\label{thm:main}
Let $c \neq 0$.
Suppose ${\omega d}/{c}=1+\varepsilon$ for a sufficiently small absolute constant $\varepsilon > 0$ and ${\sigma}/{(c^{2}d)}>\frac13$, the system (\ref{main second}) admits a $C^2$ solitary wave solution with velocity $c$. Moreover, the solitary wave profile $\eta$ has the asymptotic expansion
\begin{equation}
\eta(\xi)=d\, \varepsilon\, Q\bigg(\varepsilon^{\frac{1}{2}}\bigg(\frac{\sigma}{c^2d}-\frac13\bigg)^{-\frac{1}{2}}\frac{\xi}{d}\bigg)+O(\varepsilon^2)
\end{equation}
where $Q(\bar x)=-3\mathrm{sech}^2(\bar x/2)$ solves the steady the KdV equation
\begin{equation}
Q''=Q+\frac{1}{2}Q^2.
\end{equation}

The study of the motion of water waves is a classical question and one of the central
problems in fluid dynamics. Among the basic questions one can ask are the local
well-posedness of the Cauchy problem, the regularity of solutions and formation of
singularities, and the existence and stability of special solutions. In this paper, we consider two-dimensional pure capillary solitary water waves in finite depth with constant vorticity. More precisely, we study an inviscid, incompressible fluid separated from the air by a free surface, evolving under surface tension in the absence of gravity and with constant vorticity. We construct a family of solitary wave solutions in such setting, and, to the best of our knowledge, this fills a remaining gap in the finite-depth pure-capillary
solitary-wave theory in the presence of vorticity.

\subsection{Free-boundary Euler equations and capillary water waves with constant vorticity}
Let the fluid occupy a time-dependent domain $\mathcal{D}(t)\subset\mathbb{R}^2$ of finite depth,
with flat bottom $\{y=-d\}$ and free surface $\partial\mathcal{D}(t)$. We assume unit density
and non-zero constant vorticity. The velocity field $\mathbf{u}$ and pressure $p$ solve
the free-boundary incompressible Euler equations
\begin{equation}\label{eq:euler-freebdry}
\begin{aligned}
&\mathbf{u}_t+\mathbf{u}\cdot\nabla \mathbf{u}=-\nabla p && \text{in }\mathcal{D}(t),\\
&\nabla\cdot \mathbf{u}=0,\ \mathrm{curl}\,\mathbf{u}=\omega && \text{in }\mathcal{D}(t).\\
\end{aligned}
\end{equation}
In addition, we have the kinematic boundary conditions
\begin{align}\label{kinematic_top}
\mathcal{V}(\partial\mathcal{D}(t)) &= \mathbf{u}\cdot \mathbf{n}\,\quad \text{on }\partial\mathcal{D}(t),\\
\label{kinematic_bottom}
\mathbf u\cdot e_2 &= 0\quad\quad\ \ \text{on }\{y=-d\},
\end{align}
where $\mathbf{n}$ denotes the outward unit normal to the free surface and
$\mathcal{V}(\partial\mathcal{D}(t))$ is its normal velocity. We also have the dynamic boundary condition
\begin{align}\label{Young-Laplace}
p = \sigma H\quad \text{on }\partial\mathcal{D}(t),
\end{align}
where $H$ is the mean curvature of the interface, and $\sigma > 0$ is the surface tension coefficient.

The system \eqref{eq:euler-freebdry}--\eqref{Young-Laplace} is often studied in the
Zakharov--Craig--Sulem framework, which we briefly recall below. A more detailed derivation can be seen in, for example, \cite{Wahlen2007HamiltonianConstantVorticity}.  We assume that our free boundary is a graph $\partial\mathcal D(t)=\{(x,y): y=\eta(t,x)\}$. Since the vorticity $\mathrm{curl}\,\mathbf u = \omega$ is a constant, the velocity can be written as a shear flow plus an irrotational perturbation as
\begin{equation}\label{velocity}
\mathbf u = (-\omega y,0)+\nabla\phi.
\end{equation}
Then, in terms of the velocity potential $\phi$, its harmonic conjugate $\psi$, and the free boundary function $\eta$, \eqref{eq:euler-freebdry}--\eqref{Young-Laplace} can be recast as
\begin{equation}\label{main second}
\begin{aligned}
&\phi_{xx}+\phi_{yy}=0 && \text{in } -d<y<\eta(t,x),\\
&\phi_y=0 && \text{on } y=-d,\\
&\eta_t=\phi_y-(\phi_x-\omega \eta)\eta_x && \text{on } y=\eta(t,x),\\
&\phi_t+\frac12(\phi_x^2+\phi_y^2)-\omega\eta\,\phi_x+\omega\psi
-\frac{\sigma\,\eta_{xx}}{(1+\eta_x^2)^{3/2}}=0 && \text{on } y=\eta(t,x).
\end{aligned}
\end{equation}

\subsection{Previous work on solitary waves and our the main result}
Among the most studied special solutions of the water-wave system are solitary waves,
i.e.\ localized traveling disturbances propagating at constant speed over an otherwise flat
surface. The scientific interest in such waves dates back at least to the experiments of
J.\ S.\ Russell in 1844 \cite{russell1844report}, whose observations anticipated key features
of the modern theory.

The existence theory for solitary waves in two-dimensional irrotational water flows has a
long history. For finite-depth pure-gravity waves, the first rigorous constructions were
obtained in the classical works of Friedrichs--Hyers \cite{friedrichs1954existence}, Beale
\cite{thomas1977existence}, and Amick--Toland \cite{amick1981periodic}. When both gravity
and capillarity are present, small-amplitude solitary waves in finite depth were constructed
using a variety of techniques beginning with Kirchg\"assner \cite{kirchgassner1988nonlinearly}
and Amick--Kirchg\"assner \cite{amick1989theory}, and extended in, for example,
\cite{buffoni1996plethora, iooss1992water, buffoni1999multiplicity, constantin2011nonlinear,
groves2004steady, groves2002dimension, groves2008fully, puaruau2005nonlinear, haragus2002finite,
deng2009three}. In infinite depth, solitary waves exist in the mixed gravity--capillary regime;
see \cite{buffoni2004existence, buffoni2004existence2, groves2011existence, iooss1996capillary}.

In recent years, existence/non-existence theories in pure-gravity (infinite depth) and pure-capillary (finite/infinite depth) regimes have seen significant progress. Hur \cite{hur2012no} proved nonexistence of two-dimensional irrotational pure-gravity solitary waves in deep water under a
decay assumption; Ifrim--Tataru \cite{ifrim2020no} established complete nonexistence in deep water
for both pure-gravity and pure-capillary cases; and Ifrim--Pineau--Tataru--Taylor \cite{ifrim2022no}
proved that irrotational pure-capillary solitary waves do not exist in finite depth. See also \cite{constantin2007rotational, groves2004steady, groves2007spatial, hur2008exact,
hur2008symmetry} for steady waves with vorticity.

In this article, we prove the existence of finite-depth solitary waves for a remaining 2D case, namely the pure capillary case with nonzero constant vorticity. To describe a wave traveling at constant speed $c$, we introduce the moving-frame variable $\xi:= x - ct$ and seek solutions of the form
\begin{equation}\label{traveling_wave_ansatz}
(\phi(t,x,y),\psi(t,x,y),\eta(t,x))=(\phi(\xi,y),\psi(\xi,y),\eta(\xi)).
\end{equation}

We state our main theorem as follows.

The study of the motion of water waves is a classical question and one of the central
problems in fluid dynamics. Among the basic questions one can ask are the local
well-posedness of the Cauchy problem, the regularity of solutions and formation of
singularities, and the existence and stability of special solutions. In this paper, we consider two-dimensional pure capillary solitary water waves in finite depth with constant vorticity. More precisely, we study an inviscid, incompressible fluid separated from the air by a free surface, evolving under surface tension in the absence of gravity and with constant vorticity. We construct a family of solitary wave solutions in such setting, and, to the best of our knowledge, this fills a remaining gap in the finite-depth pure-capillary
solitary-wave theory in the presence of vorticity.

\subsection{Free-boundary Euler equations and capillary water waves with constant vorticity}
Let the fluid occupy a time-dependent domain $\mathcal{D}(t)\subset\mathbb{R}^2$ of finite depth,
with flat bottom $\{y=-d\}$ and free surface $\partial\mathcal{D}(t)$. We assume unit density
and non-zero constant vorticity. The velocity field $\mathbf{u}$ and pressure $p$ solve
the free-boundary incompressible Euler equations
\begin{equation}\label{eq:euler-freebdry}
\begin{aligned}
&\mathbf{u}_t+\mathbf{u}\cdot\nabla \mathbf{u}=-\nabla p && \text{in }\mathcal{D}(t),\\
&\nabla\cdot \mathbf{u}=0,\ \mathrm{curl}\,\mathbf{u}=\omega && \text{in }\mathcal{D}(t).\\
\end{aligned}
\end{equation}
In addition, we have the kinematic boundary conditions
\begin{align}\label{kinematic_top}
\mathcal{V}(\partial\mathcal{D}(t)) &= \mathbf{u}\cdot \mathbf{n}\,\quad \text{on }\partial\mathcal{D}(t),\\
\label{kinematic_bottom}
\mathbf u\cdot e_2 &= 0\quad\quad\ \ \text{on }\{y=-d\},
\end{align}
where $\mathbf{n}$ denotes the outward unit normal to the free surface and
$\mathcal{V}(\partial\mathcal{D}(t))$ is its normal velocity. We also have the dynamic boundary condition
\begin{align}\label{Young-Laplace}
p = \sigma H\quad \text{on }\partial\mathcal{D}(t),
\end{align}
where $H$ is the mean curvature of the interface, and $\sigma > 0$ is the surface tension coefficient.

The system \eqref{eq:euler-freebdry}--\eqref{Young-Laplace} is often studied in the
Zakharov--Craig--Sulem framework, which we briefly recall below. A more detailed derivation can be seen in, for example, \cite{Wahlen2007HamiltonianConstantVorticity}. We assume that our free boundary is a graph $\partial\mathcal D(t)=\{(x,y): y=\eta(t,x)\}$. Since the vorticity $\mathrm{curl}\,\mathbf u = \omega$ is a constant, the velocity can be written as a shear flow plus an irrotational perturbation as
\begin{equation}\label{velocity}
\mathbf u = (-\omega y,0)+\nabla\phi.
\end{equation}
Then, in terms of the velocity potential $\phi$, its harmonic conjugate $\psi$, and the free boundary function $\eta$, \eqref{eq:euler-freebdry}--\eqref{Young-Laplace} can be recast as
\begin{equation}\label{main second}
\begin{aligned}
&\phi_{xx}+\phi_{yy}=0 && \text{in } -d<y<\eta(t,x),\\
&\phi_y=0 && \text{on } y=-d,\\
&\eta_t=\phi_y-(\phi_x-\omega \eta)\eta_x && \text{on } y=\eta(t,x),\\
&\phi_t+\frac12(\phi_x^2+\phi_y^2)-\omega\eta\,\phi_x+\omega\psi
-\frac{\sigma\,\eta_{xx}}{(1+\eta_x^2)^{3/2}}=0 && \text{on } y=\eta(t,x).
\end{aligned}
\end{equation}

\subsection{Previous work on solitary waves and our the main result}
Among the most studied special solutions of the water-wave system are solitary waves,
i.e.\ localized traveling disturbances propagating at constant speed over an otherwise flat
surface. The scientific interest in such waves dates back at least to the experiments of
J.\ S.\ Russell in 1844 \cite{russell1844report}, whose observations anticipated key features
of the modern theory.

In this article, we prove the existence of finite-depth solitary waves for a remaining 2D case, namely the pure capillary case with nonzero constant vorticity. To describe a wave traveling at constant speed $c$, we introduce the moving-frame variable $\xi:= x - ct$ and seek solutions of the form
\begin{equation}\label{traveling_wave_ansatz}
(\phi(t,x,y),\psi(t,x,y),\eta(t,x))=(\phi(\xi,y),\psi(\xi,y),\eta(\xi)).
\end{equation}

We state our main theorem as follows.

We state our main theorem as follows.

\subsection{Main ideas and comparison with related work}
Our analysis uses a spatial-dynamics and Hamiltonian formulation of the traveling-wave Euler system,
followed by a nonlinear transformation that rectifies the free-boundary conditions and the symplectic
structure. This leads to an infinite-dimensional reversible Hamiltonian system which, along a
distinguished curve in parameter space, has a two-dimensional center subspace. We verify the resolvent
bounds needed for Mielke's center-manifold theorem, obtain a finite-dimensional reduced canonical system,
and identify a KdV-type normal form admitting a homoclinic orbit after a long-wave scaling and a cubic expansion of the reduced Hamiltonian. The persistence of this orbit yields solitary waves for the
original free-boundary Euler equations.

