\begin{introtheorem}\label{intthm:character}
    If $H$ is a type B or type C Hessenberg space, there exist non-negative integers $a,b \in \mathbb{N}$, integers $c,d \in \{0,1\}$, and a subset $I \subseteq [n]$, each determined by $S(H)$, such that
    \[
     \lrep{H}_1 = a\,\mathbb{1} + \sum_{i\in I} \fh_i + b\,\fh_1 + c\,\fs + d\,\bm{\delta}
    \]
    and
    \[
     \rrep{H}_1 = \bm{\chi} + \sum_{i\in I} \left(\fh_i - \mathbb{1}\right) + b\left(\fh_1-\mathbb{1}\right) + c \left(
    \fs-\mathbb{1}\right) + d\,\bm{\delta},
    \]
    where
    \begin{itemize}
        \item $\bm{\chi}$ is the character of the defining representation of $\Wn$ on $\C^n$
        \item $\fh_i$ is the character of the action of $\Wn$ on the cosets of $\mathfrak{S}_i\times \mathfrak{W}_{n-i}$,
        \item $\fs$ is the character of the action of $\Wn$ on the cosets of $\mathfrak{W}_1\times \mathfrak{W}_{n-1}$,
        \item $\mathbb{1}$ is the character of the trivial representation, and
        \item $\bm{\delta}$ is the character $w\mapsto (-1)^{\abs{\Neg(w)}}$ where $\Neg(w) = \{w(i) \mid i \in [n],\; w(i) < 0\}$.
    \end{itemize}
    Moreover, $c$ and $d$ are never simultaneously $1$, and $d$ is always $0$ in type B.
\end{introtheorem}

\begin{proposition}\label{prop:permutohedral}
    Let $\bm{\chi}$ be the character of the defining representation of $\Wn$ (i.e. the action on $\C^n$). Let $\fh_i$ be the character of the action of $\Wn$ on the cosets of $\mathfrak{S}_i \times \mathfrak{W}_{n-i}$. Let $\mathbb{1}$ be the character of the trivial representation. Then
   \[
     \lrep{\Delta}_1 = \sum_{i=1}^n \fh_i - \bm{\chi}
    \hspace{1cm}\text{and}\hspace{1cm}
    \rrep{\Delta}_1 = \sum_{i=1}^n \fh_i - n\,\mathbb{1}.
    \]
\end{proposition}

\begin{definition}\label{def:itypes}
Consider $S(H)$ and $i \in [n]$. We call $i$\begin{itemize}
    \item \emph{uncovered} if $\begin{cases} \{t_{i-1},t_i\} \cap S(H) = \emptyset &\text{and } i \neq n-1\text{, or} \\ \{t_{n-1},t_{n-1},t_{n}\} \cap S(H) = \emptyset &\text{and } i = n-1\end{cases}$
    \item \emph{surrounded} if $i \in [n-2]$ and $\{t_{i-1},t_i\} \cap S(H) = \{t_{i-1}\}$ and $t_m \in S(H)\text{ for some } m>i$,
    \item \emph{shaded} if $t_i \in S(H)$ or if $t_n \in S(H)$ and $i=n-1$.
\end{itemize}
\end{definition}

\begin{theorem}\label{thm:character}
    Say $\Delta \subsetneq H$. Let $\bm{\chi}$ be the character of the defining representation of $\Wn$ (i.e. the action on $\C^n$). Let $\fh_i$ be the character of the action of $\Wn$ on the cosets of $\mathfrak{S}_i\times \mathfrak{W}_{n-i}$. Let $\fs$ be the character of the action of $\Wn$ on cosets of $\mathfrak{W}_1 \times \mathfrak{W}_{n-1}$. Let $\mathbb{1}$ be the character of the trivial representation and $\bm{\delta}$ be the character $w\mapsto (-1)^{\abs{\Neg(w)}}$. Then
    \begin{align*}
     \lrep{H}_1 = \abs{\{i \in [n] \mid i\text{ shaded}\,\}}\mathbb{1} &+ \sum_{i\text{ uncovered}} \fh_i \\[6pt]&+ \abs{\{i \in [n-2] \mid i\text{ surrounded}\,\}}\fh_1 \\[6pt]&+ \begin{cases}
    \fs &\text{if $t_n \notin S(H)$}\\
    \bm{\delta} &\text{if }\{t_{n-1},t_n\} \cap S(H) = \{t_{n}\}\\
    0 &\text{otherwise.}
\end{cases}
    \end{align*}
    and
        \begin{align*}
     \rrep{H}_1 = \bm{\chi} &+ \sum_{i\text{ uncovered}} \left(\fh_i - \mathbb{1}\right) \\[6pt]&+ \abs{\{i \in [n-2] \mid i\text{ surrounded}\,\}}\left(\fh_1-\mathbb{1}\right) \\[6pt]&+ \begin{cases}
    \fs-\mathbb{1} &\text{if $t_n \notin S(H)$}\\
    \bm{\delta} &\text{if }\{t_{n-1},t_n\} \cap S(H) = \{t_{n}\}\\
    0 &\text{otherwise.}
\end{cases}
    \end{align*}
\end{theorem}

\label{sec:intro}
Let $\mathfrak{W}_n$ be the group of signed permutations, which are permutations $w$ on the set $\{1,\ldots,n,-n,\ldots,-1\}$ such that $w(-i) = -w(i)$. The group $\mathfrak{W}_n$ is also the Weyl group of types B and C, and so is associated with two distinct root systems. A Hessenberg space $H$ is a particular subset of roots, which determines a set $S(H)$ of transpositions in $\mathfrak{W}_n$. Given a Hessenberg space, one can define the \emph{graded ring of splines} $\mathcal{M}_{H}$ on $\mathfrak{W}_n$. This ring has two $\C[x_\bullet]\coloneqq \C[x_1,\ldots,x_{n}]$-module structures and a $\mathfrak{W}_n$-module structure. This paper determines the algebraic and combinatorial properties of the degree-one graded piece of $\mathcal{M}_{H}$ from the combinatorics of $S(H)$.\par
Let $(i,j)$ be the unique transposition in $\mathfrak{W}_n$ that switches $i$ with $j$ and $-i$ with $-j$ (note $i=-j$ is possible). If $H$ is a Hessenberg space, then the ring of splines is
 \[
 \mathcal{M}_{H} = \bigoplus_{i \geq 0} \mathcal{M}_{H}^i \coloneqq \left\{ \left.\bm{\rho} \in \prod_{w \in \mathfrak{W}_n} \C[x_\bullet]\; \right|\; \bm{\rho}(w) - \bm{\rho}(w(i,j)) \in \left\langle x_{w(i)}-x_{w(j)} \right\rangle\text{ if } (i,j) \in S(H) \right\}\!,
 \]
with (graded) $\mathfrak{W}_n$-module structure $w\cdot \bm{\rho}(v) = w\bm{\rho}(w^{-1}v)$ and (graded) $\C[x_\bullet]$-module structure given by multiplication. \par
The ring $\mathcal{M}_{H}$ is isomorphic to the $T$-equivariant cohomology of a smooth subvariety of the full flag variety $G/B$ called a \emph{regular semisimple Hessenberg variety}. The connection between this combinatorial ring and equivariant cohomology is called \emph{GKM Theory} \cite{GKM_theory}, and it allows for the study of this cohomology in purely combinatorial and algebraic terms. \par
The $\mathfrak{W}_n$-module structure on $\mathcal{M}_{H}$ was first defined as the \emph{dot action} on equivariant cohomology by Tymoczko in \cite{tymoczko2008permutation}. There are two natural $\mathfrak{W}_n$-equivariant quotients, called the left $\mathrm{L}_{H}$ and right $\mathrm{R}_{H}$ quotients, of $\mathcal{M}_{H}$ that are in fact graded $\mathbb{C}$-vector spaces (the left quotient corresponds to ordinary cohomology). The graded $\mathfrak{W}_n$-module structure of $\mathcal{M}_{H}$ induces graded $\mathfrak{W}_n$-representations on the quotients. The primary goal of this paper is to compute the characters of the degree-one pieces of
\[
\mathrm{L}_{H} = \bigoplus_{i\geq 0} \left(\mathrm{L}_{H}\right)_i \;\; \text{ and }\;\;\mathrm{R}_{H} = \bigoplus_{i\geq 0} \left(\mathrm{R}_{H}\right)_i
\] from the combinatorial data of $S(H)$. In doing so, we compute the dot action on second (equivariant and ordinary) cohomology for all regular semisimple Hessenberg varieties in types B and C. This generalizes what is known in type A \cite{chow_linearp,chohonglee_second_cohom,ayzenberg2022second}.\par
These graded representations are of interest to algebraic combinatorists in part because they are the $\mathfrak{W}_n$-equivalents of very well-studied $\mathfrak{S}_n$-representations, which have connections to chromatic symmetric functions \cite{guaypaquet2016shar_wachs_conj,SW2016chromaticquasisymmetric,brosnan_chow_dotactn_is_chromsym} and LLT polynomials \cite{guaypaquet2016shar_wachs_conj,Ayzenberg2018isospectral, ALEXANDERSSON2018LLTchromsym}. We detail some of these connections in the Appendix. From these works, in type A there are combinatorial formulas to compute the characters of both $\mathrm{L}_{H}$ and $\mathrm{R}_{H}$ in at least one basis for the character space of $\mathfrak{S}_n$. This is \textbf{not} the case in types B or C.\par
For one specific family of $H$ (those corresponding to simple roots), the character of $\mathrm{L}_{H}$ was computed by Stembridge \cite{Stembridge_Permuto} in general type. Stembridge's results are via a connection to \emph{permutohedra}, as the associated Hessenberg varieties are in fact toric varieties. While it is possible to compute the left and right characters for general $H$ in types B and C, even partial results are rare and rely on sophisticated geometric tools \cite{BalibanuCrooks_sheaves,PrecupSommers_sheaves}. Formulas are non-combinatorial (and usually non-positive), require the computation of type B/C Green polynomials and Poincar\'e polynomials of other varieties, and proofs require the geometry of perverse sheaves. \par
In contrast, for the degree-one graded piece, the main result of this paper applies to all Hessenberg spaces $H$ and is entirely computable from the combinatorics of roots and transpositions in $\mathfrak{W}_n$. Moreover, we compute the degree-one pieces of the characters $\mathfrak{ch}\left(\mathrm{L}_{H}\right)$ and $\mathfrak{ch}\left(\mathrm{R}_{H}\right)$ in a positive manner.

\setcounter{tocdepth}{1}

Our theorem is the first type B/C result to provide an explicit non-negative expansion for a graded piece of $\lrep{H}$ or $\rrep{H}$ in terms of characters of irreducible- and/or permutation-representations for all Hessenberg spaces. It is also the first to compute the characters for any graded piece of $\lrep{H}$ and $\rrep{H}$ by directly providing a basis for the representation, and in doing so gives a family of characters with which one can attempt to expand the catalog of results and connections that exist in type A to types B and C.\par

The right character follows from the general procedure of ``removing" from any permutation representation with permutation basis $\{v_i\}$ the trivial sub-representation $\C\left\{ \sum v_i \right\}$. In particular, if $\mathfrak{p}$ is the character of a representation with permutation basis $\{v_i\}$, then the character on the complementary subspace $\C\{v_i-v_{i+1}\}$ is $\mathfrak{p}-\mathbb{1}$.
\end{proof}
In type B, $t_n \in S(H)$ implies that $t_{n-1} \in S(H)$, and so it is impossible to get the $\bm{\delta}$ term from Theorem \ref{thm:character}. In type C the opposite is true: $t_{n-1} \in S(H)$ implies that $t_n \in S(H)$, and so $\delta$ may appear. This is the ``moreover" part of Theorem \ref{intthm:character}.
\begin{example}
 The table below gives every $\lrep{H}_1$ and $\lrep{H}_1$ for $n=4$.
\begin{table}[H]
\centering
\renewcommand{\arraystretch}{1.5}
\begin{tabular}{|c|p{2in}|p{2in}|p{0.3in}|}\hline
 { $t_i \in S(H)$} & {\large $\lrep{H}_1$} & \large $\rrep{H}_1$ &\large dim\\\hline
 $\emptyset$ & $\fh_1 +\fh_2+\fh_3+\fh_4-\bchi$ & $\fh_1 +\fh_2+\fh_3+\fh_4-4\triv$ & $76$ \\\hline
 $\{t_1\}$ & $\triv + \fh_3 + \fh_4 + \fs$ &$\bchi + \fh_3 + \fh_4 + \fs -3\triv$ & $53$\\\hline
 $\{t_2\}$ & $\triv+\fh_1+\fh_4+\fs $ &$\bchi+\fh_1+\fh_4+\fs -3\triv$ &$29$\\\hline
 $\{t_3\}$ & $\triv+\fh_1+\fh_2+\fs $ &$\bchi+\fh_1+\fh_2+\fs -3\triv$ &$37$\\\hline
 $\{t_4\}$ & $2\triv+\fh_1+\fh_2+\bm{\delta}$ &$\bchi+\fh_1+\fh_2+\bm{\delta}-2\triv$ &$35$\\\hline
 $\{t_1,t_2\}$ & $2\triv+\fh_4+\fs $ &$\bchi+\fh_4+\fs -2\triv$ &$22$\\\hline
 $\{t_1,t_3\}$ & $2\triv+\fh_1+\fs $ &$\bchi+\fh_1+\fs -2\triv$ &$14$\\\hline
 $\{t_1,t_4\}$ & $3\triv+\fh_1 +\bm{\delta}$ &$\bchi+\fh_1 +\bm{\delta}-\triv$ &$12$\\\hline
 $\{t_2,t_3\}$ & $2\triv+\fh_1+\fs $ &$\bchi+\fh_1+\fs -2\triv$ &$14$\\\hline
 $\{t_2,t_4\}$ & $3\triv+\fh_1+\bm{\delta} $ &$\bchi+\fh_1+\bm{\delta} -\triv$ &$12$\\\hline
 $\{t_3,t_4\}$ & $2\triv+\fh_1+\fh_2 $ &$\bchi+\fh_1+\fh_2 -2\triv$ &$34$\\\hline
 $\{t_1,t_2,t_3\}$ &$3\triv+\fs $ &$\bchi+\fs -\triv$ &$7$\\\hline
 $\{t_1,t_2,t_4\}$ &$4\triv+\bm{\delta} $ &$\bchi+\bm{\delta} $ &$5$\\\hline
 $\{t_1,t_3,t_4\}$ &$3\triv+\fh_1$ &$\bchi+\fh_1-\triv$ &$11$\\\hline
 $\{t_2,t_3,t_4\}$ & $3\triv+\fh_1 $ &$\bchi+\fh_1 -\triv$ &$11$\\\hline
 $\{t_1,t_2,t_3,t_4\}$ & $4\mathbb{1}$ &$\bchi$ &$4$\\\hline

