\label{s:data}

\Cref{table:triples} contains \emph{all} triples of pairs of integers, $(p_1,q_1)$, $(p_2,q_2)$, $(p_3,q_3)$ such that:
\begin{itemize}
\item there exist rational homology balls $B_1$,

\begin{proof}[Proof of Corollaries \ref{cor:Bpq} and \ref{cor:Lpq}]
For this we note that by \Cref{thm:2Farey}, either $\dfrac{p}{q}$ or $\dfrac{p}{p-q}$ appears in a 2-Farey triple.  It follows that $B_{p,q}$ embeds smoothly in $\CPb$ and hence so does its boundary $L(p^2,pq-1)$.  Finally note that an embedding of a 3-manifold into a 4-manifold does not depend on the orientation of either.
\end{proof}

\section{A common framework for known results}\label{s:compare}

In this section we make an observation which attempts to put \Cref{thm:ANN} in a common framework with previous results of Hacking--Prokhorov \cite{hp}, Evans--Smith \cite{es}, and Lisca--Parma \cite{HD2}.  Each of these results gives a binary tree, together with an embedding of a triple of rational balls in $\CP$ for each node of the tree.  The first example of this is the famous Markov tree which contains all triples of solutions of
\begin{equation*}
p_1^2+p_2^2+p_3^2=3p_1p_2p_3.
\end{equation*}
This features in each of \cite{es,hp,HD2}.  Two further trees of triples arise from \cite{HD2}, and the 2-Farey tree underlies \Cref{thm:ANN}.  The following definition may be interpreted in terms of the proofs of \Cref{prop:2Femb} and  \cite[Proposition 4.6]{HD1}.  In each case we have in mind three curves lying on or near nested tori, and the recursive rule is given by handlesliding one curve over another.

\begin{defn}
A \emph{signed slide triple tree} is an infinite rooted binary tree with a triple $\left(\left(\dfrac{p_1}{q_1},\delta_1\right),\left(\dfrac{p_2}{q_2},\delta_2\right),\left(\dfrac{p_3}{q_3},\delta_3\right)\right)$ at each node and the recursive rule

{\centering
\includegraphics[width=\textwidth]{./figures/sliderule.pdf}
\par}
\noindent
where $\delta_i\in\{\pm1\}$, $x_1=p_2q_3-p_3q_2$ and $x_3=p_1q_2-p_2q_1$.
\end{defn}

We note that the following gives new recursive descriptions of the triples appearing in \cite[Theorem 1.2(2),(3)]{HD2}.
\begin{thm}\label{thm:trees}
Let $\left(\left(\dfrac{p_1}{q_1},\delta_1\right),\left(\dfrac{p_2}{q_2},\delta_2\right),\left(\dfrac{p_3}{q_3},\delta_3\right)\right)$ be a triple appearing at a node of a signed slide triple tree whose root is labeled by one of
\begin{enumerate}
\item \triple{1}{-1}{1}{5}{1}{1}{2}{1}{1},
\item \triple{1}{1}{-1}{-3}{-1}{-1}{2}{1}{1},
\item \triple{1}{-1}{1}{-3}{-1}{-1}{2}{1}{1},
\item \triple{1}{0}{-1}{3}{2}{-1}{2}{2}{-1}.
\end{enumerate}
Then the disjoint union $\bigsqcup\delta_i B_{p_i,q_i}$ embeds smoothly in $\CP$.  The first root above gives the triples arising from the Markov tree, the second and third are the second and third families of embeddings in \cite[Theorem 1.2]{HD2}, and the fourth gives the triples in \Cref{thm:ANN}.
\end{thm}

\begin{proof}[Proof sketch]
For the case of the first root, which yields the Markov tree, we appeal to \cite{perling}, which refers to $\{q_1,q_2,q_3\}$ as the \emph{T-weights} of the Markov triple $\{p_1,p_2,p_3\}$.  The identities \cite[(1)]{perling} show that $x_1=3p_1$ and $x_3=3p_3$, and then the recursive rule above agrees with the mutation rules for Markov triples and their T-weights \cite[\S2 and Lemma 3.1]{perling}.  The first three rows of the tree are shown below.  We omit the $\delta_i$ which are all equal to $+1$.

{\centering
\includegraphics[width=10cm]{./figures/MarkovTree.pdf}
\par}
\noindent

The second root above yields the triples of rational balls embedding in $\CP$ given by Lisca--Parma in \cite[Theorem 1.2(2)]{HD2}.
One can show that $x_1=\delta_1p_1$ and $x_3=\delta_3p_3$ and that, as for the standard Markov equation, the recursive rule given above agrees with the mutation rule for the triples of solutions and corresponding weights of the Markov-type equation
\begin{equation*}
\delta_1 p_1^2+\delta_2 p_2^2 +\delta_3 p_3^2=p_1 p_2 p_3,\end{equation*}
with $\{\delta_1, \delta_2, \delta_3\}=\{1,-1,-1\}$ and weights
\begin{align*}
q_1&\equiv\delta_2p_2/p_3\equiv-\delta_3p_3/p_2\pmod{p_1},\\
q_2&\equiv\delta_3p_3/p_1\equiv-\delta_1p_1/p_3\pmod{p_2},\\
q_3&\equiv\delta_1p_1/p_2\equiv-\delta_2p_2/p_1\pmod{p_3}.
\end{align*}
Each weight $q_i$ is strictly between $0$ and $p_i$ unless $p_i=1$, in which case we take $q_i=1$.

The first entries of the resulting tree of triples are shown below.  In each triple the underlined entry has $\delta_i=1$.  Note that $B_{-p,-q}=B_{p,q}$, but the negative signs in the tree below are needed for the recursion.

{\centering
\includegraphics[width=10cm]{./figures/HD2secondtree.pdf}
\par}
\noindent

The first few entries of the tree arising from the third root in the statement are shown below, as well as two extra nodes before the root.  In each triple the underlined entry has $\delta_i=-1$.

{\centering
\includegraphics[width=10cm]{./figures/HD2thirdtree.pdf}
\par}
\noindent
We claim that this contains precisely the triples of rational balls embedding in $\CP$ given by Lisca--Parma in \cite[Theorem 1.2(3)]{HD2}.  To see this, for each node, let $\gamma_i=(p_i,q_i)$,  and take
$$(x_1,x_2,x_3)=(\gamma_2\cdot\gamma_3,\gamma_1\cdot\gamma_3,\gamma_1\cdot\gamma_2),$$
where $(p,q)\cdot(r,s)=ps-qr$.  For example, if we consider the first triple
$\left(\frac{1}{-1},\frac{1}{\underline{0}},\frac{-1}{-1}\right)$, we get
$$\gamma_1=(1,-1), \gamma_2=(1,0), \gamma_3=(-1,-1) \text{ and } (x_1,x_2,x_3)=(-1,-2,1),$$
and for the second triple $\left(\frac{1}{-1},\frac{2}{1},\frac{1}{\underline{0}}\right)$ we obtain
$$\gamma_1=(1,-1), \gamma_2=(2,1), \gamma_3=(1,0) \text{ and } (x_1,x_2,x_3)=(-1,-1,-3).$$

We claim that every triple in this tree satisfies the following Markov-type equation
\begin{equation}
\delta_1 x_1^2+\delta_2 x_2^2 +\delta_3 x_3^2=x_1 x_2 x_3-4,\label{eq:3-2}
\end{equation}
and also
\begin{equation}
\delta_1 p_1^2+\delta_2 p_2^2 +\delta_3 p_3^2-\delta_1\delta_2p_1 p_2 x_3-\delta_1\delta_3p_1p_3x_2-\delta_2\delta_3p_2p_3x_1-p_1p_3x_1x_3=0.\label{eq:3-3}
\end{equation}
These are equations (3-2) and (3-3) in \cite{HD2}.

It is straightforward to check these equations are satisfied for the triple $\left(\frac{1}{-1},\frac{1}{\underline{0}},\frac{-1}{-1}\right)$, which forms the base case for a structural induction.  We then check that triples given by the recursive rule continue to satisfy \eqref{eq:3-2} and \eqref{eq:3-3}.

Suppose that we have $\delta_2=-1$ and that
$$\gamma_1=(p_1,q_1), \gamma_2=(p_2,q_2), \gamma_3=(p_3,q_3),$$
and thus
$$(x_1,x_2,x_3)=(p_2q_3-p_3q_2,p_1q_3-p_3q_1,p_1q_2-p_2q_1).$$
By induction, we assume that \eqref{eq:3-2} and \eqref{eq:3-3} hold for these variables.
Moving left down the tree, the recursive rule leads to $\delta_3=-1$,
$$\widehat{\gamma}_1=(p_1,q_1), \widehat{\gamma}_2=(-p_3-x_1p_2,-q_3-x_1q_2), \widehat{\gamma}_3=(p_2,q_2),$$
and
\begin{equation}
(\widehat{x}_1,\widehat{x}_2,\widehat{x}_3)=(x_1,x_3,-x_2-x_1x_3).\label{eq:mut}
\end{equation}
We verify that \eqref{eq:3-2} holds:
\begin{align*}
&\widehat{x}_1^2+\widehat{x}_2^2-\widehat{x}_3^2-\widehat{x}_1\widehat{x}_2\widehat{x}_3+4\\
&=x_1^2+x_3^2-x_2^2-2x_1x_2x_3-x_1^2x_3^2+x_1x_2x_3+x_1^2x_3^2+4\\
&=x_1^2-x_2^2+x_3^2-x_1x_2x_3+4\\
&=0,
\end{align*}
and also \eqref{eq:3-3}:
\begin{align*}
&\widehat{p}_1^2+\widehat{p}_2^2-\widehat{p}_3^2-\widehat{p}_1 \widehat{p}_2 \widehat{x}_3+\widehat{p}_1\widehat{p}_3\widehat{x}_2+\widehat{p}_2\widehat{p}_3\widehat{x}_1-\widehat{p}_1\widehat{p}_3\widehat{x}_1\widehat{x}_3\\
&=p_1^2+p_3^2+2p_2p_3x_1+p_2^2x_1^2-p_2^2-p_1p_3x_2-p_1p_2x_1x_2-p_1p_3x_1x_3-p_1p_2x_1^2x_3\\
&\quad+p_1p_2x_3-p_2p_3x_1-p_2^2x_1^2+p_1p_2x_1x_2+p_1p_2x_1^2x_3\\
&=p_1^2-p_2^2+p_3^2+p_1p_2x_3-p_1p_3x_2+p_2p_3x_1-p_1p_3x_1x_3\\
&=0,
\end{align*}
in both cases using induction for the final equality.

The other cases (moving right down the tree from $\delta_2=-1$, or starting with $\delta_2=1$) are minor variants of this.

The proof of \cite[Theorem 1.2(3)]{HD2} shows that the triples therein come from solutions to \eqref{eq:3-2} and \eqref{eq:3-3} such that the triples $(x_1,x_2,x_3)$ are obtained from the minimal solution $(1,2,1)$ by mutations, sign changes, and permutations as in \eqref{eq:mut}.  By comparison, we conclude that our  signed slide triple tree contains all of the triples of rational balls in the family given in \cite[Theorem 1.2(3)]{HD2}.

For the last case, one can show by induction that $x_1=x_3=2$ for all nodes, and then the recursion rule is easily seen to agree with that of the 2-Farey tree.
\end{proof}

\label{s:balls}
We use Hirzebruch--Jung continued fractions, with the following notation:
$$[a_1,a_2,\ldots,a_k]
:=a_1-\frac{1}{a_2-\raisebox{-3mm}{$\ddots$
\raisebox{-2mm}{${-\frac{1}{\displaystyle{a

\label{s:homotopyCP2s}

In this section we describe some methods for finding rational balls bounded by lens spaces which embed in a homotopy $\CP$ or $\CPb$.
These methods are suitable for use in a co

\begin{quest}\label{q:exoticCP2s}
Are all homotopy $\CP$'s constructed in this section diffeomorphic to the standard $\CP$?
\end{quest}

\begin{proof}
First note that by reversing the first diagram in either \Cref{fig:Bpq} or \ref{fig:Apq}, we see that $B_{p,q}\cong B_{p,p-q}$.

Now take $k\in\Z$.  Start with $B_{p,kp+q}$ as in the second diagram of either \Cref{fig:Bpq} or \ref{fig:Apq}, with the dotted circle replaced by a $\langle0\rangle$-framed circle as in the proof of Lemma $\ref{lem:Bpq}$.  Then the  diffeomorphism $B_{p,q}\cong B_{p,kp+ q}$ follows by performing Rolfsen twists \cite[Section 5.3]{GS} along this $\langle0\rangle$-framed circle.
\end{proof}

\section{The 2-Farey tree}\label{s:2Fareytree}

We define a variant of the classical Farey tree \cite{aigner}, which we refer to as the 2-Farey tree.  This is a binary tree with ordered triples of fractions, called 2-Farey triples, at each node.  The triple at the root is $\left(\frac10,\frac32,\frac22\right)$ and the recursive rule is given by

{\centering
\includegraphics[width=7cm]{./figures/2Fareyrule.pdf}
\par}
\noindent

The first three rows of the tree, and two further nodes, are shown below.

{\centering
\includegraphics[width=10cm]{./figures/2Fareytree.pdf}
\par}
\noindent
Observe that each triple contains two reduced fractions and one fraction $p/q$ with $\gcd(p,q)=2$.

For comparison, part of the standard Farey tree is shown below.  It uses the same recursive rule as above but the triple at the root is $\left(\frac10,\frac11,\frac01\right)$.

{\centering
\includegraphics[width=10cm]{./figures/Fareytree.pdf}
\par}
\noindent
Note that the 2-Farey tree is obtained from the subtree of the Farey tree on the left, beginning at $\left(\frac10,\frac31,\frac21\right)$, by multiplying all denominators by 2.

The following theorem  follows immediately from the corresponding facts for the classical Farey tree found in the wonderful book by Aigner \cite[Section 3.2]{aigner}.

\begin{thm}\label{thm:2Farey}
Every fraction $p/q$ with $q$ even, $p\ge q$, and $\gcd(p,q)\le2$ appears in the $2$-Farey tree.  Every such fraction with $p>2$ is generated as a mediant exactly once.  For each $2$-Farey triple $\left(\dfrac{p_1}{q_1},\dfrac{p_2}{q_2},\dfrac{p_3}{q_3}\right)$, we have
\begin{equation}\label{eqn:2Farey}
p_2=p_1+p_3,\quad q_2=q_1+q_3,\quad \mathrm{and}\quad p_iq_{i+1}-p_{i+1}q_i=2\ \mathrm{for}\  i\in\{1,2\}.
\end{equation}
Furthermore:

\begin{enumerate}[(i)]
\item If $p_1>q_1$ and $p_2>q_2$ satisfy $q_i$ even and $p_1q_2-p_2q_1=\pm2$, then $p_1/q_1$ and $p_2/q_2$ are contained in a $2$-Farey triple.
\item For any coprime natural numbers $p_1, p_2$, there exist unique even numbers $q_1$ and $q_2$ satisfying
$0\le q_i\le p_i$ and $p_1q_2-p_2q_1=\pm2$.  After possibly relabeling, we may take $p_1q_2-p_2q_1=2$, and then $\left(\dfrac{p_1}{q_1},\dfrac{p_1+p_2}{q_1+q_2},\dfrac{p_2}{q_2}\right)$ is a $2$-Farey triple.
\end{enumerate}
\end{thm}

\begin{rmk}
One could alternatively choose the triple $\left(\frac20,\frac31,\frac11\right)$ at the root of the 2-Farey tree.  The resulting tree would be obtained from that above by replacing each fraction $p/q$ by $p/(p-q)$, and reflecting the tree and each triple from left to right.  It contains all triples $\left(\dfrac{p_1}{q_1},\dfrac{p_2}{q_2},\dfrac{p_3}{q_3}\right)$ with $p_i\equiv q_i \pmod2$ and  satisfying \eqref{eqn:2Farey}.
Since by \Cref{lem:addp} we have $B_{p,q}=B_{p,p-q}$, this would result in the same embeddings of rational balls in $\CPb$ in the next section.