\section{The derivation of Hamiltonian formulation}\label{sec:formulation}
The goal of this section is to formulate the traveling wave problem of (\ref{main second}) as a Hamiltonian system. As a first step, we use the moving-frame variable $\xi:= x - ct$ and plug in ansatz as in (\ref{traveling_wave_ansatz}) to the system \eqref{main second}, obtaining
\begin{equation}\label{main third}
\begin{aligned}
&\phi_{\xi\xi}+\phi_{yy}=0 && \text{in } -d<y<\eta(\xi),\\
&\phi_y=0 && \text{on } y=-d,\\
&-c\,\eta_{\xi}=\phi_y-(\phi_{\xi}-\omega \eta)\eta_{\xi} && \text{on } y=\eta(\xi),\\
&-c\,\phi_{\xi}+\frac12(\phi_{\xi}^2+\phi_y^2)-\omega\eta\,\phi_{\xi}+\omega\psi
-\frac{\sigma\,\eta_{\xi\xi}}{(1+\eta_{\xi}^2)^{3/2}}=0 && \text{on } y=\eta(\xi).
\end{aligned}
\end{equation}
It is convenient to nondimensionalize. Under the scalings
\[
\xi\mapsto \xi/d,\quad y\mapsto y/d,\quad \omega\mapsto \omega d/c,\quad
\sigma\mapsto \sigma/(c^2 d),
\qquad
\phi\mapsto \phi/(cd),\quad \psi\mapsto \psi/(cd),\quad \eta\mapsto \eta/d,
\]
we may normalize $c=d=1$ in \eqref{main third}. In this normalization, the third equation
in \eqref{main third} implies
\begin{equation}\label{psi-eta=0 1}
\frac{d}{d\xi}\Bigl(\psi(\xi,\eta(\xi))-\frac{\omega}{2}\eta(\xi)^2-\eta(\xi)\Bigr)=0.
\end{equation}
and thus
\begin{equation}\label{psi-eta=0 2}
\psi(\xi,\eta(\xi))=\frac{\omega}{2}\eta(\xi)^2+\eta(\xi)+\mathrm{const}.
\end{equation}
For solitary waves the profile decays, $\eta(\xi)\to 0$ as $|\xi|\to\infty$, and hence the
constant must be zero. Substituting \eqref{psi-eta=0 2} into \eqref{main third} yields
\begin{equation}\label{main fourth}
\begin{aligned}
&\phi_{\xi\xi}+\phi_{yy}=0 && \text{in } -1<y<\eta(\xi),\\
&\phi_y=0 && \text{on } y=-1,\\
&-\eta_{\xi}=\phi_y-(\phi_{\xi}-\omega \eta)\eta_{\xi} && \text{on } y=\eta(\xi),\\
&-\phi_{\xi}+\frac12\phi_{\xi}^2+\frac12\phi_y^2-\omega\eta\,\phi_{\xi}
+\omega\eta+\frac{\omega^2}{2}\eta^2
-\frac{\sigma\,\eta_{\xi\xi}}{(1+\eta_{\xi}^2)^{3/2}}=0 && \text{on } y=\eta(\xi).
\end{aligned}
\end{equation}

Note that $\tilde f_1^\varepsilon$ transforms the boundary conditions \eqref{BC first} into
\begin{align}\label{105 boundary}
\Phi_y(x,0)=\Phi_y(x,1)=0 .
\end{align}
We therefore introduce
\begin{align}\label{domain of L}
\mathrm{dom}(\mathbf{L})
:=\{(\Phi,\theta,Z,\eta)\in N_0:\ \Phi_y(0)=\Phi_y(1)=0\},
\end{align}
equipped with the $\|\cdot\|_{X^2}$ norm, and the operator $\mathbf L$ will be defined shortly. Recalling \eqref{def of u_H}, let $m\in \mathrm{dom}(\mathbf{L})$ be sufficiently small in $\|\cdot\|_{X^2}$.
Then for every $v'_m\in M_0$ we compute
\begin{align*}
d\tilde H^\varepsilon_m[v'_m]
&= d H^{\omega_0+\varepsilon_1,\sigma_0+\varepsilon_2}_{(f_1^\varepsilon)^{-1}(m)}
\bigl[(d f_1^\varepsilon)^{-1}_m[v'_m]\bigr] \\
&=\Omega^{\omega_0+\varepsilon_1,\sigma_0+\varepsilon_2}_{(f_1^\varepsilon)^{-1}(m)}
\Bigl[v_H^{\omega_0+\varepsilon_1,\sigma_0+\varepsilon_2}\bigl((f_1^\varepsilon)^{-1}(m)\bigr),
(d f_1^\varepsilon)^{-1}_m[v'_m]\Bigr] \\
&=\tilde\Omega^\varepsilon_m\Bigl[
d f_1^\varepsilon\bigl((f_1^\varepsilon)^{-1}(m)\bigr)
\,v_H^{\omega_0+\varepsilon_1,\sigma_0+\varepsilon_2}\bigl((f_1^\varepsilon)^{-1}(m)\bigr),
\,v'_m\Bigr].
\end{align*}
It follows that the Hamiltonian vector field associated with $(\tilde H^\varepsilon,\tilde\Omega^\varepsilon)$ is
\begin{align}\label{newvH}
\widetilde{v_H^\varepsilon}(m)
=
d f_1^\varepsilon\bigl((f_1^\varepsilon)^{-1}(m)\bigr)
\Bigl[v_H^{\omega_0+\varepsilon_1,\sigma_0+\varepsilon_2}\bigl((f_1^\varepsilon)^{-1}(m)\bigr)\Bigr],
\end{align}
for all $m\in \mathrm{dom}(\mathbf{L})$ with sufficiently small $\|m\|_{X^2}$. Using \eqref{newvH}, a straightforward but tedious computation
(see \cite[Appendix~A]{buffoni1996plethora}) yields the explicit formula for
$\widetilde{v_H^\varepsilon}=(\widetilde{\Phi_H^\varepsilon},\widetilde{\theta_H^\varepsilon},
\widetilde{Z_H^\varepsilon},\widetilde{\eta_H^\varepsilon})$.
Writing $m=(\Phi,\theta,Z,\eta)$, we obtain
\begin{equation}\label{B'K equation 2}
\begin{aligned}
\widetilde{\Phi_H^\varepsilon}(m)
&=\frac{1}{\eta+1}\Bigl(\theta_y+R\bigl(2y\Phi_y+Ry^2(\theta_y-1)-\Phi-\Phi(1)+Z\bigr)\Bigr) \\
&\quad-\frac{\bar Z(1+R^2)^{3/2}}{\sigma_0+\varepsilon_2}
\left(\theta_p-\frac12\left(y^2-\frac13\right)\right) \\
&\quad+\frac{(\omega_0+\varepsilon_1)R^2y^2(4y-3)}{6(\eta+1)}
+(\omega_0+\varepsilon_1)(\eta+1)\left(y-\frac12\right)
-\frac{\omega_0+\varepsilon_1}{\eta+1}\left(y-\frac12\right),\\
\widetilde{\theta_H^\varepsilon}(m)
&=-\frac{1}{\eta+1}\Bigl(\Phi_y-(\omega_0+\varepsilon_1)R(1-y)y\Bigr),\\
\widetilde{Z_H^\varepsilon}(m)
&=-\frac{1}{\eta+1}\Bigl(\theta_y(1)+R\bigl(R(\theta_y(1)-1)+Z\bigr)\Bigr)
-\frac{(\omega_0+\varepsilon_1)(\eta+1)}{2}
+\frac{\omega_0+\varepsilon_1}{2(\eta+1)},\\
\widetilde{\eta_H^\varepsilon}(m)
&=R,
\end{aligned}
\end{equation}
where
\[
\bar Z=\frac{(1+R^2)\,(\theta_y(1)-1)^2}{2(\eta+1)^2}-\frac12.
\]

\subsubsection*{Step 6. Leading-order asymptotics for $\eta$ in nondimensional variables}
Using (\ref{eq:QPX-scaling}), we get
\begin{equation}\label{eq:qPiEtae}
\begin{aligned}
{q(x)}&=\varepsilon_1\,\sqrt{\sigma_0-\tfrac13}\, Q(\bar x);\\
p(x) &= \varepsilon_1^{3/2}P(\bar x).
\end{aligned}
\end{equation}
Therefore, we have $q=O(\varepsilon_1)$ and $p=O(\varepsilon_1^{3/2})$, and thus $|(q,p)|=O(\varepsilon_1)$ and $|(q,p,\varepsilon_1)|=O(\varepsilon_1)$, and by \eqref{eq:r-estimate} we obtain
\begin{equation}\label{eq:r-order}
\|\tilde r(v(x),\varepsilon_1)\|=O(\varepsilon_1^2),
\qquad\Rightarrow\qquad
\Pi_\eta(\tilde r(v(x),\varepsilon_1))=O(\varepsilon_1^2).
\end{equation}
Combining \eqref{eq:eta-q-leading}, \eqref{eq:qPiEtae}, and \eqref{eq:r-order} yields the nondimensional leading-order expansion
\begin{equation}\label{eq:eta-dimless-asympt}
\eta(x)=\varepsilon_1\,Q(\bar x)+O(\varepsilon_1^2),
\qquad
\bar x=\sqrt{\varepsilon_1}(\sigma_0-\tfrac13)^{-1/2}x,
\end{equation}
where $Q$ is the stationary KdV soliton \eqref{eq:homoclinic-QP}. Identifying $\varepsilon_1=\varepsilon$ via \eqref{eq:eps-def} in the regime $\omega_0=1$, we may rewrite \eqref{eq:eta-dimless-asympt} as
\begin{equation}\label{eq:eta-dimless-eps}
\eta(x)=\varepsilon\,Q(\bar x)+O(\varepsilon^2),
\qquad
\bar x=\sqrt{\varepsilon}\,(\sigma_0-\tfrac13)^{-1/2}x,
\qquad
Q(\bar x)=-3\,\sech^2(\bar x/2).
\end{equation}

Using $x=\xi=\xi_{\mathrm{phys}}/d$ and \eqref{eq:eta-phys-from-dimless}, together with $\sigma_0=\sigma=\sigma_{\mathrm{phys}}/(c^2 d)$ from \eqref{eq:nondim-params}, we compute
\[
\bar x=\sqrt{\varepsilon}\,(\sigma_0-\tfrac13)^{-1/2}x
=\sqrt{\varepsilon}\,
\Bigl(\frac{\sigma_{\mathrm{phys}}}{c^2 d}-\frac13\Bigr)^{-1/2}\,
\frac{\xi_{\mathrm{phys}}}{d}.
\]
Therefore, the leading-order solitary-wave profile in physical variables is
\begin{equation}\label{eq:eta-phys-asympt}
\eta_{\mathrm{phys}}(\xi_{\mathrm{phys}})
=
d\,\varepsilon\,
Q\!\left(\varepsilon^{\frac{1}{2}}\bigg(\frac{\sigma_{\mathrm{phys}}}{c^2 d}-\frac13\bigg)^{-\frac{1}{2}}\frac{\xi_{\mathrm{phys}}}{d}\right)
+O(\varepsilon^2),
\end{equation}
as in Theorem \ref{thm:main}.

\subsection{Main ideas and comparison with related work}
Our analysis uses a spatial-dynamics and Hamiltonian formulation of the traveling-wave Euler system,
followed by a nonlinear transformation that rectifies the free-boundary conditions and the symplectic
structure. This leads to an infinite-dimensional reversible Hamiltonian system which, along a
distinguished curve in parameter space, has a two-dimensional center subspace. We verify the resolvent
bounds needed for Mielke's center-manifold theorem, obtain a finite-dimensional reduced canonical system,
and identify a KdV-type normal form admitting a homoclinic orbit after a long-wave scaling and a cubic expansion of the reduced Hamiltonian. The persistence of this orbit yields solitary waves for the
original free-boundary Euler equations.

Our approach is complementary to Rowan--Wan \cite{rowan2024two}, who constructed pure-capillary solitary waves in infinite depth via a Babenko-type boundary-integral equation; the infinite-depth geometry leads to a different nonlocal operator and dispersive structure. We also note that Kharif--Abid--Chen--Hsu \cite{kharif2025nonlinear} derived amplitude equations predicting finite-depth capillary solitary waves with vorticity by formal asymptotics and matching arguments.

\subsection{Organization of the paper}
The rest of the paper is organized as follows. In Section~\ref{sec:formulation} we introduce the traveling-wave
reduction and the Hamiltonian/spatial-dynamics formulation. In Section~\ref{sec:center} we carry out the center-manifold
reduction and derive the reduced canonical system. In Section~\ref{sec:normalform} we compute the normal form and prove
the existence of a homoclinic orbit, and we lift the construction to a solitary wave of the full Euler system.

\subsection*{Acknowledgement}
(i)~T.-Y. Hsiao is supported by the European Union  ERC CONSOLIDATOR GRANT 2023 GUnDHam, Project Number: 101124921. T.-Y. Hsiao would like to express sincere gratitude to Vera Hur, Zhao Yang, and Alberto Maspero for their invaluable guidance and continuous encouragement. Special thanks are also due to Chongchun Zeng and Erik Wahlén for their insightful advice and generous help during the completion of this project; (ii)~Z. Liang was supported in part by NSF grant DMS-2153992 through his advisor during winter 2025 semester when part of this work was completed. Z. Liang would like to thank his advisor Sijue Wu for her continuous support, and would like to thank Zach Deiman, Noah Stevenson, and Yuchuan Yang for helpful discussions;  (iii)~G. To was supported by the Swedish Research Council (grant no. 2020-00440); (iv)~Y. Zhang is supported by  the European Union  ERC STARTING GRANT 2020 GeoSub, Project Number: 945655.