\clearpage
\begin{table}[H]
 \centering
 \renewcommand{\arraystretch}{2.5}
 \begin{tabular}{|p{3.5in}|p{2.3in}|}\hline
 {\large Representations or Characters of $\Wn$} & {\large Expansions in $\Lambda_n(x,y)$} \\\hline
 The trivial character $\mathbb{1}$ & $s_{(n),\emptyset} = h_{(n),\emptyset}$ \\\hline
 The sign character $\mathrm{sgn}$ & $s_{\emptyset,(1^n)}$ \\\hline
 $\bm{\delta}$ where $\bm{\delta}(w) = (-1)^{\abs{\Neg(w)}}$ & $s_{\emptyset,(n)} = h_{\emptyset,(n)}$ \\\hline
 The irreducible character $\chi^{(\lambda,\mu)}$ & $s_{\lambda,\mu}$ \\\hline
 $\mathrm{sgn} \otimes \chi^{(\lambda,\mu)} $ & $s_{\mu^t,\lambda^t}$ \\\hline
 $\bm{\delta} \otimes \chi^{(\lambda,\mu)} $ & $s_{\mu,\lambda}$ \\\hline
 The defining representation on $\C^n$ (so $\bm{\chi}$) & $h_{(n-1),(1)}$ \\\hline
 Action on cosets of $\mathfrak{S}_k\times \mathfrak{W}_{n-k}$ (so $\fh_k$) & $\ds h_{(n-k),\emptyset} \sum_{j=0}^{\lambda_i} h_{(j),(k-j)}$ \\\hline
 Action on cosets of $\mathfrak{W}_1\times \mathfrak{W}_{n-1}$ (so $\fs$) & $\ds h_{(n-1,1),\emptyset}$ \\\hline
 Action on cosets of $\mathfrak{S}_\lambda \times \mathfrak{W}_\mu$ & $\ds h_{\mu,\emptyset} \prod_{\lambda_i \in \lambda} \sum_{j=0}^{\lambda_i} h_{(j),(\lambda_i-j)}$ \\\hline
 Action on cosets of the type D${}_n$ subgroup & $\ds h_{(n),\emptyset} + h_{\emptyset,(n)} $ \\\hline
 An induced character $\left(\chi \times \theta\right)_{\mathfrak{W}_k \times \mathfrak{W}_\ell}^{\Wn}$ & $\Frob_{BC}(\chi)\Frob_{BC}(\theta)$\\\hline
 The induced character $\left(\chi^{(\lambda,\mu)} \times \chi^{(\alpha,\beta)}\right)_{\mathfrak{W}_k \times \mathfrak{W}_\ell}^{\Wn}$ & $(s_\lambda(x)\cdot s_\alpha(x))(s_\mu(y)\cdot s_\beta(y))$\\\hline
 The induced character $\left(\mathbb{1}\right)_{\Sn}^{\Wn}$ & ${\ds \sum_{k=0}^n s_{(k),(n-k)} = \sum_{k=0}^n h_{(k),(n-k)}}$ \\\hline
 The induced character $\left(\mathbb{1} \times \bm{\delta} \right)_{\mathfrak{W}_\lambda \times \mathfrak{W}_\mu}^{\Wn}$ & $\ds h_{\lambda,\mu} = \sum_{\gamma}\sum_{\nu} K_{\gamma,\lambda}K_{\nu,\mu}s_\gamma(x)s_\nu(y)$ \\\hline
 \end{tabular}\vspace*{4pt}
 \captionsetup{width=6.2in}
 \caption{Representations of $\Wn$ and Frobenius characteristics \cite{GeissKinch_hyperoctChars,Macdonald_symfuncs, Zelevinsky_reps_classicalgroups, Stembridge_Permuto, Stembridge_guide, Skandera_hyperoct}.}
 \label{tab:reps}
\end{table}
\clearpage

Theorem \ref{intthm:character} is Proposition \ref{prop:permutohedral} if $H = \Delta$ and Theorem \ref{thm:character} otherwise, which use language from Definition \ref{def:itypes} to describe the coefficients $a,b,c$, and $d$. The representation ring of $\mathfrak{W}_n$ is isomorphic to the \emph{type B/C symmetric functions} via the Frobenius characteristic map \cite{Macdonald_symfuncs}. The tools to translate Theorem \ref{intthm:character} into type B/C symmetric functions are in the Appendix. There, we make the observation that $\mathfrak{ch}\left(\mathrm{L}_{H}\right)_1$ is in fact $h_{\lambda,\mu}$-positive. This is particularly interesting, as it generalizes the behavior seen in type A, where the left graded character is conjectured to be $h_\lambda$-positive and is intimately connected to \emph{chromatic symmetric functions}. This is known as the graded Stanley-Stembridge conjecture \cite{StanleyStembridge,SW2016chromaticquasisymmetric}, and has been the subject of much research \cite{STANLEY1995chromsym, Gasharov96, GuayPaquet13,  harada2017cohomology, brosnan_chow_dotactn_is_chromsym,Dahlberg19, Abreu_Nigro20}. A proof of the ungraded conjecture was recently given by Hikita \cite{Hikita_stanstem}.\par

Our theorem is the first type B/C result to provide an explicit non-negative expansion for a graded piece of $\mathfrak{ch}\left(\mathrm{L}_{H}\right)$ or $\mathfrak{ch}\left(\mathrm{R}_{H}\right)$ in terms of characters of irreducible- and/or permutation-representations for all Hessenberg spaces. It is also the first to compute the characters for any graded piece of $\mathfrak{ch}\left(\mathrm{L}_{H}\right)$ and $\mathfrak{ch}\left(\mathrm{R}_{H}\right)$ by directly providing a basis for the representation, and in doing so gives a family of characters with which one can attempt to expand the catalog of results and connections that exist in type A to types B and C.\par

The paper is structured as follows. Section \ref{sec:background} provides some of the necessary background on signed permutations, Hessenberg spaces, and splines. Section \ref{sec:ideals_of_transpositions} shows how to translate from Hessenberg spaces $H$ to sets $S(H)$ of signed transpositions in a manner that unifies some type B and type C calculations. Section \ref{sec:trivial} reduces the $S(H)$ that one must consider in order to compute $\mathcal{M}_{H}^1$ to a much smaller collection, one that completely unifies the type B and type C calculations. Section \ref{sec:one_inversion} determines which elements of $\mathfrak{W}_n$ have exactly one $H$-inversion for each $H$. Section \ref{sec:linear_splines} defines several sets of splines contained in $\mathcal{M}_{H}^1$, provides linear relations between them, and computes the dot action on them. Section \ref{sec:generators} argues, using the elements from \S \ref{sec:one_inversion}, that the splines from \S \ref{sec:linear_splines} form a $\mathbb{C}$-generating set for $\mathcal{M}_{H}^1$, and uses the linear relations between them to reduce the size of this generating set. Section \ref{sec:bases_reps} reduces this generating set further in two different ways, one that results in a basis for $\mathcal{M}_{H}^1$ that is conducive to computing $\mathfrak{ch}\left(\mathrm{L}_{H}\right)_1$ and one that is conducive to computing $\mathfrak{ch}\left(\mathrm{R}_{H}\right)_1$. Then, we prove the main theorem. Appendix \ref{sec:symmetricfunctions} describes how one can translate this result to the language of type B/C symmetric functions.

Theorem~\ref{intthm:character} is obtained by giving an explicit basis for the relevant graded piece and computing the dot action on it: the paper “directly provid[es] a basis for the representation,” and thereby computes “the characters for any graded piece of $\mathfrak{ch}(\mathrm{L}_{H})$ and $\mathfrak{ch}(\mathrm{R}_{H})$” (in particular degree $1$ here). The outlined proof route is:
\begin{itemize}
\item Translate from Hessenberg spaces $H$ to sets $S(H)$ of signed transpositions (Section~\ref{sec:ideals_of_transpositions}).
\item Reduce the $S(H)$ that must be considered to compute $\mathcal{M}_{H}^1$ to “a much smaller collection” (Section~\ref{sec:trivial}).
\item Determine which elements of $\mathfrak{W}_n$ have “exactly one $H$-inversion for each $H$” (Section~\ref{sec:one_inversion}).
\item Define “several sets of splines contained in $\mathcal{M}_{H}^1$,” give “linear relations between them,” and “compute the dot action on them” (Section~\ref{sec:linear_splines}).
\item Use the one-$H$-inversion elements to argue these splines form a $\mathbb{C}$-generating set for $\mathcal{M}_{H}^1$, then use the linear relations to “reduce the size of this generating set” (Section~\ref{sec:generators}).
\item Further reduce to bases “conducive to computing $\mathfrak{ch}(\mathrm{L}_{H})_1$” and “conducive to computing $\mathfrak{ch}(\mathrm{R}_{H})_1$,” and then “prove the main theorem” (Section~\ref{sec:bases_reps}).
\end{itemize}
Additionally, the Appendix develops the translation to type B/C symmetric functions and notes the positivity phenomenon: “we make the observation that $\mathfrak{ch}(\mathrm{L}_{H})_1$ is in fact $h_{\lambda,\mu}$-positive.”

In establishing the main theorem, one gives an explicit basis for the relevant graded piece and computes the dot action on it: the paper “directly provid[es] a basis for the representation,” and thereby computes “the characters for any graded piece of $\mathfrak{ch}(\mathrm{L}_{H})$ and $\mathfrak{ch}(\mathrm{R}_{H})$” (in particular degree $1$ here). The outlined proof route is:
\begin{itemize}
\item Translate from Hessenberg spaces $H$ to sets $S(H)$ of signed transpositions.
\item Reduce the $S(H)$ that must be considered to compute $\mathcal{M}_{H}^1$ to “a much smaller collection”.
\item Determine which elements of $\mathfrak{W}_n$ have “exactly one $H$-inversion for each $H$”.
\item Define “several sets of splines contained in $\mathcal{M}_{H}^1$,” give “linear relations between them,” and “compute the dot action on them”.
\item Use the one-$H$-inversion elements to argue these splines form a $\mathbb{C}$-generating set for $\mathcal{M}_{H}^1$, then use the linear relations to “reduce the size of this generating set”.
\item Further reduce to bases “conducive to computing $\mathfrak{ch}(\mathrm{L}_{H})_1$” and “conducive to computing $\mathfrak{ch}(\mathrm{R}_{H})_1$,” and then prove the main theorem.
\end{itemize}
Additionally, the Appendix develops the translation to type B/C symmetric functions and notes the positivity phenomenon: “we make the observation that $\mathfrak{ch}(\mathrm{L}_{H})_1$ is in fact $h_{\lambda,\mu}$-positive.”