We could also consider an expanded 2-Farey tree obtained by multiplying denominators by 2 in all nodes of the full Farey tree shown above.  It again follows from \Cref{lem:addp} that this does not yield any more embedding results than those coming from the 2-Farey tree we have selected, though we will see that for the base of our induction, it is helpful to include two more nodes at the base of the tree.
\end{rmk}

\section{Embeddings of 2-Farey triples}\label{s:ANN}

Given an ordered $n$-tuple $\left(\dfrac{p_1}{q_1},\dfrac{p_2}{q_2},\dots,\dfrac{p_n}{q_n}\right)$, with $\gcd(p_i,q_i)\in\{1,2\}$, we define a Kirby diagram of a manifold $X=X_{\frac{p_1}{q_1},\frac{p_2}{q_2},\dots,\frac{p_n}{q_n}}$ as follows.  Let $r_i=1-i/(n+1)$ for $i=1,\dots,n$, so that
$$1>r_1>r_2>\dots>r_n>0,$$
and $\varepsilon=1/(4n)$, so that the intervals $[r_i-\varepsilon,r_i+\varepsilon]$ are pairwise disjoint in $(0,1)$.

The manifold $X$ is a handlebody with one $0$-handle, one $1$-handle, and $n$ $2$-handles.  To draw the diagram, start with a dotted circle drawn as a circle of diameter 3.  For each $1\le i\le n$, draw the framed knot $K_{p_i,q_i}$ from \Cref{s:balls} in an $\varepsilon$-neighbourhood of the torus which is distance $r_i$ from the dotted circle.  An example is shown in Figure \ref{fig:Base}.

\begin{figure}[htbp]
\centering
\includegraphics[width=14cm]{./figures/Base.pdf}
\caption{{\bf The manifold $X_{\frac10,\frac12,\frac02}$.} The black arrow indicates a handle slide that may be used to simplify the diagram.}
\label{fig:Base}
\end{figure}

\begin{lemma}\label{lem:LPembed}
For any $\left(\dfrac{p_1}{q_1},\dfrac{p_2}{q_2},\dots,\dfrac{p_n}{q_n}\right)$, with $\gcd(p_i,q_i)\in\{1,2\}$, the disjoint union
$$\bigsqcup_{i=1}^n B_{p_i,q_i}$$
embeds smoothly in $X_{\frac{p_1}{q_1},\frac{p_2}{q_2},\dots,\frac{p_n}{q_n}}$.
\end{lemma}
\begin{proof}
This follows exactly as in the proof of \cite[Lemma 4.1]{HD2}.  We give a sketch here for convenience.  Consider the 4-manifold given by a single dotted circle, as found in each of $X_{\frac{p_1}{q_1},\frac{p_2}{q_2},\dots,\frac{p_n}{q_n}}$ and $B_{p_i,q_i}$.  This is a copy of $S^1\times B^3$.  For convenience, view $B^3$ as the unit ball in $\R^3$.  Then a torus of constant distance $r$ from the dotted circle corresponds to $S^1\times S_\zeta$, where $S_\zeta$ is the intersection of the unit sphere with the plane $z=\zeta$.  Here we  take $\zeta(r)$ to be a monotone decreasing bijection from $(0,\infty)$ to $(-1,1)$.

For each $1\le i\le n$, the set of points $(x,y,z)\in B^3$ with $z\in [\zeta(r_i-\varepsilon),\zeta(r_i+\varepsilon)]$ is again a 3-ball, after smoothing corners, which we may denote by $B^3_i$.  The disjoint union $\bigsqcup_{i=1}^n S^1\times B^3_i$ is embedded in $S^1\times B^3$, and since the 2-handle for each $B_{p_i,q_i}$ is attached to the boundary of $S^1\times B^3_i$, the statement follows.
\end{proof}

\label{s:intro}
A basic, yet fundamental, question in 4-manifold topology is the following.
Given a smooth 4-manifold $X$ and a class $\mathcal{Y}$ of 3-manifolds, which $Y \in \mathcal{Y}$ smoothly embed in $X$?
In this paper, we focus on the case when $\mathcal{Y}$ is the class of lens spaces, which are the simplest 3-manifolds after the sphere, and on the simplest possible 4-manifold in which they can embed\footnote{It is known since Hantzsche's work~\cite{Hantzsche} that lens spaces cannot embed in $S^4$.}: the complex projective plane $\CP$.

Whenever a lens space $L$ embeds in $\CP$, it splits it into two components, one of which is a \emph{rational homology ball}, a 4-manifold with the rational homology of a point.
Therefore, asking for the embedding of $L$ into $\CP$ is equivalent to asking for the embedding into $\CP$ of a rational ball whose boundary is $L$.
Note that lens spaces that bound rational homology balls have been classified by Lisca~\cite{Lisca-ribbon}.

Early results of the existence of this type of embedding come from algebraic geometry~\cite{Manetti, hp} and symplectic topology~\cite{es}, as well as from~\cite{nonsymp,LPStein,HD1, HD2,EMPR}; see below for a more detailed historical account.
These results indicate that it is interesting to look at embeddings of \emph{triples} of rational homology balls in $\CP$, a question which is related (but not equivalent) to asking about disjoint embeddings of lens spaces.

Here we push the ideas of~\cite{HD2} and of~\cite{EMPR} further, and we obtain many more embeddings of \emph{triples} of rational homology balls with lens space boundary in $\CPb$;
reversing all orientations gives embeddings in $\CP$.
Our triples give previously unknown examples, except possibly when $\min\{p_i\} \le 2$.

Early results of the existence of this type of embedding come from algebraic geometry~\cite{Manetti, hp} and symplectic topology~\cite{es}, as well as from~\cite{nonsymp,LPStein,HD1, HD2,EMPR}; see below for a more detailed historical account.
These results indicate that it is interesting to look at embeddings of \emph{triples} of rational homology balls in $\CP$, a question which is related (but not equivalent) to asking about disjoint embeddings of lens spaces.

When $p$ and $q$ are coprime or have greatest common divisor 2, $B_{p,q}$ is a rational homology ball constructed with only one 1-handle and one 2-handle, whose boundary is $L(p^2,pq-1)$.
Kirby diagrams for these rational balls are given in Figures~\ref{fig:Bpq} and~\ref{fig:Apq}.

\begin{maincor}\label{cor:Bpq}
Let $p\ge q$ be nonnegative integers. If $\gcd(p,q)=2$ or if $p$ is odd and $\gcd(p,q)=1$, then the rational ball $B_{p,q}$ admits a smooth orientation-preserving embedding in $\CPb$.
\end{maincor}

\begin{maincor}\label{cor:Lpq}
Let $p\ge q$ be nonnegative integers. If $\gcd(p,q)=2$ or if $p$ is odd and $\gcd(p,q)=1$, then the lens space $L(p^2,pq-1)$ admits a smooth embedding in $\CP$.
\end{maincor}

\begin{enumerate}[(i)]
\item If $p_1>q_1$ and $p_2>q_2$ satisfy $q_i$ even and $p_1q_2-p_2q_1=\pm2$, then $p_1/q_1$ and $p_2/q_2$ are contained in a $2$-Farey triple.
\item For any coprime natural numbers $p_1, p_2$, there exist unique even numbers $q_1$ and $q_2$ satisfying
$0\le q_i\le p_i$ and $p_1q_2-p_2q_1=\pm2$. After possibly relabeling, we may take $p_1q_2-p_2q_1=2$, and then $\left(\dfrac{p_1}{q_1},\dfrac{p_1+p_2}{q_1+q_2},\dfrac{p_2}{q_2}\right)$ is a $2$-Farey triple.
\end{enumerate}
\end{thm}

\begin{lemma}\label{lem:LPembed}
For any $\left(\dfrac{p_1}{q_1},\dfrac{p_2}{q_2},\dots,\dfrac{p_n}{q_n}\right)$, with $\gcd(p_i,q_i)\in\{1,2\}$, the disjoint union
$$\bigsqcup_{i=1}^n B_{p_i,q_i}$$
embeds smoothly in $X_{\frac{p_1}{q_1},\frac{p_2}{q_2},\dots,\frac{p_n}{q_n}}$.
\end{lemma}
\begin{proof}
This follows exactly as in the proof of \cite[Lemma 4.1]{HD2}. We give a sketch here for convenience. Consider the 4-manifold given by a single dotted circle, as found in each of $X_{\frac{p_1}{q_1},\frac{p_2}{q_2},\dots,\frac{p_n}{q_n}}$ and $B_{p_i,q_i}$. This is a copy of $S^1\times B^3$. For convenience, view $B^3$ as the unit ball in $\R^3$. Then a torus of constant distance $r$ from the dotted circle corresponds to $S^1\times S_\zeta$, where $S_\zeta$ is the intersection of the unit sphere with the plane $z=\zeta$. Here we take $\zeta(r)$ to be a monotone decreasing bijection from $(0,\infty)$ to $(-1,1)$.

We note that the following gives new recursive descriptions of the triples appearing in \cite[Theorem 1.2(2),(3)]{HD2}.
\begin{thm}\label{thm:trees}
Let $\left(\left(\dfrac{p_1}{q_1},\delta_1\right),\left(\dfrac{p_2}{q_2},\delta_2\right),\left(\dfrac{p_3}{q_3},\delta_3\right)\right)$ be a triple appearing at a node of a signed slide triple tree whose root is labeled by one of
\begin{enumerate}
\item \triple{1}{-1}{1}{5}{1}{1}{2}{1}{1},
\item \triple{1}{1}{-1}{-3}{-1}{-1}{2}{1}{1},
\item \triple{1}{-1}{1}{-3}{-1}{-1}{2}{1}{1},
\item \triple{1}{0}{-1}{3}{2}{-1}{2}{2}{-1}.
\end{enumerate}
Then the disjoint union $\bigsqcup\delta_i B_{p_i,q_i}$ embeds smoothly in $\CP$. The first root above gives the triples arising from the Markov tree, the second and third are the second and third families of embeddings in \cite[Theorem 1.2]{HD2}, and the fourth gives the triples in \Cref{thm:ANN}.
\end{thm}

\begin{prop}\label{prop:Addc}
Suppose that we have two rational numbers, $p/q = [a_1,\dots,a_m]$ and $r/s = [b_1,\dots,b_n]$, and an integer $c$ such that:
\begin{itemize}
\item $L(p,q)$ and $L(r,s)$ bound rational homology balls,
\item letting $t/u = [a_m,\dots,a_1,c,b_1,\dots,b_n]$, $L(t,u)$ also bounds a rational homology ball, and
\item $p$, $r$, and $t$ are pairwise coprime.
\end{itemize}
Then there exists a homotopy $\CP$ or $\CPb$, $X$, and an embedding $B \sqcup B' \sqcup (-B'') \hookrightarrow X$, where $B$, $B'$, and $B''$ are three rational homology balls bounding $L(p,q)$, $L(r,s)$, and $L(t,u)$, respectively, constructed using handles of index at most $2$.

\begin{figure}[htbp]
\centering
\includegraphics[width=14cm]{./figures/Apq.pdf}
\caption{{\bf Two Kirby diagrams of the rational homology ball $B_{p,q}$ when $\gcd(p,q)=2$.} The blue attaching circle for the 2-handle in the second diagram is the $(2,-1)$ cable, relative to the torus framing, of the torus knot $T_{-p/2,q/2}$.}
\label{fig:Apq}
\end{figure}

\begin{figure}[htbp]
\centering
\includegraphics[width=14cm]{./figures/Bpq.pdf}
\caption{{\bf Two Kirby diagrams of the rational homology ball $B_{p,q}$ when $\gcd(p,q)=1$.} The second diagram should be closed like a braid closure, so that there is one dotted circle in black and one 2-handle attaching circle, which is a copy of the torus knot $T_{-p,q}$, in blue.  The framing on the 2-handle is $-1$ relative to the torus framing.}
\label{fig:Bpq}
\end{figure}

\begin{mainthm}\label{thm:ANN}
If $p_1,p_2,q_1,q_2$ are nonnegative integers with $q_1, q_2$ even, $p_i\ge q_i$, and $p_1q_2-p_2q_1=\pm2$, then the disjoint union
\[
B_{p_1,q_1}\sqcup B_{p_2,q_2}\sqcup B_{p_1+p_2,q_1+q_2}
\]
admits a smooth orientation-preserving embedding in $\CPb$.
\end{mainthm}

When $p$ and $q$ are coprime or have greatest common divisor 2, $B_{p,q}$ is a rational homology ball constructed with only one 1-handle and one 2-handle, whose boundary is $L(p^2,pq-1)$.
Kirby diagrams for these rational balls are given in Figures~\ref{fig:Bpq} and~\ref{fig:Apq}.

An immediate corollary of Theorem~\ref{thm:ANN} above is the following.

\begin{maincor}\label{cor:Bpq}
Let $p\ge q$ be nonnegative integers.  If $\gcd(p,q)=2$ or if $p$ is odd and $\gcd(p,q)=1$, then the rational ball $B_{p,q}$ admits a smooth orientation-preserving embedding in $\CPb$.
\end{maincor}

In terms of embeddings of lens spaces, the following corollary shows that, among lens spaces of the form $L(p^2,pq-1)$ that bound a rational homology ball,
roughly three-quarters  embed in $\CP$.

\begin{maincor}\label{cor:Lpq}
Let $p\ge q$ be nonnegative integers.  If $\gcd(p,q)=2$ or if $p$ is odd and $\gcd(p,q)=1$, then the lens space $L(p^2,pq-1)$ admits a smooth embedding in $\CP$.
\end{maincor}

The inspiration for the main theorem comes from the data extracted from the output of two more general constructions, that we describe in detail in Section~\ref{s:homotopyCP2s}, which give a large collection of embeddings of triples of rational homology balls in  homotopy $\CP$'s.
We expect that all these homotopy $\CP$'s are in fact standard (see \Cref{q:exoticCP2s} below).

In a companion paper~\cite{GOobst},  we provide obstructions to the existence of embeddings of such triples, and work in the direction of  a conjecture of Koll\'ar \cite{Kollar}; see also~\cite{JPP} for related recent work. It is worth noting that the embeddings of triples of Theorem~\ref{thm:ANN} account for a very large fraction of the unobstructed triples.

The proof of Theorem~\ref{thm:ANN} is inductive, and relies on a tricky, but rather direct, Kirby calculus argument.
It uses structural induction on a simple variant of the Farey tree, which we will define in Section~\ref{s:2Fareytree}.
The Kirby calculus argument is related to Lisca and Parma's horizontal decompositions, and is informed by the Kirby diagrams of~\cite{EMPR}.
The main idea is to reduce all computations to handle slides in the neighbourhood of a 2-torus in the boundary of the 1-handlebody $S^1\times D^3$.

The handlesliding used to prove Theorem~\ref{thm:ANN} suggests another point of view on the results of Lisca and Parma in \cite{HD2}.  In \Cref{thm:trees} we use this point of view to derive new recursive descriptions of the triples of rational balls embedded in $\CP$ obtained in that paper.

