{for all }1\le i\le r.\]

The stem does not explicitly reveal the correct statement or uniquely signal choice A. It asks which statement holds, and the alternatives differ by subtle quantifier and strength changes rather than by obvious cues.

The item is not tautological from the stem alone, but it does function largely as a theorem-identification question: the correct option appears to be the exact intended theorem statement, while the others are nearby strengthenings or weakenings.

Answering correctly requires genuine reasoning about quantifier order, existence versus universality, and the difference between positivity, exact measure constraints, and cover-dependent rotations. The strongest valid conclusion is not obvious.

The distractors are mathematically meaningful and plausible. They reflect common failure modes: overstrengthening to exact densities, weakening to a single partition cell, replacing existential with universal quantification, and mishandling dependence on the cover.

A strong MCQ with little answer leakage and high-quality distractors that test careful reasoning about quantifiers and statement strength, though it is somewhat theorem-recognition based rather than fully non-tautological.