\begin{remark}\label{rk-mini}If $w=0$, then it follows from \eqref{eq-vw} that $v$ only depends on the variable $u_1$, and satisfies
\begin{equation}\label{eq-ellip}
\frac{d^2 v}{d u_1 ^2}=-2\sinh(2v).
\end{equation}
The solution to this ordinary differential equation can be expressed using an elliptic function,
\[
v = \operatorname{arcsinh}\Bigl( k \; \mathrm{sn}\bigl( 2u_1 + \delta,\; i k \bigr) \Bigr),
\]
where $\delta$ and $k>0$ are integration constants, and $\mathrm{sn}(\,\cdot,\,ik)$ denotes the Jacobian elliptic sine function with  modulus $i k$.
Applying the classical Jacobi imaginary modulus transformation, the solution also admits the equivalent form
\[
v = \operatorname{arcsinh}\Bigl( \frac{k}{\sqrt{1+k^2}} \; \mathrm{sn}\bigl( \sqrt{1+k^2}\,(2u_1 + \delta),\; \frac{k}{\sqrt{1+k^2}} \bigr) \Bigr).\]

Geometrically, given a solution of~\eqref{eq-ellip}, one first obtains a minimal surface $\psi:\mathbb{C}\rightarrow \mathbb{S}^3$ with induced metric $e^{2v}|dz|^2$
and Hopf differential $\frac i 2 dz\otimes dz$.
Such a surface should be necessarily homogeneous or of cohomogeneity one and hence belongs to the $T_{m,k,a}$ family constructed and classified by Hsiang and Lawson in \cite{Hsiang-Lawson}. This family contains infinitely many closed minimal surfaces, whose topology is either a torus or a Klein bottle. Denote by $N$ the Gauss map of such a surface. Then the map \emph{(}see the proof of Corollary $1$ in \cite{Urbano}\emph{)}
\[
\phi(z)=(V_\psi, e^{2i u_2}):\mathbb{C}\rightarrow \mathbb{S}^2\times\mathbb{S}^1
\]
defines a minimal surface in $\mathbb{S}^2\times\mathbb{S}^1$, where
\[
V_\psi=\frac{1}{\sqrt{2}}(-2ie^{-2v}\psi_{z}\wedge\psi_{\overline{z}}+\psi\wedge N):\mathbb{C}\rightarrow \mathbb{S}^2\subset \Lambda^2\mathbb{R}^4.
\]
Apart from the obvious slices, these surfaces constitute all minimal-Willmore surfaces in $\mathbb{S}^2\times\mathbb{S}^1$.
\end{remark}

\begin{thm}\label{thm1}
   Let \( x: \Sigma \rightarrow \mathbb{S}^2 \times \mathbb{S}^2 \) be a minimal-Willmore surface. Then either \( x \) is a special complex curve given by a slice or a diagonal; or, up to an isometry, is contained in a totally geodesic submanifold $\mathbb{S}^2 \times \mathbb{S}^1$, and can be described by a solution of the sinh-Gordon equation in one variable.
\end{thm}

For a closed surface $x:\Sigma\rightarrow (N^n,g)$ in a Riemannian manifold, the squared $L^2$-norm of the trace-free second fundamental form is a fundamental conformal invariant; that is, it is preserved under conformal changes of the metric. This functional, referred to as the conformally invariant Willmore functional \cite{Michelat,Mondino}, will henceforth be called simply the Willmore functional. The Willmore functional is equivalent, up to a topological invariant, to the integral
$$\int_{\Sigma}(|\vec{H}|^2+K_{1212})dA,$$
where $\vec{H}$ is the mean curvature vector and $K_{1212}$ denotes the sectional curvature of $(N^n,g)$ restricted to the tangent plane of the surface. In what follows, we will denote this integral by $\mathcal{W}$ and also refer to it as the Willmore functional. This functional exhibits notable links with other fundamental quantities, including, among others, the renormalized area functional within the AdS/CFT correspondence \cite{Alexakis, Graham}. Surfaces that satisfy the Euler-Lagrange equation of this functional are called {\em Willmore surfaces}.

When the ambient space is a real space form, both the Willmore functional and Willmore surfaces have been extensively investigated. Notable advances include the resolution of the Willmore conjecture in $\mathbb{S}^3$, originally proposed by Willmore in \cite{Willmore} and proved by Marques and Neves in \cite{Marques}. Regarding the construction and classification of Willmore surfaces, we refer to \cite{Bryant, Ejiri, WD, MWW} and references therein. A basic fact is that every minimal surface in a real space form is automatically Willmore.
When the ambient space is a non-space form Riemannian manifold, the Euler–Lagrange equation of the Willmore functional has been derived independently by several groups of geometers from different viewpoints; see, for example, \cite{Hu-Li, Mondino, Pedit, Wang-Xie}. Owing to the presence of ambient curvature terms in the Euler–Lagrange equation, not every minimal surface is Willmore in a non-space form Riemannian manifold. In \cite{Montiel-Urbano}, Montiel and Urbano proved that the only surfaces in $\mathbb{CP}^2$ that are both minimal and Willmore are the superminimal surfaces of positive spin, i.e., complex curves and minimal Lagrangian surfaces. Recently, the last two authors of this paper have generalized this result to the self-dual K\"ahler-Einstein surfaces (see Proposition 2.3 in \cite{Wang-Xie}). Note that $\mathbb{S}^2 \times \mathbb{S}^2$ (equipped with the standard product metric and complex structure), being neither self-dual nor anti-self-dual, is another canonical example of a K\"ahler-Einstein surface. This motivates the study of which minimal surfaces in $\mathbb{S}^2\times\mathbb{S}^2$ are Willmore. We refer to a surface as minimal-Willmore if it is both minimal and Willmore.

\section{Introduction}
For a closed surface $x:\Sigma\rightarrow (N^n,g)$ in a Riemannian manifold, the squared $L^2$-norm of the trace-free second fundamental form is a fundamental conformal invariant; that is, it is preserved under conformal changes of the metric. This functional, referred to as the conformally invariant Willmore functional \cite{Michelat,Mondino}, will henceforth be called simply the Willmore functional. The Willmore functional is equivalent, up to a topological invariant, to the integral
$$\int_{\Sigma}(|\vec{H}|^2+K_{1212})dA,$$
where $\vec{H}$ is the mean curvature vector and $K_{1212}$ denotes the sectional curvature of $(N^n,g)$ restricted to the tangent plane of the surface. In what follows, we will denote this integral by $\mathcal{W}$ and also refer to it as the Willmore functional. This functional exhibits notable links with other fundamental quantities, including, among others, the renormalized area functional within the AdS/CFT correspondence \cite{Alexakis, Graham}. Surfaces that satisfy the Euler-Lagrange equation of this functional are called {\em Willmore surfaces}.

In their work \cite{Urbano}, Tobarro and Urbano gave a beautiful local characterization of minimal surfaces in $\mathbb{S}^2\times\mathbb{S}^2$ that possess no complex points. Their description is formulated in terms of a pair of solutions $v$ and $w$ to the sinh‑Gordon equation
$$v_{z\bar z}+\frac{1}{2}\sinh (2v)=0.$$
Within this framework, the Willmore condition introduces a new partial differential equation linking $v$ and $w$, see \eqref{eq-mwill}. By analyzing this coupled system, we establish the theorem above.

\begin{example}\label{ex-slice} For any point $p\in\mathbb{S}^2$, the corresponding slices
\[
\mathbb{S}^2\times\{p\}=\{(x,p)\in\mathbb{S}^2\times\mathbb{S}^2 \mid x\in\mathbb{S}^2\},
\]
\[
\{p\}\times\mathbb{S}^2=\{(p,x)\in\mathbb{S}^2\times\mathbb{S}^2 \mid x\in\mathbb{S}^2\}.
\]
are totally geodesic with the first one satisfying $C_1=C_2=1$ and the second one satisfying $C_1=-C_2=1$. That is, the slices are complex with respect to both $J_1$ and $J_2$.
\end{example}
\begin{example}\label{ex-diagonal} The diagonal
$$D\triangleq\{(x,x)\in\mathbb{S}^2\times\mathbb{S}^2 \mid x\in\mathbb{S}^2\}$$
is totally geodesic and satisfies $C_1=1$ and $C_2=0$, which means that $D$ is complex with respect to $J_1$ and Lagrangian with respect to $J_2$.
\end{example}
\begin{example}\label{ex-Clifford}
 The torus defined by
 $$T\triangleq\{(x,y)\in\mathbb{S}^2\times\mathbb{S}^2 \mid x_1=y_1=0\}$$
 is totally geodesic and satisfies $C_1=C_2=0$, which means that $T$ is Lagrangian for both $J_1$ and $J_2$. This surface is known as the Clifford torus in $\mathbb{S}^2 \times \mathbb{S}^2$, characterized by the flatness of both its tangent and normal bundles.
\end{example}
\begin{example}
Let $\Sigma=\mathbb{C}/\Lambda$ be a torus generated by the lattice $\Lambda=\{m+n\tau \mid m,n\in\mathbb{Z}\}$, where $\tau$ is a complex number with $\operatorname{Im}\tau>0$. Let $\wp:\Sigma\rightarrow\mathbb{S}^2$ be the Weierstrass $\wp$-function with a second-order pole at the origin, and $z_0\in\Sigma$ be a point at which $\wp$ is not ramified. Then $x=\big(\wp(z),\wp(z-z_0)\big):\Sigma\rightarrow(\mathbb{S}^2\times\mathbb{S}^2)$ is a holomorphic embedding with the complex structure $J_1$. Such minimal tori are referred to as Weierstrass tori.
\end{example}

\begin{proposition}\label{pro-mw}
 Let $x:\Sigma \rightarrow \mathbb{S}^2\times \mathbb{S}^2$ be a complex curve or minimal Lagrangian surface. Then $x$ is Willmore if and only if it is one of the surfaces given in Example~\ref{ex-slice} $\sim$ Example~\ref{ex-Clifford}.
\end{proposition}
\begin{proof}
Without loss of generality, we assume that $x$ is complex or minimal Lagrangian with respect to the complex structure $J_1$. Consequently, $C_1$ is a constant which must be either $0$ or $\pm 1$. It follows from \cite[Proposition 3]{Urbano} (see also \cite{Chen}) that we only need to prove that $C_2$ is a constant.

By \eqref{eq-C1z}, we have $f_1\bar{\gamma_1}=0$. Substituting this into \eqref{eq-mW} yields $\bar{f_1}{\gamma_2}=0$, which, by \eqref{eq-C2z}, implies that $C_2$ is holomorphic. Since $C_2$ is real-valued, it must be constant.
\end{proof}
\begin{theorem}\label{thm-mw}
 Let $x:\Sigma \rightarrow \mathbb{S}^2\times \mathbb{S}^2$ be a minimal surface without complex points. Then $x$ is Willmore if and only if up to an isometry it is a minimal-Willmore surface in $\mathbb{S}^2\times\mathbb{S}^1$.
\end{theorem}

Let $x:\Sigma \rightarrow \mathbb{S}^2\times \mathbb{S}^2$ be a minimal immersion of a simply connected surface without complex points and $z$ be a complex coordinate such that the Hopf differential $\Theta(z)\triangleq\frac{\langle J_1 x_z, J_2 x_z\rangle}{2}dz^2$ (which is holomorphic) satisfies $\Theta(z)=dz^2$.
 Then the functions
 $$v\triangleq\frac{1}{2}\ln \sqrt{\frac{(1+C_1)(1+C_2)}{(1-C_1)(1-C_2)}},~~~w\triangleq\frac{1}{2}\ln \sqrt{\frac{(1-C_1)(1+C_2)}{(1+C_1)(1-C_2)}}$$
 satisfy the sinh-Gordon equation, i.e.,
 $$v_{z\bar z}+\frac{1}{2}\sinh (2v)=0,~~~w_{z\bar z}+\frac{1}{2}\sinh (2w)=0.$$
 Conversely, given two solutions $v, w:\mathbb{C}\rightarrow \mathbb{R}$ of the sinh-Gordon equation, one can construct a 1-parameter family of minimal immersions $X_t: \mathbb{S}^2 \times \mathbb{S}^2$ by taking the following quantities as the fundamental data,
$$\sigma=\frac{1}{2}\ln\big(4\cosh(v+w)\cosh(v-w)\big),~~~ \rho=\left(\ln \sqrt{\frac{\cosh(v+w)}{\cosh(v-w)}}\right)_{\!\!z},~~~$$
$$C_1=\tanh(v-w),~~~C_2=\tanh(v+w),$$
$$\gamma_1=\sqrt{2}e^{\frac{it}{2}}\sqrt{\frac{\cosh(v+w)}{\cosh(v-w)}},~~~\gamma_2=\sqrt{2}e^{\frac{it}{2}}\sqrt{\frac{\cosh(v-w)}{\cosh(v+w)}},~~~f_1=-i \gamma_1 (v-w)_z,~~~f_2=-i\gamma_2(v+w)_z.$$
Geometrically, this construction originates from the Gauss maps of two minimal surfaces in \( \mathbb{S}^3 \), whose induced metrics are \( e^{2v} |dz|^2 \) and \( e^{2w} |dw|^2 \), respectively, and which share the same Hopf differential \( \theta(z) = \frac{i}{2} dz^2 \). For further details, we refer to Section~5 of \cite{Urbano}.