\label{thm:main}
Let $c \neq 0$.
Suppose ${\omega d}/{c}=1+\varepsilon$ for a sufficiently small absolute constant $\varepsilon > 0$ and ${\sigma}/{(c^{2}d)}>\frac13$, the system
\begin{equation}\label{main second}
\begin{aligned}
&\phi_{xx}+\phi_{yy}=0 && \text{in } -d<y<\eta(t,x),\\
&\phi_y=0 && \text{on } y=-d,\\
&\eta_t=\phi_y-(\phi_x-\omega \eta)\eta_x && \text{on } y=\eta(t,x),\\
&\phi_t+\frac12(\phi_x^2+\phi_y^2)-\omega\eta\,\phi_x+\omega\psi
-\frac{\sigma\,\eta_{xx}}{(1+\eta_x^2)^{3/2}}=0 && \text{on } y=\eta(t,x).
\end{aligned}
\end{equation}
admits a $C^2$ solitary wave solution with velocity $c$. Moreover, the solitary wave profile $\eta$ has the asymptotic expansion
\begin{equation}
\eta(\xi)=d\, \varepsilon\, Q\bigg(\varepsilon^{\frac{1}{2}}\bigg(\frac{\sigma}{c^2d}-\frac13\bigg)^{-\frac{1}{2}}\frac{\xi}{d}\bigg)+O(\varepsilon^2)
\end{equation}
where $Q(\bar x)=-3\mathrm{sech}^2(\bar x/2)$ solves the steady the KdV equation
\begin{equation}
Q''=Q+\frac{1}{2}Q^2.
\end{equation}

Question:
Consider the two-dimensional pure-capillary water-wave system with constant vorticity in finite depth, where the fluid occupies the region \(-d<y<\eta(t,x)\), the free surface is \(y=\eta(t,x)\), \(\phi\) is the velocity potential, and \(\psi\) is the harmonic conjugate of \(\phi\). Assume \(c\neq 0\),
\[
\frac{\omega d}{c}=1+\varepsilon
\]
for a sufficiently small absolute constant \(\varepsilon>0\), and
\[
\frac{\sigma}{c^2d}>\frac13.
\]
Suppose \((\phi,\psi,\eta)\) satisfies
\[
\begin{aligned}
&\phi_{xx}+\phi_{yy}=0 && \text{in } -d<y<\eta(t,x),\\
&\phi_y=0 && \text{on } y=-d,\\
&\eta_t=\phi_y-(\phi_x-\omega \eta)\eta_x && \text{on } y=\eta(t,x),\\
&\phi_t+\frac12(\phi_x^2+\phi_y^2)-\omega\eta\,\phi_x+\omega\psi
-\frac{\sigma\,\eta_{xx}}{(1+\eta_x^2)^{3/2}}=0 && \text{on } y=\eta(t,x).
\end{aligned}
\]
A solitary wave with velocity \(c\) means a traveling solution of the form
\[
(\phi(t,x,y),\psi(t,x,y),\eta(t,x))=(\phi(\xi,y),\psi(\xi,y),\eta(\xi)),\qquad \xi=x-ct,
\]
with localized surface profile. Under these assumptions, which statement about the existence and leading-order profile of such a wave is valid?

The system admits a \(C^2\) solitary wave solution traveling with velocity \(c\). Moreover, its surface profile satisfies
\[
\eta(\xi)=d\,\varepsilon\,Q\!\left(\varepsilon^{1/2}\left(\frac{\sigma}{c^2d}-\frac13\right)^{-1/2}\frac{\xi}{d}\right)+O(\varepsilon^2),
\]
where
\[
Q(\bar x)=-3\,\operatorname{sech}^2(\bar x/2)
\]
and \(Q\) solves the steady KdV equation
\[
Q''=Q+\frac12 Q^2.
\]

Choices:
B.
The system admits a \(C^2\) solitary wave solution traveling with velocity \(c\). Moreover, its surface profile satisfies
\[
\eta(\xi)=d\,\varepsilon\,Q\!\left(\varepsilon^{1/2}\left(\frac{\sigma}{c^2d}-\frac13\right)^{-1/2}\frac{\xi}{d}\right)+O(\varepsilon^2),
\]
where
\[
Q(\bar x)=3\,\operatorname{sech}^2(\bar x/2)
\]
and \(Q\) solves the steady KdV equation
\[
Q''=Q+\frac12 Q^2.
\]
C.
The system admits a solitary wave solution traveling with velocity \(c\), and its surface profile has leading-order behavior
\[
\eta(\xi)=d\,\varepsilon\,Q\!\left(\varepsilon^{1/2}\left(\frac{\sigma}{c^2d}-\frac13\right)^{-1/2}\frac{\xi}{d}\right)+o(\varepsilon),
\]
for some localized profile \(Q\) solving
\[
Q''=Q+\frac12 Q^2.
\]
D.
The system admits a \(C^2\) solitary wave solution traveling with velocity \(c\) for every \(\sigma/(c^2d)>0\). Moreover, its surface profile satisfies
\[
\eta(\xi)=d\,\varepsilon\,Q\!\left(\varepsilon^{1/2}\left(\frac{\sigma}{c^2d}-\frac13\right)^{-1/2}\frac{\xi}{d}\right)+O(\varepsilon^2),
\]
where
\[
Q(\bar x)=-3\,\operatorname{sech}^2(\bar x/2)
\]
and \(Q\) solves the steady KdV equation
\[
Q''=Q+\frac12 Q^2.
\]
E.
For each sufficiently small \(\varepsilon>0\), there exists an absolute constant \(C>0\), independent of \(\sigma\), \(c\), and \(d\), such that whenever \(\omega d/c=1+\varepsilon\) and \(\sigma/(c^2d)>1/3\), the system admits a \(C^2\) solitary wave solution traveling with velocity \(c\), and its surface profile satisfies
\[
\eta(\xi)=d\,\varepsilon\,Q\!\left(\varepsilon^{1/2}\left(\frac{\sigma}{c^2d}-\frac13\right)^{-1/2}\frac{\xi}{d}\right)+R(\xi),
\qquad \|R\|_{L^\infty}\le C\varepsilon^2,
\]
where
\[
Q(\bar x)=-3\,\operatorname{sech}^2(\bar x/2)
\]
and \(Q\) solves
\[
Q''=Q+\frac12 Q^2.
\]

drops explicit formula for the unique leading soliton and weakens the remainder from O(\varepsilon^2) to o(\varepsilon)

non-uniform error bound promoted to a parameter-independent effective estimate


\label{thm:main1}
		Let $n,s,k$ be positive integers, $n \geq sk$, $k \geq 1$.
		Let $\mathcal{F} \subseteq \binom{[n]}{k}$ be a family with no $s$ pairwise disjoint sets. Then, $|\mathcal{F}| \leq \max\left\{\binom{n}{k}-\binom{n-s+1}{k}, \binom{sk-1}{k}\right\}$.

A central problem in extremal combinatorics is to determine the maximum possible size of a set family subject to a given constraint. We consider families $\mathcal{F} \subseteq 2^{[n]}$ on the ground set $[n] = \{1, \dots, n\}$. A \emph{matching} in $\mathcal{F}$ is a collection of pairwise disjoint members of $\mathcal{F}$. The \emph{matching number} of $\mathcal{F}$, denoted $\nu(\mathcal{F})$, is the cardinality of a maximum matching.

	This paper focuses on $k$-uniform families, $\mathcal{F} \subseteq \binom{[n]}{k}$, that are $s$-matching-free, i.e., $\nu(\mathcal{F}) \le s-1$. We denote the maximum possible size of such a family by $f(n,k,s)$. In 1965, Erd\H{o}s \cite{erdos1965} posed a conjecture for the exact value of $f(n,k,s)$, which has become a celebrated open problem in the field.

	\begin{conjecture}[Erd\H{o}s Matching Conjecture \cite{erdos1971}] \label{emc}
		For $n \ge sk$,
		\[f(n,k,s) = \max\left\{\binom{sk-1}{k}, \binom{n}{k}-\binom{n-s+1}{k}\right\}.\]
	\end{conjecture}

	The two expressions in the maximum correspond to two canonical extremal families.
	\begin{enumerate}
		\item The family $\mathcal{G}^* = \binom{[sk-1]}{k}$, which consists of all $k$-subsets of a fixed set of $sk-1$ elements.
		\item The family $\mathcal{F}^* = \{F \in \binom{[n]}{k} : F \cap S \neq \emptyset\}$ for a fixed set $S \subset [n]$ with $|S| = s-1$. This family consists of all $k$-subsets that intersect a fixed set of $s-1$ elements.
	\end{enumerate}
	The conjecture is trivially true for $n < sk$, as in this case $\nu(\mathcal{F}) \le s-1$ holds for the entire family $\binom{[n]}{k}$.
	The conjecture has a rich history and has been verified in several important cases. For $s=2$, the problem is to find the largest intersecting family, and the conjecture follows from the Erd\H{o}s-Ko-Rado theorem \cite{erdos1961}. Kleitman \cite{kleitman1968} proved the conjecture for the boundary case $n=sk$.

	Significant progress has been made on the conjecture for sufficiently large $n$. Erd\H{o}s \cite{erdos1965} first established the conjecture for $n \ge n_0(k,s)$. This bound was subsequently sharpened by Bollob\'{a}s, Daykin, and Erd\H{o}s \cite{bollobas1976} to $n \ge 2sk^3$, and later by Huang, Lo, and Sudakov \cite{huang2012} to $n \ge 3sk^2$.

	Other results have affirmed the conjecture for specific parameters or ranges. Frankl \cite{frankl1987} proved the conjecture for $n \geq (2s+1)k-s$ and also provided the general upper bound $f(n,k,s) \leq (s-1)\binom{n-1}{k-1}$. Frankl, R{\"{o}}dl, and Ruci{\'{n}}ski \cite{frankl2012} established the case $k=3$ for $n \geq 4s$. More recently, Kolupaev and Kupavskii \cite{kupavskii2023} proved the conjecture for $k \geq 5$, $s > 100k^3$, and $sk \leq n < s(k+100/k)$. For a comprehensive survey of recent results, see \cite{frankl2022}.

	Despite this extensive body of work, the conjecture has remained open in its entirety. In this paper, we provide a complete proof.

	\subsection{Our results}
	We establish the conjecture for all $n \ge sk$. Our proof is algorithmic and relies on a key distinction between trivial and non-trivial families.

	\begin{definition}
		A family is said to be \emph{trivial} if there exists at least one element $x \in [n]$ that is not contained in any $F \in \mathcal{F}$. A family that is not trivial is \emph{non-trivial}.
	\end{definition}

	By analyzing the behavior of families under shift operations, and paying close attention to whether intermediate families in our algorithm become trivial, we establish the following theorem in Section \ref{sec:4}.

A central problem in extremal combinatorics is to determine the maximum possible size of a set family subject to a given constraint. We consider families $\mathcal{F} \subseteq 2^{[n]}$ on the ground set $[n] = \{1, \dots, n\}$. A \emph{matching} in $\mathcal{F}$ is a collection of pairwise disjoint members of $\mathcal{F}$. The \emph{matching number} of $\mathcal{F}$, denoted $\nu(\mathcal{F})$, is the cardinality of a maximum matching.

This paper focuses on $k$-uniform families, $\mathcal{F} \subseteq \binom{[n]}{k}$, that are $s$-matching-free, i.e., $\nu(\mathcal{F}) \le s-1$. We denote the maximum possible size of such a family by $f(n,k,s)$. In 1965, Erd\H{o}s \cite{erdos1965} posed a conjecture for the exact value of $f(n,k,s)$, which has become a celebrated open problem in the field.

\begin{conjecture}[Erd\H{o}s Matching Conjecture \cite{erdos1971}] \label{emc}
 For $n \ge sk$,
 \[f(n,k,s) = \max\left\{\binom{sk-1}{k}, \binom{n}{k}-\binom{n-s+1}{k}\right\}.\]
 \end{conjecture}

The two expressions in the maximum correspond to two canonical extremal families.
 \begin{enumerate}
 \item The family $\mathcal{G}^* = \binom{[sk-1]}{k}$, which consists of all $k$-subsets of a fixed set of $sk-1$ elements.
 \item The family $\mathcal{F}^* = \{F \in \binom{[n]}{k} : F \cap S \neq \emptyset\}$ for a fixed set $S \subset [n]$ with $|S| = s-1$. This family consists of all $k$-subsets that intersect a fixed set of $s-1$ elements.
 \end{enumerate}
 The conjecture is trivially true for $n < sk$, as in this case $\nu(\mathcal{F}) \le s-1$ holds for the entire family $\binom{[n]}{k}$.
 The conjecture has a rich history and has been verified in several important cases. For $s=2$, the problem is to find the largest intersecting family, and the conjecture follows from the Erd\H{o}s-Ko-Rado theorem \cite{erdos1961}. Kleitman \cite{kleitman1968} proved the conjecture for the boundary case $n=sk$.

\begin{definition}
 A family is said to be \emph{trivial} if there exists at least one element $x \in [n]$ that is not contained in any $F \in \mathcal{F}$. A family that is not trivial is \emph{non-trivial}.
 \end{definition}

By analyzing the behavior of families under shift operations, and paying close attention to whether intermediate families in our algorithm become trivial, we establish the following theorem in Section \ref{sec:4}.