\label{intthm:character}
    If $H$ is a type B or type C Hessenberg space, there exist non-negative integers $a,b \in \mathbb{N}$, integers $c,d \in \{0,1\}$, and a subset $I \subseteq [n]$, each determined by $S(H)$, such that
    \[
     \mathfrak{ch}\left(\mathrm{L}_{H}\right)_1 = a\,\mathbb{1} + \sum_{i\in I} \fh_i + b\,\fh_1 + c\,\mathfrak{s} + d\,\bm{\delta}
    \]
    and
    \[
     \mathfrak{ch}\left(\mathrm{R}_{H}\right)_1 = \bm{\chi} + \sum_{i\in I} \left(\fh_i - \mathbb{1}\right) + b\left(\fh_1-\mathbb{1}\right) + c \left(
    \mathfrak{s}-\mathbb{1}\right) + d\,\bm{\delta},
    \]
    where
    \begin{itemize}
        \item $\bm{\chi}$ is the character of the defining representation of $\mathfrak{W}_n$ on $\mathbb{C}^n$
        \item $\fh_i$ is the character of the action of $\mathfrak{W}_n$ on the cosets of $\mathfrak{S}_i\times \mathfrak{W}_{n-i}$,
        \item $\mathfrak{s}$ is the character of the action of $\mathfrak{W}_n$ on the cosets of $\mathfrak{W}_1\times \mathfrak{W}_{n-1}$,
        \item $\mathbb{1}$ is the character of the trivial representation, and
        \item $\bm{\delta}$ is the character $w\mapsto (-1)^{\left| \Neg(w) \right|}$ where $\Neg(w) = \{w(i) \mid i \in [n],\; w(i) < 0\}$.
    \end{itemize}
    Moreover, $c$ and $d$ are never simultaneously $1$, and $d$ is always $0$ in type B.,

Question: Let \(\mathfrak{W}_n\) be the group of signed permutations, i.e. permutations \(w\) of \(\{1,\ldots,n,-n,\ldots,-1\}\) satisfying \(w(-i)=-w(i)\). Let \(H\) be a type B or type C Hessenberg space, and let \(S(H)\) be its associated set of transpositions in \(\mathfrak{W}_n\). Let \((\mathrm{L}_H)_1\) and \((\mathrm{R}_H)_1\) denote the degree-one graded pieces of the left and right \(\mathfrak{W}_n\)-representation quotients associated to \(H\), and let \(\mathfrak{ch}((\mathrm{L}_H)_1)\) and \(\mathfrak{ch}((\mathrm{R}_H)_1)\) be their characters. Here \(\mathbb{1}\) is the trivial character, \(\bm{\chi}\) is the character of the defining representation of \(\mathfrak{W}_n\) on \(\mathbb{C}^n\), \(\mathfrak{h}_i\) is the character of the action of \(\mathfrak{W}_n\) on the cosets of \(\mathfrak{S}_i\times \mathfrak{W}_{n-i}\), \(\mathfrak{s}\) is the character of the action of \(\mathfrak{W}_n\) on the cosets of \(\mathfrak{W}_1\times \mathfrak{W}_{n-1}\), and \(\bm{\delta}\) is the character \(w\mapsto (-1)^{|\Neg(w)|}\), where \(\Neg(w)=\{w(i)\mid i\in[n],\; w(i)<0\}\). Which of the following conclusions about these degree-one characters holds?

There exist non-negative integers \(a,b\in\mathbb{N}\), integers \(c,d\in\{0,1\}\), and a subset \(I\subseteq [n]\), each determined by \(S(H)\), such that
\[
\mathfrak{ch}\big((\mathrm{L}_H)_1\big)=a\,\mathbb{1}+\sum_{i\in I}\mathfrak{h}_i+b\,\mathfrak{h}_1+c\,\mathfrak{s}+d\,\bm{\delta},
\]
and
\[
\mathfrak{ch}\big((\mathrm{R}_H)_1\big)=\bm{\chi}+\sum_{i\in I}(\mathfrak{h}_i-\mathbb{1})+b(\mathfrak{h}_1-\mathbb{1})+c(\mathfrak{s}-\mathbb{1})+d\,\bm{\delta}.
\]
Moreover, \(c\) and \(d\) are never simultaneously equal to \(1\), and in type B one always has \(d=0\).

Choices:
B.
There exist non-negative integers \(a,b\in\mathbb{N}\), integers \(c,d\in\{0,1\}\), and a subset \(I\subseteq [n]\), each determined by \(S(H)\), such that
\[
\mathfrak{ch}\big((\mathrm{L}_H)_1\big)=a\,\mathbb{1}+\sum_{i\in I}\mathfrak{h}_i+b\,\mathfrak{h}_1+c\,\mathfrak{s}+d\,\bm{\delta},
\]
and
\[
\mathfrak{ch}\big((\mathrm{R}_H)_1\big)=\bm{\chi}+\sum_{i\in I}(\mathfrak{h}_i-\mathbb{1})+b(\mathfrak{h}_1-\mathbb{1})+c(\mathfrak{s}-\mathbb{1})+d\,\bm{\delta}.
\]
Moreover, one always has \(c=d=0\) in type B, and in type C the coefficients \(c\) and \(d\) may both be equal to \(1\).
C.
There exist non-negative integers \(a,b\in\mathbb{N}\), integers \(c,d\in\{0,1\}\), and a subset \(I\subseteq [n]\), each determined by \(S(H)\), such that
\[
\mathfrak{ch}\big((\mathrm{L}_H)_1\big)=a\,\mathbb{1}+\sum_{i\in I}\mathfrak{h}_i+b\,\mathfrak{h}_1+c\,\mathfrak{s}+d\,\bm{\delta},
\]
and
\[
\mathfrak{ch}\big((\mathrm{R}_H)_1\big)=\bm{\chi}+\sum_{i\in I}(\mathfrak{h}_i-\mathbb{1})+b(\mathfrak{h}_1-\mathbb{1})+c(\mathfrak{s}-\mathbb{1})+d\,\bm{\delta}.
\]
D.
There exist non-negative integers \(a,b\in\mathbb{N}\), integers \(c,d\in\{0,1\}\), and a subset \(I\subseteq [n]\), each determined by \(S(H)\), such that
\[
\mathfrak{ch}\big((\mathrm{L}_H)_1\big)=a\,\mathbb{1}+\sum_{i\in I}\mathfrak{h}_i+b\,\mathfrak{h}_1+c\,\mathfrak{s}+d\,\bm{\delta},
\]
and
\[
\mathfrak{ch}\big((\mathrm{R}_H)_1\big)=\bm{\chi}+\sum_{i\in I}(\mathfrak{h}_i-\mathbb{1})+b(\mathfrak{h}_1-\mathbb{1})+c(\mathfrak{s}-\mathbb{1})+d(\bm{\delta}-\mathbb{1}).
\]
Moreover, \(c\) and \(d\) are never simultaneously equal to \(1\), and in type B one always has \(d=0\).
E.
There exist non-negative integers \(a,b\in\mathbb{N}\), integers \(c,d\in\{0,1\}\), and a subset \(I\subseteq [n-1]\), each determined by \(S(H)\), such that
\[
\mathfrak{ch}\big((\mathrm{L}_H)_1\big)=a\,\mathbb{1}+\sum_{i\in I}\mathfrak{h}_i+b\,\mathfrak{h}_1+c\,\mathfrak{s}+d\,\bm{\delta},
\]
and
\[
\mathfrak{ch}\big((\mathrm{R}_H)_1\big)=\bm{\chi}+\sum_{i\in I}(\mathfrak{h}_i-\mathbb{1})+b(\mathfrak{h}_1-\mathbb{1})+c(\mathfrak{s}-\mathbb{1})+d\,\bm{\delta}.
\]
Moreover, \(c\) and \(d\) are never simultaneously equal to \(1\), and in type B one always has \(d=0\).

The stem defines notation and context but does not explicitly reveal the correct conclusion. There is no direct textual leakage of the extra conditions that distinguish the correct option.

The item is essentially a theorem-identification question: the correct choice states a full structural conclusion, and the task is to recognize the exact version. It is not a pure tautology, since alternatives differ in meaningful details, but it is close to restating a known result.

Selecting the answer requires careful comparison of subtle representation-theoretic details, but mostly tests precise recall/discrimination rather than genuine generative derivation or synthesis.

The distractors are plausible and mathematically targeted: one weakens the theorem, others alter boundary conditions or a specific character term, and they align with realistic failure modes such as dropping constraints or confusing basis terms.

A solid but theorem-recognition-heavy MCQ: little answer leakage and strong distractors, but only moderate success at avoiding tautological recall and inducing genuine generative reasoning.


\begin{theorem}\label{Thm: main1}
    Assume that $(M^n,g,E)$ is a spin generalized asymptotically flat manifold with nonnegative scalar curvature. Then we have the following mass-capacity inequality
    $$m(M,g,E)\geq 2\mathfrak c(M,g,E).$$
    If the equality holds, then $(M,g)$ is harmonically conformal  to $\mathbb R^n\setminus S$, where $S$ is a bounded closed subset of $\mathbb R^n$ with Hausdorff dimension no greater than $\frac{n-2}{2}$. In particular, we have
    $$\pi_i(M)=0\mbox{ for all }1\leq i\leq n-1- \left\lfloor\frac{n}{2}\right\rfloor.$$
\end{theorem}

The investigation of mass constitutes a central topic in  general relativity. Notably, the basic concept of mass for isolated gravitational systems was first formulated by Arnowitt-Deser-Misner \cite{Arnowitt-Deser-Misner}, with such systems modeled as asymptotically flat manifolds. Later, Bartnik \cite{Bartnik1986} and also Chru\'sciel \cite{Chru86} showed that the mass is indeed a geometric quantity of asymptotically flat manifolds. We recall the definition of asymptotically flat manifolds and mass as follows.
\begin{definition}
    A complete Riemannian $n$-manifold $(M,g)$ with dimension $n\geq 3$ is asymptotically flat if there is a compact subset $K\subset M$ such that
\begin{itemize}
\item[(i)] the complement $M-K$ consists of finitely many ends $\{E_l\}_{l=1}^k$, where each end $E_l$ is diffeomorphic to $\mathbb R^n-\bar B_1$ with $$\bar B_1=\{x\in \mathbb R^n:|x|\leq 1\},$$
\item[(ii)]  the metric $g$ on each end $E_l$ has the expression $g=g_{ij}\mathrm dx_i\otimes\mathrm dx_j$ in the Euclidean coordinate chart, where the metric components satisfy the decay condition
\begin{equation*}\label{Eq: decay}
|g_{ij}-\delta_{ij}|+|x||\partial g_{ij}|+|x|^2|\partial^2 g_{ij}|=O\left(|x|^{-\tau}\right),
\end{equation*}
as $x\to\infty$, where $\tau$ is a positive constant greater than $\frac{n-2}{2}$,
\item[(iii)] the scalar curvature $R(g)$ belongs to $L^1(M,g)$.
\end{itemize}
For convenience, each end $E_l$ will be called an asymptotically flat end of $(M,g)$.
\end{definition}

\begin{definition}
Let $(M,g)$ be an asymptotically flat manifold and $E$ be an asymptotically flat end of $(M,g)$.
    The mass of the end $E$ is defined as
    $$m(M,g,E)=\frac{1}{2n(n-1)\omega_n}\lim_{\rho\to+\infty}\int_{S_\rho}(\partial_j g_{ij}-\partial_i g_{jj})\frac{x_i}{|x|}\,\mathrm d\sigma,$$
    where $\omega_n$ denotes the volume of the Euclidean unit ball $B_1^n$, $S_\rho$ denotes the coordinate sphere $\{|x|=\rho\}$, and $\mathrm d\sigma$ is the Euclidean area element of $S_\rho$.
\end{definition}

Many important works have revealed the source of mass. The positive mass conjecture, with the view that nonnegative energy density contributes to nonnegative mass, states that asymptotically flat manifolds with nonnegative scalar curvature have nonnegative mass on each asymptotically flat end, where the mass vanishes on one end if and only if it is the Euclidean space. This conjecture was finally proved by Schoen-Yau \cite{SY79,SY17} with the minimal surface method, and also by Witten \cite{Witten81} with the harmonic spinor method under the extra assumption that the asymptotically flat manifold is spin. Recently, the three-dimensional positive mass theorem was given new proofs by Bray-Kazaras-Khuri-Stern \cite{BKKS22} using level-set method and also by Agostiniani-Mazzieri-Oronzio \cite{AMO24} using potential method. With black holes taken into consideration, the Penrose inequality states that if an asymptotically flat manifold with only one asymptotically flat end $E$ has nonnegative scalar curvature and the boundary is an outer-minimizing horizon $\Sigma$, then we have the inequality
$$
m(M,g, E)\geq \frac{1}{2}\left(\frac{|\Sigma|_g}{|\mathbb S^{n-1}|}\right)^{\frac{n-2}{n-1}},
$$
where the equality holds if and only if $(M,g)$ is isometric to the half spatial Schwarzschild manifold
$(M_m,g_m)$ with
$$M_m=\mathbb R^n\setminus B_{m/2}\mbox{ and }g_m=\left(1+\frac{m}{2}|x|^{2-n}\right)^{\frac{4}{n-2}}g_{euc}.$$
The Penrose inequality in dimension three was first proved independently by Huisken-Ilmanen \cite{HI01} and Bray \cite{Bray01} using different flow methods. Notice that the potential approach  developed by Agostiniani-Mazzieri-Oronzio in \cite{AMO24} also works for the three-dimensional Penrose ineqality. Based on Bray's original proof, Bray-Lee \cite{BL09} removed the technical use of the Gauss-Bonnet formula  and generalized the Penrose inequality up to dimension seven. In order to prove the Penrose inequality, Bray developed a mass-capacity inequality
$$ m(M,g,E)\geq \mathfrak c(M,g,\Sigma),$$
where $\mathfrak c(M,g,\Sigma)$ denotes the capacity of $\Sigma$ in $(M,g)$. For more works related to mass-capacity inequalities for asymptotically flat manifolds, the audience can refer to the works \cite{BM08, Miao24, Miao25} and the references therein.

The study of mass is closely related to conformal geometry. Based upon the fundamental work of Trudinger \cite{Trudinger68} and Aubin \cite{Aubin76}, Schoen ultimately resolved the Yamabe problem by leveraging the positive mass theorem. As another link between conformal geometry and general relativity, Schoen-Yau \cite{Schoen-Yau} reduced the Liouville theorem in conformal geometry to a consequence of the positive mass theorem for a class of generalized asymptotically flat manifolds, which was recently established in the works \cite{Lesourd-Unger-Yau,LLU23, Zhu23,BC2005,CZ2024}.