Therefore, we establish the claim that $p=\pm q$, which means either $v=0$ or $w=0$. It then follows from Proposition 4 in \cite{Urbano} that $x$ is non-full in $\mathbb{S}^2 \times \mathbb{S}^2$. This means it lies in a totally geodesic hypersurface of the ambient space, which, up to an isometry, is an open subset of $\mathbb{S}^2 \times \mathbb{S}^1$.
\end{proof}
\begin{remark}\label{rk-mini}If $w=0$, then it follows from \eqref{eq-vw} that $v$ only depends on the variable $u_1$, and satisfies
\begin{equation}\label{eq-ellip}
\frac{d^2 v}{d u_1 ^2}=-2\sinh(2v).
\end{equation}
The solution to this ordinary differential equation can be expressed using an elliptic function,
\[
v = \operatorname{arcsinh}\Bigl( k \; \mathrm{sn}\bigl( 2u_1 + \delta,\; i k \bigr) \Bigr),
\]
where $\delta$ and $k>0$ are integration constants, and $\mathrm{sn}(\,\cdot,\,ik)$ denotes the Jacobian elliptic sine function with modulus $i k$.
Applying the classical Jacobi imaginary modulus transformation, the solution also admits the equivalent form
\[
v = \operatorname{arcsinh}\Bigl( \frac{k}{\sqrt{1+k^2}} \; \mathrm{sn}\bigl( \sqrt{1+k^2}\,(2u_1 + \delta),\; \frac{k}{\sqrt{1+k^2}} \bigr) \Bigr).\]

\noindent In other words, when the ambient space is $\mathbb{S}^2\times\mathbb{S}^2$,  within the minimal class the Willmore condition turns out to be  rigid: it forces special holomorphicity, or reduces the problem to a totally geodesic hypersurface.
 Geometrically, the last class of minimal surfaces in Theorem~\ref{thm1} arises from a certain Gauss-map construction applied to a cohomogeneity {one} minimal surface in $\mathbb{S}^3\subset\mathbb{R}^4$, see Remark~\ref{rk-mini}.

In their work \cite{Urbano}, Tobarro and Urbano gave a beautiful local characterization of minimal surfaces in $\mathbb{S}^2\times\mathbb{S}^2$ that possess no complex points. Their description is formulated in terms of a pair of solutions $v$ and $w$ to the sinh‑Gordon equation
$$v_{z\bar z}+\frac{1}{2}\sinh (2v)=0.$$
Within this framework, the Willmore condition introduces a new partial differential equation linking $v$ and $w$, see \eqref{eq-mwill}. By analyzing this coupled system, we establish the theorem above.

Except for seeking Willmore surfaces among minimal surfaces, another natural approach is to use the product structure of  $\mathbb{S}^2\times\mathbb{S}^2$ to produce Willmore surfaces.
\begin{thm}\label{thm2}
   All Willmore surfaces of product type in $\mathbb{S}^2 \times \mathbb{S}^2$ are exhausted by products of an elastic curve in $\mathbb{S}^2$ with a great circle.
\end{thm}

Note that both of these two classes of examples yield Willmore surfaces that are non-linearly full in $\mathbb{S}^2 \times \mathbb{S}^2$. No linearly full examples are currently known. We point out that constructing explicit examples, or even addressing the existence problem, in non-space-form Riemannian manifolds is particularly challenging; see the efforts in \cite{Modino,Ikoma,Michelat} and the references therein.

The paper is organized as follows. {Section~\ref{sec2}  begins with the  preliminaries on  $\mathbb{S}^2\times \mathbb{S}^2$ and on surfaces within it. In Section~\ref{sec3}, we first derive the fundamental equations for surfaces in $\mathbb{S}^2\times \mathbb{S}^2$, and subsequently compute the corresponding Willmore equation. Section~\ref{sec4} is devoted to the classification of minimal-Willmore surfaces in $\mathbb{S}^2\times\mathbb{S}^2$. Finally, in Section~\ref{sec5}, we classify Willmore surfaces of product type.

Using the local characterization of Tobarro--Urbano \cite{Urbano} for minimal surfaces in $\mathbb{S}^2\times\mathbb{S}^2$ with no complex points, the surface is described "in terms of a pair of solutions $v$ and $w$ to the sinh-Gordon equation" $$v_{z\bar z}+\tfrac12\sinh(2v)=0.$$ "Within this framework, the Willmore condition introduces a new partial differential equation linking $v$ and $w$, see \eqref{eq-mwill}." The proof then proceeds by "analyzing this coupled system" (the sinh-Gordon equation together with the additional PDE imposed by the Willmore condition), which yields the classification stated in Theorem~\ref{thm1}.

Using the local characterization of Tobarro--Urbano \cite{Urbano} for minimal surfaces in $\mathbb{S}^2\times\mathbb{S}^2$ with no complex points, the surface is described "in terms of a pair of solutions $v$ and $w$ to the sinh-Gordon equation" $$v_{z\bar z}+\tfrac12\sinh(2v)=0.$$ "Within this framework, the Willmore condition introduces a new partial differential equation linking $v$ and $w," namely
\begin{equation}\label{eq-mwill}
    e^{it}\cosh^3(v-w)(v+w)_{\bar{z}} = \cosh^3(v+w)(v-w)_z.
    \end{equation}
The proof then proceeds by "analyzing this coupled system" (the sinh-Gordon equation together with the additional PDE imposed by the Willmore condition). To prove the main theorem, one shows that this analysis yields the stated classification.

Question: A minimal-Willmore surface in \(\mathbb{S}^2\times\mathbb{S}^2\) means a surface \(x:\Sigma\to \mathbb{S}^2\times\mathbb{S}^2\) that is both minimal and Willmore, where a Willmore surface satisfies the Euler-Lagrange equation of the Willmore functional \(\mathcal W=\int_\Sigma (|\vec H|^2+K_{1212})\,dA\). A slice is a surface of the form \(\mathbb{S}^2\times\{p\}\) or \(\{p\}\times\mathbb{S}^2\), and a diagonal means a diagonal copy of \(\mathbb{S}^2\) in \(\mathbb{S}^2\times\mathbb{S}^2\). Which statement holds for every minimal-Willmore surface \(x:\Sigma\to \mathbb{S}^2\times\mathbb{S}^2\)?

Either \(x\) is a special complex curve, namely a slice or a diagonal; or, after applying an isometry of \(\mathbb{S}^2\times\mathbb{S}^2\), the image of \(x\) is contained in a totally geodesic submanifold \(\mathbb{S}^2\times\mathbb{S}^1\), and the surface is described by a solution of the sinh-Gordon equation depending on a single variable.

Choices:
B. Either \(x\) is a special complex curve, namely a slice or a diagonal; or, after applying an isometry of \(\mathbb{S}^2\times\mathbb{S}^2\), the image of \(x\) is contained in a totally geodesic submanifold \(\mathbb{S}^2\times\mathbb{S}^1\), and the surface is described by a pair of independent solutions of the sinh-Gordon equation in two variables.
C. Either \(x\) is a special complex curve, namely a slice or a diagonal; or, after applying an isometry of \(\mathbb{S}^2\times\mathbb{S}^2\), the image of \(x\) is contained in a totally geodesic submanifold \(\mathbb{S}^2\times\mathbb{S}^1\).
D. Either \(x\) is a special complex curve, namely a slice or a diagonal; or the image of \(x\) is contained in a totally geodesic submanifold \(\mathbb{S}^2\times\mathbb{S}^1\), and every such non-complex minimal-Willmore surface is described by a solution of the sinh-Gordon equation depending on two variables.
E. Either \(x\) is a special complex curve, namely a slice or a diagonal; or, after applying an isometry of \(\mathbb{S}^2\times\mathbb{S}^2\), the image of \(x\) is contained in a totally geodesic submanifold \(\mathbb{S}^2\times\mathbb{S}^1\), and the surface is described by a solution of the sinh-Gordon equation, with no restriction that the solution depend on only one variable.

collapse from coupled system to one-variable reduction

The stem names the setting and asks for the complete classification, but it does not reveal the specific conclusion. The correct answer is not stated or strongly hinted beyond the topic itself.

This is still very close to a direct theorem-recall question: it asks for the complete classification of minimal-Willmore surfaces. However, the presence of subtly varied alternatives means it is not a pure verbatim restatement.

Selecting the correct option requires moderate reasoning/discrimination, especially distinguishing the complete statement from a weaker true version and from overgeneralized variants. Still, the task is mainly recognition of the theorem rather than genuine derivation.

The distractors are strong: one is weaker but true, others are plausible overstatements or property confusions, and they target realistic mathematical mistakes about completeness, generality, and PDE structure.

A solid theorem-classification MCQ with excellent distractors and no answer leakage, but it remains largely a recall/recognition item rather than a strongly generative reasoning question.


Question: A minimal-Willmore surface in \(\mathbb{S}^2\times\mathbb{S}^2\) means a surface \(x:\Sigma\to \mathbb{S}^2\times\mathbb{S}^2\) that is both minimal and Willmore, where a Willmore surface satisfies the Euler-Lagrange equation of the Willmore functional \(\mathcal W=\int_\Sigma (|\vec H|^2+K_{1212})\,dA\). A slice is a surface of the form \(\mathbb{S}^2\times\{p\}\) or \(\{p\}\times\mathbb{S}^2\), and a diagonal means a diagonal copy of \(\mathbb{S}^2\) in \(\mathbb{S}^2\times\mathbb{S}^2\). Which statement holds for every minimal-Willmore surface \(x:\Sigma\to \mathbb{S}^2\times\mathbb{S}^2\)?

Choices:
B. Either \(x\) is a special complex curve, namely a slice or a diagonal; or, after applying an isometry of \(\mathbb{S}^2\times\mathbb{S}^2\), the image of \(x\) is contained in a totally geodesic submanifold \(\mathbb{S}^2\times\mathbb{S}^1\), and the surface is described by a pair of independent solutions of the sinh-Gordon equation in two variables.
C. Either \(x\) is a special complex curve, namely a slice or a diagonal; or, after applying an isometry of \(\mathbb{S}^2\times\mathbb{S}^2\), the image of \(x\) is contained in a totally geodesic submanifold \(\mathbb{S}^2\times\mathbb{S}^1\).
D. Either \(x\) is a special complex curve, namely a slice or a diagonal; or the image of \(x\) is contained in a totally geodesic submanifold \(\mathbb{S}^2\times\mathbb{S}^1\), and every such non-complex minimal-Willmore surface is described by a solution of the sinh-Gordon equation depending on two variables.
E. Either \(x\) is a special complex curve, namely a slice or a diagonal; or, after applying an isometry of \(\mathbb{S}^2\times\mathbb{S}^2\), the image of \(x\) is contained in a totally geodesic submanifold \(\mathbb{S}^2\times\mathbb{S}^1\), and the surface is described by a solution of the sinh-Gordon equation, with no restriction that the solution depend on only one variable.