\newtheorem*{statement1}{Statement of Theorem 1}
\newtheorem*{statement2}{Statement of Lemma 2}
\begin{document}
 \title{Erd\H{o}s Matching (Conjecture) Theorem}
 \author{Tapas Kumar Mishra \footnote{Orcid: 0000-0002-9825-3828} \\
 Department of Computer Science and Engineering,\\
 National Institute of Technology, Rourkela
 769008, India\\
 mishrat@nitrkl.ac.in\\
 https://mishra-tapas.github.io/}
 \maketitle \begin{abstract}
 Let $\mathcal{F}$ be a family of $k$-sized subsets of $[n]$ that does not contain $s$ pairwise disjoint subsets.
 The Erd\H{o}s Matching Conjecture, a celebrated and long-standing open problem in extremal combinatorics, asserts the maximum cardinality of $\mathcal{F}$ is upper bounded by
 $\max\left\{\binom{sk-1}{k}, \binom{n}{k}-\allowbreak \binom{n-s+1}{k}\right\}$.
 These two bounds correspond to the sizes of two canonical extremal families: one in which all subsets are contained within a ground set of $sk-1$ elements, and one in which every subset intersects a fixed set of $s-1$ elements.
 In this paper, we prove the conjecture.

\begin{conjecture}[Erd\H{o}s Matching Conjecture \cite{erdos1971}] \label{emc}
 For $n \ge sk$,
 \[f(n,k,s) = \max\left\{\binom{sk-1}{k}, \binom{n}{k}-\binom{n-s+1}{k}\right\}.\]
 \end{conjecture}

By analyzing the behavior of families under shift operations, and paying close attention to whether intermediate families in our algorithm become trivial, we establish the following theorem in Section \ref{sec:4}.

The key technical ingredients in our proof are the classical $(i,j)$ shift operator of Fr\'{a}nkl. The precise effects of these operators on a family's matching number and its potential for triviality are detailed in Lemma \ref{lemma:1}, \ref{lemma:2}. This algorithmic framework, which tracks the properties of families through a sequence of shifts, can be extended to the non-uniform case as well.
 \section{Shifting operator and supporting lemmas}
 \label{sec:2}
 \begin{definition}[Fr\'{a}nkl's $(i,j)$ shift]
 Let $\mathcal{F} \subseteq 2^{[n]}$. For $1 \leq i,j \leq n$, define $\mathcal{C}_{ij}(\mathcal{F})= \{\mathcal{C}_{ij}(F): F \in \mathcal{F}\}$, where
 \begin{align*}
 \mathcal{C}_{ij}(F) = \begin{cases}
 (F \setminus \{j\})\cup \{i\} , \text{ if $i \not \in F$, $j \in F$, and $(F \setminus \{j\})\cup \{i\} \not\in \mathcal{F}$},\\
 F, \text{ otherwise.}
 \end{cases}
 \end{align*}
 \end{definition}
 We have two lemmas on the effects of a $(i,j)$ shift on a family $\mathcal{F}$.

\begin{lemma} \label{lemma:1}
 Let $\mathcal{F}$ be a family of subsets of $[n]$ and let $1 \leq i, j \leq n$. Then the following hold:
 \begin{enumerate}[label=(\roman*)]
 \item For any $F \in \mathcal{F}$, $|\mathcal{C}_{ij}(F)| = |F|$.
 \item $|\mathcal{C}_{ij}(\mathcal{F})| = |\mathcal{F}|$.
 \item The matching number does not increase, i.e., $\nu(\mathcal{C}_{ij}(\mathcal{F})) \leq \nu(\mathcal{F})$.
 \end{enumerate}
 \end{lemma}
 \begin{proof}
 The proof of (i) and (ii) are given in \cite{frankl1987}. As (iii) is probably folklore, the proof is given in the Appendix \ref{app:1}.
 \end{proof}

\section{Proof of Theorem \ref{thm:main1}}
 \label{sec:4}
For a family $\mathcal{F}$ and an element $x \in [n]$, we denote the subfamily of sets containing $x$ as $\mathcal{F}^x = \{F \in \mathcal{F} : x \in F\}$.
 \begin{statement1}
 Let $n,s,k$ be positive integers, $n \geq sk$, $k \geq 1$.
 Let $\mathcal{F} \subseteq \binom{[n]}{k}$ be a family with no $s$ pairwise disjoint sets. Then, $|\mathcal{F}| \leq \max\left\{\binom{n}{k}-\binom{n-s+1}{k}, \binom{sk-1}{k}\right\}$.
 \end{statement1}
 \begin{proof}
 The family $\mathcal{F}^* = \{F \in \binom{[n]}{k} : F \cap \{1, \dots, s-1\} \neq \emptyset\}$ satisfies the condition $\nu(\mathcal{F}^*) \le s-1$ and has size exactly $\binom{n}{k} - \binom{n-s+1}{k}$.
 Let $\mathcal{G}^*=\binom{[sk-1]}{k}$.
 We will show that no family satisfying the hypothesis can be larger in cardinality than both $\mathcal{F}^*$ and $\mathcal{G}^*$. The proof uses an iterative algorithm that transforms any valid family $\mathcal{F}$ into one where the desired structure is evident.

\begin{enumerate}
 \item If $\mathcal{F}$ is trivial, let $Y \subseteq [n']$ be the largest set of vertices with no element of $Y$ being a member of any set in $\mathcal{F}$ (i.e. $\forall y \in Y \forall F \in \mathcal{F}$, $y \not\in F$). Set $\mathcal{F}$ to be the non-trivial family on $[n']\setminus Y$ and with a relabeling (that does not change $S$), assume the ground set is exactly $[n'-|Y|]$. Set $n'=n'-|Y|$. Note that this transformation preserves $|\mathcal{F}|$ and $\nu(\mathcal{F})$.
 If $n' \leq sk-1$, $\mathcal{F} \subseteq \mathcal{G}^*$ and we terminate the algorithm.
 Otherwise, set $\mathcal{F}^* = \{F \in \binom{[n']}{k} : F \cap \{1, \dots, s-1\} \neq \emptyset\}$.
 \item If the current family $\mathcal{F}$ is not a subfamily of $\mathcal{F}^*$, then there must exist a set $A \in \mathcal{F}$ such that $A \cap S = \emptyset$.
 \item There must also exist a set $B \notin \mathcal{F}$ such that $B \cap S \neq \emptyset$. If no such $B$ exists, the algorithm terminates and proceeds to the ``Contradiction'' phase described in Section III.
 \item Let $X = A \cap B$. Define $A' = A \setminus X = \{a_1, \dots, a_r\}$ and $B' = B \setminus X = \{b_1, \dots, b_r\}$. By construction, all elements $a \in A'$ are not in $S$. We can order the elements in $B'$ such that $b_1 \in S$. The remaining elements in $B'$ and all elements in $A'$ are ordered arbitrarily.

The algorithm terminates under three conditions.\\
 \textbf{Condition 1: $n' \leq sk-1$}. In this case, $\mathcal{F} \subseteq \mathcal{G}^*$ and $|\mathcal{F}| \leq \binom{sk-1}{k}$.\\
 \textbf{Condition 2: the current family $\mathcal{F}$ is a subfamily of $\mathcal{F}^*$}. In this case, $|\mathcal{F}| \leq \binom{n'}{k}-\binom{n'-s+1}{k} \leq \binom{n}{k}-\binom{n-s+1}{k}$ and the bound follows.\\
 \textbf{Condition 3: if the algorithm reaches a state where it can find a set $A \in \mathcal{F}$ with $A \cap S = \emptyset$, but it cannot find a set $B \notin \mathcal{F}$ with $B \cap S \neq \emptyset$}. This implies that the current family $\mathcal{F}$ contains \textbf{all possible $k$-sets that intersect $S$}.
 We can now derive a contradiction. Let $A$ be the set in $\mathcal{F}$ that is disjoint from $S$. We can construct $s-1$ new sets, $B_1, \dots, B_{s-1}$, that are pairwise disjoint and also disjoint from $A$.
 \begin{itemize}
 \item Choose a set $X$ of size $(s-1)k$ from $[n']\setminus A$
 such that $\{1,\ldots,s-1\} \subset X$.
 Partition $X$ into $s-1$ sets $B_1,\ldots,B_{s-1}$
 with $B_i \cap \{1,\ldots,s-1\}=\{i\}$.
 \end{itemize}
 This construction is possible because the condition $n' \ge sk$ ensures there are enough elements available. Since each $B_i$ intersects $S$, and our family $\mathcal{F}$ contains all such sets, it follows that $\{B_1, \dots, B_{s-1}\} \subset \mathcal{F}$.
 The collection \textbf{$\{A, B_1, \dots, B_{s-1}\}$} is therefore a matching of size $s$ within $\mathcal{F}$. This contradicts the hypothesis that $\nu(\mathcal{F}) \le s-1$.

\begin{lemma} \label{lemma:1}
		Let $\mathcal{F}$ be a family of subsets of $[n]$ and let $1 \leq i, j \leq n$. Then the following hold:
		\begin{enumerate}[label=(\roman*)]
			\item For any $F \in \mathcal{F}$, $|\mathcal{C}_{ij}(F)| = |F|$.
			\item $|\mathcal{C}_{ij}(\mathcal{F})| = |\mathcal{F}|$.
			\item The matching number does not increase, i.e., $\nu(\mathcal{C}_{ij}(\mathcal{F})) \leq \nu(\mathcal{F})$.
		\end{enumerate}
	\end{lemma}

\begin{lemma}\label{lemma:2}
		Let $\mathcal{F}$ be a trivial family on the ground set $[n]$. Then for any $1 \leq i, j \leq n$, $i\neq j$, the family $\mathcal{C}_{ij}(\mathcal{F})$ is also a trivial family.
	\end{lemma}

The proof of Theorem~\ref{thm:main1} uses as key technical ingredients the classical $(i,j)$ shift operator of Fr\'{a}nkl. The effects of these operators on a family’s matching number and its potential for triviality are detailed in Lemma~\ref{lemma:1} and Lemma~\ref{lemma:2}. The argument is presented in an algorithmic framework that “tracks the properties of families through a sequence of shifts,” and this shifting framework can also be extended to the non-uniform case.

Question: Let $n,s,k$ be positive integers with $n\ge sk$ and $k\ge 1$. Write $[n]=\{1,2,\dots,n\}$, and let $\binom{[n]}{k}$ denote the family of all $k$-element subsets of $[n]$. Suppose $\mathcal F\subseteq \binom{[n]}{k}$ is a $k$-uniform family that contains no $s$ pairwise disjoint sets. Which of the following quantitative estimates holds for $|\mathcal F|$?

$|\mathcal F|\le \max\left\{\binom{n}{k}-\binom{n-s+1}{k},\;\binom{sk-1}{k}\right\}$.

Choices:
B. $|\mathcal F|\le \max\left\{\binom{n}{k}-\binom{n-s}{k},\;\binom{sk-1}{k}\right\}$.
C. $|\mathcal F|\le \binom{n}{k}-\binom{n-s+1}{k}+\binom{sk-1}{k}$.
D. $|\mathcal F|\le \min\left\{\binom{n}{k}-\binom{n-s+1}{k},\;\binom{sk-1}{k}\right\}$.
E. $|\mathcal F|\le \max\left\{\binom{n}{k}-\binom{n-s+1}{k},\;\binom{sk}{k}\right\}$.

extremal-count threshold in the intersecting-with-fixed-$(s-1)$-set term

replaced the sharp maximum of the two extremal bounds by their sum

aggregation of the two candidate extremal bounds

boundary case $sk-1$ replaced by $sk$ in the complete-support extremal term


Question: Let $n,s,k$ be positive integers with $n\ge sk$ and $k\ge 1$. Write $[n]=\{1,2,\dots,n\}$, and let $\binom{[n]}{k}$ denote the family of all $k$-element subsets of $[n]$. Suppose $\mathcal F\subseteq \binom{[n]}{k}$ is a $k$-uniform family that contains no $s$ pairwise disjoint sets. Which of the following conclusions about $|\mathcal F|$ holds under these hypotheses?

$|\mathcal F|\le \max\left\{\binom{n}{k}-\binom{n-s+1}{k},\,\binom{sk-1}{k}\right\}$.