\subsection*{History and motivation}
As mentioned above, the motivation for studying embeddings of lens spaces in $\CP$ comes from algebraic geometry.
Inspired by consequences of the orbifold Bogomolov--Miyaoka--Yau inequality, Koll\'ar asks in~\cite{Kollar} whether every compact, orientable, simply-connected 4-manifold $M$ with $H_1(M) = 0$, $H_2(M)\cong\mathbb{Z}$, and with boundary a union of spherical 3-manifolds $Y_1, \dots, Y_n$, satisfies:
\[
\sum_{i=1}^n \left(1-\frac1{|\pi_1(Y_i)|}\right) \le 3.
\]
Lens spaces are finite cyclic quotients of $S^3$, so they are spherical 3-manifolds, and indeed the complements of the rational homology balls that we embed in $\CP$ are manifolds satisfying the assumption in Koll\'ar's question.

A prominent family of examples of such manifolds originates from weighted projective planes $\mathbb{P}(a \sd b \sd c)$ by removing small open neighbourhoods of their (potentially) singular points $(0\sd 0\sd 1)$, $(0\sd 1\sd 0)$, and $(1\sd 0\sd 0)$.
When $(a,b,c)$ satisfies $a^2 + b^2 + c^2 = 3abc$, the weighted projective plane $\mathbb{P}(a^2 \sd b^2 \sd c^2)$ admits a deformation to $\CP$.
Such triples are called \emph{Markov triples}, and the deformation gives rise to an embedding of three rational homology balls, each with boundary a lens space, into $\CP$.
Whenever an embedding of this type exists, the orders of the fundamental groups are coprime perfect squares, and the inequality conjectured by Koll\'ar implies that $M$ has at most three boundary components.

In the realm of complex algebraic geometry, this is in fact the only instance of such embeddings, as was shown by Hacking and Prokhorov~\cite{hp}.
The corresponding result in the symplectic category is also known to hold, by work of Evans and Smith~\cite{es}.
Explicit Kirby diagrams for the associated 4-manifolds with boundary were exhibited by Etnyre, Min, Piccirillo, and Roy~\cite{EMPR}.
Topologically, however, the situation is quite different: earlier work of the second author~\cite{nonsymp},
 and of Lisca and Parma~\cite{LPStein,HD1, HD2}, shows that there are infinitely many embeddings of rational balls in $\CP$ that do not arise symplectically.

We would like to stress that, among the rational homology balls $\pm B_{p,q}$, the only ones that carry some `good' complex or symplectic structure are the those for which $\gcd(p,q) = 1$, with the positive sign.
By `good' here we mean that either they are diffeomorphic to a Milnor fibre in the complex setup \cite{wahl}, or that they have a symplectic structure with respect to which the boundary is convex in the symplectic setup \cite{LiscaSympFill}.
In particular, since in Theorem~\ref{thm:ANN} we have embeddings of $B_{p,q}$ in $\CPb$, none of the nontrival rational homology balls in any of the triples, with the exception of $B_{2,0}$, bear any complex or symplectic significance.

\subsection*{Notation and conventions}
Unless otherwise stated, homology is taken with integer coefficients. The lens space $L(p,q)$ is the quotient of $S^3 \subset \mathbb{C}^2$ by the action of the group generated by  $\begin{pmatrix}\zeta & 0 \\ 0 & \zeta^q\end{pmatrix}$, where $\zeta$ is a primitive $p^{\rm th}$ root of $1$, which is also the link of the cyclic quotient singularity of type $\frac1p(1,q)$. With this description, $L(p,q)$ is obtained by doing $-p/q$-surgery on the unknot in $S^3$.

\subsection*{Organisation of the paper}
In Section~\ref{s:balls} we give different descriptions of the balls $B_{p,q}$.
In Section~\ref{s:2Fareytree} we introduce the 2-Farey tree and state some of its properties, which we use in Section~\ref{s:ANN} to prove Theorem~\ref{thm:ANN}.
In Section~\ref{s:compare} we put the results from Theorem~\ref{thm:ANN} in a common framework with those of Lisca and Parma~\cite{HD2}, and also give new recursive descriptions of the examples obtained in \cite{HD2}.
In Section~\ref{s:homotopyCP2s} we describe two constructions of embeddings of triples of rational balls, coming from plumbings, and one construction of embeddings of rational balls and pairs of rational balls, coming from knots with lens space surgeries.
Finally, in Section~\ref{s:data} we share and discuss some data.

\subsection*{Acknowledgements}
We would like to warmly thank Yank{\i} Lekili for a question which initiated this collaboration.
We also thank Nikolas Adaloglou, Giulia Carfora, Johannes Hauber, Woohyeok Jo, Paolo Lisca, Marco Marengon, Jongil Park, and Kyungbae Park for stimulating conversations.
We would like to thank Andrea Parma for providing helpful data and clarifications about his papers.
The first author thanks the University of Glasgow for their hospitality.
The second thanks Oxford University and Nantes University for their hospitality.
Part of this work has been carried out at several conferences and workshops, including \emph{Surfaces in 4-manifolds} in le Croisic, \emph{4-vari\'et\'es: \`a travers les dimensions}
at CIRM in Marseilles, \emph{The low-dimensional workshop} during the \emph{Singularities and low-dimensional topology} research semester at the Erd\H{o}s Center in Budapest and \emph{Combinatorial and gauge-theoretical methods in low-dimensional topology and geometry} at the Centro De Giorgi in Pisa.

The post-theorem introduction says: "The proof of Theorem~\ref{thm:ANN} is inductive, and relies on a tricky, but rather direct, Kirby calculus argument." More precisely, it "uses structural induction on a simple variant of the Farey tree" (to be defined in Section~\ref{s:2Fareytree}). The Kirby-calculus part is "related to Lisca and Parma's horizontal decompositions" and "informed by the Kirby diagrams of~\cite{EMPR}." The "main idea" is "to reduce all computations to handle slides in the neighbourhood of a 2-torus in the boundary of the 1-handlebody $S^1\times D^3$."

The post-theorem introduction says: “The proof of the main theorem is inductive, and relies on a tricky, but rather direct, Kirby calculus argument.” More precisely, it “uses structural induction on a simple variant of the Farey tree” (to be defined next). The Kirby-calculus part is “related to Lisca and Parma's horizontal decompositions” and “informed by the Kirby diagrams of~\cite{EMPR}.” The “main idea” is “to reduce all computations to handle slides in the neighbourhood of a 2-torus in the boundary of the 1-handlebody $S^1\times D^3$.”

Question: For integers \(p,q\) with \(\gcd(p,q)\in\{1,2\}\), let \(B_{p,q}\) denote the rational homology 4-ball whose boundary is the lens space \(L(p^2,pq-1)\). Let \(\overline{\mathbb{CP}}^{\,2}\) denote the complex projective plane with reversed orientation. Suppose \(p_1,p_2,q_1,q_2\) are nonnegative integers such that \(q_1\) and \(q_2\) are even, \(p_i\ge q_i\) for \(i=1,2\), and \(p_1q_2-p_2q_1=\pm2\). Under these assumptions, which existence statement holds?

The disjoint union \(B_{p_1,q_1}\sqcup B_{p_2,q_2}\sqcup B_{p_1+p_2,\,q_1+q_2}\) admits a smooth orientation-preserving embedding in \(\overline{\mathbb{CP}}^{\,2}\).

Choices:
B. For every such choice of \(p_1,p_2,q_1,q_2\), the disjoint union \(B_{p_1,q_1}\sqcup B_{p_2,q_2}\sqcup B_{p_1+p_2,\,q_1+q_2}\) admits a smooth orientation-preserving embedding in \(\mathbb{CP}^{2}\).
C. There exists a smooth embedding of the disjoint union \(B_{p_1,q_1}\sqcup B_{p_2,q_2}\sqcup B_{p_1+p_2,\,q_1+q_2}\) in \(\overline{\mathbb{CP}}^{\,2}\).
D. If \(p_1,p_2,q_1,q_2\) are nonnegative integers with \(q_1\) and \(q_2\) even, \(p_i\ge q_i\), and \(p_1q_2-p_2q_1=\pm1\), then the disjoint union \(B_{p_1,q_1}\sqcup B_{p_2,q_2}\sqcup B_{p_1+p_2,\,q_1+q_2}\) admits a smooth orientation-preserving embedding in \(\overline{\mathbb{CP}}^{\,2}\).
E. If \(p_1,p_2,q_1,q_2\) are nonnegative integers with \(q_1\) and \(q_2\) even, \(p_i\ge q_i\), and \(p_1q_2-p_2q_1=\pm2\), then the disjoint union \(B_{p_1,q_1}\sqcup B_{p_2,q_2}\sqcup B_{p_1+p_2,\,q_1+q_2}\) admits a smooth embedding in \(\overline{\mathbb{CP}}^{\,2}\) whose image can be chosen so that each of the three summands embeds separately into a standard 4-ball in \(\overline{\mathbb{CP}}^{\,2}\).


\begin{align}\label{eq+1}
\sum_{\substack{bm\equiv \pm an\ppmod d\\(mn,q)=1}}\lambda(m) \tau(n)W\left(\frac mM\right)W\left(\frac {n}{N}\right),
\end{align}

\begin{align}
&E_{M,N}\ll q^{-1+\ve}(MN)^{\frac12}+q^\ve(M/N)^{\frac12}, \label{eqtbA}\\
&E_{M,N}\ll M^{\theta_f} \left(q^{-1+\ve}(MN)^{\frac12}+q^\ve(N/M)^{\frac12}\r). \label{eqtbB}
\end{align}

\begin{remark}\label{rem1}
Observe that $\ve(f,\chi)$ is independent of $\chi$ when $f$ is a Maa\ss~form.
When $\ve(f,\chi)=\ve(f)=-1$, summing both sides of equation \eqref{eqfe} over all primitive characters leads to the vanishing of $M_{f,E}(q;a,b)$ by symmetry. For holomorphic $f$, the root number $\ve(f,\chi)$ depends on $\chi$ at most though its parity $\chi(-1)$. The symmetry also implies that the total contribution of such characters $\chi$ satisfying $\ve(f,\chi)=-1$ to $M_{f,E}(q;a,b)$ is zero. Therefore, we restrict our consideration to the following cases:
\begin{itemize}
  \item for a Maa\ss~form $f$, we take $\ve(f)=1$,
  \item for a holomorphic $f$, we consider only characters with $\chi(-1)=\ve(f)$.
\end{itemize}
In both cases, it holds that
\[
\ve(f,\chi)=1,
\]
which we assume henceforth.
\end{remark}

\begin{theorem}\label{themain}
Let $f$ be a cuspidal Hecke eigenform of level $1$ with Hecke eigenvalues $\lambda(n)\ll \tau(n)n^{\theta_f}$. If $f$ is Maa\ss~form we further assume  that its root number satisfies $\ve(f)=1$. For $q\nequiv 2 \pmod4$ and $1\le a,b\le q$ satisfying $(a,b)=(ab,q)=1$, we have
\begin{align*}
M_{f,E}(q;a,b)=&c_f\prod_{p\mid qab}\left(1-\frac{\lambda(p)}{p}+\frac1{p^2}\r)\left(1-\frac1{p^2}\r)^{-2}\\
&\times\frac{c_{a,b}+c_{b,a}}{(ab)^{\frac12}}\frac{L(1,f)^2}{\zeta(2)} +O\left(q^{-\frac{1}{20}+\ve}a^{\frac{3}{10}}+q^{-\frac{1-2\theta_f}{22+16\theta_f}+\ve}b^{\frac{3+2\theta_f }{11+8\theta_f}}\r),
\end{align*}
where the coefficients $c_f$, $c_{a,b}$ are defined by
\begin{align}
c_f=
\begin{cases}
1/2,  &\text{if} \ f\ \text{is holomorphic},\\
1, &\text{if}\ f\ \text{is a Maa\ss\ form},
\end{cases}
\quad c_{a,b}=\sum_{a_1\mid a^\infty}\sum_{b_1\mid b^\infty} \frac{\lambda(aa_1b_1)\tau(ba_1b_1)}{a_1b_1}. \label{eqc1c2}
\end{align}
\end{theorem}

\subsection{Moments of twisted $L$-functions}
The study of moments in families of $L$-functions constitutes a central problem in modern number theory. As emphasized in the elegant work of Young \cite{You11}, these moments are not only pivotal for their wide-ranging applications but also serve as fundamental objects that unveil deep structural properties and inherent symmetries within the family.

Asymptotic formulae for moments of $L$-function with a power-saving error term are particularly crucial, as they play vital roles in amplification techniques, mollification methods, and resonance techniques. The complexity of such moment calculations can be quantified by the ratio $r=\log \mathcal{C}/\log |\mathcal{F}|$, where $|\mathcal{F}|$ denotes the size of the family and $\mathcal{C}$ its analytic conductor. Notably, computational difficulty increases with growing complexity. The threshold $r=4$ is the precise boundary, where most current analytic techniques fall just short of producing an asymptotic formula. In the few successful cases, some deep input is typically indispensable; see \cite{CLMR24} \cite{CIS12} \cite{IS00} \cite{Kha12} \cite{KMV00} \cite{Li24} for example.

Let $f$ and $g$ denote two fixed (holomorphic or non-holomorphic) Hecke eigenforms of level $1$, not necessarily cuspidal. The following second moment is defined in the context $r=4$: for $q\nequiv 2\pmod4$,
\begin{align}\label{eqgmoment}
M_{f,g}(q)=\frac1{\vp^*(q)}\ssum_{\chi\!\!\!\!\!\pmod q}L\left(\tfrac12,f\otimes\chi\r)\ol{L\left(\tfrac12,g\otimes\chi\r)},
\end{align}
where the sum runs over all primitive characters modulo $q$, and $\vp^*(q)$ denotes the number of such characters. Asymptotics for the moment \eqref{eqgmoment} with a power-saving error term are crucial to amplification and related analytic techniques in problems such as subconvexity, nonvanishing, and extreme values of $L$-functions; see \cite{BFKMMS23} for a rich sample of applications.

When both $f$ and $g$ correspond to non-cuspidal Eisenstein series, the moment \eqref{eqgmoment} reduces to
\[
M_{E,E}(q)=\frac1{\vp^*(q)}\ssum_{\chi\!\!\!\!\!\pmod q}\left|L\left(\tfrac12,\chi\r)\right|^4,
\]
the fourth moment of Dirichlet $L$-functions, which has a very long history and has been extensively studied; see \cite{BFK+17a} \cite{HB81} \cite{Sou07} \cite{Wu23} \cite{You11} for example. The first asymptotic formula with a power-savings error term was established by Young \cite{You11} for prime moduli. Wu \cite{Wu23} proved the asymptotic formula for all admissible moduli, and the best-known error term to date is $O\left(q^{-\frac1{20}+\ve}\r)$; see \cite{BFK+17b} \cite{GWZ25}.

When both $f$ and $g$ are cuspidal (holomorphic or Maa\ss) in \eqref{eqgmoment}, some special cases (ie. $f=g$) were studied earlier by Stefanicki \cite{Ste96} and Gao, Khan, and  Ricotta \cite{GKR09}. Blomer and Mili\'cevi\'c \cite{BM15} established a power-saving asymptotic for most moduli, specifically when $q$ is not close to a prime or to a product of two primes of comparable size. The main obstacle in these cases has been the lack of power-saving estimates for bilinear forms involving Kloosterman sums in the P\'olya-Vinogradov range.
The case of prime moduli was later addressed by Kowalski, Michel, and Sawin \cite{KMS17}, who proved the asymptotic with a power-saving error term $O\left(p^{-\frac1{144}+\ve}\r)$. The remaining case was recently resolved by  Mili\'cevi\'c, Qin, and Wu \cite{MQW25} and independently by  Pascadi~\cite{Pascadi2025}. Both works \cite{MQW25} and \cite{Pascadi2025} established a power-saving asymptotic for all admissible moduli $q$, with the former also removing the dependence on the Ramanujan--Petersson conjecture and achieving a sharper error term $O\left(q^{-\frac1{216}+\ve}\r)$.