The stem defines the setting and terminology but does not reveal the classification result or explicitly hint at the one-variable sinh-Gordon reduction. The correct answer is not leaked by the wording of the question itself.

The item is essentially a theorem-identification question: the correct option states a precise classification theorem rather than requiring derivation from the definitions in the stem. However, it is not a pure tautology because the distractors alter meaningful logical strength and technical details.

Some reasoning is needed to choose the strongest true statement over weaker or overextended variants, especially regarding the one-variable sinh-Gordon condition. Still, this mainly tests precise recall/discrimination of a known result, not substantial generative mathematical reasoning.

The distractors are plausible and mathematically structured: one is weaker but true, while others introduce realistic confusions about dimensional dependence or strength of the conclusion. They are distinct and align with common failure modes in reading classification theorems.

A reasonably strong theorem-discrimination MCQ with no answer leakage and good distractors, but it leans more toward recall of a classification result than genuine generative reasoning.


\label{thm:A}
	Let $\Sigma$ be an $S$-integral reduction type with cuspidal part $\Sigma^{\csp}$ and let $\mathfrak p \not\in S$ be a prime of $K$.
	If $(a_1,\ldots,a_{g+n-1}) \in K_{\mathfrak p}^{g+n-1}$ is contained in the kernel of the matrix $M(\Sigma^{\csp})$ defined in \eqref{eq:MSigma}--\eqref{eq:matrixD}, let $\omega = \omega(\Sigma^{\csp}) \coloneqq a_1 \omega_1 + \dots + a_{g+n-1} \omega_{g+n-1}$ and let $c = c(P_0,\Sigma,\omega)$ be the constant defined by
	\eqref{eq:constants} and \eqref{eq:b-lambda}.
	Then
	\[
		\int_{P_0}^P \omega = c \quad \text{ for all } P \in \mathcal{Y}(\cO_{K,S})_\Sigma.
	\]

\begin{equation}
\label{eq:chabauty-condition-over-number-field-intro}
r + \# S + ([K : \bQ]-1)n < g + \rank \cO_K^\times + \#|D| + n_2(D) - 1,
\end{equation}

\begin{enumerate}[label=(\arabic*)]
	\item \label{item:algo-D-transversal-model}
	Extend $\cX$ to a regular model over~$\cO_K$. Ensure that the model is $D$-transversal over primes in~$S$ via blow-ups if necessary (see Section~\ref{sec:redtypes}).
	\item \label{item:algo-intersection-numbers}
	For each prime~$\fq$ of bad reduction for~$\cX$, determine the set of components and the intersection matrix of the mod-$\fq$ fibre~$\cX_{\fq}$.
	\item \label{item:algo-mw-basis}
	Find divisors $G_i\in \Div^0(Y)$ generating a full rank subgroup of the Mordell--Weil group $J(K)$.
	\item \label{item:algo-correction-divisors}
	Using \cite[Lemma~2.7]{affchab1}, determine the vertical correction divisors $\Phi_{\fq}(G_i)$ for each prime $\fq$ of $K$. This involves the Moore--Penrose pseudoinverse of the intersection matrix of the mod-$\fq$ fibre computed in step~\ref{item:algo-intersection-numbers}.
	\item \label{item:algo-log-differentials}
	Find a basis $\omega_1,\ldots,\omega_{g+n-1}$ of the space of log differentials $\rH^0(X,\Omega^1(D))$ and compute their residues $\Res_Q(\omega_j) \in k(Q)$ at all cusps $Q \in |D|$.
	\item \label{item:algo-units}
	For each $Q\in |D|$, find $\varepsilon_i \in \cO_{k(Q)}^\times$ generating a full rank subgroup of $\cO_{k(Q)}^\times$.
	\item \label{item:algo-prime-ideal-generators}
	For finitely many ``special'' primes $\fq$ of $K$, determine generators $\rho_{\fq}$ of $\fq$ in $K^\times\otimes {\bQ}$.
	Similarly, for $Q\in |D|$ and finitely many ``special'' primes $\lambda$ of $k(Q)$, determine generators $\pi_{\lambda}$ of $\lambda$ in $k(Q)^\times\otimes {\bQ}$.
	\item \label{item:algo-integrals}
	Find a method for calculating $\fp$-adic integrals of log differentials on $X$.
	\item \label{item:algo-chabauty-function}
	For each $S$-integral reduction type~$\Sigma$, determine the matrix $M(\Sigma^{\csp})$ and find a non-trivial element $(a_1,\ldots,a_{g+n-1})$ in the kernel. Compute the constant $c=c(P_0,\Sigma,\omega)$ for the  Chabauty differential $\omega = \sum_j a_j \omega_j$.
	\item \label{item:algo-chabauty-locus}
	On each residue disc, expand the $\fp$-adic analytic function $P \mapsto \int_{P_0}^P \omega$ into a power series and determine the locus where $\int_{P_0}^P \omega=c$.
\end{enumerate}

\begin{thmABC}\label{thm:A}
	Let $\Sigma$ be an $S$-integral reduction type with cuspidal part $\Sigma^{\csp}$ and let $\fp \not\in S$ be a prime of $K$.
	If $(a_1,\ldots,a_{g+n-1}) \in K_{\fp}^{g+n-1}$ is contained in the kernel of the matrix $M(\Sigma^{\csp})$ defined in \eqref{eq:MSigma}--\eqref{eq:matrixD}, let $\omega = \omega(\Sigma^{\csp}) \coloneqq a_1 \omega_1 + \dots + a_{g+n-1} \omega_{g+n-1}$ and let $c = c(P_0,\Sigma,\omega)$ be the constant defined by
	\eqref{eq:constants} and \eqref{eq:b-lambda}.
	Then
	\[
		\int_{P_0}^P \omega = c \quad \text{ for all } P \in \cY(\cO_{K,S})_\Sigma.
	\]
\end{thmABC}

\label{sec: introduction}

Let $K$ be a number field and $S$ a finite set of primes of $\mathcal{O}_K$.
Let $Y/K$ be a smooth affine curve and let $\mathcal{Y}/\cO_{K,S}$ be a
regular model of $Y$ over the ring of $S$-integers of $K$.
If $Y$ is hyperbolic, the set of $S$-integral points $\mathcal{Y}(\cO_{K,S})$ is finite by the theorems of Siegel, Mahler, and Faltings.
However, finding the points in $\mathcal{Y}(\cO_{K,S})$ remains a difficult open problem in general.
In \cite{affchab1} we presented a new approach addressing this question for curves satisfying the hypothesis
\begin{equation}
\label{eq:chabauty-condition-over-number-field-intro}
r + \# S + ([K : \mathbb Q]-1)n < g + \rank \cO_K^\times + \#|D| + n_2(D) - 1,
\end{equation}
an inequality involving the genus $g$, the Mordell--Weil rank $r$, the cardinality of~$S$, the degree and unit rank of $K$, and invariants $n=\# D(\overline{K})$ and $n_2(D) = \frac12\# (D(\mathbb C) \smallsetminus  D(\mathbb{R}))$ coming from the boundary $D$ of $Y$.
Our method is an $S$-integral analogue of the method of Chabauty--Coleman \cite{Cha41, Col85}.
For an auxiliary prime $\mathfrak p \not\in S$, the set $\mathcal{Y}(\cO_{K,S})$ is cut out inside $\mathcal{Y}(\cO_{\mathfrak p})$ by finitely many $\mathfrak p$-adic integrals of \emph{logarithmic differentials}, i.e.\ meromorphic differentials on a compactification of~$Y$ with at worst simple poles at $D$.
In this article we present an algorithm to determine these log differentials and the zeros of their integrals in practice, thus computing a finite subset of $\mathcal{Y}(\cO_{\mathfrak p})$ containing $\mathcal{Y}(\cO_{K,S})$.

\subsection{Main results}

Let $Y/K$ be given as $X\smallsetminus D$ where $X/K$ is a smooth projective curve and $D \neq \emptyset$ is a finite set of closed points called \emph{cusps}.
Let $S$ be a finite set of primes.
Let~$\mathcal{X}$ be a regular model of $X$ over the ring~$\cO_{K,S}$ of $S$-integers, i.e., a regular, flat, projective $\cO_{K,S}$-scheme with an isomorphism $\cX_{K} \cong X$ \cite[Definition~10.1.1]{liu2006algebraic}.
Let $\cD$ be the closure of~$D$ in~$\mathcal{X}$ and set $\mathcal{Y} \coloneqq \mathcal{X} \smallsetminus \cD$. Assume that $Y(K) \neq\emptyset$ and fix a base point $P_0 \in Y(K)$.
We use the following notation, which will be kept throughout this paper:
\begin{itemize}
	\item $r \coloneqq \rank J(K)$ the Mordell--Weil rank of the Jacobian $J$ of~$X$;
	\item $g$ the genus of~$X$;
	\item $\#|D| > 0$ the number of cusps;
	\item $n \coloneqq \#D(\overline{K})>0$ the number of geometric cusps;	\item $[K:\mathbb Q]n = n_1(D) + 2n_2(D)$ with $n_1(D) \coloneqq \#D(\mathbb{R})$ the number of real cusps and $n_2(D) = \frac12\# (D(\mathbb C) \smallsetminus  D(\mathbb{R}))$ the number of conjugate pairs of complex cusps.
	Here $D$ is viewed as a scheme over $\mathbb Q$ (rather than over $K$).
\end{itemize}

We partition the $S$-integral points
\[
	\mathcal{Y}(\cO_{K,S}) = \coprod_{\Sigma} \mathcal{Y}(\cO_{K,S})_{\Sigma},
\]
where $\Sigma$ runs through the finitely many $S$-integral \emph{reduction types} of $\mathcal{Y}$, see \Cref{sec:redtypes}.
In \cite[Proposition 3.13]{affchab1} we showed that the Abel--Jacobi image of $\mathcal{Y}(\cO_{K,S})_{\Sigma}$ inside the generalised Jacobian $J_Y(K)$ is contained in the Selmer set $\Sel(P_0,\Sigma)$, which is (empty or) a translate of a finitely generated abelian group of controllable rank and is defined using arithmetic intersection theory and the $D$-intersection map $\sigma\colon J_Y(K) \to V_D$, see \Cref{sec:affchab1}.
Let $\omega_1,\ldots,\omega_{g+n-1}$ be a basis of the space $\mathrm H^0(X,\Omega^1(D))$ of logarithmic differentials on $(X,D)$ with $\omega_1,\ldots,\omega_g$ holomorphic. Assume $n \geq 2$.
Our main result is:

\label{sec: introduction}

Let $K$ be a number field and $S$ a finite set of primes of $\mathcal{O}_K$.
Let $Y/K$ be a smooth affine curve and let $\mathcal{Y}/\cO_{K,S}$ be a
regular model of $Y$ over the ring of $S$-integers of $K$.
If $Y$ is hyperbolic, the set of $S$-integral points $\mathcal{Y}(\cO_{K,S})$ is finite by the theorems of Siegel, Mahler, and Faltings.
However, finding the points in $\mathcal{Y}(\cO_{K,S})$ remains a difficult open problem in general.
In \cite{affchab1} we presented a new approach addressing this question for curves satisfying the hypothesis
\begin{equation}
\label{eq:chabauty-condition-over-number-field-intro}
r + \# S + ([K : \mathbb Q]-1)n < g + \rank \cO_K^\times + \#|D| + n_2(D) - 1,
\end{equation}
an inequality involving the genus $g$, the Mordell--Weil rank $r$, the cardinality of~$S$, the degree and unit rank of $K$, and invariants $n=\# D(\overline{K})$ and $n_2(D) = \frac12\# (D(\mathbb C) \smallsetminus D(\mathbb{R}))$ coming from the boundary $D$ of $Y$.
Our method is an $S$-integral analogue of the method of Chabauty--Coleman \cite{Cha41, Col85}.
For an auxiliary prime $\mathfrak p \not\in S$, the set $\mathcal{Y}(\cO_{K,S})$ is cut out inside $\mathcal{Y}(\cO_{\mathfrak p})$ by finitely many $\mathfrak p$-adic integrals of \emph{logarithmic differentials}, i.e.\ meromorphic differentials on a compactification of~$Y$ with at worst simple poles at $D$.
In this article we present an algorithm to determine these log differentials and the zeros of their integrals in practice, thus computing a finite subset of $\mathcal{Y}(\cO_{\mathfrak p})$ containing $\mathcal{Y}(\cO_{K,S})$.