Choices:
B. $|\mathcal F|\le \max\left\{\binom{n}{k}-\binom{n-s}{k},\,\binom{sk-1}{k}\right\}$.
C. $|\mathcal F|\le \binom{n}{k}-\binom{n-s+1}{k}+\binom{sk-1}{k}$.
D. $|\mathcal F|\le \min\left\{\binom{n}{k}-\binom{n-s+1}{k},\,\binom{sk-1}{k}\right\}$.
E. $|\mathcal F|\le \max\left\{\binom{n}{k}-\binom{n-s+1}{k},\,\binom{sk}{k}\right\}$.

outer max structure replaced by a larger sum

selection of extremal branch between the two canonical bounds

critical support size $sk-1$ in the compact extremal family


\begin{thm}\label{theorem:main}
    Let $f$ be a cuspidal modular form of weight $k$ on $\SL_2(\BZ)$. For any integers $l\geq2$ and $d\geq1$ and $dl>4$, one has
    \[\sum_{n\leq x}  \lambda^l_{\Sym^d f}(n)=xP_{d,l}(\log x)+O\left(x^{\theta_{d,l}+\epsilon}\right).\]
    Here $P_{d,l}$ is a polynomial of degree $K_{0,d,l}-1$, where $K_{i,d,l}$ denote the Kostka number defined in sections \ref{section:Weylmod} and \ref{section:Kostka number}. The exponent $\theta_{d, l}$ is given by
    $$\theta_{d,l}=1-1/\left((d+1)^l/2-4K_{0,d,l}/21-K_{1,d,l}/3-5K_{2,d,l}/14\right).$$
\end{thm}

\begin{prop}\label{proposition:dirichletseriescomp}
The following identities hold:
\begin{align*}
   &D\left(f^{\otimes l},s\right)=L\left(f^{\otimes l},s\right)U_{f,l}(s),D\left(\left(\Sym^d f\right)^{\otimes l},s\right)=L\left(\left(\Sym^df\right)^{\otimes l},s\right)U_{\Sym^d f,l}(s),\\
   &D\left(f^{\otimes l},\chi,s\right)=L\left(f^{\otimes l},\chi,s\right)U_{f,l,\chi}(s),D\left(\left(\Sym^d f\right)^{\otimes l},\chi,s\right)=L\left(\left(\Sym^df\right)^{\otimes l},\chi,s\right)U_{\Sym^d f,l,\chi}(s).
\end{align*}
       where each  $U_*(s)$ is an Euler product that is absolutely convergent for  $\Re(s)>1/2$ and uniformly convergent in the region $\Re(s)>1/2+\epsilon$ for any $\epsilon>0$.
\end{prop}

Let $f$ be a cuspidal eigenform of weight $k$ on $\SL_2(\BZ)$, and let
\[f(z)=\sum_{n=1}^\infty \lambda_f(n)n^{\frac{k-1}{2}}q^n\]
be its $q$-expansion. Estimates of $\lambda_f(n)$ and their average behavior are fundamental problems in number theory.
 Estimating the sum
$\sum\limits_{n\leq x}\lambda_f(n),$ has a long history, with contributions from
 \cite{kloosterman1927asymptotische}, \cite{rankin1983sums}, \cite{weil1948some}, \cite{wu2009power}, among others. The sum of higher power moments
$\sum\limits_{n\leq x}\lambda_f^l(n)$
has also attracted considerable attention. If $l=2$, Rankin\cite{rankin1939contributions} and Selberg\cite{selberg1940bemerkungen} already studied this problem and recently Huang's work \cite{huang2021rankin} provides improved estimates with
\[\sum_{n\leq x}\lambda_f^2(n)=Cx+O\left(x^{\frac{3}{5}-\frac{1}{560}+\epsilon}\right).\]

For $l\geq3$, symmetric power $L$-functions of modular forms are required. Kim and Kim \& Shahidi \cite{kim2003functoriality}, \cite{kim2002cuspidality},\cite{kim2002functorial} showed that $\Sym^l f$ is automorphic for $l\leq 4$. Using these results, the estimates for $l\leq 8$ are derived in \cite{lau2011integral}, \cite{lu2009average} and \cite{lu2011higher}. Later Zhai\cite{zhai2013average} considered the sum over sums of two squares
\[\sum_{n_1^2+n_2^2\leq x}\lambda_f^l(n_1^2+n_2^2),\]
and provided estimates for $l\leq 8$.

Recently, Newton \& Thorne \cite{newton2021symmetric1},\cite{newton2021symmetric2} proved the automorphy and cuspidality of $\Sym^l f$ for all $l\geq1$. Using this celebrated work, Xu\cite{xu2022general} derived a general formula for $\sum\limits_{n\leq x}\lambda_f^l(n)$ and $\sum\limits_{n_1^2+n_2^2\leq x}\lambda_f^l(n_1^2+n_2^2)$ for every $l\geq1$. This result was later refined by Liu\cite{liu2023asymptotic} as
\begin{equation}\label{equation:introduction1}
 \sum_{n\leq x} \lambda^l_f(n)=xQ_l(\log x)+O\left(x^{\theta_l+\epsilon}\right),
\end{equation}
\begin{equation}\label{equation:introduction2}
 \sum_{n_1^2+n_2^2\leq x}\lambda_f^l(n_1^2+n_2^2)=xQ'_l(\log x)+O\left(x^{{\theta}'_l+\epsilon}\right),
\end{equation}
where $Q_l$ and $Q'_l$ are polynomials of degree $\binom{l}{m}-\binom{l}{m-1}-1$ if $l=2m$ and equals 0 if $l$ is odd. Here $\theta_l, {\theta}'_l$ are explicit constants less than 1.

Another direction of research concerns the average behavior of Fourier coefficients of symmetric powers of $f$. Several results have been obtained in this direction, but they only focus on some special cases. For example, in \cite{fomenko2006identities} and \cite{fomenko2008mean}, Fomenko proved asymptotic formulas for
\[\sum_{n\leq x}\lambda_{\Sym^2 f}(n)\ \text{and}\ \sum_{n\leq x}\lambda^2_{\Sym^2 f}(n).\]
In \cite{he2019integral} and \cite{luo2021asymptotics}, the summations \[\sum_{n\leq x}\lambda^l_{\Sym^2 f}(n)\ \text{and}\ \sum_{n\leq x}\lambda^2_{\Sym^d f}(n)\]
with $d\geq 2$ and $l\leq 8$ were studied. Later in \cite{liu2023average}, Liu obtained the following results with better error terms:
\begin{equation} \label{equation:introduction3}
 \sum_{n\leq x}\lambda^l_{\Sym^2 f}(n)=xP_l(\log x)+O\left(x^{\theta_l+\epsilon}\right)
\end{equation}
for $2\leq l\leq 8$, and
\begin{equation}\label{equation:introduction4}
 \sum_{n\leq x}\lambda^2_{\Sym^d f}(n)=c_dx+O\left(x^{\tilde{\theta}_d+\epsilon}\right),
\end{equation}
for any $d\geq 2$.
Here $P_l$ are polynomials with explicitly computed degrees, $c_d$ is a constant and $\theta_l, \tilde{\theta}_d$ are positive constants less than 1.

Extending these results to arbitrary integers $d\geq1$ and $l\geq1$ presents significant challenges, as the associated Dirichlet series become increasingly complex. In this paper, we address this problem by studying the underlying Galois representations and applying combinatorial results concerning their decompositions.
Our work generalizes the average formulas (\ref{equation:introduction1}),(\ref{equation:introduction2}),(\ref{equation:introduction3}),(\ref{equation:introduction4}) in the following aspects.
\begin{itemize}
 \item We obtain asymptotic formulas with improved error terms.
 \item We establish a universal asymptotic formula for $\sum\limits_{n\leq x}\lambda^l_{\Sym^d f}(n)$ valid for any $d\geq 1$ and $l\geq 1$,which generalizes (\ref{equation:introduction3}) and (\ref{equation:introduction4}).
 \item We prove an average formula for sums over values of a binary quadratic form, namely $ \sum\limits_{Q(n_1,n_2)\leq x}\lambda^l_{\Sym^d f}(Q(n_1,n_2))$ for any definite binary quadratic form $Q$ and any $d,l\geq1$.
\end{itemize}
\subsection{Main theorems}
We now state our main results for general $d$ and $l$.

\begin{equation}\label{equation:introduction1}
  \sum_{n\leq x}  \lambda^l_f(n)=xQ_l(\log x)+O\left(x^{\theta_l+\epsilon}\right),
\end{equation}

\begin{equation} \label{equation:introduction3}
    \sum_{n\leq x}\lambda^l_{\Sym^2 f}(n)=xP_l(\log x)+O\left(x^{\theta_l+\epsilon}\right)
\end{equation}

\begin{equation}\label{equation:introduction4}
    \sum_{n\leq x}\lambda^2_{\Sym^d f}(n)=c_dx+O\left(x^{\tilde{\theta}_d+\epsilon}\right),
\end{equation}

The estimation in the general case is new. And our estimates in the error terms are better than those in the work of \cite{liu2023average} for small $d$ and $l$. We list the cases with $d=2,3\leq l\leq8$ and $l=2,3\leq d\leq8$ in the following tables.

We also have the following theorem for summation over binary quadratic forms.
\begin{thm}\label{thm:binary}
 Let $f$ be a cuspidal modular form of weight $k$ on $\SL_2(\BZ)$. For any integers $l\geq2,d\geq1$ with $dl>4$ and any binary quadratic form $Q(n_1,n_2)$, one has
 \[\sum_{Q(n_1,n_2)\leq x} \lambda^l_{\Sym^d f}(Q(n_1,n_2))=xP_{d,l,Q}(\log x)+O\left(x^{\theta_{d,l,Q}+\epsilon}\right).\]
 Here $P_{d,l,Q}$ is a polynomial of degree $K_{0,d,l}-1$. If the class number of $Q$ is not 1, then
 $$\theta_{d,l,Q}=1-\frac{3}{3(d+1)^l-K_{0,d,l}};$$ if the class number is 1, then
 $$\theta_{d,l,Q}=1-1/\left((d+1)^l-8K_{0,d,l}/21-2K_{1,d,l}/3-5K_{2,d,l}/7\right).$$
\end{thm}

We now provide an alternative explicit description of these Kostka numbers. Let $K_{i,d,l}=K_{\mu,\lambda}$ (as defined above). By Proposition \ref{prop:decomposition}, $K_{i,d,l}$ is the multiplicity of $\Sym^i V$ in $\left(\Sym^d V\right)^{\otimes l}$ for a two dimensional representation $V$. We extend the definition of $K_{i,d,l}$ to all $i\in\BZ$ by setting $K_{i,d,l}=0$ for $i\notin [0,dl]$, or when $i$ and $dl$ have different parity. From Proposition \ref{prop:decomposition} (4), we derive a recursive formula for $K_{i,d,l}$:
\[K_{i,d,l}=K_{i-d,d,l-1}+K_{i-d+2,d,l-1}+\cdots+K_{i+d,d,l-1}.\]
On the other hand, consider the polynomial:
\[(1+x+x^2+\cdots+x^d)^{l}=\sum_{j=-\infty}^{+\infty}C_{j,d,l}x^j.\]
Define $A_{i,d,l}=C_{\lfloor\frac{dl-i}{2}\rfloor,d,l}-C_{\lfloor\frac{dl-i-1}{2}\rfloor,d,l}.$
\begin{prop} We have
 $ K_{i,d,l}=A_{i,d,l}.$
 \end{prop}
\begin{proof}
We only need to show that $A_{i,d,l}$ and $K_{i,d,l}$ satisfy the same recursive formula and boundary conditions.
 First, the coefficients $C_{i,d,l}$ satisfy the recursion:
 \[C_{i,d,l}=C_{i-d,d,l-1}+C_{i-d+1,d,l-1}+\cdots+C_{i,d,l-1}.\]
 Thus
 \begin{align*}
 A_{i,d,l}=&C_{\lfloor\frac{dl-i}{2}\rfloor,d,l}-C_{\lfloor\frac{dl-i-1}{2}\rfloor,d,l}\\
 =&\sum_{t=j-d}^jC_{t,d,l-1}-\sum_{t=j'-d}^{j'}C_{t,d,l-1}.
 \end{align*}
 where $j=\lfloor\frac{dl-i}{2}\rfloor, j'=\lfloor\frac{dl-i-1}{2}\rfloor.$
 On the other hand,
 \begin{align*}
 \sum_{t=0}^dA_{i-d+2t,d,l-1}&=\sum_{t=0}^d\left(C_{\lfloor\frac{dl-d-(i-d+2t)}{2}\rfloor,d,l-1}-C_{\lfloor\frac{dl-d-(i-d+2t)-1}{2}\rfloor,d,l-1}\right)\\
 &=\sum_{t=0}^d\left(C_{\lfloor\frac{dl-i-2t)}{2}\rfloor,d,l-1}-C_{\lfloor\frac{dl-i-2t-1}{2}\rfloor,d,l-1}\right)
 \end{align*}
 By telescoping, this sum equals $A_{i,d,l}$.
 Thus
 \[A_{i,d,l}=A_{i-d,d,l-1}+A_{i-d+2,d,l-1}+\cdots+A_{i+d,d,l-1}.\]
 For the boundary condition $l=1$:
 \begin{itemize}
 \item $K_{d,d,1}=1$ and $K_{i,d,1}=0$ for $i\neq d$
 \item $A_{d,d,1}=C_{\lfloor \frac{d-d}{2}\rfloor,d,1}-C_{\lfloor \frac{d-d-1}{2}\rfloor,d,1}=C_{0,d,1}-C_{-1,d,1}=1-0=1$ and $A_{i,d,1}=0$ for $i\neq d$.
 \end{itemize}
 Thus $A_{i,d,d}$ and $K_{i,d,d}$ satisfy the same recursion and boundary conditions, so $A_{i,d,l}=K_{i,d,l}$.
\end{proof}
From this proposition, the Kostka number $K_{i,d,l}$ can be expressed in terms of binomial coefficients. This gives the following formula.
\begin{cor}\label{corollary:Kostka}
 \begin{equation}
 K_{i,d,l}=\begin{cases}
 0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{if $2\nmid dl-i$}\\
 \sum\limits_{j=0}^{\lfloor(dl-i)/(2d+2)\rfloor}(-1)^j\binom{l}{j}\binom
 {(dl-i)/2-j(d+1)+l-2}{l-2}
 \ \text{otherwise}
 \end{cases}.
 \end{equation}
\end{cor}
\begin{proof}
Since
 \[(1+x+x^2+\cdots+x^d)^{l}=(1-x^{d+1})^{l}(1-x)^{-l},\]
 the coefficients $K_{i,d,l}=A_{i,d,l}$ can be computed as
\begin{align*}
 A_{i,d,l}=&\sum_{j=0}^{\lfloor(dl-i)/(2d+2)\rfloor}(-1)^j\binom{l}{j}\binom{\lfloor(dl-i)/2\rfloor-j(d+1)+l-1}{l-1}\\
 &-\sum_{j=0}^{\lfloor(dl-i-1)/(2d+2)\rfloor}(-1)^j\begin{pmatrix}
 l\\j
 \end{pmatrix}\begin{pmatrix}
 \lfloor(dl-i-1)/2\rfloor-j(d+1)+l-1\\l-1
 \end{pmatrix}.
\end{align*}