In this paper, we continue the study of mass inequalities for generalized asymptotically flat manifolds, which are initiated from Schoen-Yau's work \cite{Schoen-Yau}. Such manifolds  are also well-known in \cite{Lesourd-Unger-Yau} as {\it asymptotically flat manifolds with arbitrary ends} with the following precise definition.
\begin{definition}
    A triple $(M,g,E)$ is said to be a generalized asymptotically flat manifold if
    \begin{itemize}
        \item $(M,g)$ is a complete Riemannian $n$-manifold {\it without boundary}, whose dimension $n$ is no less than three;
        \item $E$ is an asymptotically flat end of $(M,g)$;
\item the scalar curvature $R(g)$ belongs to $L^1(E,g)$.
    \end{itemize}
\end{definition}

The positive mass theorem for generalized asymptotically flat manifolds was systematically researched in \cite{Lesourd-Unger-Yau,LLU23, Zhu23,BC2005,CZ2024}. Soon after, the Penrose inequality for generalized asymptotically flat manifolds was initiated by the second-named author \cite{Zhu2024} to include spatial slices of the {\it extreme} Reissner-Nordstr\"om spacetime into application, where the ultimate goal is to show the following {\it mass-systole conjecture}:
\begin{conjecture}
Let $(M,g, E)$ be a generalized asymptotically flat manifold with nonnegative scalar curvature. Then we have
$$
m(M,g, E)\geq \frac{1}{2}\left(\frac{\sys(M,g,E)}{|\mathbb S^{n-1}|}\right)^{\frac{n-2}{n-1}},
$$
where
\begin{equation*}
\sys(M,g,E)=\inf\left\{|\Sigma|_g:\left.
\begin{array}{c}\text{$\Sigma$ is a smoothly embedded hypersurface }\\
\text{homologous to $\partial E$ in $M$}
\end{array}\right.\right\}.
\end{equation*}
Moreover, the equality holds if and only if
\begin{itemize}
    \item $(M,g)$ is isometric to the Euclidean space;
    \item or there is a strictly outer-minimizing minimal $(n-1)$-sphere $\Sigma_h$ homologous to $\partial E$ such that the region outside $\Sigma_h$ is isometric to the half spatial Schwarzschild manifold with mass $m>0$.
\end{itemize}
\end{conjecture}
The conjecture is now only known to be true when $M$ has the underlying topology $\mathbb R^3\setminus\{O\}$ concerning the work \cite{Zhu2024}. As a further exploration, we are going to consider the mass-capacity inequality for generalized asymptotically flat manifolds.

To present our theorem, we need to introduce the {\it capacity} for generalized asymptotically flat manifolds.
Given any constant $s>1$ we shall use $E_s$ to  denote the open subset of $E$ diffeomorphic to $\mathbb R^n\setminus \bar B_s$.
Let $\eta$ denote a fixed smooth function satisfying $\eta\equiv 0$ in $E_2$ and $\eta\equiv 1$ outside $E_1$.
\begin{definition}\label{def:capacity}
    The capacity of a generalized asymptotically flat manifold $(M,g,E)$ is defined to be
    $$\mathfrak c(M,g,E)=\inf_{\phi\in\mathcal C}\frac{1}{n(n-2)\omega_n}\int_M|\nabla_g\phi|^2\,\mathrm d\mu_g,$$
    where  $\mathcal C$ denotes the collection of all smooth functions $\phi$ such that $\phi-\eta$ has compact support.
\end{definition}
\begin{remark}
  It is a standard fact that we can relax the test function $\phi$ to be any function in $W^{1,2}_{loc}(M,g)$ such that $\phi-\eta$ has compact support.
\end{remark}

In this paper, we are going to prove the following theorem.

Many important works have revealed the source of mass. The positive mass conjecture, with the view that nonnegative energy density contributes to nonnegative mass, states that asymptotically flat manifolds with nonnegative scalar curvature have nonnegative mass on each asymptotically flat end, where the mass vanishes on one end if and only if it is the Euclidean space. This conjecture was finally proved by Schoen-Yau \cite{SY79,SY17} with the minimal surface method, and also by Witten \cite{Witten81} with the harmonic spinor method under the extra assumption that the asymptotically flat manifold is spin. Recently, the three-dimensional positive mass theorem was given new proofs by Bray-Kazaras-Khuri-Stern \cite{BKKS22} using level-set method and also by Agostiniani-Mazzieri-Oronzio \cite{AMO24} using potential method. With black holes taken into consideration, the Penrose inequality states that if an asymptotically flat manifold with only one asymptotically flat end $E$ has nonnegative scalar curvature and the boundary is an outer-minimizing horizon $\Sigma$, then we have the inequality
$$
m(M,g, E)\geq \frac{1}{2}\left(\frac{|\Sigma|_g}{|\mathbb S^{n-1}|}\right)^{\frac{n-2}{n-1}},
$$
where the equality holds if and only if $(M,g)$ is isometric to the half spatial Schwarzschild manifold
$(M_m,g_m)$ with
$$M_m=\mathbb R^n\setminus B_{m/2}\mbox{ and }g_m=\left(1+\frac{m}{2}|x|^{2-n}\right)^{\frac{4}{n-2}}g_{euc}.$$
The Penrose inequality in dimension three was first proved independently by Huisken-Ilmanen \cite{HI01} and Bray \cite{Bray01} using different flow methods. Notice that the potential approach developed by Agostiniani-Mazzieri-Oronzio in \cite{AMO24} also works for the three-dimensional Penrose ineqality. Based on Bray's original proof, Bray-Lee \cite{BL09} removed the technical use of the Gauss-Bonnet formula and generalized the Penrose inequality up to dimension seven. In order to prove the Penrose inequality, Bray developed a mass-capacity inequality
$$ m(M,g,E)\geq \mathfrak c(M,g,\Sigma),$$
where $\mathfrak c(M,g,\Sigma)$ denotes the capacity of $\Sigma$ in $(M,g)$. For more works related to mass-capacity inequalities for asymptotically flat manifolds, the audience can refer to the works \cite{BM08, Miao24, Miao25} and the references therein.

In this paper, we continue the study of mass inequalities for generalized asymptotically flat manifolds, which are initiated from Schoen-Yau's work \cite{Schoen-Yau}. Such manifolds are also well-known in \cite{Lesourd-Unger-Yau} as {\it asymptotically flat manifolds with arbitrary ends} with the following precise definition.
\begin{definition}
 A triple $(M,g,E)$ is said to be a generalized asymptotically flat manifold if
 \begin{itemize}
 \item $(M,g)$ is a complete Riemannian $n$-manifold {\it without boundary}, whose dimension $n$ is no less than three;
 \item $E$ is an asymptotically flat end of $(M,g)$;
\item the scalar curvature $R(g)$ belongs to $L^1(E,g)$.
 \end{itemize}
\end{definition}

The positive mass theorem for generalized asymptotically flat manifolds was systematically researched in \cite{Lesourd-Unger-Yau,LLU23, Zhu23,BC2005,CZ2024}. Soon after, the Penrose inequality for generalized asymptotically flat manifolds was initiated by the second-named author \cite{Zhu2024} to include spatial slices of the {\it extreme} Reissner-Nordstr\"om spacetime into application, where the ultimate goal is to show the following {\it mass-systole conjecture}:
\begin{conjecture}
Let $(M,g, E)$ be a generalized asymptotically flat manifold with nonnegative scalar curvature. Then we have
$$
m(M,g, E)\geq \frac{1}{2}\left(\frac{\sys(M,g,E)}{|\mathbb S^{n-1}|}\right)^{\frac{n-2}{n-1}},
$$
where
\begin{equation*}
\sys(M,g,E)=\inf\left\{|\Sigma|_g:\left.
\begin{array}{c}\text{$\Sigma$ is a smoothly embedded hypersurface }\\
\text{homologous to $\partial E$ in $M$}
\end{array}\right.\right\}.
\end{equation*}
Moreover, the equality holds if and only if
\begin{itemize}
 \item $(M,g)$ is isometric to the Euclidean space;
 \item or there is a strictly outer-minimizing minimal $(n-1)$-sphere $\Sigma_h$ homologous to $\partial E$ such that the region outside $\Sigma_h$ is isometric to the half spatial Schwarzschild manifold with mass $m>0$.
\end{itemize}
\end{conjecture}
The conjecture is now only known to be true when $M$ has the underlying topology $\mathbb R^3\setminus\{O\}$ concerning the work \cite{Zhu2024}. As a further exploration, we are going to consider the mass-capacity inequality for generalized asymptotically flat manifolds.

To present our theorem, we need to introduce the {\it capacity} for generalized asymptotically flat manifolds.
Given any constant $s>1$ we shall use $E_s$ to denote the open subset of $E$ diffeomorphic to $\mathbb R^n\setminus \bar B_s$.
Let $\eta$ denote a fixed smooth function satisfying $\eta\equiv 0$ in $E_2$ and $\eta\equiv 1$ outside $E_1$.
\begin{definition}\label{def:capacity}
 The capacity of a generalized asymptotically flat manifold $(M,g,E)$ is defined to be
 $$\mathfrak c(M,g,E)=\inf_{\phi\in\mathcal C}\frac{1}{n(n-2)\omega_n}\int_M|\nabla_g\phi|^2\,\mathrm d\mu_g,$$
 where $\mathcal C$ denotes the collection of all smooth functions $\phi$ such that $\phi-\eta$ has compact support.
\end{definition}
\begin{remark}
 It is a standard fact that we can relax the test function $\phi$ to be any function in $W^{1,2}_{loc}(M,g)$ such that $\phi-\eta$ has compact support.
\end{remark}

In this paper, we are going to prove the following theorem.

\begin{abstract}
In the spin case, we can establish a mass-capacity inequality for generalized asymptotically flat manifolds $(M,g,E)$ with nonnegative scalar curvature, where the equality implies that $(M,g)$ is harmonically conformal to $\mathbb R^n\setminus S$ for a closed bounded subset $S$ of $\mathbb R^n$ with Hausdorff dimension no greater than $\frac{n-2}{2}$.
\end{abstract}

To present our theorem, we need to introduce the {\it capacity} for generalized asymptotically flat manifolds.
Given any constant $s>1$ we shall use $E_s$ to denote the open subset of $E$ diffeomorphic to $\mathbb R^n\setminus \bar B_s$.
Let $\eta$ denote a fixed smooth function satisfying $\eta\equiv 0$ in $E_2$ and $\eta\equiv 1$ outside $E_1$.
\begin{definition}\label{def:capacity}
 The capacity of a generalized asymptotically flat manifold $(M,g,E)$ is defined to be
 $$\mathfrak c(M,g,E)=\inf_{\phi\in\mathcal C}\frac{1}{n(n-2)\omega_n}\int_M|\nabla_g\phi|^2\,\mathrm d\mu_g,$$
 where $\mathcal C$ denotes the collection of all smooth functions $\phi$ such that $\phi-\eta$ has compact support.
\end{definition}
\begin{remark}
 It is a standard fact that we can relax the test function $\phi$ to be any function in $W^{1,2}_{loc}(M,g)$ such that $\phi-\eta$ has compact support.
\end{remark}

To include previous Bray's mass-capacity inequality as a special case, we also extend our main theorem to the setting of Riemannian manifolds with corner defined as follows.
\begin{definition}
 A triple $(M,g,\Sigma)$ will be called a Riemannian manifold with corner if $M$ is a smooth manifold equipped with a continuous metric $g$, and $\Sigma$ is a closed separating smooth hypersurface $\Sigma$ in $M$ such that the metric $g$ is actually smooth to boundary in the closure of every component of $M\setminus \Sigma$. For convenience, $\Sigma$ will be called the corner of $(M,g)$.
\end{definition}
In particular, we can consider generalized asymptotically flat manifolds with corner.
\begin{definition}
 A generalized asymptotically flat manifold $(M,g,E)$ will be called a generalized asymptotically flat manifold with corner, if there exists a closed separating smooth hypersurface $\Sigma$ such that $(M,g,\Sigma)$ is a Riemannian manifold with corner.
\end{definition}

