\label{thm:main1}
Let $\mathcal {A}$ be an admissible affine arrangement in $\mathbb R^n$ which is invariant under the action of a discrete translation subgroup $\mathbb Z^n$ of $\mathbb R^n$ (this does not have to be the usual embedding of $\mathbb Z^n$). Suppose $n\le 4$. Then $\mathcal {A}$ is a $K(\pi,1)$ arrangement. More generally, modulo a group theoretical conjecture on the spherical Artin group of type $D_n$ (Conjecture~\ref{conj:dn}), $\mathcal {A}$ is a $K(\pi,1)$ arrangement for any $n$.

\begin{theorem}
\label{thm:main1}
Let $\ca$ be an admissible affine arrangement in $\mathbb R^n$ which is invariant under the action of a discrete translation subgroup $\mathbb Z^n$ of $\mathbb R^n$ (this does not have to be the usual embedding of $\mathbb Z^n$). Suppose $n\le 4$. Then $\ca$ is a $K(\pi,1)$ arrangement. More generally, modulo a group theoretical conjecture on the spherical Artin group of type $D_n$ (Conjecture~\ref{conj:dn}), $\ca$ is a $K(\pi,1)$ arrangement for any $n$.
\end{theorem}

\label{subsec:deligne complex}

\subsection{Falk complexes}

In \cite{falk1995k}, for each affine arrangement in $\mathbb C^2$ which is the complexification of a real arrangement, Falk described a loc

\begin{thm}
\label{thm:main}
Suppose $\ca$ is a complete admissible affine arrangement in $\mathbb R^n$ with finite shape. Suppose Conjecture~\ref{conj:dn} holds. Then $(\falk_\ca,d_\infty)$ is an injective metric space and $\ca$ is a $K(\pi,1)$ arrangement.
\end{thm}

\begin{thm}
\label{thm:examples}
Suppose Conjecture~\ref{conj:dn} holds in dimension $n$. Then $(\falk_{\ch_{k,n}},d_\infty)$ and $(\falk_{\ck_{k,n}},d_\infty)$ are injective metric spaces, and
$\ch_{k,n}$ and $\ck_{k,n}$ are $K(\pi,1)$ arrangements for any $k\ge 1$.

Thus by Theorem~\ref{thm:dn dim 3 and 4}, for $n=2,3,4$ and any $k\ge 1$, the arrangements $\ch_{k,n}$ and $\ck_{k,n}$ are $K(\pi,1)$ arrangements.
\end{thm}

\begin{remark}
\label{rmk:contrast}
    There is an interesting contrast between this proposition and Theorem~\ref{thm:bowtie free An}, as the poset in Theorem~\ref{thm:bowtie free An} is not upward flag.
\end{remark}

Let $\mathcal {A}$ be an \emph{affine hyperplane arrangement} in $\mathbb R^n$, i.e., a locally finite collection of affine hyperplanes in $\mathbb R^n$. We consider the complex manifold which is the complement of the following collection of hyperplanes in $\mathbb C^n$:
$$M(\mathcal {A})=\mathbb C^n-\bigcup_{H\in \mathcal {A}}(H\otimes \mathbb C).$$
It is an important question to understand the topology of $M(\mathcal {A})$, see e.g.~\cite{falk1986homotopy,falk1998homotopy}. We will be specifically interested in the asphericity of $M(\mathcal {A})$.
If the manifold $M(\mathcal {A})$ is aspherical, we call $\mathcal {A}$ a \emph{$K(\pi,1)$ arrangement}.

Unlike the situation of knot complements in $\mathbb S^3$, asphericity of $M(\mathcal {A})$ is a relatively rare phenomenon. However, there are some specific classes of $\mathcal {A}$ where asphericity is known, for example:
\begin{enumerate}
 \item $\mathcal {A}$ is central and simplicial by Deligne \cite{deligne};
 \item $\mathcal {A}$ is supersolvable by Terao \cite{terao1986modular};
 \item $\mathcal {A}$ is certain type of line arrangement in $\mathbb R^2$ by Falk \cite{falk1995k};
 \item $\mathcal {A}$ is the collection of reflection hyperplanes associated with an affine Coxeter group by Paolini and Salvetti \cite{paolini2021proof}.
\end{enumerate}
These results are obtained through different means: (1) and (4) rely heavily on Garside theory; (2) is obtained through a fibration argument; (3) uses a form of conformal non-positive curvature for 2-dimensional complexes, allowing one to compute the second homotopy group directly.
Given that there are relatively few methods and examples of aspherical arrangements when $n\ge 3$, it is desirable to extend Falk's method over dimension 2, which is the goal of this article.
In higher dimensions, we must use a different notion of non-positive curvature in place of the conformal non-positive curvature in \cite{falk1995k} which can only be used in dimension 2.