Let $Y/K$ be given as $X\smallsetminus D$ where $X/K$ is a smooth projective curve and $D \neq \emptyset$ is a finite set of closed points called \emph{cusps}.
Let $S$ be a finite set of primes.
Let~$\mathcal{X}$ be a regular model of $X$ over the ring~$\cO_{K,S}$ of $S$-integers, i.e., a regular, flat, projective $\cO_{K,S}$-scheme with an isomorphism $\cX_{K} \cong X$ \cite[Definition~10.1.1]{liu2006algebraic}.
Let $\cD$ be the closure of~$D$ in~$\mathcal{X}$ and set $\mathcal{Y} \coloneqq \mathcal{X} \smallsetminus \cD$. Assume that $Y(K) \neq\emptyset$ and fix a base point $P_0 \in Y(K)$.
We use the following notation, which will be kept throughout this paper:
\begin{itemize}
 \item $r \coloneqq \rank J(K)$ the Mordell--Weil rank of the Jacobian $J$ of~$X$;
 \item $g$ the genus of~$X$;
 \item $\#|D| > 0$ the number of cusps;
 \item $n \coloneqq \#D(\overline{K})>0$ the number of geometric cusps; \item $[K:\mathbb Q]n = n_1(D) + 2n_2(D)$ with $n_1(D) \coloneqq \#D(\mathbb{R})$ the number of real cusps and $n_2(D) = \frac12\# (D(\mathbb C) \smallsetminus D(\mathbb{R}))$ the number of conjugate pairs of complex cusps.
 Here $D$ is viewed as a scheme over $\mathbb Q$ (rather than over $K$).
\end{itemize}

We partition the $S$-integral points
\[
 \mathcal{Y}(\cO_{K,S}) = \coprod_{\Sigma} \mathcal{Y}(\cO_{K,S})_{\Sigma},
\]
where $\Sigma$ runs through the finitely many $S$-integral \emph{reduction types} of $\mathcal{Y}$, see \Cref{sec:redtypes}.
In \cite[Proposition 3.13]{affchab1} we showed that the Abel--Jacobi image of $\mathcal{Y}(\cO_{K,S})_{\Sigma}$ inside the generalised Jacobian $J_Y(K)$ is contained in the Selmer set $\Sel(P_0,\Sigma)$, which is (empty or) a translate of a finitely generated abelian group of controllable rank and is defined using arithmetic intersection theory and the $D$-intersection map $\sigma\colon J_Y(K) \to V_D$, see \Cref{sec:affchab1}.
Let $\omega_1,\ldots,\omega_{g+n-1}$ be a basis of the space $\mathrm H^0(X,\Omega^1(D))$ of logarithmic differentials on $(X,D)$ with $\omega_1,\ldots,\omega_g$ holomorphic. Assume $n \geq 2$.
Our main result is:

\begin{equation}
	\label{eq:MSigma}
	M(U) \coloneqq \begin{pmatrix}
		A & B  \\
		0 & C  \\
		0 & D(U)
	\end{pmatrix}
\end{equation}

\begin{equation}
\label{eq:b-lambda}
b_{\lambda} = -i_{\lambda}(\cP_0,\tilde\cQ) + i_{\lambda}(\Phi_{\fq}(\cpt(\Sigma_{\fq}) - \cpt_{\fq}(P_0)), \tilde\cQ),
\end{equation}

\begin{equation}
	\label{eq:constants}
	c = c(b, \omega) \coloneqq \sum_{\substack{Q \in |D|,\\ \varphi\colon k(Q) \to \Kpbar}} \varphi(\Res_Q \omega) \sum_{\textnormal{$\lambda$ prime of $\cO_{k(Q)}$}} b_{\lambda} \log \varphi(\pi_{\lambda}).
	\end{equation}

\begin{equation}
		\label{eq:matrixD}
		D(U)_{i,j} \coloneqq \sum_{\substack{Q \in |D|,\\ \varphi\colon k(Q) \to \Kpbar}} \varphi(\Res_{Q} \omega_{g+j}) \sum_{\substack{\text{$\fq$ prime,}\\ \text{$\lambda|\fq$ in $\cO_{k(Q)}$}}} u_{i,\lambda} \log \varphi(\pi_{\lambda}).
	\end{equation}

Thus $\cY(\cO_{K,S})_{\Sigma}$ is contained in the set of zeros of the $\fp$-adic analytic function $\cY(\cO_{\fp}) \to K_{\fp}$ that sends $P$ to $\int_{P_0}^P \omega - c$.
The constant $c(P_0,\Sigma,\omega)$ and the entries of the matrix $M(\Sigma^{\csp})$ are given by explicit formulas in terms of $\fp$-adic integrals of the basis differentials $\omega_j$, the residues of the $\omega_j$ at the cusps, the $\fp$-adic logarithm of certain elements of the residue fields of the cusps, and intersection numbers on the regular model~$\cX$.

\begin{thm}
 \label{thm:explicit-function}
 If $(a_1,\ldots,a_{g+n-1}) \in K_{\fp}^{g+n-1}$ is contained in the kernel of the matrix $M(U)$ defined in equations~\eqref{eq:MSigma}--\eqref{eq:matrixD}, let $\omega = \omega(U) \coloneqq a_1 \omega_1 + \dots + a_{g+n-1} \omega_{g+n-1}$.
 Define
 \begin{equation}
 \label{eq:constants}
 c = c(b, \omega) \coloneqq \sum_{\substack{Q \in |D|,\\ \varphi\colon k(Q) \to \Kpbar}} \varphi(\Res_Q \omega) \sum_{\textnormal{$\lambda$ prime of $\cO_{k(Q)}$}} b_{\lambda} \log \varphi(\pi_{\lambda}).
 \end{equation}
 Then
 \[
 \int_F \omega = c \quad \text{ for all } F \in b_0 + W.
 \]
\end{thm}

We choose the base point $P_0 = (-1,1) \in \cY(\bZ)$ and the auxiliary prime~$p = 7$.
Setting $P_0 = (0,3)$ and $P_1 = (1,3)$, the classes of the divisors $G_i = P_i - P_0$ ($i = 1,2$) generate a subgroup of finite index in the Mordell--Weil group. The intersection numbers $i_{\ell}(\Psi_{\ell}(G_i), \cQ_+' - \cQ_-')$ in~\eqref{eq:hyp-M-entries} are all zero: for the horizontal part $i_{\ell}(\cP_i - \cP_0, \cQ_+' - \cQ_-')$ this follows from $\cP_0,\cP_1,\cP_2$ being integral points. The vertical divisors~$V$ in $\cX_{\ell}'$ we have $i_{\ell}(V, \cQ_+' - \cQ_-') = 0$ since the points $Q_{+}$ and $Q_-$ reduce onto the same component of~$\cX_{\ell}'$. For $\ell \neq 3$ this is clear since~$\cX'_{\ell}$ has only one component; for $\ell = 3$ this follows from the fact that the points at infinity reduce to smooth points of the mod-3 fibre of the non-regular model~$\cX$, and since the desingularisation $\cX' \to \cX$ is an isomorphism over the non-singular locus, they still reduce onto the same component of~$\cX_3'$. We find that the matrix~$M$ is given by
\begin{equation*}
 M = \begin{pmatrix}
 \int_{P_0}^{P_1} \omega_0 & \int_{P_0}^{P_1} \omega_1 & \int_{P_0}^{P_1} \omega_2 \\[1mm]
 \int_{P_0}^{P_2} \omega_0 & \int_{P_0}^{P_2} \omega_1 & \int_{P_0}^{P_2} \omega_2
 \end{pmatrix},
\end{equation*}
without the correction terms in the last column. A nontrivial element of the kernel is $(a_0,a_1,a_2) \in \bQ_7^3$ with
\begin{align*}
 a_0 &= 1 + O(7^8), \\
 a_1 &= 5 + 3\cdot 7 + 3\cdot 7^2 + 5\cdot 7^3 + 3\cdot 7^4 + 2\cdot 7^6 + 2\cdot 7^7 + O(7^8), \\
 a_2 &= 5 + 6\cdot 7 + 6\cdot 7^2 + 7^3 + 4\cdot 7^4 + 6\cdot 7^5 + 5\cdot 7^6 + 3\cdot 7^7 + O(7^8).
\end{align*}
Setting $\omega \coloneqq a_0 \omega_0 + a_1 \omega_1 + a_2 \omega_2$, the function $\rho\colon \cY(\bZ_p) \to \bQ_p$, $\rho(P) = \int_{P_0}^P \omega$ takes a constant value $c(P_0,\Sigma,\omega)$ on all integral points of a fixed reduction type~$\Sigma$. By the same argument as above, the intersection numbers in~\eqref{eq:split-hyperelliptic-constants-final} vanish, so the constants $c(P_0,\Sigma,\omega)$ are all zero. The function~$\rho$ thus vanishes on all of~$\cY(\bZ)$.
We can analyse the zeros of~$\rho$ on each residue disc by computing the power series expansion in a uniformising parameter. For example, consider the point $P_0 = (-1,1)$. Its residue disc is parametrised by $Q(t) = (x(t), y(t))$ with
\begin{align*}
 x(t) = -1 + 7t,\quad
 y(t) = \sqrt{f(x(t))} = \sum_{n=0}^\infty \binom{1/2}{n}(f(-1 + 7t)-1)^n \end{align*}
and in terms of the parameter~$t$, the function~$\rho$ is given by
\begin{align*}
 \rho(Q(t)) = \sum_{j=0}^2 a_j \int_{P_0}^{Q(t)} x(t)^j \frac{\rd x(t)}{y(t)} = (7 + 3\cdot 7^2) t + 6\cdot 7^2 \cdot t^2 + O(7^3,t^3)
\end{align*}
The sequence of valuations of the coefficients starts $(\infty, 1, 2, 3, 4, 6, 7, \ldots)$, so by Strassmann's Theorem (see for example \cite[Theorem~5.6.1]{gouvea}), the only zero of $\rho(Q(t))$ in~$\bZ_7$ is at $t = 0$. This shows that $P_0 = (-1,1)$ is the only integral point of~$\cY$ in its residue disc. Each of the remaining residue discs contains one of the known integral points, and in each case one can show with Strassmann's Theorem that there are no other integral points in that disc, proving \Cref{thm:example-split}. Sage code for computing the vanishing locus of the function~$\rho$ in this example, for any auxiliary prime~$p$, can be found at \url{https://github.com/martinluedtke/AffChab2}.