Throughout this paper, we always take the convention that $\mathbb S^2$ has mean curvature two in $\mathbb R^3$ with respect to the outward unit normal. With extra mean-curvature condition along the corner, we can prove
\begin{theorem}\label{Thm: main2}
 Assume that $(M^n,g,E)$ is a spin generalized asymptotically flat manifold with corner, which has nonnegative scalar curvature. Assume that the sum of the mean curvatures of the corner on both sides with respect to the outward unit normal is nonnegative. Then we have the following mass-capacity inequality
 $$m(M,g,E)\geq 2\mathfrak c(M,g,E).$$
 If the equality holds, then $(M,g)$ is smooth and harmonically conformal to $\mathbb R^n\setminus S$, where $S$ is a bounded closed subset of $\mathbb R^n$ with Hausdorff dimension no greater than $\frac{n-2}{2}$. In particular, we have
 $$\pi_i(M)=0\mbox{ for all }1\leq i\leq n-1- \left\lfloor\frac{n}{2}\right\rfloor.$$
\end{theorem}
As a direct consequence, we can show
\begin{corollary}[Bray's mass capacity inequality]\label{Cor: Bray}
 Assume that $(M^n,g)$ is a spin asymptotically flat manifold with only one asymptotically flat end $E$, non-empty boundary $\Sigma$, and nonnegative scalar curvature.
Assume further that the boundary $\Sigma$ is minimal. Then we have
\begin{equation}\label{Eq: Bray's mass-capacity}
 m(M,g,E)\geq \mathfrak c(M,g,\Sigma),
\end{equation}
where $\mathfrak c(M,g,\Sigma)$ denotes the capacity of $\Sigma$ in $(M,g)$ given by
$$\mathfrak c(M,g,\Sigma)=\inf_{\phi\in\mathcal C}\frac{1}{n(n-2)\omega_n}\int_M|\nabla_g\phi|^2\,\mathrm d\mu_g,$$
 where $\mathcal C$ denotes the collection of all smooth functions $\phi$ with compact support such that $\phi\equiv 1$ on $\Sigma$.
The equality holds in \eqref{Eq: Bray's mass-capacity} if and only if $(M,g)$ is isometric to the half spatial Schwarzschild manifold.
\end{corollary}

\begin{proposition}\label{Prop: mass capacity}
 If $(M,g)$ has nonnegative scalar curvature, then we have
 $$m(M,g,E)\geq 2 \mathfrak c(M,g,E).$$
\end{proposition}
\begin{proof}
 Given any constant $\varepsilon>0$ we define
 \begin{equation}\label{Eq: approximation metric}
 g_\varepsilon=u_\varepsilon^{\frac{4}{n-2}}g\mbox{ with } u_\varepsilon=\frac{u+\varepsilon}{1+\varepsilon},
 \end{equation}
 where $u$ is the positive harmonic function from Corollary \ref{Cor: function u}.
 Note that $u_\varepsilon$ has a positive lower bound, so $(M,g_\varepsilon)$ remains to be a complete Riemannian manifold without boundary. It follows from the expansion of $u$ in Corollary \ref{Cor: function u} that $(M,g_\varepsilon,E)$ remains to be a generalized asymptotically flat manifold and that we have
$$m(M,g_\varepsilon,E)=m(M,g,E)-2(1+\varepsilon)^{-1}\mathfrak c(M,g,E).$$
 Clearly, $u_\varepsilon$ is a harmonic function on $(M,g)$ since $u$ is. We see that $(M,g_\varepsilon)$ has nonnegative scalar curvature. Then it follows from the spin positive mass theorem (see \cite{BC2005, CZ2024} for instance) for generalized asymptotically flat manifolds that we have
 $$m(M,g_\varepsilon,E)\geq 0,$$ which yields
 $$m(M,g,E)\geq 2(1+\varepsilon)^{-1}\mathfrak c(M,g,E).$$
 The proof is now completed by letting $\varepsilon\to 0$.
\end{proof}
\begin{remark}
 If one simply takes the function $u$ to be the conformal factor, then the positive mass theorem may not be applied since the Riemannian manifold after conformal deformation can be incomplete. This explains the reason why we have to use an approximation argument for the mass-capacity inequality.
\end{remark}

\begin{proof}[Proof of Theorem \ref{Thm: main2} ]
 By repeating the proof of Corollary \ref{Cor: function u} we are able to construct a positive weakly harmonic $C^{1,\alpha}$-function $u$ on $(M,g)$ such that we have the expansion
 $$u(x)=1-\mathfrak c(M,g,E)\cdot|x|^{2-n}+w,$$
 with $w=O_2(|x|^{2-n-\tau})$ as $x\to\infty$. For any constant $\varepsilon>0$, the conformal manifold
 $(M,g_\varepsilon,E)$ with $$g_\varepsilon=\left(\frac{u+\varepsilon}{1+\varepsilon}\right)^{\frac{4}{n-2}}g$$ is a spin generalized asymptotically flat manifold with corner satisfying the assumption of Proposition \ref{Prop: spin PMT with corner}. As a consequence, we have
 $$0\leq m(M,g_\varepsilon,E)=m(M,g,E)-2(1+\varepsilon)^{-1}\mathfrak c(M,g,E).$$
 Letting $\varepsilon\to 0$, we obtain the desired mass-capacity inequality
 $$m(M,g,E)\geq 2\mathfrak c(M,g,E).$$
 If the equality holds, then the same argument in the proof of Proposition \ref{Prop: spin PMT with corner} yields that $(M,\bar g)$ with $\bar g=u^{\frac{4}{n-2}}g$ is smooth and flat up to a change of the smooth structure. Notice that $u^{-1}$ is a weakly harmonic function on $(M,\bar g)$. We conclude that $u$ is smooth and so $g$ is also smooth. Now we return to the smooth case and the desired conclusions come from Theorem \ref{Thm: main1}.
\end{proof}

\begin{remark}
In the non-spin case, the mass-capacity inequality is still true from the exactly same proof assuming the validity of the positive mass theorem. However, the rigidity seems out of reach with current techniques due to the issue of incompleteness.
\end{remark}

To include previous Bray's mass-capacity inequality as a special case, we also extend our main theorem to the setting of Riemannian manifolds with corner defined as follows.
\begin{definition}
    A triple $(M,g,\Sigma)$ will be called a Riemannian manifold with corner if $M$ is a smooth manifold equipped with a continuous metric $g$, and $\Sigma$ is a closed separating smooth hypersurface $\Sigma$ in $M$ such that the metric $g$ is actually smooth to boundary in the closure of every component of $M\setminus \Sigma$. For convenience, $\Sigma$ will be called the corner of $(M,g)$.
\end{definition}
In particular, we can consider generalized asymptotically flat manifolds with corner.
\begin{definition}
    A generalized asymptotically flat manifold $(M,g,E)$ will be called a generalized asymptotically flat manifold with corner, if there exists a closed separating smooth hypersurface $\Sigma$ such that $(M,g,\Sigma)$ is a Riemannian manifold with corner.
\end{definition}

Throughout this paper, we always take the convention that $\mathbb S^2$ has mean curvature two in $\mathbb R^3$ with respect to the outward unit normal. With extra mean-curvature condition along the corner, we can prove
\begin{theorem}\label{Thm: main2}
  Assume that $(M^n,g,E)$ is a spin generalized asymptotically flat manifold with corner, which has nonnegative scalar curvature. Assume that the sum of the mean curvatures of the corner on both sides with respect to the outward unit normal is nonnegative.  Then we have the following mass-capacity inequality
    $$m(M,g,E)\geq 2\mathfrak c(M,g,E).$$
    If the equality holds, then $(M,g)$ is smooth and harmonically conformal  to $\mathbb R^n\setminus S$, where $S$ is a bounded closed subset of $\mathbb R^n$ with Hausdorff dimension no greater than $\frac{n-2}{2}$. In particular, we have
    $$\pi_i(M)=0\mbox{ for all }1\leq i\leq n-1- \left\lfloor\frac{n}{2}\right\rfloor.$$
\end{theorem}
As a direct consequence, we can show
\begin{corollary}[Bray's mass capacity inequality]\label{Cor: Bray}
    Assume that $(M^n,g)$ is a spin asymptotically flat manifold with only one asymptotically flat end $E$, non-empty boundary $\Sigma$, and nonnegative scalar curvature.
Assume further that the boundary $\Sigma$ is minimal. Then we have
\begin{equation}\label{Eq: Bray's mass-capacity}
    m(M,g,E)\geq \mathfrak c(M,g,\Sigma),
\end{equation}
where $\mathfrak c(M,g,\Sigma)$ denotes the capacity of $\Sigma$ in $(M,g)$ given by
$$\mathfrak c(M,g,\Sigma)=\inf_{\phi\in\mathcal C}\frac{1}{n(n-2)\omega_n}\int_M|\nabla_g\phi|^2\,\mathrm d\mu_g,$$
    where  $\mathcal C$ denotes the collection of all smooth functions $\phi$ with compact support such that $\phi\equiv 1$ on $\Sigma$.
The equality holds in \eqref{Eq: Bray's mass-capacity} if and only if $(M,g)$ is isometric to the half spatial Schwarzschild manifold.
\end{corollary}

\subsection*{Arrangements}
The remaining part of this paper is arranged as follows.
Sections \ref{Sec: inequality} and \ref{Sec: rigidity} are devoted to proving Theorem \ref{Thm: main1}. In Section \ref{Sec: inequality}, we prove the mass-capacity inequality based on the conformal method. In Section \ref{Sec: rigidity}, we establish the rigidity in the equality case of the mass-capacity inequality by constructing sufficiently many parallel spinors from an approximation procedure. In Section \ref{Sec: corner}, we show necessary modifications in the corner case and present the proofs of Theorem \ref{Thm: main2} and Corollary \ref{Cor: Bray}.

\bigskip

Sections \ref{Sec: inequality} and \ref{Sec: rigidity} are devoted to proving Theorem~\ref{Thm: main1}. In Section \ref{Sec: inequality}, the mass-capacity inequality is proved "based on the conformal method." In Section \ref{Sec: rigidity}, the rigidity in the equality case is established "by constructing sufficiently many parallel spinors from an approximation procedure."

Next we prove the main theorem. First, we establish the mass-capacity inequality, “based on the conformal method.” Then we establish the rigidity in the equality case, “by constructing sufficiently many parallel spinors from an approximation procedure.”

Question: Let $(M^n,g,E)$ be a spin generalized asymptotically flat manifold with $n\ge 3$, meaning that $(M,g)$ is a complete boundaryless Riemannian $n$-manifold, $E$ is an asymptotically flat end of $(M,g)$, and the scalar curvature satisfies $R(g)\in L^1(E,g)$. Assume moreover that the scalar curvature is nonnegative. Let $m(M,g,E)$ denote the ADM mass of the end $E$, and let $\mathfrak c(M,g,E)$ denote the capacity of the end $E$ (equivalently, the constant appearing in the asymptotic expansion $u(x)=1-\mathfrak c(M,g,E)|x|^{2-n}+o(|x|^{2-n})$ on $E$ for the associated positive harmonic function tending to $1$ at infinity). Which of the following conclusions is valid under these hypotheses?

One has the mass-capacity inequality $$m(M,g,E)\ge 2\mathfrak c(M,g,E).$$ If equality holds, then $(M,g)$ is harmonically conformal to $\mathbb R^n\setminus S$, meaning that for some bounded closed set $S\subset \mathbb R^n$ there is a conformal identification with Euclidean space off $S$ whose conformal factor is positive and harmonic; moreover $S$ has Hausdorff dimension at most $\frac{n-2}{2}$. In particular, $$\pi_i(M)=0\quad\text{for all }1\le i\le n-1-\left\lfloor\frac n2\right\rfloor.$$

Choices:
B. One has the mass-capacity inequality $$m(M,g,E)\ge \mathfrak c(M,g,E).$$ If equality holds, then $(M,g)$ is harmonically conformal to $\mathbb R^n\setminus S$ for some bounded closed set $S\subset \mathbb R^n$ with Hausdorff dimension at most $\frac{n-2}{2}$, and in particular $$\pi_i(M)=0\quad\text{for all }1\le i\le n-1-\left\lfloor\frac n2\right\rfloor.$$
C. One has the mass-capacity inequality $$m(M,g,E)\ge 2\mathfrak c(M,g,E).$$
D. One has the mass-capacity inequality $$m(M,g,E)\ge 2\mathfrak c(M,g,E).$$ If equality holds, then $(M,g)$ is globally conformal to $\mathbb R^n\setminus S$ for some bounded closed set $S\subset \mathbb R^n$ with Hausdorff dimension at most $\frac{n-2}{2}$; moreover the conformal factor extends smoothly across $S$. In particular, $$\pi_i(M)=0\quad\text{for all }1\le i\le n-1.$$
E. There exists a bounded closed set $S\subset \mathbb R^n$ with Hausdorff dimension at most $\frac{n-2}{2}$, depending only on $n$, such that every spin generalized asymptotically flat manifold $(M^n,g,E)$ with nonnegative scalar curvature and satisfying $$m(M,g,E)=2\mathfrak c(M,g,E)$$ is harmonically conformal to $\mathbb R^n\setminus S$. In particular, $$\pi_i(M)=0\quad\text{for all }1\le i\le n-1-\left\lfloor\frac n2\right\rfloor.$$

sharp factor 2 from conformal mass formula

entire equality-case rigidity and homotopy conclusion dropped

equality gives harmonic conformality off a singular set, not smooth extension across it nor full homotopy vanishing

dependence of the singular set on the manifold; no universal set S depending only on n

The stem states only the hypotheses and definitions; it does not explicitly reveal the sharp mass-capacity inequality, the equality case, or the topological consequences. There is no direct answer leakage.

This is very close to a theorem-recall item: the correct choice is essentially the full theorem statement under its exact hypotheses. However, the alternatives differ in sharp constants, rigidity strength, and quantifier structure, so it is not a pure verbatim restatement.

Some reasoning is required to distinguish the strongest valid conclusion from a weaker true statement and from subtly overstated false variants. Still, success depends largely on recalling the theorem rather than generating a conclusion from first principles.

The distractors are strong: one is a weaker true statement, others modify the sharp factor, overstate regularity/topological rigidity, or introduce an incorrect universal quantifier. These are plausible and mathematically meaningful failure modes.

A solid MCQ with no answer leakage and strong distractors, but it leans toward theorem recognition rather than deep generative reasoning.


\label{main result}
Let $k$ be a positive integer such that $k\in \{7, 8, 10,$ $11,12,13\}.$ If $\{a, b, c, d\}$ is a $D(4)$-quadruple with $b=ka$, then it is regular. In other words, we have $d=d_{\pm}.$

\label{intr}

\begin{definition}Let $n\neq0$ be an integer. We call a set of $m$ distinct positive integers a $D(n)$-$m$-tuple, or $m$-tuple with the property $D(n)$, if the product of any two of its distinct elements increased by $n$ is a perfect square.
\end{definition}
We research the $n=4$ case, which has many similarities to the classical $n=1$ case. First author and Filipin proved in \cite{blizfil} the nonexistence of $D(4)$-quintuples.