Given an affine arrangement $\mathcal {A}$, an \emph{$\mathcal {A}$-vertex} is a point in $\mathbb R^n$ which can be realized as intersection of elements of $\mathcal {A}$. The \emph{local arrangement} at an $\mathcal {A}$-vertex $x$ is the collection of all hyperplanes in $\mathcal {A}$ that contain $x$. An interesting feature of Falk's result, is the local-to-global phenonmenon that for certain classes of arrangements $\mathcal {A}$, one can detect the asphericity of $M(\mathcal {A})$ by looking at the combinatorial features of its local arrangements. Motivated by this, we consider the following class of arrangements characterized by their local arrangements.
\begin{definition}
 \label{def:admissible1}
We say an affine hyperplane arrangement $\mathcal {A}$ in $\mathbb R^n$ is \emph{admissible}, if at each $\mathcal A$-vertex $x$, the local arrangement at $x$ is a translate of the following four types:
\begin{enumerate}
 \item (type $B_n$) $x_i\pm x_j=0$ for $1\le i\neq j\le n$ and $x_i=0$ for $1\le i\le n$;
 \item (type $D_n$) $x_i\pm x_j=0$ for $1\le i\neq j\le n$;
 \item (skewed type $A_n$) $x_i=0$ for $1\le i\le n$ and $x_i=x_j$ for $1\le i\neq j\le n$, or any image of this this arrangement under the $(\bb{Z}/2\bb{Z})^n$ action on $\bb[n]{R}$ by reflections about the coordinate hyperplanes;
 \item or a product of the previous types.
\end{enumerate}
\end{definition}

Note that any affine Coxeter arrangement associated with a non-exceptional affine Coxeter group (i.e., types $\widetilde A_n,\widetilde B_n,\widetilde C_n,\widetilde D_n$) is an admissible arrangement\footnote{We use a different description of the $\widetilde A_n$ arrangement, where the hyperplanes are $x_i\in \mathbb Z$ for $1\le i\le n$ and $x_i-x_j\in \mathbb Z$ for $1\le i\neq j\le n$. This does not affect the topology of $M(\mathcal {A})$.}. Although most of the arrangements in Definition~\ref{def:admissible1} are not Coxeter arrangements.

\begin{conj}(Haettel)
\label{conj:dn}
Suppose $\Lambda$ is of type $D_n$ for $n\geq 3$. Then $((\Delta'_\Lambda)^0,<)$ is a poset that is bowtie free and upward flag.
\end{conj}

\begin{definition}
 \label{def:admissible1}
We say an affine hyperplane arrangement $\ca$ in $\mathbb R^n$ is \emph{admissible}, if at each $\mathcal A$-vertex $x$, the local arrangement at $x$ is a translate of the following four types:
\begin{enumerate}
    \item (type $B_n$) $x_i\pm x_j=0$ for $1\le i\neq j\le n$ and $x_i=0$ for $1\le i\le n$;
    \item (type $D_n$) $x_i\pm x_j=0$ for $1\le i\neq j\le n$;
    \item (skewed type $A_n$) $x_i=0$ for $1\le i\le n$ and $x_i=x_j$ for $1\le i\neq j\le n$, or any image of this this arrangement under the $(\bb{Z}/2\bb{Z})^n$ action on $\bb[n]{R}$ by reflections about the coordinate hyperplanes;
    \item or a product of the previous types.
\end{enumerate}
\end{definition}