Choose the auxiliary prime $p=7$, a primitive third root of unity $\zeta_3\in \bQ_p$ and put $\sqrt{-3}\coloneqq 1+2\zeta_3\in\bQ_p$.
 Then the matrix $M(\Sigma^{\csp})$ of \eqref{eq:MSigma} is equal to
 \begin{equation*}
 M(\Sigma^{\csp}) = \begin{pmatrix}
 \int_{P_0}^A \omega_1 & \int_{P_0}^A \omega_2 - \beta_2 & \int_{P_0}^A \omega_3 - \beta_3\\
 0 & D(\Sigma^{\csp})_2 & D(\Sigma^{\csp})_3
 \end{pmatrix},
 \end{equation*}
 where using \eqref{eq:H} we have
 \begin{align*}
 \beta_2 &=
(-1) \left(i_{\pi_2}(\cA - \cP_0, \cQ_1) \log(2) + i_{\pi_3}(\cA - \cP_0, \cQ_1) \log(3)\right) \\
 &\qquad + (-\zeta_3) i_{\pi_{(\sqrt{-3})}}(\cA - \cP_0, \cQ_2) \log(\sqrt{-3}) + (-\zeta_3^{-1}) i_{\pi_{(\sqrt{-3})}}(\cA - \cP_0, \cQ_2) \log(-\sqrt{-3}) \\
 &= -\log(2) - \frac12 \log(3)
 \end{align*}
 because $A$ reduces onto $\cQ_1$ modulo $2$ and onto $\cQ_1 \cap \cQ_2$ modulo $3$; $P_0$ never reduces to a cusp; $i_{\pi_\bullet}(\cA,\cQ_i)=1$ in all cases; and $\log(\sqrt{-3})=\log(3)/2$.
 A similar calculation shows $\beta_3 = \beta_2$.
 The entries $D(\Sigma^{\csp})_i$ are computed using \eqref{eq:matrixD}:
 \begin{align*}
 D(\Sigma^{\csp})_2 &= -\zeta_3 \log(2\zeta_3 +23) - \zeta_3^{-1} \log(2\zeta_3^{-1} + 23),\\
 D(\Sigma^{\csp})_3 &= -\zeta_3^{-1} \log(2\zeta_3 +23) - \zeta_3 \log(2\zeta_3^{-1} + 23).
 \end{align*}
 We also calculate the integrals $\int_{P_0}^A \omega_j$ with the method outlined above resulting in the matrix
 \begin{equation*}
 M(\Sigma^{\csp})= \footnotesize{
 \begin{pmatrix}
 2\cdot 7 + 5\cdot 7^2 + 4 \cdot 7^4 + 5 \cdot 7^5 & 6\cdot 7 + 7^2 + 3\cdot 7^4 & 2\cdot 7 + 6\cdot 7^2 + 2\cdot 7^3 \\
 0 & 6\cdot 7^2 + 2\cdot 7^3 + 6\cdot 7^4 & 7 + 4\cdot 7^2 + 7^3 + 5\cdot 7^4 + 4\cdot 7^5
 \end{pmatrix}
 }
 \end{equation*}
 displayed with precision $O(7^6)$.
 The element
 \[
 (a_1,a_2,a_3) = (1, 2 + 6\cdot 7 + 2\cdot 7^2 + 3\cdot 7^3 + 4\cdot 7^5, 2\cdot 7 + 6\cdot 7^2 + 2\cdot 7^5) + O(7^6)
 \]
 lies in the kernel of $M(\Sigma^{\csp})$, so the differential $\omega = a_1\omega_1 + a_2 \omega_2 + a_3\omega_3$ vanishes on all $\{487\}$-integral points of reduction type $\Sigma$; note that the constant $c(P_0,\Sigma,\omega)$ is zero by \eqref{eq:constants} and $b=0$ by (5) above.
 The point $(x,y)=(\tfrac{216}{487},\tfrac{438}{487})$ is a $\{487\}$-integral point on $\cY$ which has reduction type $\Sigma$.
 We checked numerically that indeed
 \[
 \int_{P_0}^P \omega = O(7^{14}).
 \]
 In order to fully determine the Chabauty locus $\cY(\bZ_7)_{S,\Sigma}$, one now needs to determine the zeros of $P\mapsto \int_{P_0}^{P} \omega$ on each residue disc.
 To give the full Chabauty locus $\cY(\bZ_7)_{S}$, one then has to similarly deal with the other $\{487\}$-integral reduction types.
 One can apply the same procedure for $S$-integral points for any singleton set $S$.
 The code for this example can be found on \url{https://github.com/martinluedtke/AffChab2}.
\end{example}

Thus $\mathcal{Y}(\cO_{K,S})_{\Sigma}$ is contained in the set of zeros of the $\mathfrak p$-adic analytic function $\mathcal{Y}(\cO_{\mathfrak p}) \to K_{\mathfrak p}$ that sends $P$ to $\int_{P_0}^P \omega - c$.
The constant $c(P_0,\Sigma,\omega)$ and the entries of the matrix $M(\Sigma^{\csp})$ are given by explicit formulas in terms of $\mathfrak p$-adic integrals of the basis differentials $\omega_j$, the residues of the $\omega_j$ at the cusps, the $\mathfrak p$-adic logarithm of certain elements of the residue fields of the cusps, and intersection numbers on the regular model~$\mathcal{X}$.

To prove \Cref{thm:A}, let $\log_{J_Y}\colon J_Y(K_{\mathfrak p}) \to \mathrm H^0(X_{K_{\mathfrak p}},\Omega^1(D))^\vee$ be the logarithm of the $\mathfrak p$-adic Lie group $J_Y(K_{\mathfrak p})$, which we restrict to the direction $W$ of $\Sel(P_0,\Sigma)$.
We show that the matrix $M(\Sigma^{\csp})$ represents the dual map $\log_{J_Y}^\sharp \colon \mathrm H^0(X_{K_{\mathfrak p}}, \Omega^1(D)) \to W^\vee\otimes_{\mathbb Q} K_{\mathfrak p}$ with respect to a certain explicit basis of $W$.
This basis is constructed in \eqref{eq:F-and-Gi}, \eqref{eq:f-of-Q} and \eqref{eq:t-ell} and consists of dual bases of the Mordell--Weil group $J(K)$ of the Jacobian of $X$, of the unit groups at the cusps, and of a basis of $\sigma(W)=U$.
For $F\in W$ and $\omega$ a logarithmic differential, the fundamental calculation of $\log_{J_Y}^\sharp(\omega)(F) = \int_F \omega$ reduces to calculating $\int_{\mathrm{div}(f)} \omega$ for a certain rational function~$f$.
A $p$-adic residue theorem, proved in \Cref{sec:integration}, expresses $\int_{\mathrm{div}(f)} \omega$ in terms of the residues of $\omega$ and the values of $f$ at the cusps.
The latter can be read off from $\sigma(F)$, allowing us to conclude.
The main insight is how to combine the $D$-intersection map $\sigma$ with $\mathfrak p$-adic integration.

\subsection{Affine Chabauty algorithm}
\label{sec:algorithm}

\Cref{thm:A} is the central step in building an algorithm based on the Affine Chabauty method.
That is, on input an affine curve $Y=X\smallsetminus D$, a finite set of primes $S$ satisfying \eqref{eq:chabauty-condition-over-number-field-intro}, a regular model $\mathcal{Y} = \mathcal{X} \smallsetminus \cD$ over $\cO_{K,S}$, and an auxiliary prime $\mathfrak p \not\in S$, we want to determine a finite subset of $\mathcal{Y}(\cO_{\mathfrak p})$ containing $\mathcal{Y}(\cO_{K,S})$.
Let us give a sketch of how such an algorithm proceeds:
\begin{enumerate}[label=(\arabic*)]
	\item \label{item:algo-D-transversal-model}
	Extend $\mathcal{X}$ to a regular model over~$\cO_K$. Ensure that the model is $D$-transversal over primes in~$S$ via blow-ups if necessary (see Section~\ref{sec:redtypes}).
	\item \label{item:algo-intersection-numbers}
	For each prime~$\mathfrak q$ of bad reduction for~$\mathcal{X}$, determine the set of components and the intersection matrix of the mod-$\mathfrak q$ fibre~$\cX_{\mathfrak q}$.
	\item \label{item:algo-mw-basis}
	Find divisors $G_i\in \Div^0(Y)$ generating a full rank subgroup of the Mordell--Weil group $J(K)$.
	\item \label{item:algo-correction-divisors}
	Using \cite[Lemma~2.7]{affchab1}, determine the vertical correction divisors $\Phi_{\mathfrak q}(G_i)$ for each prime $\mathfrak q$ of $K$. This involves the Moore--Penrose pseudoinverse of the intersection matrix of the mod-$\mathfrak q$ fibre computed in step~\ref{item:algo-intersection-numbers}.
	\item \label{item:algo-log-differentials}
	Find a basis $\omega_1,\ldots,\omega_{g+n-1}$ of the space of log differentials $\mathrm H^0(X,\Omega^1(D))$ and compute their residues $\Res_Q(\omega_j) \in k(Q)$ at all cusps $Q \in |D|$.
	\item \label{item:algo-units}
	For each $Q\in |D|$, find $\varepsilon_i \in \cO_{k(Q)}^\times$ generating a full rank subgroup of $\cO_{k(Q)}^\times$.
	\item \label{item:algo-prime-ideal-generators}
	For finitely many ``special'' primes $\mathfrak q$ of $K$, determine generators $\rho_{\mathfrak q}$ of $\mathfrak q$ in $K^\times\otimes {\mathbb Q}$.
	Similarly, for $Q\in |D|$ and finitely many ``special'' primes $\lambda$ of $k(Q)$, determine generators $\pi_{\lambda}$ of $\lambda$ in $k(Q)^\times\otimes {\mathbb Q}$.
	\item \label{item:algo-integrals}
	Find a method for calculating $\mathfrak p$-adic integrals of log differentials on $X$.
	\item \label{item:algo-chabauty-function}
	For each $S$-integral reduction type~$\Sigma$, determine the matrix $M(\Sigma^{\csp})$ and find a non-trivial element $(a_1,\ldots,a_{g+n-1})$ in the kernel. Compute the constant $c=c(P_0,\Sigma,\omega)$ for the  Chabauty differential $\omega = \sum_j a_j \omega_j$.
	\item \label{item:algo-chabauty-locus}
	On each residue disc, expand the $\mathfrak p$-adic analytic function $P \mapsto \int_{P_0}^P \omega$ into a power series and determine the locus where $\int_{P_0}^P \omega=c$.
\end{enumerate}

\Cref{thm:A} guarantees that the union of the sets obtained in step~\ref{item:algo-chabauty-locus} over all $S$-integral reduction types~$\Sigma$ is a finite subset of $\mathcal{Y}(\cO_{\mathfrak p})$ containing $\mathcal{Y}(\cO_{K,S})$. The central step and main focus of this article is~\ref{item:algo-chabauty-function}. The results of all prior steps enter in the computation of the matrices $M(\Sigma^{\csp})$ and the constants $c(P_0,\Sigma,\omega)$, as described in detail in Section~\ref{sec:explicit}.

In individual examples, calculating intersection numbers \ref{item:algo-intersection-numbers} and a basis of log differentials \ref{item:algo-log-differentials} can be done by hand. For hyperelliptic curves and the superelliptic curves considered in Section~\ref{sec:examples}, we are able to compute $\mathfrak p$-adic integrals of logarithmic differentials \ref{item:algo-integrals} by reducing it to integrals of the particular differential $x^g \, \mathrm d x/y$ on various hyperelliptic curves of genus~$g$, for which we can use existing code \cite{BBK10, BB12}.
There are methods in Magma and Sagemath for computing regular models, Mordell--Weil bases, unit groups and generators of primes ideals, as required in steps \ref{item:algo-D-transversal-model}, \ref{item:algo-intersection-numbers}, \ref{item:algo-mw-basis}, \ref{item:algo-units}, \ref{item:algo-prime-ideal-generators}.