These are termed mixed moments when one eigenform is cuspidal while the other corresponds to a non-cuspidal Eisenstein series. The first power-saving asymptotic formula for this case was established by Blomer, Fouvry, Kowalski, Michel, and Mili\'cevi\'c \cite{BFK+17a}, who proved that for prime moduli $p$
\[
M_{f,E}(p)=\frac1{\vp^*(p)}\ssum_{\chi\!\!\!\!\!\pmod p}L\left(\tfrac12,f\otimes\chi\r)\ol{L\left(\tfrac12,\chi\r)}^2=\frac{L(1,f)^2}{\zeta(2)}+O\left(p^{-\frac1{68}+\ve}\r).
\]
The error term was subsequently improved to $O\left(p^{-\frac1{64}+\ve}\r)$ by Shparlinski \cite{Shp19}, and more recently to $O\left(p^{-\frac1{22}+\ve}\r)$ for holomorphic $f$ and to $O\left(p^{-\frac5{152}+\ve}\r)$ for Maa\ss\ forms $f$ by Khan and Zhang \cite{KZ23}.

In this paper, we investigate the mixed moment for arbitrary moduli. More precisely, we study the following twisted mixed moment
\[
M_{f,E}(q;a,b)=\frac1{\varphi^*(q)}\ssum_{\chi\!\!\!\!\!\pmod q} L\left(\tfrac12,f\otimes\chi\r)L\left(\tfrac12,\ol\chi\r)^2\chi(\ol{a}b),
\]
for any integers $a,b\le q$ satisfying $(a,b)=(ab,q)=1$.

Asymptotic formulae for moments of $L$-function with a power-saving error term are particularly crucial, as they play vital roles in amplification techniques, mollification methods, and resonance techniques. The complexity of such moment calculations can be quantified by the ratio $r=\log \mathcal{C}/\log |\mathcal{F}|$, where $|\mathcal{F}|$ denotes the size of the family and $\mathcal{C}$ its analytic conductor. Notably, computational difficulty increases with growing complexity. The threshold $r=4$ is the precise boundary, where most current analytic techniques fall just short of producing an asymptotic formula. In the few successful cases, some deep input is typically indispensable; see \cite{CLMR24} \cite{CIS12} \cite{IS00} \cite{Kha12} \cite{KMV00} \cite{Li24} for example.

Let $f$ and $g$ denote two fixed (holomorphic or non-holomorphic) Hecke eigenforms of level $1$, not necessarily cuspidal. The following second moment is defined in the context $r=4$: for $q\nequiv 2\pmod4$,
\begin{align}\label{eqgmoment}
M_{f,g}(q)=\frac1{\vp^*(q)}\ssum_{\chi\!\!\!\!\!\pmod q}L\left(\tfrac12,f\otimes\chi\r)\ol{L\left(\tfrac12,g\otimes\chi\r)},
\end{align}
where the sum runs over all primitive characters modulo $q$, and $\vp^*(q)$ denotes the number of such characters. Asymptotics for the moment \eqref{eqgmoment} with a power-saving error term are crucial to amplification and related analytic techniques in problems such as subconvexity, nonvanishing, and extreme values of $L$-functions; see \cite{BFKMMS23} for a rich sample of applications.

When both $f$ and $g$ correspond to non-cuspidal Eisenstein series, the moment \eqref{eqgmoment} reduces to
\[
M_{E,E}(q)=\frac1{\vp^*(q)}\ssum_{\chi\!\!\!\!\!\pmod q}\left|L\left(\tfrac12,\chi\r)\right|^4,
\]
the fourth moment of Dirichlet $L$-functions, which has a very long history and has been extensively studied; see \cite{BFK+17a} \cite{HB81} \cite{Sou07} \cite{Wu23} \cite{You11} for example. The first asymptotic formula with a power-savings error term was established by Young \cite{You11} for prime moduli. Wu \cite{Wu23} proved the asymptotic formula for all admissible moduli, and the best-known error term to date is $O\left(q^{-\frac1{20}+\ve}\r)$; see \cite{BFK+17b} \cite{GWZ25}.

When both $f$ and $g$ are cuspidal (holomorphic or Maa\ss) in \eqref{eqgmoment}, some special cases (ie. $f=g$) were studied earlier by Stefanicki \cite{Ste96} and Gao, Khan, and Ricotta \cite{GKR09}. Blomer and Mili\'cevi\'c \cite{BM15} established a power-saving asymptotic for most moduli, specifically when $q$ is not close to a prime or to a product of two primes of comparable size. The main obstacle in these cases has been the lack of power-saving estimates for bilinear forms involving Kloosterman sums in the P\'olya-Vinogradov range.
The case of prime moduli was later addressed by Kowalski, Michel, and Sawin \cite{KMS17}, who proved the asymptotic with a power-saving error term $O\left(p^{-\frac1{144}+\ve}\r)$. The remaining case was recently resolved by Mili\'cevi\'c, Qin, and Wu \cite{MQW25} and independently by Pascadi~\cite{Pascadi2025}. Both works \cite{MQW25} and \cite{Pascadi2025} established a power-saving asymptotic for all admissible moduli $q$, with the former also removing the dependence on the Ramanujan--Petersson conjecture and achieving a sharper error term $O\left(q^{-\frac1{216}+\ve}\r)$.

These are termed mixed moments when one eigenform is cuspidal while the other corresponds to a non-cuspidal Eisenstein series. The first power-saving asymptotic formula for this case was established by Blomer, Fouvry, Kowalski, Michel, and Mili\'cevi\'c \cite{BFK+17a}, who proved that for prime moduli $p$
\[
M_{f,E}(p)=\frac1{\vp^*(p)}\ssum_{\chi\!\!\!\!\!\pmod p}L\left(\tfrac12,f\otimes\chi\r)\ol{L\left(\tfrac12,\chi\r)}^2=\frac{L(1,f)^2}{\zeta(2)}+O\left(p^{-\frac1{68}+\ve}\r).
\]
The error term was subsequently improved to $O\left(p^{-\frac1{64}+\ve}\r)$ by Shparlinski \cite{Shp19}, and more recently to $O\left(p^{-\frac1{22}+\ve}\r)$ for holomorphic $f$ and to $O\left(p^{-\frac5{152}+\ve}\r)$ for Maa\ss\ forms $f$ by Khan and Zhang \cite{KZ23}.

In this paper, we investigate the mixed moment for arbitrary moduli. More precisely, we study the following twisted mixed moment
\[
M_{f,E}(q;a,b)=\frac1{\varphi^*(q)}\ssum_{\chi\!\!\!\!\!\pmod q} L\left(\tfrac12,f\otimes\chi\r)L\left(\tfrac12,\ol\chi\r)^2\chi(\ol{a}b),
\]
for any integers $a,b\le q$ satisfying $(a,b)=(ab,q)=1$.

\begin{align}\label{eqgmoment}
M_{f,g}(q)=\frac1{\vp^*(q)}\ssum_{\chi\ppmod q}L\left(\tfrac12,f\otimes\chi\r)\ol{L\left(\tfrac12,g\otimes\chi\r)},
\end{align}

\subsection{Twisted $L$-functions}
We review some basic facts on twisted $L$-functions in this section; for further details, see \cite[Section 2.3]{BFK+17a}.
Let $\chi$ be a primitive Dirichlet character modulo $q$ and let $f$ be a cuspidal Hecke eigenform with Hecke eigenvalues $\lambda(n)$. The twisted form $f\otimes\chi$ remains cuspidal for the congruence subgroup $\Gamma_0(q^2)$ with nebentypus $\chi^2$ (see e.g \cite[Propositions 14.19 \& 14.20]{IK04}). Its $L$-function is given by
\[
L(s,f\otimes\chi)=\sum_{n\ge1}\frac{\lambda(n)\chi(n)}{n^{s}}=\prod_p\left(1-\frac{\lambda(p)\chi(p)}{p^s}+\frac{\chi^2(p)}{p^{2s}}\r)^{-1},\ \ \text{for}\ \ \re(s)>1,
\]
where the product is over primes $p$.
The completed $L$-function is defined by
\begin{align*}
\Lambda(s,f\otimes\chi)=q^s L_\infty(s,f\otimes\chi)L(s,f\otimes\chi),
\end{align*}
with
\begin{align*}
L_\infty(s,f\otimes\chi)=\begin{cases}
(2\pi)^{-\frac{k-1}2-s}\Gamma\left(\frac{k-1}2+s\r), &\text{if} \ f\ \text{is holomorphic of weight}\ k,\\
\pi^{-s-\ma}\Gamma\left(\frac{s+i\kappa+\ma}2\r)\Gamma\left(\frac{s-i\kappa+\ma}2\r), &\text{if}\ f\ \text{is a Maa\ss\ form with eigenvalue}\ \tfrac14+\kappa^2.
\end{cases}
\end{align*}
This completed $L$-function admits analytic continuation to $\mathbb{C}$ and satisfies the functional equation (see e.g \cite[Theorem 14.17, Proposition 14.20]{IK04}):
\[
\Lambda(s,f\otimes\chi)=\ve(f\otimes\chi)\Lambda(1-s,f\otimes\ol{\chi}),
\]
where the root number is given by
\begin{align*}
\ve(f\otimes\chi)=\begin{cases}
\ve(f)\ve_\chi^2, &\text{if} \ f\ \text{is holomorphic},\\
\chi(-1)\ve(f)\ve_\chi^2, &\text{if}\ f\ \text{is a Maa\ss\ form},
\end{cases}
\end{align*}
and $\ve(f)=\pm 1$ denotes the root number of $L(s,f)$.

Using the identity
\[
S(hq\ol{b}, m;a_1k)=e\left(-m\frac{\ol{a}_1}{b}\right)S_{\infty,1/a_1}(hq,m;\gammaup),
\]
where $\gammaup=a_1\ssqrt{b}k$ and $Q=a_1b$, we derive from \eqref{eqdefmS} that
\begin{align*}
\mS(a_1,r)=&\frac{\sqrt{b}}{aN_1}\sum_{h\sim H}\sum_{m\sim M^*} \lambda(m)e\left(-m\frac{\ol{a}_1}{b}\right)\\
&\times\sum_{\gammaup}\frac 1{\gammaup}S_{\infty,1/a_1}(hq, m;\gammaup)W\left(\frac {r\gammaup}{a_1\ssqrt{b}N_1}\r)\mathring{V}_{+}\left(\frac{bm}{\gammaup^2},h\right).
\end{align*}
Performing the variable substitution $(bx-hq)/N\rightarrow x$ in \eqref{eqV0} yields
\[
\mathring{V}_{+}\left(\frac{bm}{\gammaup^2},h\right)=\frac{N}{b}\Omega\Bigg(\frac{4\pi\ssqrt{hqm}}{\gammaup},\frac{4\pi\ssqrt{mN}}{\gammaup},h,m\Bigg),
\]
where
\[
\Omega(y,z,h,m)=\int_{0}^\infty W(x)W\left(\frac {hq+xN}M\r) \mJ_{+}\left(\ssqrt{y^2+xz^2}\right)\d x
\]
with
\begin{equation}\label{eqyz}
y\asymp Y:=\frac{r\ssqrt{MM^*}}{a_1\ssqrt{b}N_1}\gg q^\ve,\quad z\asymp \frac{r\ssqrt{M^*N}}{a_1\ssqrt{b}N_1},\quad h\asymp H,\quad m\asymp M^*.
\end{equation}
For $M\gg q^\ve N$, it is easy to see from \eqref{eqlargeM} and \eqref{eqyz} that
\[
y\gg z\quad \text{and}\quad y\gg z^2q^{-\ve}.
\]
Then Lemma \ref{lemdecomJ} yields the decomposition
\[
\Omega(y,z,h,m)=\Omega_+(y,z,h,m)e^{iy}+\Omega_-(y,z,h,m)e^{-iy}+O_A\left(q^{-A}\r),
\]
where the functions $\Omega_{\pm}$ satisfy
\begin{align}\label{eqWpmd}
y^{j_1}z^{j_2}h^{j_3}m^{j_4}\frac{\partial^{j_1+j_2+j_3+j_4}}{\partial y^{j_1}\partial z^{j_2}\partial h^{j_3}\partial m^{j_4}}\Omega_{\pm}(y,z,h,m)\ll_{j_1,j_2,j_3,j_4}q^{(j_1+j_2+j_3+j_4)\ve} Y^{-\frac12}.
\end{align}
After separating variables in $\Omega_\pm$ via the Mellin transform, we bound $\mS(a_1,r)$ by sums of the form
\begin{equation*}
\mS(a_1,r)\ll q^\ve\frac{N Y^{-\frac12}}{a\ssqrt{b}N_1}\B|\sum_{H<h\le 2H}\alpha_h\sum_{M^*<m\le 2M^*} \beta_m\lambda(m)\sum_{\gammaup}\frac1{\gammaup}S_{\infty,1/a_1}(hq,m;\gammaup)\Theta \B(\frac{4\pi\ssqrt{hqm}}{\gammaup}\B)\B|
\end{equation*}
for some coefficients satisfying $\alpha_h\ll h^\ve$, $\beta_m\ll m^\ve$ and a test function
\[
\Theta (y)=W\B(\frac yY\B)\exp(\pm iy)y^{-u}
\]
with $\re(u)=\ve$.