Now let $f$ be a cuspidal eigenform on $\SL_2(\BZ)$. For any integer $d\geq0$, the symmetric power lifting $\Sym^d f$ is an automorphic form on $\GL_{d+1}$. For a character $\chi$ of $\fh$, we define the associated $L$-function as follows. If $\chi$ is non-trivial,
let
\[L\left(\Sym^d f, \chi,s\right):=\sum_{n=1}^\infty\frac{\lambda_{\Sym^df}(n)a_\chi(n)}{n^s}\]
be the Rankin-Selberg $L$-function associated to $\Sym^d f$ and $f_\chi$. If $\chi$ is trivial, define $$L\left(\Sym^d f, \chi,s\right)=L\left(\Sym^df,s\right)L\left(\Sym^df,\chi_F,s\right).$$ Similarly, we can define the $L$-functions $L\left(f^{\otimes l},\chi,s\right)$ and $L\left(\left(\Sym^d f\right)^{\otimes l},\chi,s\right)$ by incorporating the coefficients $a_\chi(n)$. Leveraging the tensor product decompositions from previous sections, we obtain the following factorizations:
 \begin{equation}
 L\left(f^{\otimes l},\chi,s\right)=\prod_{i=0}^{[l/2]}L\left(\Sym^{l-2i}f,\chi,s\right)^{c_{l,i}},
 \end{equation}
 \begin{equation}\label{equation:chilfactor}
 L\left(\left(\Sym^d f\right)^{\otimes l},\chi,s\right)=\prod_{i=0}^{dl}L\left(\Sym^i f,\chi,s\right)^{K_{i,d,l}}.
 \end{equation}
\section{The estimations via Perron's formula}
\subsection{The Dirichlet series}
Let $f$ be a cuspidal eigenform of weight $k$ on $\SL_2(\BZ)$, let $Q$ be a binary quadratic form and let $\chi$ be a character of $\fh$. We define the following Dirichlet series, which are central to our analysis:
\begin{align*}
&D\left(f^{\otimes l},s\right):=\sum_{n=1}^\infty \frac{\lambda_f^l(n)}{n^s},&D&\left(\left(\Sym^df\right)^{\otimes l},s\right):=\sum_{n=1}^\infty \frac{\lambda_{\Sym^df}^l(n)}{n^s},\\
&D\left(f^{\otimes l},Q,s\right):=\sum_{n=1}^\infty \frac{\lambda_f^l(n)r(n,Q)}{n^s},&D&\left(\left(\Sym^d f\right)^{\otimes l},Q,s\right):=\sum_{n=1}^\infty \frac{\lambda_{\Sym^d f}^l(n)r(n,Q)}{n^s},\\
&D\left(f^{\otimes l},\chi,s\right):=\sum_{n=1}^\infty \frac{\lambda_f^l(n)a_{\chi}(n)}{n^s},&D&\left(\left(\Sym^d f\right)^{\otimes l},\chi,s\right):=\sum_{n=1}^\infty \frac{\lambda_{\Sym^d f}^l(n)a_\chi(n)}{n^s}.
\end{align*}
From the expression for $r(n,Q)$ given in (\ref{equation:binarydecom}), we immediately obtain the relations
\begin{align*}
 D\left(f^{\otimes l},Q,s\right)&=\frac{w_D}{h(D)}\sum_{\chi}\overline{\chi(\fa_Q)}D\left(f^{\otimes l},\chi,s\right),\\
 D\left(\left(\Sym^d f\right)^{\otimes l},Q,s\right)&=\frac{w_D}{h(D)}\sum_{\chi}\overline{\chi(\fa_Q)}D\left(\left(\Sym^d f\right)^{\otimes l},\chi,s\right).
\end{align*}
The connection between these Dirichlet series and the complete $L$-functions is provided by the following proposition, which is an analogue of Theorem \ref{theorem:l-functions} for the twisted case.

The parameter $T$ is now chosen to balance the error terms.
 We first consider the case where the class number of $Q$ is one. In this case, only the trivial character contributes. Choosing
$$T=x^{1/\left((d+1)^l-\frac{8}{21}K_{0,d,l}-\frac{2}{3}K_{1,d,l}-\frac{5}{7}K_{2,d,l}\right)},$$
we obtain the asymptotic formula
\[\sum_{n\leq x}\lambda^l_{\Sym^df}(n)a_\chi(n)=xP_{0,d,l}(\log(x))+O\left(x^{1-1/\left((d+1)^l-\frac{8}{21}K_{0,d,l}-\frac{2}{3}K_{1,d,l}-\frac{5}{7}K_{2,d,l}\right)+\epsilon}\right).\]
Next assume the class number is greater than 1. The main term comes only from the trivial character, but the error terms include contributions from all characters. Choosing
 $$T=x^{\frac{3}{3(d+1)^l-K_{0,d,l}}},$$ we obtain
\[\sum_{n\leq x}\lambda^l_{\Sym^df}(n)a_\chi(n)=xP_{0,d,l}(\log(x))+O\left(x^{1-\frac{3}{3(d+1)^l-K_{0,d,l}}+\epsilon}\right).\]
Summing over all characters $\chi$, and using (\ref{equation:binarydecom}) completes the proof of Theorem \ref{thm:binary}.
\bibliography{bib/ref}
\bibliographystyle{plain}

In Corollary \ref{corollary:Kostka},  we  provide an explicit formula for the Kostka number $K_{i,d,l}$, which determines the degrees of the main terms and the exponents in the error terms. The formula is
\begin{align*}        K_{i,d,l}=\begin{cases}
            0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  \ \ \ \ \ \ \   \ \ \ \ \ \ \ \ \ \ \ \ \text{if $2\nmid dl-i$}\\
            \sum\limits_{j=0}^{\lfloor(dl-i)/(2d+2)\rfloor}(-1)^j\binom{l}{j}\binom
        {(dl-i)/2-j(d+1)+l-2}{l-2}
    \ \text{otherwise}
        \end{cases}.
    \end{align*}
Note that  if $dl$ is odd, then $K_{0,d,l}=0$ and $P_{d,l}$ is 0.

The estimation in the general case is new. And our estimates in the error terms are better than those in the work of \cite{liu2023average} for small $d$ and $l$. We list the cases with $d=2,3\leq l\leq8$ and $l=2,3\leq d\leq8$ in the following tables.

\begin{center}
\begin{tabular}{|c|c|c|}
\hline
$d=2, l=$ &  Exponents in \cite{liu2023average} & $\theta_{d, l}$ \\
\hline
3 & 0.9193... & 0.918287938
 \\
\hline
4 & 0.9737... & 0.973534972
  \\
\hline
5 & 0.99136... & 0.991304348
\\
\hline
6 & 0.99714... & 0.99713291
 \\
\hline
7 & 0.9990558... & 0.999051362
 \\
\hline
8 & 0.9996868... & 0.999685565
  \\
\hline
\end{tabular}
\end{center}

\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
$l=2, d=$ & Exponents in \cite{liu2023average} & $\theta_{d, l}$ \\
\hline
3 & 0.866... & 0.865814696
 \\
\hline
4 & 0.9166... & 0.916334661
  \\
\hline
5 & 0.9428... & 0.942701228
 \\
\hline
6 & 0.9583... & 0.958250497
 \\
\hline
7 & $0.96824\ldots$ &0.968205905
  \\
\hline
8 & 0.97499... & 0.974970203
 \\
\hline
\end{tabular}
\end{center}

We also have the following theorem for summation over binary quadratic forms.
\begin{thm}\label{thm:binary}
      Let $f$ be a cuspidal modular form of weight $k$ on $\SL_2(\BZ)$. For any integers $l\geq2,d\geq1$ with $dl>4$ and any binary quadratic form $Q(n_1,n_2)$, one has
       \[\sum_{Q(n_1,n_2)\leq x}  \lambda^l_{\Sym^d f}(Q(n_1,n_2))=xP_{d,l,Q}(\log x)+O\left(x^{\theta_{d,l,Q}+\epsilon}\right).\]
        Here $P_{d,l,Q}$ is a polynomial of degree $K_{0,d,l}-1$.  If the class number of $Q$ is not 1,  then
        $$\theta_{d,l,Q}=1-\frac{3}{3(d+1)^l-K_{0,d,l}};$$ if the class number is 1, then
        $$\theta_{d,l,Q}=1-1/\left((d+1)^l-8K_{0,d,l}/21-2K_{1,d,l}/3-5K_{2,d,l}/7\right).$$
\end{thm}

\subsection{Proof Sketch }
The proofs of our theorems follow a framework that can be viewed as an $\ell$-adic analogue of the methods for Artin representations developed in \cite{Yang2024second} , \cite{Yang2024first} and \cite{Yang2026higher}.

The starting point is Perron's formula, which translates the estimation of the sum $\sum\limits_{n\leq x}\lambda^l_{\Sym^d f}(n)$ into the problem of evaluating a contour integral of the associated Dirichlet series $\sum\limits_{n=1}^\infty\frac{\lambda^l_{\Sym^d f}(n)}{n^s}$. This Dirichlet series is intimately related via its Euler product to the $L$-function $L\left(\left(\Sym^d f\right)^{\otimes l},s\right)$, which corresponds to  the $\ell$-adic representation $\Sym^d(\rho_f)^{\otimes l}$ (see Proposition \ref{proposition:dirichletseriescomp}).

A key insight is the decomposition of this high-dimensional tensor representation into a direct sum of irreducible constituents, which are shown to be of the form $\Sym^i \rho_f$. Consequently, the $L$-function factors into a product of symmetric power $L$-functions:
$$L\left(\left(\Sym^d f\right)^{\otimes l},s\right)=\prod_{i=0}^{ld}L\left(\Sym^i f,s\right)^{K_{i,d,l}}$$
The automorphy and entirety of these symmetric power $L$-functions, established by Newton and Thorne \cite{newton2021symmetric1},\cite{newton2021symmetric2}, are crucial here. The final asymptotic formulas are then derived by combining this factorization with established subconvexity bounds for the involved $L$-functions.

For sums over values of a binary quadratic form $Q$, the approach is similar but involves additional ingredients. The generating Dirichlet series becomes $\sum\limits_{n=1}^\infty\frac{\lambda^l_{\Sym^d f}(n)r(n,Q)}{n^s}$, where $r(n,Q)$ counts representations of $n$ by $Q$. Let $F=\BQ(\sqrt{D})$ be the corresponding imaginary quadratic field. When the class number $h(D)=1$, the series $\sum\limits_{n=1}^\infty \frac{r(n,Q)}{n^s}$ is essentially the Dedekind zeta function $\zeta_F(s)$.  In general, it can be expressed as a finite sum of Hecke $L$-functions associated to characters of the class group of $F$. Each such $L$-function corresponds to a modular form $f_\chi$
 of weight $1$. Thus, the analysis reduces to studying Rankin-Selberg type $L$-functions of the form $L\left(\left(\Sym^d f\right)^{\otimes l}\otimes\rho_{f_\chi},s\right)$ which again admit a decomposition into products of simpler $L$-functions. The theorem is proved by applying subconvexity estimates to these components.

\begin{remark}
    While this paper has been submitted and review, we noticed that recently Venkatasubbareddy submitted a paper\cite{arXiv:2601.17079} working on a similar problem and obtained similar results as our Theorem \ref{theorem:main}. But our work is different from \cite{arXiv:2601.17079} in the following aspects. First we use a representation  point of view to deal with the Dirichlet series, this allows us to connect our work with combination. Such ideas are similar with our work \cite{Yang2024first} and \cite{Yang2026higher}. They can be used to deal with more complicated cases and play a key role in the following work \cite{Yang2026second}. Next we also consider average results over binary quadratic forms, which are not considered in \cite{arXiv:2601.17079}. Finally, the estimation for the error terms in the two works are different.

\end{remark}

The post-theorem introduction contains a \subsection{Proof Sketch} explaining the strategy for Theorem~\ref{theorem:main}:

- **Perron's formula / contour integral reduction.** The “starting point is Perron's formula,” which converts estimating \(\sum_{n\le x}\lambda^l_{\Sym^d f}(n)\) into evaluating “a contour integral of the associated Dirichlet series” \(\sum_{n\ge1}\lambda^l_{\Sym^d f}(n)n^{-s}\).

- **Relate the Dirichlet series to an \(L\)-function via Euler product.** This Dirichlet series is “intimately related via its Euler product” to \(L((\Sym^d f)^{\otimes l},s)\), corresponding to the \(\ell\)-adic representation \(\Sym^d(\rho_f)^{\otimes l}\) (cf. Proposition~\ref{proposition:dirichletseriescomp}).