\subsection{Structure of the paper}

\Cref{sec:affchab1} reviews the Affine Chabauty method developed in \cite{affchab1}.
We define intersection numbers, the $D$-intersection map, reduction types, and the Selmer set.
In \Cref{sec:integration} we prove the residue theorem for $p$-adic integrals of logarithmic differentials.
In \Cref{sec:explicit} we define the matrix $M(\Sigma^{\csp})$ and prove the main result.
We illustrate our method with several examples in \Cref{sec:examples}.

\subsection*{Acknowledgements}

The first author is supported by a Walter Benjamin Scholarship from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation), project number LE 5634/1, and also acknowledges support from the DFG through TRR 326 Geometry and Arithmetic of Uniformized Structures, project number 444845124. The second author is supported by a Minerva Fellowship of the Minerva Stiftung Gesellschaft für die Forschung mbH and also acknowledges support through a guest postdoc fellowship at the Max Planck Institute for Mathematics in Bonn and through an NWO Grant, project number VI.Vidi.192.106.

To prove \Cref{thm:A}, consider the \(\mathfrak p\)-adic Lie group logarithm \(\log_{J_Y}\colon J_Y(K_{\mathfrak p}) \to \mathrm H^0(X_{K_{\mathfrak p}},\Omega^1(D))^\vee\), restricted to the direction \(W\) of \(\Sel(P_0,\Sigma)\). Show that the matrix \(M(\Sigma^{\csp})\) represents the dual map \(\log_{J_Y}^\sharp\colon \mathrm H^0(X_{K_{\mathfrak p}}, \Omega^1(D)) \to W^\vee\otimes_{\mathbb Q} K_{\mathfrak p}\) with respect to an explicit basis of \(W\). This basis (from \eqref{eq:F-and-Gi}, \eqref{eq:f-of-Q}, \eqref{eq:t-ell}) consists of dual bases coming from the Mordell--Weil group \(J(K)\), unit groups at the cusps, and a basis of \(\sigma(W)=U\). For \(F\in W\) and a logarithmic differential \(\omega\), the key computation \(\log_{J_Y}^\sharp(\omega)(F)=\int_F\omega\) is reduced to computing \(\int_{\mathrm{div}(f)}\omega\) for a suitable rational function \(f\). A \(p\)-adic residue theorem (\Cref{sec:integration}) expresses \(\int_{\mathrm{div}(f)}\omega\) in terms of residues of \(\omega\) and values of \(f\) at the cusps; these values can be read off from \(\sigma(F)\), which allows one to conclude. The stated relation \(\int_{P_0}^P\omega=c\) for \(P\in\mathcal{Y}(\cO_{K,S})_\Sigma\) then follows from taking \((a_1,\ldots,a_{g+n-1})\) in the kernel of the representing matrix \(M(\Sigma^{\csp})\), i.e. combining the \(D\)-intersection map \(\sigma\) with \(\mathfrak p\)-adic integration.

To prove the main theorem, consider the \(\mathfrak p\)-adic Lie group logarithm \(\log_{J_Y}\colon J_Y(K_{\mathfrak p}) \to \mathrm H^0(X_{K_{\mathfrak p}},\Omega^1(D))^\vee\), restricted to the direction \(W\) of \(\Sel(P_0,\Sigma)\). Show that the matrix \(M(\Sigma^{\csp})\) represents the dual map \(\log_{J_Y}^\sharp\colon \mathrm H^0(X_{K_{\mathfrak p}}, \Omega^1(D)) \to W^\vee\otimes_{\mathbb Q} K_{\mathfrak p}\) with respect to an explicit basis of \(W\). This basis (from
\begin{equation}
		\label{eq:F-and-Gi}
		F = \sum_{i=1}^r x_i(F) G_i + \div(f)
	\end{equation}
,
\begin{equation}
		\label{eq:f-of-Q}
		(f(Q))_Q = \prod_{i=1}^{k} e_{i}^{a_{i}(F)} \cdot  \left(\prod_{\text{$\lambda$ prime of $\cO_{k(Q)}$}} \pi_{\lambda}^{v_\lambda(f(Q))}\right)_Q \quad \text{in} \quad \left.\left(\prod_Q k(Q)^\times_{\bQ}\right)\right/K_{\bQ}^\times,
	\end{equation}
,
\begin{equation}
		\label{eq:t-ell}
		\sigma(F) = \sum_{i=1}^s t_{i}(F) u_i \qquad \text{in $V_D$},
	\end{equation}
) consists of dual bases coming from the Mordell--Weil group \(J(K)\), unit groups at the cusps, and a basis of \(\sigma(W)=U\). For \(F\in W\) and a logarithmic differential \(\omega\), the key computation \(\log_{J_Y}^\sharp(\omega)(F)=\int_F\omega\) is reduced to computing \(\int_{\mathrm{div}(f)}\omega\) for a suitable rational function \(f\). A \(p\)-adic residue theorem (We discuss this later.) expresses \(\int_{\mathrm{div}(f)}\omega\) in terms of residues of \(\omega\) and values of \(f\) at the cusps; these values can be read off from \(\sigma(F)\), which allows one to conclude. The stated relation \(\int_{P_0}^P\omega=c\) for \(P\in\mathcal{Y}(\cO_{K,S})_\Sigma\) then follows from taking \((a_1,\ldots,a_{g+n-1})\) in the kernel of the representing matrix \(M(\Sigma^{\csp})\), i.e. combining the \(D\)-intersection map \(\sigma\) with \(\mathfrak p\)-adic integration.

\label{thm:A}
	Let $\Sigma$ be an $S$-integral reduction type with cuspidal part $\Sigma^{\csp}$ and let $\mathfrak p \not\in S$ be a prime of $K$.
	If $(a_1,\ldots,a_{g+n-1}) \in K_{\mathfrak p}^{g+n-1}$ is contained in the kernel of the matrix
	\begin{equation}
	\label{eq:MSigma}
	M(U) \coloneqq \begin{pmatrix}
		A & B  \\
		0 & C  \\
		0 & D(U)
	\end{pmatrix}
\end{equation}
	with
	\begin{equation}
		\label{eq:matrixD}
		D(U)_{i,j} \coloneqq \sum_{\substack{Q \in |D|,\\ \varphi\colon k(Q) \to \Kpbar}} \varphi(\Res_{Q} \omega_{g+j}) \sum_{\substack{\text{$\fq$ prime,}\\ \text{$\lambda|\fq$ in $\cO_{k(Q)}$}}} u_{i,\lambda} \log \varphi(\pi_{\lambda}).
	\end{equation}
	(evaluated at $U=\Sigma^{\csp}$), let $\omega = \omega(\Sigma^{\csp}) \coloneqq a_1 \omega_1 + \dots + a_{g+n-1} \omega_{g+n-1}$ and let $c = c(P_0,\Sigma,\omega)$ be the constant defined by
	\begin{equation}
	\label{eq:constants}
	c = c(b, \omega) \coloneqq \sum_{\substack{Q \in |D|,\\ \varphi\colon k(Q) \to \Kpbar}} \varphi(\Res_Q \omega) \sum_{\textnormal{$\lambda$ prime of $\cO_{k(Q)}$}} b_{\lambda} \log \varphi(\pi_{\lambda}).
	\end{equation}
	and
	\begin{equation}
\label{eq:b-lambda}
b_{\lambda} = -i_{\lambda}(\cP_0,\tilde\cQ) + i_{\lambda}(\Phi_{\fq}(\cpt(\Sigma_{\fq}) - \cpt_{\fq}(P_0)), \tilde\cQ),
\end{equation}
	Then
	\[
		\int_{P_0}^P \omega = c \quad \text{ for all } P \in \mathcal{Y}(\cO_{K,S})_\Sigma.
	\],

Question:
Let $K$ be a number field, $S$ a finite set of primes of $\mathcal O_K$, $Y=X\setminus D$ with $X/K$ a smooth projective curve and $D\neq\varnothing$ a finite set of closed points, and let $\mathcal X/\mathcal O_{K,S}$ be a regular model with $\mathcal Y=\mathcal X\setminus \mathcal D$. Fix a base point $P_0\in Y(K)$, write $g$ for the genus of $X$ and $n=\#D(\overline K)$, and choose a basis $\omega_1,\dots,\omega_{g+n-1}$ of $H^0(X,\Omega^1(D))$ (the logarithmic differentials on $(X,D)$). For a fixed $S$-integral reduction type $\Sigma$, let $\mathcal Y(\mathcal O_{K,S})_\Sigma$ denote the set of $S$-integral points of reduction type $\Sigma$, and let $\Sigma^{\mathrm{csp}}$ be its cuspidal part. Let $\mathfrak p\notin S$ be a prime of $K$. Suppose $(a_1,\ldots,a_{g+n-1})\in K_{\mathfrak p}^{g+n-1}$ lies in the kernel of the matrix
\[
M(U)=\begin{pmatrix}A&B\\0&C\\0&D(U)\end{pmatrix}
\]
evaluated at $U=\Sigma^{\mathrm{csp}}$, where
\[
D(U)_{i,j}=\sum_{\substack{Q\in |D|,\\ \varphi:k(Q)\to \overline{K_{\mathfrak p}}}} \varphi(\operatorname{Res}_Q\omega_{g+j})\sum_{\substack{\mathfrak q\text{ prime},\\ \lambda\mid \mathfrak q\text{ in }\mathcal O_{k(Q)}}} u_{i,\lambda}\log \varphi(\pi_\lambda).
\]
Set
\[
\omega=a_1\omega_1+\cdots+a_{g+n-1}\omega_{g+n-1},
\]
and define
\[
c=c(P_0,\Sigma,\omega)=\sum_{\substack{Q\in |D|,\\ \varphi:k(Q)\to \overline{K_{\mathfrak p}}}} \varphi(\operatorname{Res}_Q\omega)\sum_{\lambda\text{ prime of }\mathcal O_{k(Q)}} b_\lambda\log \varphi(\pi_\lambda),
\]
with
\[
b_\lambda=-i_\lambda(\mathcal P_0,\widetilde{\mathcal Q})+i_\lambda\bigl(\Phi_{\mathfrak q}(\operatorname{cpt}(\Sigma_{\mathfrak q})-\operatorname{cpt}_{\mathfrak q}(P_0)),\widetilde{\mathcal Q}\bigr).
\]
Which statement holds for every point $P\in \mathcal Y(\mathcal O_{K,S})_\Sigma$?

For every $P\in \mathcal Y(\mathcal O_{K,S})_\Sigma$, the $\mathfrak p$-adic integral of $\omega$ from $P_0$ to $P$ equals the constant $c$; equivalently, \[\int_{P_0}^P \omega = c(P_0,\Sigma,\omega).\]

Choices:
B. For every $P\in \mathcal Y(\mathcal O_{K,S})_\Sigma$, the $\mathfrak p$-adic integral of $\omega$ from $P_0$ to $P$ vanishes; equivalently, \[\int_{P_0}^P \omega = 0.\]
C. For every $P\in \mathcal Y(\mathcal O_{K,S})_\Sigma$, the value of the $\mathfrak p$-adic integral $\int_{P_0}^P \omega$ is independent of the choice of $P$; equivalently, there exists a constant in $K_{\mathfrak p}$, depending on $P_0$, $\Sigma$, and $\omega$, such that \[\int_{P_0}^P \omega\] is the same for all $P\in \mathcal Y(\mathcal O_{K,S})_\Sigma$.
D. For every $P\in \mathcal Y(\mathcal O_{K,S})$, the $\mathfrak p$-adic integral of $\omega$ from $P_0$ to $P$ equals the constant $c(P_0,\Sigma,\omega)$; equivalently, \[\int_{P_0}^P \omega = c(P_0,\Sigma,\omega).\]
E. If $(a_1,\ldots,a_{g+n-1})\in K_{\mathfrak p}^{g+n-1}$ lies in the kernel of $M(U)$ for some $U$, then for every $P\in \mathcal Y(\mathcal O_{K,S})_\Sigma$ one has \[\int_{P_0}^P \omega = c(P_0,\Sigma,\omega),\] regardless of whether $U=\Sigma^{\mathrm{csp}}$.

explicit nonzero constant c from residue computation

dropped identification of the common value with the explicit formula c(P_0,\Sigma,\omega)

dependence on the fixed reduction type \Sigma

evaluation of the matrix at U=\Sigma^{\mathrm{csp}}

The stem does not explicitly state the conclusion that the integral equals c(P_0,\Sigma,\omega). It introduces the relevant objects and hypotheses, but the correct answer is not directly leaked.