An application of the Kuznetsov formula (Lemma \ref{lemKf}) followed by spectral truncation at parameter $Y^{\frac12+\ve}$ ( Lemma \ref{lemBe}) produces
\begin{equation*}
\mS(a_1,r)\ll q^\ve\frac{N Y^{-\frac12}}{a\ssqrt{b}N_1}\B|\sum_{H<h\le 2H}\alpha_h\sum_{M^*<m\le 2M^*} \beta_m\lambda(m)\left(\mH+\mM+\mE\r)\B|,
\end{equation*}
where the spectral contributions decompose as
\[
\mH=\sum_{\substack{2\le k\le Y^{\frac12+\ve}\\ k ~\text{even}}}\sum_{f\in \mathcal{B}_k(Q)}\tilde{\Theta} (k)\ssqrt{hqm}~ \bar\rho_f(hq)~\rho_f(a_1^{-1},m),
\]
\[
\mM=\sum_{\substack{f\in \mathcal{B}(Q)\\ \kappa_f\ll Y^{\frac12+\ve}}}\hat{\Theta} (\kappa_f)\frac{\ssqrt{hqm}~ \bar\rho_f(hq)~\rho_f(a_1^{-1},m)}{\cosh(\pi \kappa_f)},
\]
\[
\mE=\frac1{4\pi}\sum_{\mc}\int_{|\kappa|\ll Y^{\frac12+\ve}}\hat{\Theta} (\kappa)\ssqrt{hqm} ~\frac{\bar\rho_{\mc}(hq,\kappa)~\rho_{1/{a_1},\mc}(m,\kappa)}{\cosh(\pi\kappa)}\d \kappa,
\]
representing the holomorphic cusp forms, Maa\ss\ cusp forms, and the Eisenstein series respectively.
We foucus our detailed analysis on the Maa\ss\ case (the holomorphic and Eisenstein cases are identical). Since $\hat{\Theta} (\kappa_f)\ll Y^{-\frac12+\ve}$ (see \eqref{eqphi1}), the contribution of the Maa\ss\ case is bounded by
\[
\ll q^\ve\frac{NY^{-1}}{a\ssqrt{b}N_1}\sum_{\substack{f\in \mathcal{B}(Q)\\ \kappa_f\ll Y^{\frac12+\ve}}}\B|\sum_{H<h\le 2H}\alpha_h\sum_{M^*<m\le 2M^*} \beta_m\lambda(m)\frac{\ssqrt{hqm} ~\bar\rho_f(hq)~\rho_f(a_1^{-1},m)}{\cosh(\pi \kappa_f)}\B|,
\]
which, after applying the Cauchy-Schwarz inequality, can be bounded by
\begin{align*}
\ll q^\ve\frac{NY^{-1}}{a\ssqrt{b}N_1}&\B(\sum_{\substack{f\in \mathcal{B}(Q)\\ \kappa_f\ll Y^{\frac12+\ve}}}\frac{1}{\cosh(\pi \kappa_f)}\B|\sum_{H<h\le 2H}\alpha_h \ssqrt{hq} ~\rho_f(hq)\B|^2\B)^{\frac12}\\
&\times \B(\sum_{\substack{f\in \mathcal{B}(Q)\\ \kappa_f\ll Y^{\frac12+\ve}}}\frac{1}{\cosh(\pi \kappa_f)}\B|\sum_{M^*<m\le 2M^*}\beta_m\lambda(m)\ssqrt{m} ~\rho_f(a_1^{-1},m)\B|^2\B)^{\frac12}.
\end{align*}
We may handle the second spectral sum via Lemma \ref{lemsls}, but for the first spectral sum, an application of Lemma \ref{lemsls} leads to an extra factor of $q$ to the final estimate. To address this issue, we apply the following estimate, established in \cite[Theorems 11.3-11.6]{GWZ25}.
\begin{lemma}\label{thmsls}
Let $Q,s\in\mathbb{N}$, $K, N\ge 1$, and let $(a_n)_{n\in[N,2N]}$ be complex coefficients. Then we have
\[
\sum_{\substack{2\le k\le K\\ k~ \text{even}}}\Gamma(k)\sum_{f\in\mathcal{B}_k(Q)}\B|\sum_{n}a_n\ssqrt{sn}~\rho_f(sn)\B|^2, \ \ \ \sum_{\mc}\int_{-K}^K\frac1{\cosh(\pi\kappa)}\B|\sum_n a_n\ssqrt{sn}~\rho_{\mc}(\pm sn,\kappa)\B|^2\d\kappa,
\]
\[
\sum_{\substack{f\in\mathcal{B}(Q)\\ |\kappa_f|\le K}}\frac1{\cosh(\pi\kappa_f)}\B|\sum_{n}a_n\ssqrt{sn}~\rho_f(sn)\B|^2
\]
are bounded by
\[
\ll (sQKN)^\ve\B(K+\frac{s^{\frac12}}{Q^{\frac12}}\B)\B(K+\frac{(s,Q)^{\frac12}N}{Q^{\frac12}}\B)N\|a_n\|_\infty.
\]
\end{lemma}
Applying Lemma \ref{thmsls} to the first spectral sum and Lemma \ref{lemsls} to the second one reduces the Maa\ss\ contribution to
\begin{equation*}
\ll q^\ve\frac{NY^{-1}}{a\ssqrt{b}N_1}\B(H^{\frac12}{M^*}^{\frac12}\left(Y+\frac{q}{a_1b}\r)^{\frac14} \left(Y+\frac{(a_1b,q)H^2}{a_1b}\r)^{\frac14}\left(Y+\frac{M^*}{a_1b}\r)^{\frac12}\B).
\end{equation*}
Taking
\[
 H=\frac{M}q,\ \ \ M^*= q^\ve\frac{a_1^2N_1^2}{r^2}\frac{bM}{N^2}, \ \ \ Y=\frac{r\ssqrt{MM^*}}{a_1\ssqrt{b}N_1}=\frac{M}{N},
\]
and observing that
$\frac{M^*}{a_1b}\ll Y$ ( since $N_1^2\ll N/a$), we obtain
for the Maa\ss\ contribution
\begin{align*}
&\ll q^\ve\frac{N}{r\ssqrt{q}}\left(Y+\frac{q}{ab}\r)^{\frac14} \left(Y+\frac{(ab,q)H^2}{ab}\r)^{\frac14}Y^{\frac12}\\
&\ll q^\ve r^{-1}\B(\frac{M}{q^{\frac12}}+\frac{(ab,q)^{\frac14}M^{\frac 54}N^{\frac14}}{(ab)^{\frac14}q} +\frac{M^{\frac 34}N^{\frac14}}{(ab)^{\frac14}q^{\frac14}}+\frac{(ab,q)^{\frac14}MN^{\frac12}}{(ab)^{\frac12}q^{\frac 34}}\B).
\end{align*}

\begin{remark}
The dependence on  $\theta_f$ in the error term of Theorem \ref{themain} arises from the trivial estimate \eqref{eqtbB}, which constitutes the sole component requiring an upper bound for the Hecke eigenvalues of $f$. The assumption $\ve(f)=1$ for a Maa\ss~form is natural, as otherwise $M_{f,E}(q;a,b)=0$ for parity reasons (see Remark \ref{rem1}). In the Maa\ss~case, the coefficient $c_f$ addresses a literature discrepancy originating from a typographical oversight in \cite[Theorem 1.2]{BFK+17a} and was subsequently replicated in \cite[Thoerem 1.1]{KZ23}.
\end{remark}

Setting $a=b=1$ in Theorem \ref{themain} results in a power-saving asymptotic for $M_{f,E}(q)$ across all admissible moduli $q$.

\begin{corollary}\label{cor}
Let $f$ be a cuspidal Hecke eigenform of level $1$ with Hecke eigenvalues $\lambda(n)\ll \tau(n)n^{\theta_f}$. If $f$ is Maa\ss~form we further assume that its root number satisfies $\ve(f)=1$. For $q\nequiv 2 \pmod4$, we have
\begin{align*}
M_{f,E}(q)=c_f\prod_{p\mid q}\left(1-\frac{\lambda(p)}{p}+\frac1{p^2}\r)\left(1-\frac1{p^2}\r)^{-2}\frac{L(1,f)^2}{\zeta(2)}+O\left(q^{-\frac{1-2\theta_f }{22+16\theta_f }+\ve}\r),
\end{align*}
where $c_f$ is the same constant as previously defined.
\end{corollary}
For the holomorphic case, Deligne's well-known work \cite{Del74} gives $\theta_f=0$, whereas for the Maa\ss~form we may take $\theta_f= 7/{64}$ due to Kim--Sarnak \cite{Kim03}. This yields an error term as sharp as the best bound \cite{KZ23} known for prime moduli.

Through the functional equations of $L$-functions and the orthogonality of characters, the evaluation of the moment $M_{f,E}(q,a,b)$ reduces to analyzing the shifted convolution sum
\begin{align}\label{eq+1}
\sum_{\substack{bm\equiv \pm an\!\!\!\!\!\pmod d\\(mn,q)=1}}\lambda(m) \tau(n)W\left(\frac mM\right)W\left(\frac {n}{N}\right),
\end{align}
where $a, b,M,N\ge1$ and $W$ is smooth test functions. The classical approach to such shifted convolution sum involves either the $\delta$-method or Jutila's circle method to eliminate the congruence condition (see \cite{BM15}). For \eqref{eq+1}, a more efficient approach is to expand the divisor function using the multiplicative decomposition $\tau(n)=\sum_{n_1n_2=n}1$, which decomposes the $n$-sum into two sums with smooth coefficients, making them amenable to Fourier-analytic techniques.

The diagonal terms (where $bm=an$) in \eqref{eq+1} contribute to the main term, which can be evaluated explicitly using standard techniques. The primary challenge arise from the off-diagonal terms, which we classify into two distinct cases based on the relative magnitudes of $bM$ and $aN$: the balanced case and the unbalanced case. These two cases demand fundamentally different approaches.

In the balanced case where $bM$ and $aN$ are of comparable size, we first remove the coprimality condition and reduce the problem to estimating the sum
\begin{align}\label{eqAdef}
A_q(a,b,M,N)=\sum_{\substack{bm\equiv \pm an \!\!\!\!\!\pmod q\\ bm\neq an}}\lambda(m) \tau(n) W\left(\frac {bm}M\r) W\left(\frac {an}N\r),
\end{align}
where $W$ are some smooth functions.
For $M\ge N$, we establish the bound
\[
A_q(a,b,M,N)\ll  q^\ve\Bigg(\frac{M}{q^{\frac12}}+\frac{(ab,q)^{\frac14}M^{\frac54}N^{\frac14}}{(ab)^{\frac14}q} +\frac{M^{\frac34}N^{\frac14}}{(ab)^{\frac14}q^{\frac14}} +\frac{(ab,q)^{\frac14}MN^{\frac12}}{(ab)^{\frac12}q^{\frac34}}\Bigg)
\]
with an analogous result holding for $N\ge M$. Compared to the bound in \cite[(8.13)]{BM15}, our result achieves two key improvements: reducing the influence of the factor $(ab,q)^{\frac12}$ and recovering a saving $(ab)^{-\frac14}$ in the third term. This improvements stem from two innovation: the elimination of dependence on the Ramanujan--Petersson conjecture and the minimization of the impact of $a, b$ through new upper bounds for sums involving Kloosterman sums, as established in the recent work of Gao, Wu, and Zhao \cite{GWZ25}.

For the unbalanced case where $BM$ and $AN$ differ significantly in size, the theory of algebraic trace functions becomes inapplicable due to its restriction to prime moduli. To address this, we employ the multiplicative decomposition of the divisor function $\tau(n)$, which allows us to reformulate the problem as analyzing a bilinear sum involving incomplete Kloosterman sum. The estimate of this bilinear form relies primarily on a recent result by Kerr, Shparlinski, Wu, and Xi \cite{KSWX23} concerning arbitrary moduli.
The approach also requires the Weil bound for Kloosterman sums and several Fourier-analytic tools, including the Poisson summation formula and the Voronoi summation formula.

\subsection{Notation}
We adopt the standard convention that $\ve$ denotes an arbitrarily small positive constant whose values may vary from one occurrence to another. All implied constants in our estimates may depend on $\ve$ as well as on the fixed cusp form $f$.
Throughout this paper, $W(x)$ denotes a smooth complex-valued function with compact support in $[1/2,3]$, satisfying
\begin{align}\label{eqW}
W^{(j)}(x)\ll_{j}1,
\end{align}
for all integers $j\ge0$. The function $W(x)$ may also vary at each occurrence. We call a modulus $q$ admissible if $q\nequiv 2\pmod4$, which is a natural assumption since no primitive characters exist modulo $q$ otherwise.

Through the functional equations of $L$-functions and the orthogonality of characters, the evaluation of the moment $M_{f,E}(q,a,b)$ (hence Theorem~\ref{themain}) “reduces to analyzing the shifted convolution sum”
\[
\sum_{\substack{bm\equiv \pm an\!\!\!\!\!\pmod d\\(mn,q)=1}}\lambda(m) \, \tau(n)\,W\!\left(\frac mM\right)W\!\left(\frac {n}{N}\right).
\]
Instead of using “the $\delta$-method or Jutila's circle method,” the argument uses “a more efficient approach” by expanding the divisor function via “the multiplicative decomposition $\tau(n)=\sum_{n_1n_2=n}1$,” which splits the $n$-sum into two smooth sums “amenable to Fourier-analytic techniques.”

The “diagonal terms (where $bm=an$)” give the main term and “can be evaluated explicitly using standard techniques.” The remaining “off-diagonal terms” are treated by splitting into two regimes “based on the relative magnitudes of $bM$ and $aN$”:

1) **Balanced case** ($bM$ comparable to $aN$): “we first remove the coprimality condition and reduce the problem” to bounding
\[
A_q(a,b,M,N)=\sum_{\substack{bm\equiv \pm an \!\!\!\!\!\pmod q\\ bm\neq an}}\lambda(m)\tau(n) W\!\left(\frac {bm}M\right) W\!\left(\frac {an}N\right),
\]
and establish the stated upper bound for $A_q(a,b,M,N)$ (and its analogue when $N\ge M$). The text attributes the improvement to “two innovation: the elimination of dependence on the Ramanujan--Petersson conjecture and the minimization of the impact of $a, b$ through new upper bounds for sums involving Kloosterman sums” from \cite{GWZ25}.

2) **Unbalanced case** ($BM$ and $AN$ very different): since “the theory of algebraic trace functions becomes inapplicable due to its restriction to prime moduli,” one again uses the decomposition of $\tau(n)$ to “reformulate the problem as analyzing a bilinear sum involving incomplete Kloosterman sum.” This bilinear form is estimated using \cite{KSWX23} (for “arbitrary moduli”), together with “the Weil bound for Kloosterman sums” and Fourier-analytic tools including “the Poisson summation formula and the Voronoi summation formula.”

Finally, the remark notes that the $\theta_f$-dependence in the error term “arises from the trivial estimate \eqref{eqtbB},” which is “the sole component requiring an upper bound for the Hecke eigenvalues of $f$.”

Through the functional equations of $L$-functions and the orthogonality of characters, the evaluation of the moment $M_{f,E}(q,a,b)$ (hence, in establishing the main theorem) “reduces to analyzing the shifted convolution sum”
\[
\sum_{\substack{bm\equiv \pm an\!\!\!\!\pmod d\\(mn,q)=1}}\lambda(m) \, \tau(n)\,W\!\left(\frac mM\right)W\!\left(\frac {n}{N}\right).
\]
Instead of using “the $\delta$-method or Jutila's circle method,” the argument uses “a more efficient approach” by expanding the divisor function via “the multiplicative decomposition $\tau(n)=\sum_{n_1n_2=n}1$,” which splits the $n$-sum into two smooth sums “amenable to Fourier-analytic techniques.”

The “diagonal terms (where $bm=an$)” give the main term and “can be evaluated explicitly using standard techniques.” The remaining “off-diagonal terms” are treated by splitting into two regimes “based on the relative magnitudes of $bM$ and $aN$”:

1) **Balanced case** ($bM$ comparable to $aN$): “we first remove the coprimality condition and reduce the problem” to bounding
\[
A_q(a,b,M,N)=\sum_{\substack{bm\equiv \pm an \!\!\!\!\pmod q\\ bm\neq an}}\lambda(m)\tau(n) W\!\left(\frac {bm}M\right) W\!\left(\frac {an}N\right),
\]
and establish the stated upper bound for $A_q(a,b,M,N)$ (and its analogue when $N\ge M$). The text attributes the improvement to “two innovation: the elimination of dependence on the Ramanujan--Petersson conjecture and the minimization of the impact of $a, b$ through new upper bounds for sums involving Kloosterman sums” from \cite{GWZ25}.

2) **Unbalanced case** ($BM$ and $AN$ very different): since “the theory of algebraic trace functions becomes inapplicable due to its restriction to prime moduli,” one again uses the decomposition of $\tau(n)$ to “reformulate the problem as analyzing a bilinear sum involving incomplete Kloosterman sum.” This bilinear form is estimated using \cite{KSWX23} (for “arbitrary moduli”), together with “the Weil bound for Kloosterman sums” and Fourier-analytic tools including “the Poisson summation formula and the Voronoi summation formula.”