For a $D(4)$-triple $\{a,b,c\}, a<b<c$, we define
 \[ d_{\pm}(a,b,c)=a+b+c+\frac{1}{2}(abc \pm \sqrt{(ab+4)(ac+4)(bc+4)}).\]

$D(4)$-quadruple  $\{a,b,c,d_{+}\}$ is called a \textbf{regular quadruple}
 if $d_{-} \neq 0$, then $\{a,b,c,d_{-}\}$ is a regular $D(4)$-quadruple with $d_{-}<c$. It is easy to verify that $c=d_+(a,b,d_-)$.

In both the classical case, $n=1$, and $n=4$, conjectures about the uniqueness of an extension of a triple to a quadruple with a larger element are still open.

\begin{conjecture}\label{conj}
Any $D(4)$-quadruple is regular.
\end{conjecture}

In this paper, we study the extensibility of $D(4)$-pairs $\{a,b\}$, where $b = ka$ and $k \in \{7,8,10,11,12,13\}$ are defined by following the methods in \cite{glavni}. So, the main theorem we prove is:

\label{intr}

\begin{definition}Let $n\neq0$ be an integer. We call a set of $m$ distinct positive integers a $D(n)$-$m$-tuple, or $m$-tuple with the property $D(n)$, if the product of any two of its distinct elements increased by $n$ is a perfect square.
\end{definition}
We research the $n=4$ case, which has many similarities to the classical $n=1$ case. First author and Filipin proved in \cite{blizfil} the nonexistence of $D(4)$-quintuples.

$D(4)$-quadruple $\{a,b,c,d_{+}\}$ is called a \textbf{regular quadruple}
 if $d_{-} \neq 0$, then $\{a,b,c,d_{-}\}$ is a regular $D(4)$-quadruple with $d_{-}<c$. It is easy to verify that $c=d_+(a,b,d_-)$.

In both the classical case, $n=1$, and $n=4$, conjectures about the uniqueness of an extension of a triple to a quadruple with a larger element are still open.

\begin{conjecture}\label{conj}
Any $D(4)$-quadruple is regular.
\end{conjecture}

In this paper, we study the extensibility of $D(4)$-pairs $\{a,b\}$, where $b = ka$ and $k \in \{7,8,10,11,12,13\}$ are defined by following the methods in \cite{glavni}. So, the main theorem we prove is:

\begin{conjecture}\label{conj}
Any $D(4)$-quadruple is regular.
\end{conjecture}

Firstly, in Section $2$ we show that $D(4)$-pairs $\{a,ka\}$, $k\in \{7, 8, 10,$ $11,12,13\}$ can be extended to a $D(4)$-triple only with an element $c$, which is a member of a family of positive integers depending on $a$. Then in Sections $3-5$, we prove that these triples can only be extended to regular $D(4)$-quadruples $\{a, ka, c, d\}$.\ These examples support the Conjecture \ref{conj}.
\section{Extensions of pairs to triples}\label{section_2}

\begin{cor}\label{nova lema}
Let $\{a,ka,c\}$ be a $D(4)$-triple, $k \in \{10,11,12,13\}$. Then $c=c_{\nu}^{\pm}$.
\end{cor}
\begin{proof}
Let's show that $\{1,5,a,ka \}$ is not a $D(4)$-quadruple for $k \in \{10,11,12,13\}$. If $\{1,5,a,ka \}$ is a $D(4)$-quadruple, by using \cite[Lemma 6.1]{mbt}, Lemma \ref{lower_on_b}, and Lemma \ref{pomocna lema za lemu ispod}, we obtain $a=a_{\nu}^{\pm}$, $ka=d_+(1,5,a)$, and $d_+(1,5,a)=a_{\nu+1}^{\pm}$ for the same choice of $\pm$. We divide both sides of \begin{equation}\label{pomocna}
 ka = d_+(1,5,a)=1+5+a+\frac{1}{2}(5a+3\sqrt{5a+4}\sqrt{a+4})
\end{equation} by $a$ and, using the fact that $a \geq 12$, we obtain $k \leq 8$.
\end{proof}

Secondly, we observe the solutions of the system of equations (\ref{eqn:sol_pell_bc}) and (\ref{eqn:sol_pell_ab}) and determine the intersections $y=A_n=B_l$ of sequences $(A_n)_n$ and $(B_l)_l$ defined by
\begin{align}
 &A_0=y_1,\ A_1=\frac{1}{2}(ty_1+bz_1),\ A_{n+2}=tA_{n+1}-A_n,\label{seq:An}\\
 &B_0=y_2,\ B_1=\frac{1}{2}(ry_2+bx_2),\ B_{l+2}=rB_{l+1}-B_l, \ n,l\geq 0.\label{seq:Bl}
\end{align}
The initial terms of these sequences are described in the next lemma, whose proof follows \cite[Lemma 2.5]{glavni}.
\begin{lemma}\label{lem:lem_4}
Assume that $\{a, b, c', c\}$ is not a $D(4)$-quadruple for any $c'$ with $0<c'<c^{\pm}_{\nu-1}$ and $b\geq 556881$. Then, $A_{2n}=B_{2l+1}$ has no solution. Moreover, if $A_{2n} = B_{2l}$, then $y_2 = 2$. In other cases, we have $y_2 =\pm 2.$
\end{lemma}

For any nonzero algebraic number $\alpha$ of degree $d$ over $\QQ$ whose minimal polynomial over $\ZZ$ is $a_0 \prod_{j=1}^{d}(X-\alpha^{(j)}),$ we denote by
$$
 h(\alpha)= \frac{1}{d}\left(\log |a_0|+\sum_{j=1}^{d}\log\max\left(1,\left|\alpha^{(j)} \right| \right) \right)
$$
its absolute logarithmic height. We recall the following result due to Matveev \cite{Matveev:2000}.
\begin{lemma}\label{Matveev}
Denote by $\alpha_1,\ldots,\alpha_j$ algebraic numbers, not $0$ or $1,$ by $\log\alpha_1,\ldots,$ $\log\alpha_j$ determinations of their logarithms, by $D$ the degree over $\QQ$ of the number field $\KK=\QQ(\alpha_1,\ldots,\alpha_j),$ and by $b_1,\ldots,b_j$ integers. Define $B=\max\{|b_1|,\ldots,|b_j|\}$ and
$$A_i=\max\{Dh(\alpha_i),|\log\alpha_i|,0.16\}\;\; (1\leq i\leq j),$$
where $h(\alpha)$ denotes the absolute logarithmic Weil height of $\alpha.$
Assume that the number
$$
\Lambda= b_1\log\alpha_1+\cdots+b_n\log\alpha_j
$$
does not vanish. Then
$$
|\Lambda|\geq \exp\{-C(j,\chi)D^2A_1\cdots A_j\log(eD)\log(eB)\},
$$
where $\chi=1$ if $\KK \subset\RR$ and $\chi=2$ otherwise and
$$
C(j,\chi)=\min\left\{\frac{1}{\chi}\left( \frac{1}{2}ej\right)^{\chi}30^{j+3}j^{3.5},2^{6j+20}\right\}.
$$
\end{lemma}
By applying this result we obtain upper bound for $l$ and $m$ in terms of $a$.
\begin{proposition}\label{proposition-1}
Assume that $c\in \{c_1^+, c_2^\pm\}.$ If $P_l=Q_m,$ with $m>1$ then
\begin{align*}
\frac{l}{\log(el)}&< 3.34\cdot 10^{13}\cdot\log^2(8.09c^2),\quad \text{with solutions of Type $1$},\; \\
\frac{l}{\log(el)}&< 6.63\cdot 10^{13}\cdot\log^2(8.09c^2), \quad \text{with solutions of Type $2$}.
\end{align*}
If $Q_m=P_l,\ l\geq1$ with $c=c_1^-,$ then we get
\begin{align*}
\frac{m}{\log(em)}< 13.36\cdot 10^{13}\cdot\log^2(21.3a),\quad \text{for}\; k=7,8,10,11,12,13.
\end{align*}
\end{proposition}
\begin{proof}
 We will demonstrate the proof for the case $c=c_1^-$ since the proof for the other cases closely follows \cite[Proposition 2.9]{glavni}. We apply Lemma \ref{proposition-1}
 with $j=3$ and $\chi=1$ to the linear form \ref{Lambda'} and take \[D=4,\ b_1=m,\ b_2=-l,\ b_3=1,\ \alpha_1=\beta,\ \alpha_2=\alpha,\ \alpha_3=\gamma'.\]
 Since $l\leq m$, we can take $B=m$. Also, we have
 \[h(\alpha_1)=\frac{1}{2}\log\beta,\ h(\alpha_2)=\frac{1}{2}\log\alpha.\]
 Since $\gamma'=\gamma^{-1}$, then $h(\gamma)=h(\gamma')$ and from \cite[Proposition 2.9]{glavni} we have
 \[h(\gamma)<\frac{1}{4}\log\left[\frac{2^4r^4c^4(1+\sqrt{k})^4}{(c-a)^2}\right].\]
 We have \begin{align*}(c_1^--a)^2&=a^2(2\sqrt{k+\frac{4}{a^2}}-k)^2>a^2(2\sqrt{k}-k)^2,\\
 r&<a\sqrt{1.1k},\\ c=c_1^-&=a(k+1-2\sqrt{k+\frac{4}{a^2}})<a(\sqrt{k}-1)^2,\end{align*} and now it follows that
 \[h(\gamma')=h(\gamma)<\frac{1}{4}\log(93309063a^6)<\frac{3}{2}\log(21.3a).\]
 By applying Lemma \ref{proposition-1} with
 \[A_1=2\log\beta,\ A_2=2\log\alpha,\ A_3=6\log(21.3a)\]
 we get
 \begin{align}\label{Matveev_0}
\log|\Lambda'|>-1.3901&\cdot 10^{11}\cdot16\cdot2 \cdot\log\beta\\&\cdot2\log\alpha\cdot6\log(21.3a)\cdot\log(4e)\cdot\log(em).\nonumber
\end{align}
From $1)$ of Lemma \ref{bound-Lambda} and using $l\geq1$, it is easy to conclude that $$m\log\beta<l\log\alpha+2.6\alpha^{-2l}-\log\gamma'<2l\log\alpha.$$
Also,
$$\log|\Lambda'|<-1.9069l\log\alpha$$
and
$$\log\alpha=\log\left[a\left(\sqrt{k+\frac{4}{a^2}}+\sqrt{k}\right)\right]<\log(7.212a)<\log(21.3a).$$
Now, by combining everything, we get
\[\frac{m}{\log(em)}<1.336\cdot10^{14}\cdot\log^2(21.3a),\ \text{for}\ k=7,8,10,11,12,13. \qedhere\]
\end{proof}

In this section, we examine the equation \[x=Q_m=P_l.\] Firstly, we state a useful result from \cite{glavni} and then apply it to obtain lower bounds for the indices $m$ and $l$ in terms of $a$.
\begin{lemma}\cite[Lemma 3.1]{glavni}\label{lem_6} If $a$ is odd, then
\begin{align}
\label{cong_1} Q_{2m}&\equiv x_0+\dfrac{1}{2} a(cx_0m^2+sz_0m)\pmod{a^2},\\
P_{2l}&\equiv x_2+\dfrac{1}{2} a(bx_2l^2+ry_2l)\pmod{a^2} .
\end{align}
If $a$ is even, then
\begin{align}
 Q_{2m}&\equiv x_0+\dfrac{1}{2} a(cx_0m^2+sz_0m)\pmod{\dfrac{1}{2} a^2},\\
P_{2l}&\equiv x_2+\dfrac{1}{2} a(bx_2l^2+ry_2l)\pmod{\dfrac{1}{2} a^2} .
\end{align}
\end{lemma}
\begin{lemma}\label{lower-Type-a} \hfill
\begin{enumerate}[1)]
 \item Assume that $b=ka,\ k\in \{7,8,10,11,12,13\}$ and $c=c_1^-$. If the equation $P_l=Q_m$ has a solution $(l,m)$ (of Type $1$), then we have
$$
m \geq \frac{1}{4}\left(-2+\sqrt{4+\sqrt{a}}
\right).$$
 \item Assume that $b=ka,\ k\in \{7,8,10,11,12,13\}$ and $c \in \{c_1^+,c_2^{\pm}\}$. If the equation $P_l=Q_m$ has a solution $(l,m)$ (of Type $1$), then we have
$$
l \geq \frac{1}{12}\left(-2+\sqrt{4+3\sqrt{a}}
\right).$$
\end{enumerate}
\end{lemma}
\begin{proof}
 Proof follows \cite[Lemma 3.2]{glavni} and here we will demonstrate the proof of $1)$. From the recurrent sequence (\ref{sequence_s_nu}), we have $s \equiv 2,r\ (\bmod\ a)$. Also, $b=ka \equiv 0\ (\bmod\ a)$ and $c=c_{1}^{-}\equiv -2 r\ (\bmod\ a)$. Using the previous Lemma, with solutions in Type $1$, we get
 $$
- 4 r m^2\pm 4m\equiv \pm2rl\pmod{a},\quad \mbox{if}\quad s\equiv 2\pmod{a}
$$
and
$$
\ - 4m^2\pm 2m\mp2l\equiv 0\pmod{\dfrac{a}{\gcd (a,r)}},\quad \mbox{if}\quad s\equiv r\pmod{a}.
$$
In the first case, we multiply the congruence by $r$ and since $r^2 \equiv 4\ (\bmod\ a)$ we have $r \equiv \pm2\ (\bmod\ a')$ for some $a'$ that divides $a$ and $a' \geq \sqrt{a}$. So we get
$$-16m^2\pm8m\mp8l \equiv 0\ (\bmod\ a').$$
Using $l\leq m$ it follows that
$$16m^2+16m \geq |-16m^2\pm8m\mp8l|\geq a'\geq \sqrt{a},$$
which implies
\begin{align}\label{1}
 m \geq \frac{1}{4}\left(-2 +\sqrt{4+\sqrt{a}} \right).\end{align}
In the second case, since $\gcd(a,r)\leq2$ and $l \leq m$, we have
$$4m^2+4m \geq |- 4m^2\pm 2m\mp2l|\geq\frac{a}{\gcd(a,r)}\geq\frac{a}{2},$$
which implies
\begin{align}\label{2}
 m \geq \frac{1}{4}\left(-2+\sqrt{4+2a}\right).
\end{align}
Combining (\ref{1}) and (\ref{2}) we obtain the desired inequality.
\end{proof}

The cases $k \in \{2,3,5,6\}$ have already been observed in \cite{glavni} and \cite{Filipin}, and it is easy to verify that pairs $\{a,4a\}$ and $\{a,9a\}$ cannot satisfy the $D(4)$ property.