This is essentially a direct restatement of a theorem/proposition: under the kernel hypothesis for M(\Sigma^{\mathrm{csp}}), conclude that \int_{P_0}^P \omega equals the explicit constant c(P_0,\Sigma,\omega). The correct option reproduces that conclusion almost verbatim.

Some precision is needed to distinguish the exact theorem statement from nearby variants (weaker true statement C, overgeneralization D, and quantifier error E), but the item mainly tests recall of the theorem rather than substantial generative reasoning.

The distractors are mathematically meaningful and target realistic failure modes: forgetting the nonzero constant term (B), choosing a weaker but true conclusion (C), overextending beyond the fixed reduction type (D), and ignoring the dependence on evaluating at U=\Sigma^{\mathrm{csp}} (E).

Technically well-constructed in its distractors, but it is largely a theorem-recall item rather than a genuinely non-tautological or generative reasoning question.


\begin{remark} \label{rmk: cusp exists}
\begin{enumerate}
    \item  Every hyperbolic lattice of rank $\ge 20$ admits a cusp with infinite stabilizer. Indeed, every hyperbolic lattice of rank $\ge 20$ has an infinite symmetry group (cf. \cite{nikulin.rank.greater.than.five}), and therefore admits a cusp with infinite stabilizer by \cite[Theorem~6.4.1]{nikulin.rank.greater.than.five}.

    \item Note that \Cref{prop:exceptionalIntersection} fails if $L$ admits no cusps with infinite stabilizer. As an example, let $L$ a hyperbolic lattice of rank $3$ with $\Aut(\D_L)\cong \ZZ$ and no cusp with infinite stabilizer. Such lattices exist, and were classified in \cite{brandhorst.mezzedimi.abelian}; for instance we can take $L=U(11)\oplus A_1$. Then $\Aut(\D_L)$ is generated by a symmetry $f$ with infinite order and Salem degree $2$, and thus $f$ preserves a line in $L$. Hence $E(L)$ has rank $1$, while $E'$ is $L$ itself.
\end{enumerate}
\end{remark}

\subsection{Symmetries and exceptional lattice}

Let $L$ be an even, integral hyperbolic lattice. The quotient $\mathrm{Aut}(\D_L)=\mathrm{O}^+(L)/W(L)$ of the group of positive isometries of $L$ by the Weyl subgroup of reflections along $(-2)$-roots is called the \emph{symmetry group} of $L$. Foundational work by Nikulin \cite{nikulin.rank.greater.than.five,nikulin.rank.three} and Vinberg \cite{vinberg.rank.four} in the 80s gives a complete classification of hyperbolic lattices with a finite symmetry group. This was recently generalized by Yu \cite{yu.zero.entropy} (in the case of Picard lattices of K3 surfaces) and Brandhorst--Mezzedimi \cite{brandhorst.mezzedimi.borcherds,brandhorst.mezzedimi.abelian}, who classified hyperbolic lattices with a \emph{virtually abelian} symmetry group, that is, containing an abelian subgroup of finite index. The maximum rank of such a hyperbolic lattice is $26$, attained by the unimodular lattice ${\rm II}_{1,25} \coloneqq U\oplus \Lambda$, where $\Lambda$ denotes the Leech lattice.

In order to capture the complexity of symmetry groups of general hyperbolic lattices, Nikulin introduced in \cite{nikulin.elliptic.fibrations} the notion of \emph{exceptional lattice}: If $L$ is a hyperbolic lattice, then the exceptional lattice $E(L)\subseteq L$ is the sublattice generated by elements that have finite orbit under $\mathrm{Aut}(\D_L)$. Following \cite[§4]{nikulin.elliptic.fibrations}, there are three possibilities for the exceptional lattice if it is non-trivial:
\begin{enumerate}
 \item $E(L)$ is a hyperbolic lattice. Then $E(L)=L$, and in particular $\mathrm{Aut}(\D_L)$ is finite.
 \item $E(L)$ is negative semidefinite. Then it contains an isotropic vector, and therefore $L$ is a \emph{Borcherds lattice} (see \cite[Definition~1.1]{brandhorst.mezzedimi.borcherds}). In particular $\mathrm{Aut}(\D_L)$ is virtually abelian.
 \item $E(L)$ is negative definite.
\end{enumerate}

As explained above, the lattices satisfying (1) or (2) have already been classified. Moreover, Nikulin proves in \cite[Theorem~4.4]{nikulin.elliptic.fibrations} that there are only finitely many hyperbolic lattices of \emph{fixed} rank $\ge 3$ with non-trivial exceptional lattice.
The goal of this article is to make the next step towards a full classification also in case (3), by proving that $E(L)$ is in fact trivial in large enough rank. More precisely, we will show the following:

\begin{abstract}
For an even, integral hyperbolic lattice $L$, the symmetry group of $L$ is the quotient of the group of isometries of $L$ by the Weyl subgroup of $(-2)$-reflections. Following Nikulin, the exceptional lattice of $L$ is defined as the sublattice generated by elements that have finite orbit under the symmetry group of $L$. We prove that every hyperbolic lattice of rank at least $46$ has trivial exceptional lattice. In particular, every such lattice admits a symmetry of maximal Salem degree.
\end{abstract}

As explained above, the lattices satisfying (1) or (2) have already been classified. Moreover, Nikulin proves in \cite[Theorem~4.4]{nikulin.elliptic.fibrations} that there are only finitely many hyperbolic lattices of \emph{fixed} rank $\ge 3$ with non-trivial exceptional lattice.
The goal of this article is to make the next step towards a full classification also in case (3), by proving that $E(L)$ is in fact trivial in large enough rank. More precisely, we will show the following:

In particular, the classification of hyperbolic lattices with non-trivial exceptional lattice becomes a finite problem:

\begin{corollary} \label{cor:finitelymany}
 Up to isometry, there are only finitely many hyperbolic lattices of rank $\ge 3$ with non-trivial exceptional lattice.
\end{corollary}

\begin{corollary}
Let $L$ be a hyperbolic lattice of rank at least $46$. Then, the following hold:
\begin{enumerate}
\item There are no non-trivial $\Aut(\D_L)$-invariant $\RR$-linear subspaces of $L_{\mathbb{R}}$.
\item There exists an isometry $f \in \Aut(\D_L)$ of maximal Salem degree.
\end{enumerate}
\end{corollary}

Now let $L$ be a hyperbolic lattice of rank at least $46$. By taking an arbitrary maximal overlattice of $L$, we reduce to lattices of the form $L=U\oplus E_8^n\oplus M$, where $n\ge 5$ and the rank of $M$ is at most $11$ (cf. \Cref{prop:structure}). By using the isometry $U\oplus E_8^3\cong U\oplus \Lambda$, and the fact that the Leech lattice $\Lambda$ has no $(-2)$-root, we deduce that the exceptional lattice $E(L)$ is contained in $M$. Since there are only finitely many possibilities for the root part of $M$, we are able to show in \Cref{section:proof} that $E(L)=0$ for all of them, thus concluding the proof of \Cref{thm:maintheorem}.

\begin{proposition} \label{prop: big roots contain UE8}
Let $R$ be an ADE lattice of rank $n \geq 9$. Then, $U \oplus R$ contains $U \oplus E_8$.
\end{proposition}
\begin{proof}
From the classification of ADE lattices, we have $R \in \{A_n,D_n\}$, hence $\ell(A_R) \leq 2$ and even $\ell(A_R) = 1$ unless $R = D_n$ with $n$ even (see \cite[Table~1.1]{Eb13}).
Using \cite[Corollary 1.13.5]{Ni80} and the inequality
$$
{\rm rk}(U \oplus R) \geq 11 \geq 9 + {\ell}(A_R),
$$
we conclude that $U \oplus R \cong E_8 \oplus M$ for a hyperbolic lattice $M$ of rank $n -6 \geq 3$ with $\ell(A_M) = \ell(A_R)$. By \cite[Corollary 1.13.5]{Ni80}, we are done if $n - 6 \geq 3 + \ell(A_R)$.
It thus remains to consider the cases $R \in \{D_9,D_{10},A_9\}$. A straightforward computation shows that the discriminant group of $D_9$ (resp. $D_{10}$, $A_9$) is isomorphic as a finite quadratic module to the discriminant group of $\langle -4\rangle$ (resp. $A_1\oplus A_1$, $\langle -10\rangle$), see e.g. \cite[Table~1]{gvirtz.mezzedimi}. Therefore $D_9$ (resp. $D_{10}$, $A_9$) is in the same genus of $E_8\oplus \langle -4\rangle$ (resp. $E_8\oplus A_1\oplus A_1$, $E_8\oplus \langle -10\rangle$), from which
$$U\oplus D_9\cong U\oplus E_8\oplus \langle -4\rangle, \quad U\oplus D_{10} \cong U\oplus E_8\oplus A_1\oplus A_1, \quad U\oplus A_9\cong U\oplus E_8\oplus \langle -10\rangle. $$
\end{proof}

\begin{proposition}\label{prop:structure}
 If $L$ is a hyperbolic lattice of rank at least $6$, then there exist an overlattice
 $L'$ of $L$, an integer $n$ and a negative-definite lattice $M$ such that
 $$L'\cong U\oplus E_{8}^{n}\oplus M,$$
 with $\rk M\le 11$ and such that the root part of $M$ does not contain any $E_{8}$ or an ADE lattice of rank at least $9$.
\end{proposition}

\begin{proposition}\label{prop:structure}
	If $L$ is a hyperbolic lattice of rank at least $6$, then there exist an overlattice
    $L'$ of $L$, an integer $n$ and a negative-definite lattice $M$ such that
    $$L'\cong U\oplus E_{8}^{n}\oplus M,$$
    with $\rk M\le 11$ and such that the root part of $M$ does not contain any $E_{8}$ or an ADE lattice of rank at least $9$.
\end{proposition}

\begin{theorem}\label{thm:maintheorem}
    Let $L$ be a hyperbolic lattice of rank at least $46$. Then, the exceptional
    lattice of $L$ is trivial.
\end{theorem}

In particular, the classification of hyperbolic lattices with non-trivial exceptional lattice becomes a finite problem:

\begin{corollary} \label{cor:finitelymany}
    Up to isometry, there are only finitely many hyperbolic lattices of rank $\ge 3$ with non-trivial exceptional lattice.
\end{corollary}

Observe that in rank $2$, a hyperbolic lattice has non-trivial exceptional lattice if and only if it has finite symmetry group, or equivalently if and only if it represents $0$ or $-2$ (see \cite[Corollary~3.4]{galluzzi.lombardo.peters}).

Given the finiteness result in \Cref{cor:finitelymany}, it is natural to ask what is the largest rank of a hyperbolic lattice with non-trivial exceptional lattice. If $L\coloneqq {\rm II}_{1,25}$, then already Borcherds noted in \cite{borcherds.leech} that $E(L)$ has rank $1$, and it is generated by the unique cusp (i.e. primitive, isotropic vector in the closure of the fundamental domain $\D_L$) with infinite stabilizer. We do not know any example of larger rank. In the spirit of the classifications in \cite{nikulin.rank.greater.than.five,nikulin.rank.three,vinberg.rank.four,brandhorst.mezzedimi.borcherds}, it would be interesting to have a complete classification of hyperbolic lattices with a non-trivial exceptional lattice, thus completing Nikulin's program in \cite{nikulin.elliptic.fibrations}.