Finally, the remark notes that the $\theta_f$-dependence in the error term “arises from the trivial estimate
\begin{align}
&E_{M,N}\\ll q^{-1+\ve}(MN)^{\frac12}+q^\ve(M/N)^{\frac12}, \label{eqtbA}\\
&E_{M,N}\\ll M^{\theta_f} \left(q^{-1+\ve}(MN)^{\frac12}+q^\ve(N/M)^{\frac12}\r). \label{eqtbB}
\end{align}
,” which is “the sole component requiring an upper bound for the Hecke eigenvalues of $f$.”

Question:
Let f be a cuspidal Hecke eigenform of level 1 with Hecke eigenvalues λ(n) satisfying λ(n) ≪ τ(n)n^{θ_f}, where τ(n) is the divisor function. If f is a Maaß form, also assume that its root number satisfies ε(f)=1. For an integer q with q ≢ 2 (mod 4), and integers 1 ≤ a,b ≤ q satisfying (a,b)=(ab,q)=1, define the twisted mixed moment
M_{f,E}(q;a,b) = (1/φ*(q)) ∑_{χ primitive mod q} L(1/2,f⊗χ) L(1/2,χ̄)^2 χ(āb),
where φ*(q) is the number of primitive Dirichlet characters modulo q and ā denotes the inverse of a modulo q. Also define
c_f = 1/2 if f is holomorphic, and c_f = 1 if f is a Maaß form,
and
c_{a,b} = ∑_{a_1|a^∞} ∑_{b_1|b^∞} λ(aa_1b_1)τ(ba_1b_1)/(a_1b_1),
with c_{b,a} defined by swapping a and b. Which statement holds for every such f, q, a, and b?

One has
M_{f,E}(q;a,b) = c_f · ∏_{p∣qab} (1 − λ(p)/p + 1/p^2)(1 − 1/p^2)^(−2) · ((c_{a,b}+c_{b,a})/(ab)^(1/2)) · (L(1,f)^2/ζ(2)) + O(q^(−1/20+ε)a^(3/10) + q^(−(1−2θ_f)/(22+16θ_f)+ε) b^((3+2θ_f)/(11+8θ_f))).

Choices:
B.
One has
M_{f,E}(q;a,b) = c_f \cdot \prod_{p\mid qab} (1 - \lambda(p)/p + 1/p^2)(1 - 1/p^2)^{(-2)} \cdot \frac{c_{a,b}+c_{b,a}}{(ab)^{1/2}} \cdot \frac{L(1,f)^2}{\zeta(2)} + O\!\left(q^{-1/20+\varepsilon}(ab)^{3/10} + q^{-(1-2\theta_f)/(22+16\theta_f)+\varepsilon}(ab)^{(3+2\theta_f)/(11+8\theta_f)}\right).
C.
One has
M_{f,E}(q;a,b) = c_f \cdot \prod_{p\mid qab} (1 - \lambda(p)/p + 1/p^2)(1 - 1/p^2)^{(-2)} \cdot \frac{c_{a,b}+c_{b,a}}{(ab)^{1/2}} \cdot \frac{L(1,f)^2}{\zeta(2)} + O\!\left(q^{-1/20+\varepsilon}a^{3/10} + q^{\varepsilon} b^{(3+2\theta_f)/(11+8\theta_f)}\right).
D.
One has
M_{f,E}(q;a,b) = c_f \cdot \prod_{p\mid qab} (1 - \lambda(p)/p + 1/p^2)(1 - 1/p^2)^{(-2)} \cdot \frac{c_{a,b}+c_{b,a}}{(ab)^{1/2}} \cdot \frac{L(1,f)^2}{\zeta(2)} + O\!\left(q^{-1/20+\varepsilon}a^{3/10} + q^{-(1-2\theta_f)/(22+16\theta_f)+\varepsilon} a^{(3+2\theta_f)/(11+8\theta_f)}\right).
E.
One has
M_{f,E}(q;a,b) = c_f \cdot \prod_{p\mid qab} (1 - \lambda(p)/p + 1/p^2)(1 - 1/p^2)^{(-2)} \cdot \frac{c_{a,b}}{(ab)^{1/2}} \cdot \frac{L(1,f)^2}{\zeta(2)} + O\!\left(q^{-1/20+\varepsilon}a^{3/10} + q^{-(1-2\theta_f)/(22+16\theta_f)+\varepsilon} b^{(3+2\theta_f)/(11+8\theta_f)}\right).

separate balanced/unbalanced dependence on a and b in the error term

dropped the power-saving q-decay in the second error term

assignment of the second-regime parameter dependence to b rather than a

two-sided diagonal contribution c_{a,b}+c_{b,a}


Question:
Let \(f\) be a level-\(1\) cuspidal Hecke eigenform with Hecke eigenvalues \(\lambda(n)\) satisfying \(\lambda(n)\ll \tau(n)n^{\theta_f}\), where \(\tau(n)\) is the divisor function. If \(f\) is a Maa\ss{} form, also assume its root number satisfies \(\varepsilon(f)=1\). For an integer \(q\not\equiv 2\pmod 4\), and integers \(1\le a,b\le q\) with \((a,b)=(ab,q)=1\), define the twisted mixed moment
\[
M_{f,E}(q;a,b)=\frac1{\varphi^*(q)}\sum_{\chi\, (\mathrm{mod}\, q)}^{\mathrm{primitive}} L\!\left(\tfrac12,f\otimes\chi\right)L\!\left(\tfrac12,\overline\chi\right)^2\chi(\overline a b),
\]
where \(\varphi^*(q)\) is the number of primitive Dirichlet characters modulo \(q\), and \(\overline a\) denotes the inverse of \(a\) modulo \(q\). Also define
\[
c_f=
\begin{cases}
1/2,& \text{if }f\text{ is holomorphic},\\
1,& \text{if }f\text{ is a Maa\ss{} form},
\end{cases}
\qquad
c_{a,b}=\sum_{a_1\mid a^\infty}\sum_{b_1\mid b^\infty} \frac{\lambda(aa_1b_1)\tau(ba_1b_1)}{a_1b_1},
\]
where \(a_1\mid a^\infty\) and \(b_1\mid b^\infty\) mean that every prime divisor of \(a_1\) divides \(a\), and every prime divisor of \(b_1\) divides \(b\). Under these assumptions, which explicit asymptotic formula holds for \(M_{f,E}(q;a,b)\)?

For every \(\varepsilon>0\),
\[
\begin{aligned}
M_{f,E}(q;a,b)=&\;c_f\prod_{p\mid qab}\left(1-\frac{\lambda(p)}{p}+\frac1{p^2}\right)\left(1-\frac1{p^2}\right)^{-2}
\times \frac{c_{a,b}+c_{b,a}}{(ab)^{1/2}}\frac{L(1,f)^2}{\zeta(2)} \\
&\;+O\!\left(q^{-\frac1{20}+\varepsilon}a^{\frac3{10}}+q^{-\frac{1-2\theta_f}{22+16\theta_f}+\varepsilon}b^{\frac{3+2\theta_f}{11+8\theta_f}}\right),
\end{aligned}
\]
where the product is over primes \(p\) dividing \(qab\).

Choices:
B.
For every \(\varepsilon>0\),
\[
\begin{aligned}
M_{f,E}(q;a,b)=&\;c_f\prod_{p\mid qab}\left(1-\frac{\lambda(p)}{p}+\frac1{p^2}\right)\left(1-\frac1{p^2}\right)^{-2}
\times \frac{c_{a,b}+c_{b,a}}{(ab)^{1/2}}\frac{L(1,f)^2}{\zeta(2)} \\
&\;+O\!\left(q^{-\frac1{20}+\varepsilon}a^{\frac3{10}}+q^{-\frac1{22}+\varepsilon}b^{\frac3{11}}\right),
\end{aligned}
\]
where the product is over primes \(p\) dividing \(qab\).
C.
For every \(\varepsilon>0\),
\[
M_{f,E}(q;a,b)=c_f\prod_{p\mid qab}\left(1-\frac{\lambda(p)}{p}+\frac1{p^2}\right)\left(1-\frac1{p^2}\right)^{-2}
\times \frac{c_{a,b}+c_{b,a}}{(ab)^{1/2}}\frac{L(1,f)^2}{\zeta(2)}+O_{f,\varepsilon,a,b}\!
\left(q^{-\delta}\right)
\]
for some \(\delta=\delta(f)>0\).
D.
For every \(\varepsilon>0\),
\[
\begin{aligned}
M_{f,E}(q;a,b)=&\;c_f\prod_{p\mid qab}\left(1-\frac{\lambda(p)}{p}+\frac1{p^2}\right)\left(1-\frac1{p^2}\right)^{-2}
\times \frac{c_{a,b}}{(ab)^{1/2}}\frac{L(1,f)^2}{\zeta(2)} \\
&\;+O\!\left(q^{-\frac1{20}+\varepsilon}a^{\frac3{10}}+q^{-\frac{1-2\theta_f}{22+16\theta_f}+\varepsilon}b^{\frac{3+2\theta_f}{11+8\theta_f}}\right),
\end{aligned}
\]
where the product is over primes \(p\) dividing \(qab\).
E.
For every \(\varepsilon>0\),
\[
\begin{aligned}
M_{f,E}(q;a,b)=&\;c_f\prod_{p\mid qab}\left(1-\frac{\lambda(p)}{p}+\frac1{p^2}\right)\left(1-\frac1{p^2}\right)^{-2}
\times \frac{c_{a,b}+c_{b,a}}{(ab)^{1/2}}\frac{L(1,f)^2}{\zeta(2)} \\
&\;+O\!\left(q^{-\frac1{20}+\varepsilon}(ab)^{\frac3{10}}+q^{-\frac{1-2\theta_f}{22+16\theta_f}+\varepsilon}(ab)^{\frac{3+2\theta_f}{11+8\theta_f}}\right),
\end{aligned}
\]
where the product is over primes \(p\) dividing \(qab\).

theta_f-dependent exponent in the second error term

explicit error exponents and explicit dependence on a,b

symmetric main-term contribution c_{a,b}+c_{b,a}

separate a- and b-dependence coming from different off-diagonal regimes


\label{lem:anest}
Let $(M_i,g_i)$ be Riemannian manifolds of dimensions $m_i$ ($i=1,2$), and set $g=g_1\times g_2$. Then the following hold for all measurable $S\subset M_1\times M_2$ and $(m_1+m_2-1)$–rectifiable $\Sigma$:
\begin{enumerate}
\item[(i)] For every set $S \subset M_1 \times M_2$,
\begin{equation}\label{eq:anest1}
  \vv_{g_+^t}(S) = t^{m_2} \,\vv_g(S).
\end{equation}
\item[(ii)] If $\Sigma \subset M_1 \times M_2$ is $(m_1+m_2-1)$-rectifiable, then for $t \leq 1$,
\begin{equation}\label{eq:anest2}
  \hh_{g_+^t}^{m_1+m_2-1}(\Sigma)
  \leq t^{m_2-1}\,\hh_g^{m_1+m_2-1}(\Sigma).
\end{equation}
\item[(iii)] Equality holds in \eqref{eq:anest2} if and only if the $g$-normal of $\Sigma$ is tangent to $M_2$, up to a $\hh_g^{m_1+m_2-1}$-null set.
\end{enumerate}

\begin{proposition}
\label{prp:prpc}
\mbox{}
Under the hypotheses of Proposition~\ref{prp:pr} there holds
\begin{equation}
\label{eq:prpc}
\ptl E_i\to \ptl E\quad\text{in the Hausdorff topology}.
\end{equation}
\end{proposition}

\begin{proposition}
\label{Thm:sts}
Let $\Sigma\subset N$ be a smooth constant–mean–curvature hypersurface.
If $\Sigma$ is stable, then $tM\times\Sigma$ is stable for all sufficiently small $t>0$.
\end{proposition}

\begin{proposition}
\label{prp:reduce}
If Theorem \ref{thm:main} is true when the first factor is a round sphere, then it holds in general, i.e, when the first factor is any compact Riemannian manifold.
\end{proposition}

\begin{theorem}
\label{thm:main}
Let $M,N$ be compact Riemannian manifolds and $\be\in (0,1)$. Assume that the boundaries of the isoperimetric regions in $N$ of volume fraction $\be$ are of class $C^{2,\alpha}$.
\begin{enum}
\item Then there exists $t_0=t_0(\be)>0$ so that  for every $0<t\le t_0$
every isoperimetric region of volume $\be\vv(t M \times N)$ in $t M \times N$, is of the form  $ M \times S$, where $S$ is an isoperimetric region in $N$.

By scaling isotropically, we get equivalently.
\item
For every $s\ge s_0$
every isoperimetric region of volume $\be \vv(M \times s N)$ in $M \times s N$, is of the form  $M \times S$, where $S$ is an isoperimetric region in $s N$ and $s_0=t_0^{-1}$.
\end{enum}
\end{theorem}

\begin{equation}\label{eq:stab}
- \int_\Sg u \,\big(\Delta u + \ric(\nu,\nu)u + |\sg|^2 u\big) \,\geq 0,
\end{equation}

The classical isoperimetric problem seeks to determine, within a given ambient space, the subsets of prescribed volume that minimize boundary measure.
It is one of the most fundamental variational problems in geometry.

In products of Euclidean and hyperbolic spaces, W.–T. and W.–Y. Hsiang~\cite{hsiang} provided a complete description of isoperimetric hypersurfaces.
In products of a circle with a model space, isoperimetric regions were classified by Pedrosa and Ritoré \cite{peri}.
Morgan~\cite{Morpr} established lower bounds for the isoperimetric profile of a Riemannian product in terms of concave lower bounds for the profiles of the factors.
In the spherical case, Pedrosa~\cite{Pedrosa2004} classified the isoperimetric regions in the spherical cylinder $\mathbb{S}^n\times\mathbb{R}$.
In Riemannian cylinders $M\times\mathbb{R}$, Duzaar and Steffen~\cite{du-st} proved that large-volume minimizers are slabs.
This was later extended to $M\times\mathbb{R}^k$, where large isoperimetric regions are products of the first factor with geodesic balls in the second factor, see~\cite{manstr}.

A natural question was originally proposed by M. Hutchings — as reported by F. Morgan in~\cite{morpol} — who conjectured that under anisotropic scaling, the isoperimetric regions should align with a cylindrical product structure. More concretely, if one rescales one factor of a product, do minimizers eventually acquire a cylindrical structure of the form $M \times S$, where $S$ is an isoperimetric region in the other factor, once the scaling parameter is sufficiently small (or large)?

In this paper, we address this conjecture under the assumption that the isoperimetric hypersurfaces of the second factor \(N\) are of class \(C^{2,\alpha}\), a condition that holds, for example, when \(\dim N\leqslant 8\) or for small volumes (and their complements).