Given an affine arrangement $\ca$, an \emph{$\ca$-vertex} is a point in $\mathbb R^n$ which can be realized as intersection of elements of $\ca$. The \emph{local arrangement} at an $\ca$-vertex $x$ is the collection of all hyperplanes in $\ca$ that contain $x$. An interesting feature of Falk's result, is the local-to-global phenonmenon that for certain classes of arrangements $\ca$, one can detect the asphericity of $M(\ca)$ by looking at the combinatorial features of its local arrangements. Motivated by this, we consider the following class of arrangements characterized by their local arrangements.
\begin{definition}
 \label{def:admissible1}
We say an affine hyperplane arrangement $\ca$ in $\mathbb R^n$ is \emph{admissible}, if at each $\mathcal A$-vertex $x$, the local arrangement at $x$ is a translate of the following four types:
\begin{enumerate}
 \item (type $B_n$) $x_i\pm x_j=0$ for $1\le i\neq j\le n$ and $x_i=0$ for $1\le i\le n$;
 \item (type $D_n$) $x_i\pm x_j=0$ for $1\le i\neq j\le n$;
 \item (skewed type $A_n$) $x_i=0$ for $1\le i\le n$ and $x_i=x_j$ for $1\le i\neq j\le n$, or any image of this this arrangement under the $(\bb{Z}/2\bb{Z})^n$ action on $\bb[n]{R}$ by reflections about the coordinate hyperplanes;
 \item or a product of the previous types.
\end{enumerate}
\end{definition}

Note that any affine Coxeter arrangement associated with a non-exceptional affine Coxeter group (i.e., types $\widetilde A_n,\widetilde B_n,\widetilde C_n,\widetilde D_n$) is an admissible arrangement\footnote{We use a different description of the $\widetilde A_n$ arrangement, where the hyperplanes are $x_i\in \mathbb Z$ for $1\le i\le n$ and $x_i-x_j\in \mathbb Z$ for $1\le i\neq j\le n$. This does not affect the topology of $M(\ca)$.}. Although most of the arrangements in Definition~\ref{def:admissible1} are not Coxeter arrangements.

In the situation of the above theorem, we have a free action of $\mathbb Z^n$ on $M(\ca)$. Then the fundamental group of $M(\ca)/\mathbb Z^n$ can be viewed as a generalization of the affine Artin groups (when $\ca$ is an affine Coxeter arrangement, this gives a finite index subgroup of the corresponding affine Artin group).

\begin{thm}(=Theorem~\ref{thm:examples})
For $n\le 4$ and any $k\ge 1$, the arrangements $\ch_{k,n}$ and $\ck_{k,n}$ are $K(\pi,1)$ arrangements. More generally, modulo a group theoretical conjecture on the spherical Artin group of type $D_n$ (Conjecture~\ref{conj:dn}), $\ch_{k,n}$ and $\ck_{k,n}$ are $K(\pi,1)$ arrangements for any $n,k$.
\end{thm}

\begin{cor} \textup{(= \Cref{cor:AB})}
\label{cor:AB intro}
Suppose $\ca$ is a complete, finite shape, affine arrangement in $\mathbb R^n$ such that for each $\mathcal A$-vertex $x$, the local arrangement at $x$ is a translate of one of the following three types:
\begin{enumerate}
 \item (type $B_n$) $x_i\pm x_j=0$ for $1\le i\neq j\le n$ and $x_i=0$ for $1\le i\le n$;
 \item (skewed type $A_n$) $x_i=0$ for $1\le i\le n$ and $x_i=x_j$ for $1\le i\neq j\le n$, or any image of this this arrangement under the $(\bb{Z}/2\bb{Z})^n$ action on $\bb[n]{R}$ by reflections about the coordinate hyperplanes;
 \item or a product of the previous types.
\end{enumerate}
Then $(\falk_\ca,d_\infty)$ is an injective metric space and $\ca$ is a $K(\pi,1)$ arrangement.
\end{cor}

The Coxeter diagram $\Lambda$ of type $D_n$ (for $n\ge 3$) is shown in Figure~\ref{fig:dn}, where all edges are labeled by $3$. The associated reflection arrangement is $\{x_i=\pm x_j\}_{1\le i < j\le n}$ in $\mathbb R^n$.
We subdivide each edge of $\Delta_\Lambda$ connecting a vertex of type $\hat s_n$ and a vertex of type $\hat s_{n-1}$, and declare the middle point of such edge is of type $m$. Cut each top dimensional simplex in $\Delta_\Lambda$ into two simplices along the codimension 1 simplex spanned by vertices of type $m$ and $\{\hat s_i\}_{i=1}^{n-2}$. This gives a new simplicial complex, denoted by $\Delta'_\Lambda$. Define a map $t$ from $(\Delta'_\Lambda)^0$ to $\{1,2,\ldots,n\}$ by sending vertices of type $\hat s_i$ to $i$ for $1\le i\le n-2$, vertices of type $m$ to $n-1$, and vertices of type $\hat s_n$ and $\hat s_{n-1}$ to $n$. We define a relation $<$ on $(\Delta'_\Lambda)^0$ as follows. For two vertices $x,y$ of $\Delta'_\Lambda$, $x<y$ if $x$ and $y$ are adjacent in $\Delta'_\Lambda$ and $t(x)<t(y)$. The simplicial complex $\Delta'_{\Lambda}$, together with this relation, is called the \emph{$(s_n,s_{n-1})$-subdivision of $\Delta_\Lambda$}. Similarly, we can define $(s_n,s_{n-1})$-subdivision $\bC'_\Lambda$ of $\bC_\Lambda$. Note that $\bC'_\Lambda$ is isomorphic to the Coxeter complex of type $B_n$ via a type-preserving isomorphism. While $\Delta'_\Lambda$ is not isomophic to the Artin complex of type $B_n$, the following conjecture, due to Haettel, predicts that these two complexes share the following property -- his original motivation for this conjecture is that it leads an alternative proof that Artin group of type $\widetilde D_n$ satisfies the $K(\pi,1)$-conjecture.