- **Decompose the tensor representation and factor the \(L\)-function.** A “key insight” is decomposing the “high-dimensional tensor representation into a direct sum of irreducible constituents,” which are “of the form \(\Sym^i\rho_f\).” Hence
  \[
  L\bigl((\Sym^d f)^{\otimes l},s\bigr)=\prod_{i=0}^{ld} L(\Sym^i f,s)^{K_{i,d,l}}.
  \]

- **Use automorphy/entireness and subconvexity to get the asymptotic.** The “automorphy and entirety of these symmetric power \(L\)-functions,” due to Newton--Thorne, are “crucial,” and the “final asymptotic formulas are then derived by combining this factorization with established subconvexity bounds for the involved \(L\)-functions.”

(An additional sketch is given for the binary quadratic form theorem \(\ref{thm:binary}\): one studies \(\sum \lambda^l_{\Sym^d f}(n)r(n,Q)n^{-s}\), expresses \(\sum r(n,Q)n^{-s}\) via \(\zeta_F(s)\) when \(h(D)=1\) or as a finite sum of Hecke \(L\)-functions in general, reducing to Rankin--Selberg type \(L\)-functions \(L((\Sym^d f)^{\otimes l}\otimes \rho_{f_\chi},s)\), and then applies subconvexity to their decomposed components.)

\label{theorem:main}
    Let $f$ be a cuspidal modular form of weight $k$ on $\SL_2(\BZ)$. For any integers $l\geq2$ and $d\geq1$ and $dl>4$, one has
    \[\sum_{n\leq x}  \lambda^l_{\Sym^d f}(n)=xP_{d,l}(\log x)+O\left(x^{\theta_{d,l}+\epsilon}\right).\]
    Here $P_{d,l}$ is a polynomial of degree $K_{0,d,l}-1$, where $K_{i,d,l}$ denote the Kostka number defined in sections \ref{section:Weylmod} and \ref{section:Kostka number}. The exponent $\theta_{d, l}$ is given by
    $$\theta_{d,l}=1-1/\left((d+1)^l/2-4K_{0,d,l}/21-K_{1,d,l}/3-5K_{2,d,l}/14\right).$$,

Question:
Let $f$ be a cuspidal modular form of weight $k$ on $\mathrm{SL}_2(\mathbb{Z})$. For each integer $d\ge 1$, let $\lambda_{\mathrm{Sym}^d f}(n)$ denote the $n$th Dirichlet coefficient of the symmetric-power $L$-function $L(\mathrm{Sym}^d f,s)$, and write $\lambda_{\mathrm{Sym}^d f}^l(n)=(\lambda_{\mathrm{Sym}^d f}(n))^l$. Fix integers $l\ge 2$ and $d\ge 1$ with $dl>4$. Let $K_{i,d,l}$ be the Kostka numbers characterized by the factorization
\[
L\big((\mathrm{Sym}^d f)^{\otimes l},s\big)=\prod_{i=0}^{dl} L(\mathrm{Sym}^i f,s)^{K_{i,d,l}}.
\]
As $x\to\infty$, which asymptotic formula holds for
\[\sum_{n\le x} \lambda_{\mathrm{Sym}^d f}^l(n)?\]

For any fixed $\epsilon>0$,
\[
\sum_{n\le x} \lambda_{\mathrm{Sym}^d f}^l(n)=xP_{d,l}(\log x)+O\bigl(x^{\theta_{d,l}+\epsilon}\bigr),
\]
where $P_{d,l}$ is a polynomial of degree $K_{0,d,l}-1$, and
\[
\theta_{d,l}=1-\frac{1}{(d+1)^l/2-4K_{0,d,l}/21-K_{1,d,l}/3-5K_{2,d,l}/14}.
\]

Choices:
B.
For any fixed $\epsilon>0$,
\[
\sum_{n\le x} \lambda_{\mathrm{Sym}^d f}^l(n)=xP_{d,l}(\log x)+O\bigl(x^{\theta_{d,l}+\epsilon}\bigr),
\]
where $P_{d,l}$ is a polynomial of degree $K_{0,d,l}$, and
\[
\theta_{d,l}=1-\frac{1}{(d+1)^l/2-4K_{0,d,l}/21-K_{1,d,l}/3-5K_{2,d,l}/14}.
\]
C.
For any fixed $\epsilon>0$,
\[
\sum_{n\le x} \lambda_{\mathrm{Sym}^d f}^l(n)=xP_{d,l}(\log x)+O\bigl(x^{\theta_{d,l}+\epsilon}\bigr),
\]
for some polynomial $P_{d,l}$ and some exponent $\theta_{d,l}<1$ depending on $d$ and $l$.
D.
For any fixed $\epsilon>0$,
\[
\sum_{n\le x} \lambda_{\mathrm{Sym}^d f}^l(n)=xP_{d,l}(\log x)+O\bigl(x^{\theta_{d,l}+\epsilon}\bigr),
\]
where $P_{d,l}$ is a polynomial of degree $K_{0,d,l}-1$, and
\[
\theta_{d,l}=1-\frac{1}{(d+1)^l-4K_{0,d,l}/21-2K_{1,d,l}/3-5K_{2,d,l}/7}.
\]
E.
For any fixed $\epsilon>0$,
\[
\sum_{n\le x} \lambda_{\mathrm{Sym}^d f}^l(n)=xP_{d,l}(\log x)+O\bigl(x^{\theta_{d,l}+\epsilon}\bigr),
\]
where $P_{d,l}$ is a polynomial of degree $K_{0,d,l}-1$, and
\[
\theta_{d,l}=1-\frac{1}{(d+1)^l/2-4K_{0,d,l}/21-K_{1,d,l}/3-5K_{2,d,l}/14+\epsilon}.
\]


\begin{proposition}\label{Penrose-prop}
Let $g$ be a Lorentzian metric, and let $\gamma$ be a null geodesic. Let $g_\sigma := e^\sigma g$ be a conformal change of $g$ (up to reparametrization, $\gamma$ is also a null geodesic of $g_\sigma$). Let $\mathrm{PL}_g$ (resp.\ $\mathrm{PL}_{g_\sigma}$) denote the Penrose limit of $g$ (resp.\ of $g_\sigma$) along $\gamma$. Then there exists a conformal diffeomorphism
$ \mathrm{PL}_g \to \mathrm{PL}_{g_\sigma}.$
\end{proposition}

\begin{theorem}\label{maintheo}Let $(M, g)$ be a Lorentzian manifold of dimension at least $3$ and let $\bf P$ be its pseudo-group of local conformal transformations, that is the collection
 of all conformal diffeomorphisms between open subsets of $M$. We assume $M$ locally conformally homogeneous, i.e.~$M$ is an orbit of $\bf P$. Then one of the following possibilities holds:

 \begin{enumerate}

 \item $\bf P$ preserves a Lorentzian metric in the $g$-conformal class.

 \item $(M, g)$ is conformally flat.

 \item $(M, g)$ is locally conformally equivalent to a plane wave that is isometrically locally homogeneous.  More precisely, there exists a homogeneous plane wave $X$, such that
 $(M, [g])$ is modeled on $(\Conf(X), X)$.
 \end{enumerate}
 \end{theorem}

\label{Gromov's Theory}
 \addtocontents{toc}{\setcounter{tocdepth}{1}}

For references, see the foundational work of Gromov~\cite{Gro}, the lecture notes by D'Ambra and Gromov~\cite{DG}, and those by

A fundamental problem in conformal geometry is to classify conformal manifolds with \textit{essential} conformal transformations, that is, global conformal transformations that do not preserve a metric in the conformal class. For Riemannian conformal manifolds the situation is very rigid: up to conformal equivalence only the round sphere and Euclidean space admit essential conformal transformations. This was conjectured by Lichnerowicz in the 1960's and proved in a series of papers by Ferrand and Obata (\cite{Lelong-Ferrand71}, \cite{Obata71}, \cite{Ferrand96}, with contributions in \cite{Alekseevskii72}).

For indefinite metrics, the situation is far more flexible: there exist pseudo-Riemannian metrics that are not conformally flat yet admit essential conformal transformations, both on non-compact manifolds and, as shown in \cite{frances12}, on compact manifolds in signatures other than the Lorentzian one. Non-compact examples in Lorentzian signature consist of plane waves. These are manifolds for which the metric in dimension $n+2$ is locally given by
\begin{equation}\label{pw}
g=2\,\mathrm{d} t\, \mathrm{d} v + \boldsymbol{x}^\top Q(t) \boldsymbol{x}\ \mathrm{d} t^2 +\mathrm{d}\boldsymbol{x}^\top\mathrm{d}\boldsymbol{x},
\end{equation}
where $\boldsymbol{x}=(x^1, \ldots , x^n)^\top\in \mathbb{R}^n$, {$Q(t)$ is} a $t$-dependent symmetric $n\times n$-matrix. If $Q$ is not a scalar matrix, this metric is not conformally flat.
Plane waves for which this metric is defined on all of $\mathbb{R}^{n+2}$, for example, admit non-isometric homotheties with fixed points. Therefore these homotheties are essential (for this implication see for example \cite[Proposition 2.5]{LeistnerTeisseire22}).

\bigskip

Recently
an important rigidity result for essential Lorentzian conformal structures in the \textit{conformally homogeneous} setting has been obtained by Alekseevsky and Galaev
 in \cite{AlekseevskyGalaev25}. They showed that if the manifold is simply connected and admits a transitive group of global conformal transformations that is essential, then the manifold is either conformally flat or the conformal class contains a plane wave metric that is isometrically homogeneous and geodesically complete. In the present article we will generalize this result in the following sense: we will drop the assumption that the manifold is simply connected and we will only assume that the manifold is \textit{locally conformally homogeneous}, i.e.~the local conformal transformations act transitively. We will prove the following result.

Recently
an important rigidity result for essential Lorentzian conformal structures in the \textit{conformally homogeneous} setting has been obtained by Alekseevsky and Galaev
 in \cite{AlekseevskyGalaev25}. They showed that if the manifold is simply connected and admits a transitive group of global conformal transformations that is essential, then the manifold is either conformally flat or the conformal class contains a plane wave metric that is isometrically homogeneous and geodesically complete. In the present article we will generalize this result in the following sense: we will drop the assumption that the manifold is simply connected and we will only assume that the manifold is \textit{locally conformally homogeneous}, i.e.~the local conformal transformations act transitively. We will prove the following result.

When the pseudo-group $\mathbf{P}$ does not preserve any metric in the $g$-conformal class, we say that $(M,g)$ is \textit{weakly essential}, as introduced in Section \ref{Section: terminology}. This is a weaker condition than being essential.
 Our proof starts off similarly as in \cite{AlekseevskyGalaev25} by noting that weak essentiality yields the existence of an element in the isotropy group that is not contained in $\SO(T_pM)$.
In \cite{AlekseevskyGalaev25}, simple connectedness and global conformal homogeneity ensure that the isotropy group is connected, which in turn provides a specific element in its Lie algebra (that does not lie in the isometry algebra). In our setting, however, we cannot directly conclude the existence of such an element. Therefore, we adopt a different approach. We work only with the local isotropy of local conformal transformations, but we use powerful tools from Gromov's theory of rigid transformations (see Appendix \ref{Gromov's Theory}) and from algebraic groups, which allow us to reduce the problem to the Lie algebra level (see Section \ref{Section: isotropy}), and in particular to obtain a hyperbolic element in the isotropy algebra that is not in $\so(T_pM)$.
Since we assume only local homogeneity, we can pass to the universal cover, where weak essentiality is preserved. Moreover, on a simply connected real-analytic manifold, local conformal vector fields extend to global ones. We may therefore work with the Lie algebra of conformal vector fields on the universal cover. Loosely speaking, by weakening the assumption of homogeneity to local homogeneity, we are able to overcome the assumption of simple connectedness.
Then, in Section \ref{Section: Heis algebra}, we use the decomposition of the Lie algebra of conformal vector fields induced by the hyperbolic element to derive the existence of a codimension-one lightlike foliation of \textit{Heisenberg type}, i.e., with a transitive action of a Heisenberg algebra.
Our approach, together with the new characterization of plane waves obtained in Section \ref{Section: Determining a plane wave by Killing fields}, not only leads to the generalization in Theorem \ref{maintheo}, but also provides a more conceptual approach and simplifies the proof in \cite{AlekseevskyGalaev25} for the simply connected globally homogeneous case.