We begin by recalling the scaling identities for Hausdorff measure, volume, and perimeter, and fix notation for anisotropic homotheties on product manifolds.
In the Appendix we adapt the Ros–Morgan symmetrization on horizontal slices, showing that it suffices to treat the case $\mathbb{S}^m\times N$ (Proposition~\ref{prp:reduce})
\,(see also the related symmetrization results of Morgan--Howe--Harman~\cite{mosym}).
We then analyze \emph{cylinderoids} $M_1\times S$ and symmetrized competitors in $\mathbb{S}^m\times N$: compactness and slice estimates imply that, under the anisotropic deformation, isoperimetric sets subconverge (in both $L^1$ and Hausdorff topologies) to cylinderoids whose projections onto $N$ are isoperimetric (Propositions~\ref{prp:pr} and~\ref{prp:prpc}). Moreover, by standard regularity theory for perimeter minimizers,
all relevant competitors have $C^{2,\alpha}$ boundaries.
This allows us to restrict the analysis to $C^{2,\alpha}$ graphical
perturbations of $\Sigma_t$.
Next, exploiting the product spectral splitting and the scaling of eigenvalues, we establish a stability inheritance result: if $\Sigma\subset N$ is a stable constant mean curvature hypersurface, then $tM\times\Sigma$ is stable for all sufficiently small $t>0$ (Proposition~\ref{Thm:sts}).
Finally, assuming the $C^{2,\alpha}$ regularity of isoperimetric hypersurfaces in $N$, we adapt the Lyapunov–Schmidt reduction framework developed in~\cite{rqi} to the present product setting around $\Sg_t:=tM\times\Sigma$.
This analytic decomposition into a Jacobi–kernel component and its $L^2$–orthogonal complement, combined with stability estimates, yields that all nearby stationary graphs are parametrized by a finite-dimensional family of kernel deformations. This rigidity ultimately forces minimizers close to $\Sg_t$ to be cylinderoids (Theorem~\ref{thm:main}).

We begin by recalling the scaling relations for volume and perimeter, and by introducing anisotropic product scalings. We also summarize standard results on existence, regularity, and stability of isoperimetric regions in compact Riemannian manifolds.

Let $(M,g)$ be a Riemannian manifold. For a measurable set $S \subset M$, we denote by
\[
  \vv_g(S) \quad \text{and} \quad \pp_g(S)
\]
the Riemannian volume and perimeter, respectively. The $\alpha$-dimensional Hausdorff measure with respect to $g$ is written $\hh^\alpha_g(S)$. A direct computation from the definitions yields the following scaling relations:
\begin{equation}\label{eq:est}
  \hh_{t^2 g}^{\alpha}(S) = t^{\alpha}\,\hh_g^{\alpha}(S),
  \qquad \vv_{t^2 g}(S) = t^m\,\vv_g(S),
  \qquad \pp_{t^2 g}(S) = t^{m-1}\,\pp_g(S),
\end{equation}
where $m = \dim M$.
Throughout the manuscript, whenever no ambiguity arises, we shall drop subscripts and superscripts in the notation. We write $tM$ for the Riemannian manifold $(M,t^2 g)$, and refer to it as the \emph{$t$-homothety} of $(M,g)$.

Let $(M_i,g_i)$, $i=1,2$, be Riemannian manifolds, and set $g=g_1 \times g_2$.
We define two anisotropic product scalings:
\begin{itemize}
  \item The \emph{right anisotropic $t$-homothety} is
  \[
    (M_1 \times M_2, g_1 \times t^2 g_2),
  \]
  denoted by $(M_1 \times M_2, g_+^t)$ or simply $M_1 \times tM_2$.
  \item The \emph{left anisotropic $t$-homothety} is
  \[
    (M_1 \times M_2, t^2 g_1 \times g_2),
  \]
  denoted by $(M_1 \times M_2, g_-^t)$ or simply $tM_1 \times M_2$.
\end{itemize}
These anisotropic deformations play a central role in our analysis, as they geometrically encode the effect of collapsing or expanding one factor of a product manifold. We record the following scaling behavior for volume and perimeter under right anisotropic homotheties.

In products of Euclidean and hyperbolic spaces, W.–T. and W.–Y. Hsiang~\cite{hsiang} provided a complete description of isoperimetric hypersurfaces.
In products of a circle with a model space, isoperimetric regions were classified by Pedrosa and Ritoré \cite{peri}.
Morgan~\cite{Morpr} established lower bounds for the isoperimetric profile of a Riemannian product in terms of concave lower bounds for the profiles of the factors.
In the spherical case, Pedrosa~\cite{Pedrosa2004} classified the isoperimetric regions in the spherical cylinder $\mathbb{S}^n\times\mathbb{R}$.
In Riemannian cylinders $M\times\mathbb{R}$, Duzaar and Steffen~\cite{du-st} proved that large-volume minimizers are slabs.
This was later extended to $M\times\mathbb{R}^k$, where large isoperimetric regions are products of the first factor with geodesic balls in the second factor, see~\cite{manstr}.

In this paper, we address this conjecture under the assumption that the isoperimetric hypersurfaces of the second factor \(N\) are of class \(C^{2,\alpha}\), a condition that holds, for example, when \(\dim N\leqslant 8\) or for small volumes (and their complements).

We begin by recalling the scaling identities for Hausdorff measure, volume, and perimeter, and fix notation for anisotropic homotheties on product manifolds.
In the Appendix we adapt the Ros–Morgan symmetrization on horizontal slices, showing that it suffices to treat the case $\mathbb{S}^m\times N$ (Proposition~\ref{prp:reduce})
\,(see also the related symmetrization results of Morgan--Howe--Harman~\cite{mosym}).
We then analyze \emph{cylinderoids} $M_1\times S$ and symmetrized competitors in $\mathbb{S}^m\times N$: compactness and slice estimates imply that, under the anisotropic deformation, isoperimetric sets subconverge (in both $L^1$ and Hausdorff topologies) to cylinderoids whose projections onto $N$ are isoperimetric (Propositions~\ref{prp:pr} and~\ref{prp:prpc}). Moreover, by standard regularity theory for perimeter minimizers,
all relevant competitors have $C^{2,\alpha}$ boundaries.
This allows us to restrict the analysis to $C^{2,\alpha}$ graphical
perturbations of $\Sigma_t$.
Next, exploiting the product spectral splitting and the scaling of eigenvalues, we establish a stability inheritance result: if $\Sigma\subset N$ is a stable constant mean curvature hypersurface, then $tM\times\Sigma$ is stable for all sufficiently small $t>0$ (Proposition~\ref{Thm:sts}).
Finally, assuming the $C^{2,\alpha}$ regularity of isoperimetric hypersurfaces in $N$, we adapt the Lyapunov–Schmidt reduction framework developed in~\cite{rqi} to the present product setting around $\Sg_t:=tM\times\Sigma$.
This analytic decomposition into a Jacobi–kernel component and its $L^2$–orthogonal complement, combined with stability estimates, yields that all nearby stationary graphs are parametrized by a finite-dimensional family of kernel deformations. This rigidity ultimately forces minimizers close to $\Sg_t$ to be cylinderoids (Theorem~\ref{thm:main}).

We begin by recalling the scaling relations for volume and perimeter, and by introducing anisotropic product scalings. We also summarize standard results on existence, regularity, and stability of isoperimetric regions in compact Riemannian manifolds.

Let $(M,g)$ be a Riemannian manifold. For a measurable set $S \subset M$, we denote by
\[
 \vv_g(S) \quad \text{and} \quad \pp_g(S)
\]
the Riemannian volume and perimeter, respectively. The $\alpha$-dimensional Hausdorff measure with respect to $g$ is written $\hh^\alpha_g(S)$. A direct computation from the definitions yields the following scaling relations:
\begin{equation}\label{eq:est}
 \hh_{t^2 g}^{\alpha}(S) = t^{\alpha}\,\hh_g^{\alpha}(S),
 \qquad \vv_{t^2 g}(S) = t^m\,\vv_g(S),
 \qquad \pp_{t^2 g}(S) = t^{m-1}\,\pp_g(S),
\end{equation}
where $m = \dim M$.
Throughout the manuscript, whenever no ambiguity arises, we shall drop subscripts and superscripts in the notation. We write $tM$ for the Riemannian manifold $(M,t^2 g)$, and refer to it as the \emph{$t$-homothety} of $(M,g)$.

Let $(M_i,g_i)$, $i=1,2$, be Riemannian manifolds, and set $g=g_1 \times g_2$.
We define two anisotropic product scalings:
\begin{itemize}
 \item The \emph{right anisotropic $t$-homothety} is
 \[
 (M_1 \times M_2, g_1 \times t^2 g_2),
 \]
 denoted by $(M_1 \times M_2, g_+^t)$ or simply $M_1 \times tM_2$.
 \item The \emph{left anisotropic $t$-homothety} is
 \[
 (M_1 \times M_2, t^2 g_1 \times g_2),
 \]
 denoted by $(M_1 \times M_2, g_-^t)$ or simply $tM_1 \times M_2$.
\end{itemize}
These anisotropic deformations play a central role in our analysis, as they geometrically encode the effect of collapsing or expanding one factor of a product manifold. We record the following scaling behavior for volume and perimeter under right anisotropic homotheties.

\begin{proof}
Part (i) follows directly from computing the Jacobian of the identity map from $(M_1 \times M_2,g)$ to $(M_1 \times M_2,g_+^t)$.
For (ii), at a regular point of $\Sg$ the $g$-unit exterior normal vector $\nu$ can be decomposed as $\nu = a\nu_1 + b\nu_2$ so that $a^2+b^2=1$ and $\nu_1$, $\nu_2$ unit vectors tangent to $M_1$ and $M_2$, respectively. The Jacobian of the identity map from $(M_1 \times M_2,g)$ to $(M_1 \times M_2,g_+^t)$ restricted to $\Sg$ then equals $t^{m_2-1}(t^2a^2+b^2)^{1/2}$. This yields inequality \eqref{eq:anest2}.
Assertion (iii) follows from the fact that $\nu$ is tangent to $M_2$ precisely when $a=0$ almost everywhere.

\end{proof}
The following local isoperimetric inequalities are classical, see Duzaar and Steffen \cite{du-st}.
\begin{lemma}
\label{lem:ine}
Let $M$ be a compact $m$-dimensional Riemannian manifold. Given $0<v_0<\hh^m(M)$, there exist positive constants $\ga_1(M,v_0),\ga_2(M,v_0)$ such that for any set $S\subset M$ with $(m-1)$-rectifiable boundary and $0<\hh^m(S)<v_0$ the following isoperimetric inequalities hold
\begin{equation}
\label{eq:Ine1}
\hh^{m-1}(\ptl S)\ge \ga_1(M,v_0)\,\hh^m(S)
\end{equation}
and
\begin{equation}
\label{eq:Ine2}
\hh^{m-1}(\ptl S)\ge \ga_2 (M,v_0)\,\hh^m(S)^{(m-1)/m},
\end{equation}
\end{lemma}

We are now ready to prove the main results of the section.
\begin{proposition}
\label{prp:pr}
\mbox{}
Assume $0<\be<1$, $t_i\uparrow\infty\,$ and $\{E_i\}_{i\in\nn}$ be a sequence of compact symmetrized isoperimetric sets in $ \esf\times t_i N$ of volume $\vv_{g_+^{t_i}}(E_i)=\be \vv_{g_+^{t_i}}(\esf\times N)$. Let $t_0>0$ and $s_i=t_0\,t_i^{-1}$.
\begin{enum}
\item
Then, possibly passing to a non-relabeling subsequence, $\{E_i\}_{i\in\nn}$ converges both in $L^1$ and Hausdorff topology of $\esf\times t_0N$, to a finite perimeter set $E$ which is a cylinderoid, i.e $E=\esf\times \pi_2(E)$.
\item Moreover $\pi_2(E)$ is an isoperimetric set in $t_0N$, of volume fraction $\be$.
\item Furthermore
\begin{equation}
\label{eq:ane00}
\vv_{g_+^{t_0}}(E_i)=\vv_{g_+^{t_0}}(\esf\times \pi_2(E)) \,\,\text{and}\,\,\pp_{g_+^{t_0}}(E_i)\le\pp_{g_+^{t_0}}(\esf\times \pi_2(E))
\end{equation}
\end{enum}
\end{proposition}
\begin{proof}
Owing to \eqref{eq:anest1} we get
\begin{equation}
\label{eq:thm0}
\vv_{g_+^{t_0}}(E_i)=\be\vv_{g_+^{t_0}}(\esf\times N) :=v_0\quad\text{for every}\,\,i\in\nn.
\end{equation}
Since $N$ is compact, there exists an isoperimetric region $S\subset N$ of volume $v_0\,\vv(\esf)^{-1}$. As $E_i$ are isoperimetric by assumption, we have
\begin{equation}
\label{eq:cona}
\vv_{g_+^{t_i}}(\esf\times S)= \vv_{g_+^{t_i}}(E_i) \,\,\text{and}\,\,\pp_{g_+^{t_i}}(E_i)\le\pp_{g_+^{t_i}}(\esf\times S)
\end{equation}
Thanks to (ii) and (iii) of Lemma \ref{lem:anest}, we deduce
\begin{equation}
\label{eq:conb}
\frac{\pp_{g_+^{t_0}}(E_i)}{\pp_{g_+^{t_0}}(\esf\times S)}=\frac{\pp_{g_+^{s_it_i}}(E_i)}{\pp_{g_+^{s_it_i}}(\esf\times S)}\le\frac{s_i^{n-1}\pp_{g_+^{t_i}}(E_i)}{s_i^{n-1}\pp_{g_+^{t_i}}(\esf\times S)}\le 1
\end{equation}
Consequently, $\{\pp_{g_+^{t_0}}(E_i)\}_{i\in\nn}$ is bounded and so, possibly passing to a subsequence, $E_i\to E$ in the $L^1(\esf\times t_0 N)$ topology. Hence, by the semi-continuity of the perimeter, we get
\begin{equation}
\label{eq:con1}
\pp_{g_+^{t_0}}(E)\le\liminf\pp_{g_+^{t_0}}(E_i)\quad\text{and}\,\,\vv_{g_+^{t_0}}(E)=\be\vv_{g_+^{t_0}}(\esf\times N) :=v_0
\end{equation}
Owing to Lemma \ref{lem:h=l}, possibly passing to a non-relabeling subsequence, we get $E_i\to E$ in the Hausdorff topology of $\esf\times t_0 N$. We argue by contradiction, assume that $E\not=\esf\times \pi_2(E)$. Then by Lemma \ref{lem:ww} there are $\de>0$ and $w_0\in (0,\vv(\esf))$ so that
\begin{equation}
\label{eq:con2}
\vv_{g_+^{t_0}}(E_i{\{w_0\}})>\de\quad\text{for sufficiently large}\,\,i\in\nn.
\end{equation}
Hence, by \eqref{eq:anest1}, we get
\begin{equation}
\label{eq:con3}
\frac{\vv_{g_+^{t_i}}(E_i{\{w_0\}})}{\vv_{g_+^{t_i}}(E_i)}=\frac{\vv_{g_+^{t_0}}(E_i{\{w_0\}})}{\vv_{{g_+^{t_0}}}(E_i)}= \frac{\vv_{{g_+^{t_0}}}(E_i{\{w_0\}})}{\be\vv_{{g_+^{t_0}}}(\esf\times N)}\ge \frac{\de}{\be\vv_{{g_+^{t_0}}}(\esf\times N)} :=c_1>0,
\end{equation}
for sufficiently large $i\in\nn$. Observe that since $ E_i\{w_0\}$ is symmetrized there holds
\begin{equation}
\label{eq:3a}
\pi_2(\ptl E_i\cap E_i\{w_0\})=\pi_2( E_i\{w_0\}) := \Xi_i.
\end{equation}
Now owing to \eqref{eq:con3} and Fubini's Theorem we obtain
\begin{equation}
\label{eq:con4}
\vv_{g_+^{t_i}}(E_i)\le c_1^{-1}\int_{\Xi_i}\hh_{g_1}^m( E_i\cap \pi_2^{-1}(y))\,d\hh_{{t_i}^2g_2}^n
\end{equation}