\subsection{Falk complexes for admissible arrangements}
\label{subsec:admissible}
For $i=1,2$, let $\ca_i$ be an affine hyperplane arrangement in $\mathbb R^{m_i}$. Then the \emph{product} of $\ca_1$ and $\ca_2$ is an arrangement $\ca$ in $\mathbb R^{m_1+m_2}$ whose hyperplanes are of form $H\times \mathbb R^{m_2}$ for $H\in \ca_1$ or $\mathbb R^{m_1}\times H$ for $H\in \ca_2$.
\begin{definition}
\label{def:admissible}
We say an affine hyperplane arrangement $\ca$ in $\mathbb R^n$ is \emph{admissible}, if at each $\mathcal A$-vertex $x$, the local arrangement at $x$ is a translate of one of the following four types:
\begin{enumerate}
 \item (type $B_n$) $x_i\pm x_j=0$ for $1\le i\neq j\le n$ and $x_i=0$ for $1\le i\le n$;
 \item (type $D_n$) $x_i\pm x_j=0$ for $1\le i\neq j\le n$;
 \item (skewed type $A_n$) $x_i=0$ for $1\le i\le n$ and $x_i=x_j$ for $1\le i\neq j\le n$, or any image of this this arrangement under the $(\bb{Z}/2\bb{Z})^n$ action on $\bb[n]{R}$ by reflections about the coordinate hyperplanes;
 \item or a product of the previous types.
\end{enumerate}
\end{definition}

The next result can be proved in the same way as Theorem~\ref{thm:main}, using Proposition~\ref{prop:key} and Theorem~\ref{thm:Bnflag}.
\begin{cor}
\label{cor:AB}
Suppose $\ca$ is a complete finite shape affine arrangement in $\mathbb R^n$ such that for each $\mathcal A$-vertex $x$, the local arrangement at $x$ is a translate of one of the following three types:
\begin{enumerate}
 \item (type $B_n$) $x_i\pm x_j=0$ for $1\le i\neq j\le n$ and $x_i=0$ for $1\le i\le n$;
 \item (skewed type $A_n$) $x_i=0$ for $1\le i\le n$ and $x_i=x_j$ for $1\le i\neq j\le n$, or any image of this this arrangement under the $(\bb{Z}/2\bb{Z})^n$ action on $\bb[n]{R}$ by reflections about the coordinate hyperplanes;
 \item or a product of the previous types.
\end{enumerate}
Then $(\falk_\ca,d_\infty)$ is an injective metric space and $\ca$ is a $K(\pi,1)$ arrangement.
\end{cor}

In the situation of the above theorem, we have a free action of $\mathbb Z^n$ on $M(\mathcal {A})$. Then the fundamental group of $M(\mathcal {A})/\mathbb Z^n$ can be viewed as a generalization of the affine Artin groups (when $\mathcal {A}$ is an affine Coxeter arrangement, this gives a finite index subgroup of the corresponding affine Artin group).

\begin{cor}
Under the assumption of Theorem~\ref{thm:main1}, the manifold $M(\mathcal {A})/\mathbb Z^n$ is homotopy equivalent to a finite aspherical cell complex, which is the quotient of the Salvetti complex of $\mathcal {A}$ (\cite{s87}) by a free action of $\mathbb Z^n$. In particular, the fundamental group of $M(\mathcal {A})/\mathbb Z^n$ is of type $F$.
\end{cor}

\Cref{thm:main1} is a special case of a more general statement which does not require $\mathcal {A}$ to be $\mathbb Z^n$-invariant, see \Cref{thm:main}.