In the last section, we consider the consequence of our result for {\bf Penrose limits} of locally conformally homogeneous Lorentzian manifolds.
The Penrose limit is a famous construction in gravitational physics, which shows that any Lorentzian manifold admits a plane wave spacetime as a limit, this limit being taken along any lightlike geodesic (see \cite{Blau-PL} for instance).
As shown in Theorem \ref{maintheo}, a weakly essential locally conformally homogeneous Lorentzian manifold of dimension $\geq 3$ is either conformally flat or locally conformal to a homogeneous plane wave.
A natural question is then whether this plane wave coincides with the Penrose limit of the Lorentzian manifold along some lightlike geodesic. A priori, the Penrose limit is a plane wave associated with a metric, rather than with a conformal structure.
The underlying question is then whether two conformally related metrics have conformally equivalent Penrose limits along some lightlike geodesic (note that conformally related metrics have the same unparameterized lightlike geodesics). It turns out that the answer is affirmative, so the Penrose limit is a conformal invariant, as shown in Proposition~\ref{Penrose-prop}.
As a consequence, we obtain that the plane wave metric in the conformal class of a weakly essential locally conformally homogeneous Lorentzian manifold $(M,g)$ (as in item (3) of Theorem~\ref{maintheo}) coincides with the Penrose limit of $(M,g)$ along some null geodesic.
\begin{corollary}
Let $(M, g)$ be a non-conformally flat Lorentzian manifold of dimension at least $3$ which is locally conformally homogeneous. Assume that $(M,g)$ is weakly essential. Then, $(M,g)$ is locally conformally equivalent to its Penrose limit along some null geodesic.
\end{corollary}
This raises the question of whether a direct proof of this fact can be found, which we plan to investigate in future work.

\begin{proposition}\label{distributionprop}
Let $\g$ be the algebra of conformal vector fields of $M$, and assume that $\g$ acts transitively on $M$.
Fix a point
$p\in M$, and let $\g_p$ be the isotropy subalgebra at $p$, with image $\h_p$ under the isotropy representation $\rho_p$.
Let $V_p$ be a subspace of $T_pM$ that is invariant under $\h_p \subset \co(T_pM)$. Then
\begin{enumerate}
\item There is a vector distribution $V$ with $V_p$ as fiber at $p$ that is invariant under $\g$, i.e. $\d\psi_p(V_p)=V_{\psi(p)}$ for all local conformal transformations $\psi \in \mathbf{G}$ acting locally on $M$.
\item If there is a Lie algebra $\l\subset \g$ with stabilizer algebra $\l_p$ such that the isomorphism $T_pM\simeq \g/\g_p$ restricts to an isomorphism $V_p\simeq \l/\l_p$, then $V$ is involutive with fiber at $\psi(p)$ spanned by $\Ad_\psi (\l)|_{\psi(p)}$, for $\psi\in \mathbf{G}$ acting locally on $M$.
In particular, if $\l$ is an ideal of $\g$, then $V_q$ is spanned by $\l|_q$ for each $q\in M$.
\item If $\mathbf{L}$ is the Lie group corresponding to $\l$, then the leaves through $q=\psi(p)$ are given
by the orbits of $q$ under $\psi \mathbf L \psi^{-1}$.
In particular, if $\l$ is an ideal, then the integral manifold through $q$ is the orbit of $q$ under $\mathbf{L}$.
\end{enumerate}
\end{proposition}

\begin{proposition}\label{Prop. existence of foliation}
Let $(M,g)$ be a Lorentzian manifold of dimension $n+2$, with a transitive Lie algebra $\g$ of conformal vector fields, and stabilizer algebra $\h$ at $p\in M$.
Assume that
$\h \subset \g^0 \oplus \g^{\pm 1}$ and $\g^0=\R B \oplus \h^{00}$ with $B=\mathrm{diag}(\alpha+1,\alpha-1,\alpha,\ldots, \alpha)$.
Then there is a metric $\hat g$ in the conformal class of $g$ and
an algebra of Killing vector fields $\g_1$ for $\hat g$ that acts locally transitively at $p$. Moreover,
\begin{enumerate}
 \item If $\alpha \notin \left\{\frac{1}{2}, 1, 2\right\}$, then $(M,g)$ is conformally flat.

\begin{corollary}\label{Penrose-cor}
Let $(M, g)$ be a non-conformally flat Lorentzian manifold of dimension at least $3$ which is locally conformally homogeneous. Assume that $(M,g)$ is weakly essential. Then, $(M,g)$
 is locally conformally equivalent to its Penrose limit along some null geodesic.
\end{corollary}
Note that the proof of Theorem~\ref{maintheo} reveals more details than those stated in this corollary. The assumption implies that there is a null line bundle that is invariant under the local conformal vector fields, which is spanned by the parallel vector field of the plane wave, and that $(M,g)$ is locally conformally equivalent to the Penrose limit along all null geodesics that are \textit{transversal} to this null line bundle.

When the pseudo-group $\mathbf{P}$ does not preserve any metric in the $g$-conformal class, we say that $(M,g)$ is \textit{weakly essential}, as introduced in Section \ref{Section: terminology}. This is a weaker condition than being essential.
 Our proof starts off similarly as in \cite{AlekseevskyGalaev25} by  noting that weak essentiality yields the existence of an element in the isotropy group that is not contained in $\mathbf{SO}(T_pM)$.
In \cite{AlekseevskyGalaev25}, simple connectedness and global conformal homogeneity ensure that the isotropy group is connected, which in turn provides a specific element in its Lie algebra (that does not lie in the isometry algebra). In our setting, however, we cannot directly conclude the existence of such an element. Therefore, we adopt a different approach. We work only with the local isotropy of local conformal transformations, but we use powerful tools from Gromov's theory of rigid transformations (see Appendix \ref{Gromov's Theory}) and from algebraic groups, which allow us to reduce the problem to the Lie algebra level (see Section \ref{Section: isotropy}), and in particular to obtain a hyperbolic element in the isotropy algebra that is not in $\mathfrak{so}(T_pM)$.
Since we assume only local homogeneity, we can pass to the universal cover, where weak essentiality is preserved. Moreover, on a simply connected real-analytic manifold, local conformal vector fields extend to global ones. We may therefore work with the Lie algebra of conformal vector fields on the universal cover. Loosely speaking, by weakening the assumption of homogeneity to local homogeneity, we are able to overcome  the assumption of simple connectedness.
Then, in Section \ref{Section: Heis algebra}, we use the decomposition of the Lie algebra of conformal vector fields induced by the hyperbolic element to derive the existence of a codimension-one lightlike foliation of \textit{Heisenberg type}, i.e., with a transitive action of a Heisenberg algebra.
Our approach, together with the new characterization of plane waves obtained in Section \ref{Section: Determining a plane wave by Killing fields}, not only leads to the generalization in Theorem \ref{maintheo}, but also provides a more conceptual approach and simplifies the proof in \cite{AlekseevskyGalaev25} for the simply connected globally homogeneous case.

 \bigskip

In the last section, we consider the consequence of our result for {\bf Penrose limits} of locally conformally homogeneous Lorentzian manifolds.
The Penrose limit is a famous construction in gravitational physics, which shows that any Lorentzian manifold admits a  plane wave spacetime as a limit, this limit being taken along any lightlike geodesic (see \cite{Blau-PL} for instance).
As shown in  Theorem \ref{maintheo}, a weakly essential locally conformally homogeneous Lorentzian manifold of dimension $\geq 3$ is either conformally flat or locally conformal to a homogeneous plane wave.
A natural question is then whether this plane wave coincides with the Penrose limit of the Lorentzian manifold along some lightlike geodesic. A priori, the Penrose limit is a plane wave associated with a metric, rather than with a conformal structure.
The underlying question is then whether two conformally related metrics have conformally equivalent Penrose limits along some lightlike geodesic (note that conformally related metrics have the same unparameterized lightlike geodesics). It turns out that the answer is affirmative, so the Penrose limit is a conformal invariant, as shown  in Proposition~\ref{Penrose-prop}.
As a consequence, we obtain that the plane wave metric in the conformal class of a weakly essential locally conformally homogeneous Lorentzian manifold $(M,g)$ (as in item (3) of  Theorem~\ref{maintheo}) coincides with the Penrose limit of $(M,g)$ along some null geodesic.
\begin{corollary}
Let $(M, g)$ be a non-conformally flat Lorentzian manifold of dimension at least $3$ which is locally conformally homogeneous. Assume that $(M,g)$ is weakly essential. Then, $(M,g)$  is locally conformally equivalent to its Penrose limit along some null geodesic.
\end{corollary}
This raises the question of whether a direct proof of this fact can be found, which we plan to investigate in future work.

To prove Theorem~\ref{maintheo}, the argument is described as follows.

- Define the weakly essential case: when the pseudo-group $\mathbf{P}$ does not preserve any metric in the $g$-conformal class.

- As in \cite{AlekseevskyGalaev25}, start by using weak essentiality to get “the existence of an element in the isotropy group that is not contained in $\mathbf{SO}(T_pM)$.” The earlier approach relied on connectedness of the isotropy group to “provide a specific element in its Lie algebra,” but “in our setting … we cannot directly conclude the existence of such an element.”

- Use a different approach: “work only with the local isotropy of local conformal transformations,” and apply “powerful tools from Gromov's theory of rigid transformations … and from algebraic groups,” which “allow us to reduce the problem to the Lie algebra level … and in particular to obtain a hyperbolic element in the isotropy algebra that is not in $\mathfrak{so}(T_pM)$.”

- Because only local homogeneity is assumed, “pass to the universal cover, where weak essentiality is preserved.” On a “simply connected real-analytic manifold, local conformal vector fields extend to global ones,” so one can “work with the Lie algebra of conformal vector fields on the universal cover.”

- Then “use the decomposition of the Lie algebra of conformal vector fields induced by the hyperbolic element to derive the existence of a codimension-one lightlike foliation of \textit{Heisenberg type}, i.e., with a transitive action of a Heisenberg algebra.”

- Finally, combine this with “the new characterization of plane waves obtained in Section~\ref{Section: Determining a plane wave by Killing fields},” which “leads to the generalization in Theorem~\ref{maintheo}.”

To prove the main theorem, the argument is described as follows.

- Define the weakly essential case: when the pseudo-group $\mathbf{P}$ does not preserve any metric in the $g$-conformal class.

- As in \cite{AlekseevskyGalaev25}, start by using weak essentiality to get “the existence of an element in the isotropy group that is not contained in $\mathbf{SO}(T_pM)$.” The earlier approach relied on connectedness of the isotropy group to “provide a specific element in its Lie algebra,” but “in our setting … we cannot directly conclude the existence of such an element.”

- Use a different approach: “work only with the local isotropy of local conformal transformations,” and apply “powerful tools from Gromov's theory of rigid transformations … and from algebraic groups,” which “allow us to reduce the problem to the Lie algebra level … and in particular to obtain a hyperbolic element in the isotropy algebra that is not in $\mathfrak{so}(T_pM)$.”

- Because only local homogeneity is assumed, “pass to the universal cover, where weak essentiality is preserved.” On a “simply connected real-analytic manifold, local conformal vector fields extend to global ones,” so one can “work with the Lie algebra of conformal vector fields on the universal cover.”

- Then “use the decomposition of the Lie algebra of conformal vector fields induced by the hyperbolic element to derive the existence of a codimension-one lightlike foliation of \textit{Heisenberg type}, i.e., with a transitive action of a Heisenberg algebra.”

- Finally, combine this with “the new characterization of plane waves obtained later,” which “leads to the generalization in the main theorem.”

Question:
Which statement holds for every Lorentzian manifold $(M,g)$ of dimension at least $3$ whose pseudo-group $\mathbf P$ of local conformal transformations acts transitively on $M$? Here $\mathbf P$ means all conformal diffeomorphisms between open subsets of $M$, and “locally conformally homogeneous” means that $M$ is a single orbit of $\mathbf P$. A plane wave is a Lorentzian manifold locally given in dimension $n+2$ by
\[
g=2\,dt\,dv+\boldsymbol{x}^\top Q(t)\boldsymbol{x}\,dt^2+d\boldsymbol{x}^\top d\boldsymbol{x},
\]
where $\boldsymbol{x}\in\mathbb R^n$ and $Q(t)$ is a symmetric $n\times n$ matrix; “homogeneous plane wave” means such a plane wave is isometrically locally homogeneous.

At least one of the following holds: (1) $\mathbf P$ preserves a Lorentzian metric in the conformal class $[g]$; or (2) $(M,g)$ is conformally flat; or (3) $(M,g)$ is locally conformally equivalent to an isometrically locally homogeneous plane wave. More precisely in case (3), there exists a homogeneous plane wave $X$ such that $(M,[g])$ is modeled on $(\operatorname{Conf}(X),X)$.

Choices:
B. Exactly one of the following holds: (1) $\mathbf P$ preserves a Lorentzian metric in the conformal class $[g]$; or (2) $(M,g)$ is conformally flat; or (3) $(M,g)$ is locally conformally equivalent to an isometrically locally homogeneous plane wave. More precisely in case (3), there exists a homogeneous plane wave $X$ such that $(M,[g])$ is modeled on $(\operatorname{Conf}(X),X)$.
C. At least one of the following holds: (1) $\mathbf P$ preserves a Lorentzian metric in the conformal class $[g]$; or (2) $(M,g)$ is conformally flat; or (3) $(M,g)$ is locally conformally equivalent to a plane wave.
D. At least one of the following holds: (1) after passing to the universal cover of $M$, the lifted pseudo-group $\mathbf P$ preserves a Lorentzian metric in the lifted conformal class; or (2) $(M,g)$ is conformally flat; or (3) after passing to the universal cover, $(M,g)$ is locally conformally equivalent to an isometrically locally homogeneous plane wave. More precisely in case (3), there exists a homogeneous plane wave $X$ such that the universal cover of $(M,[g])$ is modeled on $(\operatorname{Conf}(X),X)$.
E. If $\mathbf P$ does not preserve any Lorentzian metric in the conformal class $[g]$, then $(M,g)$ is locally conformally equivalent to an isometrically locally homogeneous plane wave. More precisely, there exists a homogeneous plane wave $X$ such that $(M,[g])$ is modeled on $(\operatorname{Conf}(X),X)$.

dropped homogeneous/modeling refinement in the plane-wave alternative

conclusion only on universal cover instead of on M itself

omits conformally flat weakly essential branch