\begin{proof}
Let $\bar{g}=g_1\times\bar{g}_2$, where $\bar{g}_2=g_2|_{T\Sg\times T\Sg}$.
Note that the outer unit normal $\nu$ on $tM\times \Sg$ is tangent to the second factor. Consequently,
\begin{equation}
\label{eq:curvs}
\ric^{g_-^{t}}(\nu,\nu)=\ric^{\,g_2}(\nu,\nu),
\qquad
\sg^{\bar{g}_-^{t}}=\sg^{\,\bar{g}_2},
\end{equation}
where $\ric$ denotes Ricci tensor and $\sg$ second fundamental form. Hence
\begin{equation}
\label{eq:sts00}
q(x,\vsg):=\ric^{g_-^{t}}(\nu,\nu)+|\sg^{\bar{g}_-^{t}}|^2
= \ric^{\,g_2}(\nu,\nu)+|\sg^{\,\bar{g}_2}|^2 :=q(\vsg).
\end{equation}
Set
\begin{equation}
\label{eq:sts0}
t_0^2=\frac{\la_1(g_1)}{\|q\|_\infty}, \qquad t<t_0.
\end{equation}
As a consequence of the product spectral decomposition, it suffices to show that
\begin{equation}
\label{eq:sts1}
-\int_{tM\times\Sg} f\big(\Delta f+qf\big)\geq0 \qquad \text{for all smooth } f \text{ with } \int_{tM\times\Sg} f=0,
\end{equation}
where $f(x,\vsg)=u(x)v(\vsg)$, $x\in M$, $\vsg\in \Sg$ and $\Delta=\Delta^{\bar{g}_-^{t}}$. Owing to \eqref{eq:delprt}
\begin{equation}
\label{eq:sts2}
\begin{split}
-\int_{tM\times\Sg}f(\Delta f+qf)
&=-\int_{tM\times\Sg} v^2 u\Delta^{t^2g_1} u
 -\int_{tM\times\Sg} u^2 v \Delta^{\bar{g}_2} v
 -\int_{tM\times\Sg} q u^2 v^2.
\end{split}
\end{equation}
Applying Stokes’ theorem:
\begin{equation}
\label{eq:sts3}
-\int_{tM\times\Sg} v^2 u \Delta^{t^2g_1}u
= \int_{tM} |\nabla^{t^2g_1}u|^2_{t^2g_1}\,\int_{\Sg} v^2,
\end{equation}
and
\begin{equation}
\label{eq:sts4}
-\int_{tM\times\Sg} u^2 v \Delta^{\bar{g}_2}v
= \int_{tM} u^2 \,\int_{\Sg} |\nabla^{\bar{g}_2}v|^2_{\bar{g}_2}.
\end{equation}

\section{Appendix}
In this appendix we recall the fibrewise rearrangement used throughout the paper.
We consider here a symmetrization first introduced by Ros~\cite{ros} and later generalized by Morgan~\cite{morpol} for warped product manifolds with density.
Let $M,N$ be compact Riemannian manifolds. Let $\esf$ be a round sphere with the same volume and dimension as $M$ and with isoperimetric profile not exceeding that of $M$. Fix a point $p_0\in \esf$ and consider the foliation of the geodesic balls centered at $p_0$.
Let $E\subset M\times N$ be a set of finite perimeter. Replace each non-empty horizontal slice with the unique geodesic ball in the foliation centred at $p_0$ having the same volume. We denote by $\sym E \subset \esf\times N$ this set, which we call the symmetrized set of~$E$. Then, there holds
\begin{equation}
\label{eq:sym1}
\vv_{\esf\times N}(\sym E)=\vv_{M\times N} (E)\quad \text{and}\ \quad \pp_{\esf\times N}(\sym E)\le \pp_{M\times N}(E).
\end{equation}
Consequently,
\begin{equation}
\label{eq:sym2}
I_{M\times N}\ge I_{\esf\times N}
\end{equation}

\begin{proof}
Part (i) follows directly from computing the Jacobian of the identity map from $(M_1 \times M_2,g)$ to $(M_1 \times M_2,g_+^t)$.
For (ii), at a regular point of $\Sigma$ the $g$-unit exterior normal vector $\nu$ can be decomposed as $\nu = a\nu_1 + b\nu_2$ so that $a^2+b^2=1$ and $\nu_1$, $\nu_2$ unit vectors tangent to $M_1$ and $M_2$, respectively. The Jacobian of the identity map from $(M_1 \times M_2,g)$ to $(M_1 \times M_2,g_+^t)$ restricted to $\Sigma$ then equals $t^{m_2-1}(t^2a^2+b^2)^{1/2}$. This yields inequality \eqref{eq:anest2}.
Assertion (iii) follows from the fact that $\nu$ is tangent to $M_2$ precisely when $a=0$ almost everywhere.

\end{proof}
The following local isoperimetric inequalities are classical, see Duzaar and Steffen \cite{du-st}.
\begin{lemma}
\label{lem:ine}
Let $M$ be a compact $m$-dimensional Riemannian manifold. Given $0<v_0<\hh^m(M)$, there exist positive constants  $\ga_1(M,v_0),\ga_2(M,v_0)$ such that for any set $S\subset M$ with $(m-1)$-rectifiable boundary and $0<\hh^m(S)<v_0$ the following isoperimetric inequalities hold
\begin{equation}
\label{eq:Ine1}
\hh^{m-1}(\partial S)\geqslant \ga_1(M,v_0)\,\hh^m(S)
\end{equation}
and
\begin{equation}
\label{eq:Ine2}
\hh^{m-1}(\partial S)\geqslant \ga_2 (M,v_0)\,\hh^m(S)^{(m-1)/m},
\end{equation}
\end{lemma}

The \emph{isoperimetric profile} of a Riemannian manifold $(M,g)$ is defined by
\[
  I_g(v) = \inf \big\{ \pp_g(E) : \vv_g(E) = v \big\}.
\]
A set $E \subset M$ is called an \emph{isoperimetric region} if $\pp_g(E) = I_g(\vv_g(E))$. Existence of isoperimetric regions in compact manifolds is guaranteed by standard compactness arguments in geometric measure theory, see \cite{Maggi}. Moreover, classical regularity theory implies that the boundary of an isoperimetric
region is a smooth hypersurface except for a singular set of Hausdorff codimension at least $8$,  as proved by Morgan \cite{moreg} and by Gonzales–Massari–Tamanini \cite{gomato}. In Chapter 13.2 of \cite{Morgan2016}, Morgan, following methods of Smale \cite{Smale1999},  describes constructions of irregular isoperimetric hypersurfaces in manifolds of dimension greater than seven. Furthermore, Morgan–Johnson \cite{morjohn} and Nardulli \cite{nard} show that, for small volumes (and their complements), isoperimetric hypersurfaces are smooth.

Isoperimetric regions are considered up to sets of measure zero.
In particular, every isoperimetric set is equivalent to its closure, and its perimeter can be computed on the topological boundary:
\begin{equation}\label{eq:ww4}
\mathcal{P}(E) = \mathcal{P}(\overline{E}) = \hh^{m-1}(\partial E).
\end{equation}

Let $\Sigma\subset M$ be an isoperimetric hypersurface. Standard first and second variation computations
(see for instance Barbosa--do Carmo--Eschenburg~\cite{bd})
imply that the regular part $\reg(\Sigma)$ has constant mean curvature and satisfies a stability inequality.
Specifically, for every smooth compactly supported function $u:\reg(\Sigma)\to\mathbb{R}$ with zero mean one has
\begin{equation}\label{eq:stab}
- \int_\Sigma u \,\big(\Delta u + \ric(\nu,\nu)u + |\sigma|^2 u\big) \,\geq 0,
\end{equation}

where $\nu$ is the exterior unit normal to $\Sigma$, $\Delta$ is the Laplace operator on $\Sigma$, $|\sigma|^2$ is the squared norm of the second fundamental form, and $\ric$ denotes the Ricci tensor of the ambient manifold $M$.
We denote by $J_\Sigma$ the Jacobi operator, which acts on smooth functions $u$ as follows:
\[
  J_\Sigma u = \Delta u + \ric(\nu,\nu)\,u + |\sigma|^2 u.
\]
If the inequality in \eqref{eq:stab} is strict for all nontrivial $u$, we say that $\sigma$ is \emph{strictly stable}.

Part (i) is obtained by “computing the Jacobian of the identity map from $(M_1 \times M_2,g)$ to $(M_1 \times M_2,g_+^t)$,” giving \eqref{eq:anest1}.

For (ii), “at a regular point of $\Sigma$ the $g$-unit exterior normal vector $\nu$ can be decomposed as $\nu=a\nu_1+b\nu_2$ so that $a^2+b^2=1$ and $\nu_1$, $\nu_2$ unit vectors tangent to $M_1$ and $M_2$, respectively.” The Jacobian of the identity map restricted to $\Sigma$ “equals $t^{m_2-1}(t^2a^2+b^2)^{1/2}$,” which “yields inequality \eqref{eq:anest2}” for $t\le 1$.

For (iii), “$\nu$ is tangent to $M_2$ precisely when $a=0$ almost everywhere,” which characterizes when equality holds in \eqref{eq:anest2}.

Part (i) is obtained by “computing the Jacobian of the identity map from $(M_1 \times M_2,g)$ to $(M_1 \times M_2,g_+^t)$,” giving \eqref{eq:anest1}.

For (ii), “at a regular point of $\Sigma$ the $g$-unit exterior normal vector $\nu$ can be decomposed as $\nu=a\nu_1+b\nu_2$ so that $a^2+b^2=1$ and $\nu_1$, $\nu_2$ unit vectors tangent to $M_1$ and $M_2$, respectively.” The Jacobian of the identity map restricted to $\Sigma$ “equals $t^{m_2-1}(t^2a^2+b^2)^{1/2}$,” which for $t\le 1$ yields
\begin{equation}\label{eq:anest2}
  \hh_{g_+^t}^{m_1+m_2-1}(\Sg)
  \leq t^{m_2-1}\,\hh_g^{m_1+m_2-1}(\Sg).
\end{equation}

For (iii), “$\nu$ is tangent to $M_2$ precisely when $a=0$ almost everywhere,” which characterizes when equality holds in the equation above.

\label{lem:anest}
Let $(M_i,g_i)$ be Riemannian manifolds of dimensions $m_i$ ($i=1,2$), and set $g=g_1\times g_2$. Then the following hold for all measurable $S\subset M_1\times M_2$ and $(m_1+m_2-1)$–rectifiable $\Sigma$:
\begin{enumerate}
\item[(i)] For every set $S \subset M_1 \times M_2$,
\begin{equation}\label{eq:anest1}
  \vv_{g_+^t}(S) = t^{m_2} \,\vv_g(S).
\end{equation}
\item[(ii)] If $\Sigma \subset M_1 \times M_2$ is $(m_1+m_2-1)$-rectifiable, then for $t \leq 1$,
\begin{equation}\label{eq:anest2}
  \hh_{g_+^t}^{m_1+m_2-1}(\Sg)
  \leq t^{m_2-1}\,\hh_g^{m_1+m_2-1}(\Sg).
\end{equation}
\item[(iii)] Equality holds in the equation above if and only if the $g$-normal of $\Sigma$ is tangent to $M_2$, up to a $\hh_g^{m_1+m_2-1}$-null set.
\end{enumerate}

Inequality or Bound
Biconditional or Equivalence

Question: Let \((M_i,g_i)\) for \(i=1,2\) be Riemannian manifolds of dimensions \(m_i\), let \(g=g_1\times g_2\) on \(M_1\times M_2\), and for \(t>0\) define the right anisotropic \(t\)-homothety metric by \(g_+^t=g_1\times t^2 g_2\). For a measurable set \(S\subset M_1\times M_2\), write \(\mathrm{vol}_{g}(S)\) for its Riemannian volume, and for an \((m_1+m_2-1)\)-rectifiable set \(\Sigma\subset M_1\times M_2\), write \(\mathcal H^{m_1+m_2-1}_{g}(\Sigma)\) for its \((m_1+m_2-1)\)-dimensional Hausdorff measure with respect to \(g\). Which quantitative statement holds for volume and hypersurface measure under \(g_+^t\)?

For every measurable set \(S\subset M_1\times M_2\), one has
\[
\mathrm{vol}_{g_+^t}(S)=t^{m_2}\,\mathrm{vol}_{g}(S).
\]
Moreover, if \(\Sigma\subset M_1\times M_2\) is \((m_1+m_2-1)\)-rectifiable and \(t\le 1\), then
\[
\mathcal H^{m_1+m_2-1}_{g_+^t}(\Sigma)
\le t^{m_2-1}\,\mathcal H^{m_1+m_2-1}_{g}(\Sigma).
\]
Equality in this last inequality holds if and only if the \(g\)-normal to \(\Sigma\) is tangent to the \(M_2\)-factor, up to a \(\mathcal H^{m_1+m_2-1}_{g}\)-null set.

Choices:
B.
For every measurable set \(S\subset M_1\times M_2\), one has
\[
\mathrm{vol}_{g_+^t}(S)=t^{m_2}\,\mathrm{vol}_{g}(S).
\]
Moreover, if \(\Sigma\subset M_1\times M_2\) is \((m_1+m_2-1)\)-rectifiable, then for every \(t>0\),
\[
\mathcal H^{m_1+m_2-1}_{g_+^t}(\Sigma)
\le t^{m_2-1}\,\mathcal H^{m_1+m_2-1}_{g}(\Sigma).
\]
Equality in this last inequality holds if and only if the \(g\)-normal to \(\Sigma\) is tangent to the \(M_2\)-factor, up to a \(\mathcal H^{m_1+m_2-1}_{g}\)-null set.
C.
For every measurable set \(S\subset M_1\times M_2\), one has
\[
\mathrm{vol}_{g_+^t}(S)=t^{m_2}\,\mathrm{vol}_{g}(S).
\]
Moreover, if \(\Sigma\subset M_1\times M_2\) is \((m_1+m_2-1)\)-rectifiable and \(t\le 1\), then
\[
\mathcal H^{m_1+m_2-1}_{g_+^t}(\Sigma)
\le t^{m_2-1}\,\mathcal H^{m_1+m_2-1}_{g}(\Sigma).
\]
D.
For every measurable set \(S\subset M_1\times M_2\), one has
\[
\mathrm{vol}_{g_+^t}(S)=t^{m_1}\,\mathrm{vol}_{g}(S).
\]
Moreover, if \(\Sigma\subset M_1\times M_2\) is \((m_1+m_2-1)\)-rectifiable and \(t\le 1\), then
\[
\mathcal H^{m_1+m_2-1}_{g_+^t}(\Sigma)
\le t^{m_1-1}\,\mathcal H^{m_1+m_2-1}_{g}(\Sigma).
\]
Equality in this last inequality holds if and only if the \(g\)-normal to \(\Sigma\) is tangent to the \(M_1\)-factor, up to a \(\mathcal H^{m_1+m_2-1}_{g}\)-null set.
E.
For every measurable set \(S\subset M_1\times M_2\), one has
\[
\mathrm{vol}_{g_+^t}(S)=t^{m_2}\,\mathrm{vol}_{g}(S).
\]
Moreover, if \(\Sigma\subset M_1\times M_2\) is \((m_1+m_2-1)\)-rectifiable and \(t\le 1\), then
\[
\mathcal H^{m_1+m_2-1}_{g_+^t}(\Sigma)
= t^{m_2-1}\,\mathcal H^{m_1+m_2-1}_{g}(\Sigma).
\]
In particular, the hypersurface measure always scales exactly by \(t^{m_2-1}\), independently of the orientation of the \(g\)-normal.