\subsection{Motivation from K3 surfaces and applications}
If $L = \mathrm{Pic}(X)$ is the Picard lattice of a complex K3 surface $X$, then the classes of effective $(-2)$-curves in $L$ determine a choice of fundamental domain $\D_L$ for the action of $W(L)$ on $L$ and the associated action
\begin{eqnarray*}
\mathrm{Aut}(X) & \to & \mathrm{Aut}(\D_L) \\
g & \mapsto & g^*
\end{eqnarray*}
has finite kernel and cokernel. The \emph{cusps} of $L$ that we consider in this article correspond to elliptic fibrations of $X$ and cusps with infinite stabilizer correspond to elliptic fibrations whose Jacobian has infinite Mordell--Weil group. If $X$ admits at least one elliptic fibration with infinite automorphism group, then the exceptional lattice is the primitive sublattice of $\mathrm{Pic}(X)$ generated by all $(-2)$-curves that are contained in fibers of all such elliptic fibrations (cf. \Cref{prop:exceptionalIntersection}).

To measure the complexity of an automorphism $g$, one considers the characteristic polynomial of $f=g^*$, which factors over $L_{\mathbb{R}}$ as $S(x)C(x)$, where $S(x)$ is a Salem polynomial (or a quadratic polynomial or trivial), called the \emph{Salem factor}, and $C(x)$ is a product of cyclotomic polynomials (see \cite[Discussion before D\'efinition 1.2]{cantat.dynamique}). The Salem factor is particularly interesting, since the logarithm of its largest root coincides with the topological entropy of $X$ \cite{Gromov,Yomdin}. Thus, to measure the complexity of this number, we define the \emph{Salem degree} of $f$ as the degree of $f$ and, since this degree is always even, we say that $f$ has \emph{maximal Salem degree} if its Salem degree is $\rk(L)$ if $\rk(L)$ is even or $\rk(L)-1$ if $\rk(L)$ is odd.

Now, observe that these definitions do not use the fact that there is an underlying K3 surface $X$ at all. In fact, even the choice of $\D_L$, which is needed to consider $\mathrm{Aut}(\D_L)$ as a subgroup of $\mathrm{O}^+(L)$, is irrelevant, since any two choices lead to conjugate subgroups.
Our results on hyperbolic lattices can therefore be used to apply statements in the literature about automorphisms of K3 surfaces in terms of the exceptional lattice to the symmetry group $\mathrm{Aut}(\D_L)$. One such statement is the following, due to Yu \cite[Theorem 6.2]{yu.elliptic.fibrations}. The original proof in the case of K3 surfaces of even Picard rank extends easily to the abstract setting and we can rephrase the result in our terminology as follows:

\begin{theorem}[Yu]
Let $L$ be a hyperbolic lattice with trivial exceptional lattice $E(L)$. Then:
\begin{enumerate}
    \item The only $\mathrm{Aut}(\D_L)$-invariant $\mathbb{R}$-linear subspaces of $L_\mathbb{R}$ are $\{0\}$ and $L_\mathbb{R}$.
    \item There exists a symmetry $f\in \mathrm{Aut}(\D_L)$ of maximal Salem degree.
\end{enumerate}
If moreover $\rk(L)\ge 4$ is even, then the conditions $E(L)=0$, (1) and (2) are all equivalent.
\end{theorem}

By \Cref{rmk: cusp exists}, there always exists a cusp of infinite stabilizer if ${\rm rk}(L) \geq 20$ and by our main \Cref{thm:maintheorem}, the exceptional lattice is trivial as soon as ${\rm rk}(L) \geq 46$. We thus obtain the following application to Salem degrees:

\begin{corollary}
Let $L$ be a hyperbolic lattice of rank at least $46$. Then, the following hold:
\begin{enumerate}
\item There are no non-trivial $\mathrm{Aut}(\D_L)$-invariant $\mathbb{R}$-linear subspaces of $L_{\mathbb{R}}$.
\item There exists an isometry $f \in \mathrm{Aut}(\D_L)$ of maximal Salem degree.
\end{enumerate}
\end{corollary}

\subsection{Strategy of the proof and outline}

The starting point of the proof of \Cref{thm:maintheorem} is an explicit description of the exceptional lattice of hyperbolic lattices, originally due to Nikulin \cite[Theorem~4.2]{nikulin.elliptic.fibrations} (see also \Cref{prop:exceptionalIntersection}). More precisely, it asserts that $E(L)$ is trivial as soon as every simple $(-2)$-root $r\in L$ admits a cusp with infinite stabilizer $e\in \overline{\mathcal{D}}_L$ such that $e.r>0$.

Now let $L$ be a hyperbolic lattice of rank at least $46$. By taking an arbitrary maximal overlattice of $L$, we reduce to lattices of the form $L=U\oplus E_8^n\oplus M$, where $n\ge 5$ and the rank of $M$ is at most $11$ (cf. \Cref{prop:structure}). By using the isometry $U\oplus E_8^3\cong U\oplus \Lambda$, and the fact that the Leech lattice $\Lambda$ has no $(-2)$-root, we deduce that the exceptional lattice $E(L)$ is contained in $M$. Since there are only finitely many possibilities for the root part of $M$, we are able to show in \Cref{section:proof} that $E(L)=0$ for all of them, thus concluding the proof of \Cref{thm:maintheorem}.

Given the nature of our approach, we expect the bound in \Cref{thm:maintheorem} to be quite far from sharp. Moreover, it is probable that a (possibly computer assisted) refinement of our argument in \Cref{section:proof} via extended Dynkin diagrams leads to a much better bound in \Cref{thm:maintheorem}.

We briefly outline the structure of the paper. In Section 2 we recall basic properties of negative definite and hyperbolic lattices, and prove the structure result for maximal overlattices of hyperbolic lattices in \Cref{prop:structure}. In Section 3 we review Nikulin's definition of the exceptional lattice, and the explicit description in \Cref{prop:exceptionalIntersection}. Finally, in Section 4 we prove \Cref{thm:maintheorem}.

\begin{ack}
We gratefully acknowledge the hospitality and support of the Max-Planck-Institut f\"ur Mathematik Bonn, where we started working on this article during an internship program.
\end{ack}

The proof of \Cref{thm:maintheorem} starts from Nikulin’s explicit description of the exceptional lattice (\cite[Theorem~4.2]{nikulin.elliptic.fibrations}, see also \Cref{prop:exceptionalIntersection}): it implies that $E(L)$ is trivial “as soon as every simple $(-2)$-root $r\in L$ admits a cusp with infinite stabilizer $e\in \overline{\mathcal{D}}_L$ such that $e.r>0$.”

For a hyperbolic lattice $L$ of rank at least $46$, one “tak[es] an arbitrary maximal overlattice of $L$” to reduce to lattices of the form
\[
L = U\oplus E_8^n\oplus M,\quad n\ge 5,\ \rk(M)\le 11
\]
(cf. \Cref{prop:structure}). Using the isometry $U\oplus E_8^3\cong U\oplus \Lambda$ and that the Leech lattice $\Lambda$ has no $(-2)$-root, one deduces that “the exceptional lattice $E(L)$ is contained in $M$.” Since there are “only finitely many possibilities for the root part of $M$,” the argument in \Cref{section:proof} checks all cases and shows “that $E(L)=0$ for all of them,” which “conclud[es] the proof of \Cref{thm:maintheorem}.”

To prove the main theorem, we start from Nikulin’s explicit description of the exceptional lattice (Nikulin, “Elliptic fibrations on K3 surfaces”, Theorem~4.2), see also the following proposition.

\begin{proposition}\label{prop:exceptionalIntersection}
Let $L$ be a lattice such that $\mathcal E_\infty(L)$ is non-empty.
Then the exceptional lattice satisfies
\begin{align*}
    E(L)=\bigcap _{e\in \mathcal E_{\infty}(L)} (e^{\perp})^{(2)}_{\mathrm{pr}}.
\end{align*}
\end{proposition}

It implies that $E(L)$ is trivial “as soon as every simple $(-2)$-root $r\in L$ admits a cusp with infinite stabilizer $e\in \overline{\mathcal{D}}_L$ such that $e.r>0$.”

For a hyperbolic lattice $L$ of rank at least $46$, one “tak[es] an arbitrary maximal overlattice of $L$” to reduce to lattices of the form
\[
L = U\oplus E_8^n\oplus M,\quad n\ge 5,\ \rk(M)\le 11
\]
by the following proposition.

\begin{proposition}\label{prop:structure}
\tIf $L$ is a hyperbolic lattice of rank at least $6$, then there exist an overlattice
    $L'$ of $L$, an integer $n$ and a negative-definite lattice $M$ such that
    $$L'\cong U\oplus E_{8}^{n}\oplus M,$$
    with $\rk M\le 11$ and such that the root part of $M$ does not contain any $E_{8}$ or an ADE lattice of rank at least $9$.
\end{proposition}

Using the isometry $U\oplus E_8^3\cong U\oplus \Lambda$ and that the Leech lattice $\Lambda$ has no $(-2)$-root, one deduces that “the exceptional lattice $E(L)$ is contained in $M$.” Since there are “only finitely many possibilities for the root part of $M$,” the argument given later checks all cases and shows “that $E(L)=0$ for all of them,” which “conclud[es] the proof of the main theorem.”

\label{thm:maintheorem}
    Let $L$ be a hyperbolic lattice of rank at least $46$. Then, the exceptional
    lattice of $L$ is trivial.,

Question: Let $L$ be a hyperbolic lattice of rank at least $46$. Write $\mathrm{Aut}(\mathcal D_L)=\mathrm{O}^+(L)/W(L)$ for its symmetry group, where $\mathrm{O}^+(L)$ is the group of positive isometries of $L$ and $W(L)$ is the Weyl subgroup generated by reflections in $(-2)$-roots. The exceptional lattice $E(L)\subseteq L$ is the sublattice generated by all elements of $L$ whose orbit under $\mathrm{Aut}(\mathcal D_L)$ is finite. Which statement holds for every such lattice $L$?

The exceptional lattice is trivial; equivalently, $E(L)=0$.

Choices:
B. The exceptional lattice is always negative definite of rank at most $11$; equivalently, after passing to a maximal overlattice one has $E(L)\subseteq M$ for some negative-definite lattice $M$ with $\operatorname{rk}(M)\le 11$, and hence $E(L)\neq L$.
C. The exceptional lattice contains no nonzero hyperbolic sublattice; equivalently, $E(L)$ is not hyperbolic.
D. There exists a universal constant $N\le 46$ such that for every hyperbolic lattice $L$ of rank at least $N$, the exceptional lattice is contained in the negative-definite factor $M$ arising from some decomposition of a maximal overlattice as $U\oplus E_8^n\oplus M$ with $\operatorname{rk}(M)\le 11$, and therefore $E(L)$ is negative definite.
E. For every hyperbolic lattice $L$ of rank at least $46$, there exists a maximal overlattice $L'\cong U\oplus E_8^n\oplus M$ with $n\ge 5$ and $\operatorname{rk}(M)\le 11$ such that the exceptional lattice of $L$ coincides with the exceptional lattice of $L'$ and is equal to $M$.

The stem defines the objects involved but does not explicitly or implicitly reveal that the exceptional lattice must be trivial. The correct conclusion is not embedded in the wording.

The item is essentially a direct theorem-recall question: under the stated rank hypothesis, the correct option states the theorem's conclusion almost verbatim as 'E(L)=0.'

There is some pressure to distinguish the exact theorem from weaker true consequences and stronger false variants, but the task mainly tests recognition/recall rather than substantial mathematical generation or derivation.

The distractors are mathematically sophisticated and plausible: one is a weaker true statement, others are natural overstrengthenings or transfer claims a student might incorrectly infer from the proof structure.

A solid MCQ with no answer leakage and strong distractors, but it is largely a theorem-restatement item and only moderately tests genuine reasoning.