Firstly, in Section $2$ we show that $D(4)$-pairs $\{a,ka\}$, $k\in \{7, 8, 10,$ $11,12,13\}$ can be extended to a $D(4)$-triple only with an element $c$, which is a member of a family of positive integers depending on $a$. Then in Sections $3-5$, we prove that these triples can only be extended to regular $D(4)$-quadruples $\{a, ka, c, d\}$.\ These examples support the Conjecture \ref{conj}.

Firstly, in Section $2$ we show that $D(4)$-pairs $\{a,ka\}$, $k\in \{7, 8, 10, 11,12,13\}$ can be extended to a $D(4)$-triple only with an element $c$, which is a member of a family of positive integers depending on $a$. Then in Sections $3$--$5$, we prove that these triples can only be extended to regular $D(4)$-quadruples $\{a, ka, c, d\}$ (i.e., $d=d_{\pm}$).

Firstly, we show that $D(4)$-pairs $\{a,ka\}$, $k\in \{7, 8, 10, 11,12,13\}$ can be extended to a $D(4)$-triple only with an element $c$, which is a member of a family of positive integers depending on $a$. Then we prove that these triples can only be extended to regular $D(4)$-quadruples $\{a, ka, c, d\}$ (i.e., $d=d_{\pm}$).

Question: Let \(k\) be a positive integer with \(k\in\{7,8,10,11,12,13\}\). Suppose \(\{a,b,c,d\}\) is a set of four distinct positive integers with the property \(D(4)\), meaning that for any two distinct elements \(x,y\in\{a,b,c,d\}\), the number \(xy+4\) is a perfect square, and assume moreover that \(b=ka\). Which conclusion about \(\{a,b,c,d\}\) is valid under these hypotheses?

The quadruple \(\{a,b,c,d\}\) is regular; equivalently, the fourth element satisfies \(d=d_{\pm}\).

Choices:
B. The quadruple \(\{a,b,c,d\}\) is regular for every positive integer \(k\ge 7\) with \(b=ka\); equivalently, one always has \(d=d_{\pm}\).
C. There exists a regular \(D(4)\)-quadruple having the form \(\{a,ka,c,d\}\); in particular, for such a quadruple the fourth element satisfies \(d=d_{\pm}\).
D. The quadruple \(\{a,b,c,d\}\) can be extended from the pair \(\{a,ka\}\) only when the third element \(c\) belongs to one of the admissible families depending on \(a\); consequently, \(d\) need not equal \(d_{\pm}\).
E. For each \(k\in\{7,8,10,11,12,13\}\), every \(D(4)\)-quadruple \(\{a,b,c,d\}\) with \(b=ka\) is uniquely determined by the pair \(\{a,b\}\), so in particular there is exactly one possible value of \(d\), namely \(d=d_{+}\).

universal conclusion for every such quadruple

final step forcing regularity from admissible c-families

regularity versus uniqueness of extension and exclusion of d_-

The stem does not reveal the correct conclusion explicitly or by near-paraphrase. It states the hypotheses and asks which conclusion follows, without signaling that regularity is the intended answer.

The item is close to a theorem-recall question: the correct option essentially states the target theorem under the given hypotheses. However, it is not a pure tautology because the alternatives vary in strength, scope, and quantification.

Some reasoning is needed to distinguish the exact valid conclusion from stronger, weaker, or overgeneralized variants. Still, the item mainly tests precise theorem recognition rather than substantial generative mathematical reasoning.

The distractors are plausible and mathematically meaningful: one overgeneralizes the range of k, one gives only a weaker existential claim, one suggests a tempting but false structural caveat, and one adds an unjustified uniqueness strengthening.

A solid theorem-discrimination MCQ with strong distractors and no answer leakage, but it leans more toward recall of the exact result than toward deep generative reasoning.


\begin{theorem}\label{thm_specialcase}
    Let $C$ be a genus $2$ curve with Jacobian that has multiplication by the maximal quaternion order with discriminant $6$,
    and has field of moduli a number field $L$ with at least one real embedding.
    Assume $C$ has potentially smooth stable reduction at primes above $2$ and $3$. Then its Jacobian has superspecial reduction at infinitely many primes.
\end{theorem}

\subsection{Background}

Elkies proved in \cite{MR903384} that an elliptic curve defined over a number field of odd degree has infinitely many primes of supersingular reduction, and extended the result in \cite{MR1030140} to elliptic curves over any number field with at least one real embedding. This has been generalized to various families of abelian varieties such as abelian surfaces with multiplication by the quaternion algebra of discriminant $6$, with field of moduli $\mathbb{Q}$ and which is a Jacobian in characteristic $2$ and $3$ in \cite{MR2414789}. For a detailed discussion of related works, see \cite{chen2025infinitelysupersingularprimesmumfords}.

\subsection{The main result}
We extend the result of \cite{MR2414789} to number fields with at least one real embedding, and weaken the local conditions at the bad reduction primes $2$ and $3$. We expect that similar methods can be applied to the settings in \cites{MR2704678, MR2705896, li2025infinitelyprimesbasicreduction,chen2025infinitelysupersingularprimesmumfords}.

A special case of our main theorem is as follows.

Elkies proved in \cite{MR903384} that an elliptic curve defined over a number field of odd degree has infinitely many primes of supersingular reduction, and extended the result in \cite{MR1030140} to elliptic curves over any number field with at least one real embedding. This has been generalized to various families of abelian varieties such as abelian surfaces with multiplication by the quaternion algebra of discriminant $6$, with field of moduli $\mathbb{Q}$ and which is a Jacobian in characteristic $2$ and $3$ in \cite{MR2414789}. For a detailed discussion of related works, see \cite{chen2025infinitelysupersingularprimesmumfords}.

\subsection{The main result}
We extend the result of \cite{MR2414789} to number fields with at least one real embedding, and weaken the local conditions at the bad reduction primes $2$ and $3$. We expect that similar methods can be applied to the settings in \cites{MR2704678, MR2705896, li2025infinitelyprimesbasicreduction,chen2025infinitelysupersingularprimesmumfords}.

A special case of our main theorem is as follows.

\begin{theorem}\label{thm_main}
Let $C$ be a genus $2$ curve with Jacobian that has multiplication by the maximal quaternion order with discriminant $6$,
 and has field of moduli a number field $L$ with at least one real embedding.
 Write $j_0:=j(C)\in L$ and $\Nm_{L/\bQ}(j_0) = \frac{n}{d}$ with $n,d\in \bZ, (n,d,6) = 1, d>0$.
 Assume at least one of the following conditions:
 \begin{enumerate}
 \item $v_{\mathfrak{p}}(j_0) \leq 0$ for $\mathfrak{p}$ above $2,3$, and $[L:\bQ]+v_3(d\Nm_{L/\bQ}(27j_0+16))$ is odd or $j_0$ has at least one real conjugate in $(-\frac{16}{27},0)$;
 \item $v_{\mathfrak{p}}(j_0) \geq 0$ for $\mathfrak{p}$ above $2$, $v_{\mathfrak{p}}(j_0) \leq 0$ for $\mathfrak{p}$ above $3$, and $j_0$ has at least one real conjugate in $(0, \infty)$;
 \item $[L:\bQ]$ is even, $v_{\mathfrak{p}}(j_0) \geq 0$ for $\mathfrak{p}$ above $2,3$, $j_0$ has at least one real conjugate in $(-\infty, -\frac{16}{27})\cup (0, \infty)$.
 \end{enumerate}
 Then the Jacobian of $C$ has superspecial reduction at infinitely many primes.
\end{theorem}

An abelian surface with action of a maximal order in a rational quaternion algebra $B$ has either ordinary or superspecial reduction modulo primes not dividing the discriminant of $B$ (\cite{Clark2003}*{p.70} and \cite{MR1097624}*{p.23}). It therefore suffices to construct primes of supersingular reduction, for which we follow Elkies' general strategy. Given a genus $2$ curve $C$, we find supersingular reduction of its Jacobian from its intersection with some Heegner cycle $\cP_D$ of discriminant $D$. The intersection at a prime $p$ is captured by $v_p(P_D(j_0))>0$, where $P_D(x)$ is the integral minimal polynomial of the $j$-invariants of the points in $\cP_D$, and when this occurs, supersingular reduction at $p$ is detected by $\left(\frac{D}{p}\right) \neq 1$.

In each case, we are going to find some discriminant $D$ with a rational prime $p\notin \Nm_{L/\bQ}(S)$ such that $v_p(P_{D}(j_0)) > 0$ and $\left(\frac{D}{p}\right) \neq 1$, then as in \cite{MR2414789}, there is a prime $\fp \notin S$ above $p$, such that the Jacobian of $C$ has supersingular, and hence superspecial reduction.
 \begin{enumerate}
 \item Choose a prime $l$ satisfying the conditions:
 \begin{itemize}
 \item $l\equiv 13 \pmod{24}$,
 \item $\left(\frac{-l}{q}\right) = 1$ for every prime $q\in \Nm_{L/\bQ}(S) \backslash \{2,3\}$,
 \item $P_{-4l}(x)$ has a real root $j(a_i)$ in a subinterval of $(-\frac{16}{27},0)$ to be specified later.
 \end{itemize}
 Let $D = -4l$. By \cite{MR2414789}*{Lemma 6} and \Cref{lem:red23}, \begin{align*}
 P_{-4l}(x) &\equiv x^{h'} \pmod{4}, \\
 P_{-4l}(x) &\equiv \pm x^{h'} \pmod{3}, \\
 P_{-4l}(x) &\equiv (27x+16)S(x)^2 \pmod{l} \text{ for some }S(x) \in\bZ[x].
 \end{align*}
 Since $v_{\mathfrak{p}}(j_0) \leq 0$ for all $\mathfrak{p}$ above $2,3$, we can choose $d$ such that
 $d\prod_{\sigma\in T} \sigma(j_0)$ is integral for any $T\subset \Hom(L, \overline{\bQ})$ and any prime dividing $d$ lies in $\Nm_{L/\bQ}(S)$, then
 $N, dQ\in \bN$ and $(N, 6) = 1,\, (dQ,2) = 1$.
 Suppose $l\nmid N$, then \begin{align*}
 \left(\frac{-4l}{N}\right) &= \left(\frac{-1}{N}\right)\left(\frac{l}{N}\right)= \left(\frac{-1}{N}\right)\left(\frac{N}{l}\right) \\
 &= \left(\frac{-1}{N}\right)\left(\frac{dQ}{l}\right) = \left(\frac{-1}{N}\right)\left(\frac{l}{dQ}\right)\\
 &= \left(\frac{-1}{N}\right)\left(\frac{-1}{dQ}\right) \left(\frac{-l}{dQ}\right) \\ &= \left(\frac{-1}{s(d\Nm_{L/\bQ} j_0)^{h'}}\right)\left(\frac{-1}{s'3^{[L:\bQ]}d\Nm_{L/\bQ}(j_0)}\right)\left(\frac{-l}{3^{v_3(dQ)}}\right) \\
 &= ss'(-1)^{[L:\bQ]+v_3(dQ)}.\end{align*}
 If $[L:\bQ]+v_3(dQ)$ is odd, let \[j(a_1)\in\left(-\frac{16}{27}, -\frac{16}{27} + \min_{1\leq i \leq r} \left|j_i + \frac{16}{27}\right|\right)\]
 so that $(27j_i + 16) P_{-4l}(j_i) > 0$ for each $1\leq i\leq r$.
 If $[L:\bQ]+v_3(dQ)$ is even, by assumption let $j_t$ be the minimal real conjugate of $j_0$ in $( -\frac{16}{27} , 0)$, and \[j(a_1)\in\left(j_t, j_{t+1}\right)\] so that $(27j_i + 16) P_{-4l}(j_i) > 0$ for each $1\leq i\leq r, i\neq t$ and $(27j_t + 16) P_{-4l}(j_t) < 0$. In either case we have \[\left(\frac{-4l}{N}\right) = -1\] and there is a prime divisor $p$ of $N$ such that \[\left(\frac{-4l}{p}\right) = -1.\] The conditions on $l$ and $(N,6) = 1$ imply that $p\notin \Nm_{L/\bQ}(S)$ and $v_p(\Nm_{L/\bQ}(P_D(j_0))) = v_p(N) > 0$.
 If $l | N$, then since $l\notin \Nm_{L/\bQ}(S)$ by construction, we can choose $p = l$.
 \item Choose a prime $l$ satifying the conditions:
 \begin{itemize}
 \item $l\equiv 19\pmod{24}$,
 \item $\left(\frac{q}{l}\right)=\left(\frac{-l}{q}\right) = 1$ for every prime $q\in \Nm_{L/\bQ}(S) \backslash \{2,3\}$,
 \item $P_{-l}(x)$ has a real root $j(a_1)$ in a subinterval of $(0, \infty)$ to be specified later.
 \end{itemize} Let $D = -l$. By \Cref{lem:red23}, \Cref{lem:pair}, and \cite{MR2414789}*{Lemma 4},
 \begin{align*}
 P_{-l}(x) &\equiv 1 \pmod{2}, \\
 P_{-l}(x) &\equiv \pm x^{h'} \pmod{3}, \\
 P_{-l}(x) &\equiv 3(27x+16)S(x)^2 \pmod{l} \text{ for some }S(x) \in\bZ[x].
 \end{align*}
 Since $v_{\mathfrak{p}}(j_0) \geq 0$ for $\mathfrak{p}$ above $2$ and $v_{\mathfrak{p}}(j_0) \leq 0$ for $\mathfrak{p}$ above $3$, we can choose $d$ such that
 $d\prod_{\sigma\in T} \sigma(j_0)$ is integral for any $T\subset \Hom(L, \overline{\bQ})$ and any prime dividing $d$ lies in $\Nm_{L/\bQ}(S)\backslash\{2\}$, then
 $N, dQ\in \bN$ and $(N, 6) = 1$ .
 Suppose $l\nmid N$, then
 \begin{align*}
 \left(\frac{-l}{N}\right) &= \left(\frac{N}{l}\right) \\
 &= \left(\frac{s(d\Nm_{L/\bQ}(3(27j_0+16)))}{l}\right)\\
 &= \left(\frac{ss'3^{[L:\bQ]}dQ}{l}\right) \\
 &= ss'(-1)^{[L:\bQ]+v_3(dQ)+v_2(Q)}
 \end{align*}
 Let $j_t$ be the minimal real conjugate of $j_0$ in $(0, \infty)$ and $n$ ($0\leq n < r$) be the number of real conjugates of $j_0$ in $(-\frac{16}{27}, 0)$. If $[L:\bQ]+v_3(dQ)+v_2(Q)$ is odd, let \[\twocase{j(a_1)\in}{(j_t, j_{t+1})}{$n$ is odd}{(0, j_t)}{$n$ is even}\] so that $\Nm_{L/\bQ}(P_{-l}(j_0))\Nm_{L/\bQ}(27j_0+16)>0$. If $[L:\bQ]+v_3(dQ)+v_2(Q)$ is even, let \[\twocase{j(a_1)\in}{(j_t, j_{t+1})}{$n$ is even}{(0, j_t)}{$n$ is odd}\] so that $\Nm_{L/\bQ}(P_{-l}(j_0))\Nm_{L/\bQ}(27j_0+16)<0$. In either case we have \[\left(\frac{-l}{N}\right) = -1\] and there is a prime divisor $p$ of $N$ such that \[\left(\frac{-l}{p}\right) = -1.\] The conditions on $l$ and $(N,6) = 1$ imply that $p\notin \Nm_{L/\bQ}(S)$ and $v_p(\Nm_{L/\bQ}(P_D(j_0))) = v_p(N) > 0$.
 If $l | N$, then since $l\notin \Nm_{L/\bQ}(S)$ by construction, we can choose $p = l$.

\begin{rem} \label{rem:conclude} One can try to further weaken the local conditions by considering Heegner cycles with different forms of discriminant $D$.
For primes $p$ dividing $D$, the unpaired roots of $P_D(x)$ modulo $p$ can be predicted by checking whether a maximal order in a definite quaternion algebra ramified at $2,3,p$ contains two anticommuting CM orders of some given discriminants, as computed in \cite{MR2704678}*{3.4.2}. One can then impose conditions on the number of prime divisors of $D$ and the congruence class modulo $24$ of primes dividing $D$, so that $P_D(x)$ has no unpaired roots or $P_D(x)$ has a single unpaired root from the same elliptic point modulo each $p\nmid D$. In addition, one imposes a congruence condition on $D$ so that $\cP_{D, \cE}$ avoids intersection with one of $\cP_{-3, \cE}$ and $\cP_{-4, \cE}$ by \Cref{lem:avoidintersection}.
The local conditions on $j_0$ are determined so that it avoids intersection with the these $\cP_D$ at $p = 2,3$, and some real conjugate of $j_0$ and some real root of $P_D$ lie in the subinterval corresponding to one geodesic. For $D$ not of the form $-Np$ with $N = 1, 3, 4, 8, 12, 24$, it is possible that $P_D(x)$ has multiple real roots in each subinterval, but we still expect an equidistribution result.
\end{rem}
\bibliographystyle{amsalpha}
\bibliography{references}

More generally, the coarse moduli variety of principally polarized abelian surfaces with potential multiplication by the maximal quaternion order of discriminant $6$ is isomorphic to $\mathbb{P}^1$, and an arithmetic $j$-function (see \eqref{eq:j} in \cref{sec:coord}) is defined in \cite{MR2423455}. In terms of this coordinate, the assumption at primes $\mathfrak{p}$ above $2$ and $3$ in \Cref{thm_specialcase} corresponds to the case $v_{\mathfrak{p}}(j(C)) = 0$ by \cite{MR2414789}*{Proposition 2}, and our main theorem can be stated as follows.

\begin{theorem}\label{thm_main}
Let $C$ be a genus $2$ curve with Jacobian that has multiplication by the maximal quaternion order with discriminant $6$,
    and has field of moduli a number field $L$ with at least one real embedding.
    Write $j_0:=j(C)\in L$ and $\Nm_{L/\mathbb{Q}}(j_0) = \frac{n}{d}$ with $n,d\in \mathbb{Z}, (n,d,6) = 1, d>0$.
    Assume at least one of the following conditions:
    \begin{enumerate}
        \item $v_{\mathfrak{p}}(j_0) \leq 0$ for $\mathfrak{p}$ above $2,3$, and $[L:\mathbb{Q}]+v_3(d\Nm_{L/\mathbb{Q}}(27j_0+16))$ is odd or $j_0$ has at least one real conjugate in $(-\frac{16}{27},0)$;
        \item $v_{\mathfrak{p}}(j_0) \geq 0$ for $\mathfrak{p}$ above $2$, $v_{\mathfrak{p}}(j_0) \leq 0$ for $\mathfrak{p}$ above $3$, and $j_0$ has at least one real conjugate in $(0, \infty)$;
        \item $[L:\mathbb{Q}]$ is even, $v_{\mathfrak{p}}(j_0) \geq 0$ for $\mathfrak{p}$ above $2,3$, $j_0$ has at least one real conjugate in $(-\infty, -\frac{16}{27})\cup (0, \infty)$.
    \end{enumerate}
    Then the Jacobian of $C$ has superspecial reduction at infinitely many primes.
\end{theorem}

An abelian surface with action of a maximal order in a rational quaternion algebra $B$ has either ordinary or superspecial reduction modulo primes not dividing the discriminant of $B$ (\cite{Clark2003}*{p.70} and \cite{MR1097624}*{p.23}). It therefore suffices to construct primes of supersingular reduction, for which we follow Elkies' general strategy. Given a genus $2$ curve $C$, we find supersingular reduction of its Jacobian from its intersection with some Heegner cycle $\cP_D$ of discriminant $D$. The intersection at a prime $p$ is captured by $v_p(P_D(j_0))>0$, where $P_D(x)$ is the integral minimal polynomial of the $j$-invariants of the points in $\cP_D$, and when this occurs, supersingular reduction at $p$ is detected by $\left(\frac{D}{p}\right) \neq 1$.

As in \cite{MR2414789}, we need to consider certain elliptic point in addition to the Heegner cycle in order to pair the roots of $P_D(x)$ modulo primes dividing $D$. We use \cite{MR2704678} to generalize \cite{MR2414789} through a more detailed, case-by-case study, where in each case the discriminant $D$ is chosen in a form adapted to the conditions.
In characteristic $2$ and $3$, \cite{MR2414789} shows that the reduction of any Heegner cycle lies in the superspecial locus $\{j = 0, \infty\}$, and the intersection formula of Heegner divisors in \cite{MR2441697} provides the new input that determines the specific superspecial point for each chosen Heegner cycle. The local conditions on $j_0$ at primes above  $2$ and $3$ ensure that it avoids intersection with the chosen Heegner cycles at these primes.
For more on the choice of local conditions and Heegner cycles, and for potential further weakenings of these conditions, see \Cref{rem:conclude}.

\subsection{Notation and conventions}

Assume the following unless specified otherwise.

Let $B = B_{\Delta}$ be an indefinite quaternion algebra over $\mathbb{Q}$ of discriminant $\Delta$,  $\Lambda = \Lambda_{\Delta}$ be a maximal order of $B$, and $\Lambda^1 = \Lambda_{\Delta}^1$ be the group of units in $\Lambda$ of norm $1$. Fix an element $\mu \in \Lambda_{\Delta}$ such that $\mu^2 = -\Delta$,\footnote{The element $\mu$ exists because the field $\mathbb{Q}(\sqrt{-\Delta})$ embeds in $B$ by the local-global principle, and any two maximal orders in $B$ are conjugate to each other by strong approximation.} then the involution $\alpha \mapsto \alpha' = \mu^{-1} \bar{\alpha} \mu$ is a positive anti-involution on $B$.

An abelian surface with action of a maximal order in a rational quaternion algebra $B$ has either ordinary or superspecial reduction modulo primes not dividing the discriminant of $B$ (\cite{Clark2003}*{p.70} and \cite{MR1097624}*{p.23}), so “it therefore suffices to construct primes of supersingular reduction,” following “Elkies' general strategy.” Given $C$, one finds supersingular reduction of its Jacobian “from its intersection with some Heegner cycle $\cP_D$ of discriminant $D$.” The intersection at a prime $p$ is “captured by $v_p(P_D(j_0))>0$,” where $P_D(x)$ is “the integral minimal polynomial of the $j$-invariants of the points in $\cP_D$,” and when this occurs, supersingular reduction at $p$ is “detected by $\left(\frac{D}{p}\right) \neq 1$.”

As in \cite{MR2414789}, one must “consider certain elliptic point in addition to the Heegner cycle in order to pair the roots of $P_D(x)$ modulo primes dividing $D$.” Using \cite{MR2704678}, the argument proceeds via “a more detailed, case-by-case study,” where “in each case the discriminant $D$ is chosen in a form adapted to the conditions.” In characteristic $2$ and $3$, \cite{MR2414789} shows “the reduction of any Heegner cycle lies in the superspecial locus $\{j = 0, \infty\}$,” and “the intersection formula of Heegner divisors in \cite{MR2441697} provides the new input that determines the specific superspecial point for each chosen Heegner cycle.” Finally, “the local conditions on $j_0$ at primes above $2$ and $3$ ensure that it avoids intersection with the chosen Heegner cycles at these primes,” yielding infinitely many primes of superspecial reduction.

Question: Let $C$ be a genus $2$ curve whose Jacobian $J(C)$ admits an action of the maximal order in the rational quaternion algebra of discriminant $6$, and suppose the field of moduli of $C$ is a number field $L$ having at least one real embedding. Assume moreover that for every prime of $L$ above $2$ or $3$, the curve $C$ has potentially smooth stable reduction there (that is, after a finite extension, its stable reduction becomes smooth). Under these hypotheses, which statement about the reduction of $J(C)$ holds? Here, saying that $J(C)$ has superspecial reduction at a prime means that its reduction modulo that prime is a superspecial abelian surface.

The Jacobian $J(C)$ has superspecial reduction at infinitely many primes.

Choices:
B. The Jacobian $J(C)$ has superspecial reduction at all but finitely many primes.
C. The Jacobian $J(C)$ has superspecial reduction at at least one prime.
D. There exists a finite extension $L'/L$ such that the base change $J(C)_{L'}$ has superspecial reduction at infinitely many primes of $L'$.
E. The Jacobian $J(C)$ has infinitely many primes of supersingular reduction.

The stem does not state the conclusion outright; it only gives hypotheses and asks for the resulting reduction statement. Defining 'superspecial reduction' clarifies terminology but does not reveal which quantified claim is correct.

This is close to a theorem-recall item: the hypotheses are highly specific and the correct option matches the natural theorem conclusion. However, the options introduce meaningful quantifier and property variations, so it is not a pure verbatim restatement.

Some reasoning is needed to distinguish 'infinitely many' from weaker existence, overly strong density-type claims, base-change variants, and supersingular/superspecial confusion. Still, the task is mainly recognition of the exact theorem-level conclusion rather than deep derivation.

The distractors are plausible and mathematically well targeted: one is too strong (all but finitely many), one is a weaker true-looking statement, one alters the field/quantifier dependence, and one exploits the common superspecial vs. supersingular confusion.

A solid theorem-conclusion MCQ with strong distractors and no direct answer leakage, but it remains fairly close to theorem recall rather than testing substantial generative mathematical reasoning.