Now we give more concrete examples of arrangements where Theorem~\ref{thm:main1} (or more specifically, \Cref{thm:main}) applies. For each dimension $n$, we will construct an infinite family of finite affine arrangements $\ch_{k,n}$, and an infinite family of $\mathbb Z^n$-invariant affine arrangements $\ck_{k,n}$. These arrangements are not Coxeter arrangements, and their complexified complement do not admit iterated fibration structure in an obvious way. The finite arrangements are not simplicial. So the $K(\pi,1)$ results for these arrangements are new.

\begin{definition}
For $k\ge 1$, let $\ch_{k,n}$ be the affine hyperplane arrangement in $\mathbb R^n$ given by $x_i \in \{-2k-1, -2k+1, \dots,-3, -1, 1, 3, \dots, 2k - 1, 2k + 1\}$ for $1 \le i \le n$ and $x_i \pm x_j = 0$ for $1\le i\neq j\le n$. This family $\ch_{k,n}$ generalizes Falk's \cite[Example 3.13]{falk1995k} which is neither supersolvable or simpicial.

For $k\ge 1$, let $\ck_{k,n}$ be the affine hyperplane arrangement in $\mathbb R^n$ given by $x_i\in \mathbb Z$ for $1\le i\le n$, and $x_i+x_j \in 2k\mathbb Z+1$, $x_i-x_j \in 2k\mathbb Z$ for $1\le i\neq j\le n$.
\end{definition}

\begin{thm}(=Theorem~\ref{thm:examples})
For $n\le 4$ and any $k\ge 1$, the arrangements $\ch_{k,n}$ and $\ck_{k,n}$ are $K(\pi,1)$ arrangements. More generally, modulo a group theoretical conjecture on the spherical Artin group of type $D_n$ (Conjecture~\ref{conj:dn}), $\ch_{k,n}$ and $\ck_{k,n}$ are $K(\pi,1)$ arrangements for any $n,k$.
\end{thm}

For $n=2$, Falk \cite{falk1995k} constructed a (locally infinite) 2-dimensional complex $\falk_\mathcal {A}$ that is homotopy equivalent to the universal cover of $M(\mathcal {A})$. This construction of Falk can be generalized to higher dimensions without too much difficulty \cite{huang2024cycles}. Later, we show $\falk_\mathcal {A}$ is homotopy equivalent to the universal cover of $M(\mathcal {A})$ whenever all the local arrangements are $K(\pi,1)$. Thus all the above results rely on showing the complex $\falk_\mathcal {A}$ is contractible.

Recall that a geodesic metric space $X$ is \emph{injective} if any pairwise intersecting closed metric balls in $X$ have non-empty common intersection. For example, $\mathbb R^n$ equipped with the $\ell^\infty$ metric is injective. Injective metric spaces are contractible, and they are connected to the above theorems in the following way.

\begin{thm}
\textup{(= \Cref{thm:main} and \Cref{thm:examples})}
Under the assumptions of any of the previous theorems, the Falk complex $\falk_\mathcal {A}$ admits a metric which makes it an injective metric space.
\end{thm}

The reason for us to consider skewed type $A_n$ arrangements as local arrangements, rather than the standard $A_n$-arrangements, is for the compatibility of arranging injective metrics on $\falk_\mathcal {A}$. A substantial part of the article (Section~\ref{sec:check links}) is devoted to checking such compatibility (Proposition~\ref{prop:key} and Remark~\ref{rmk:contrast}).
It is a topic of independent interest to produce natural examples of injective metric spaces arising from group theory, and the above theorem gives many such examples. As the above theorem becomes conditional for $n\ge 5$, we also have the following variation that is unconditional for all dimensions, where we allow the local arrangements to be a mixture of $A_n$-type and $B_n$-type.
The following corollary is more interesting in terms of providing new examples of injective metric spaces, rather than $K(\pi,1)$ results, as one can prove the arrangements in the following corollary are $K(\pi,1)$ via an iterated fibration argument.

\begin{cor} \textup{(= \Cref{cor:AB})}
\label{cor:AB intro}
Suppose $\mathcal {A}$ is a complete, finite shape, affine arrangement in $\mathbb R^n$ such that for each $\mathcal A$-vertex $x$, the local arrangement at $x$ is a translate of one of the following three types:
\begin{enumerate}
    \item (type $B_n$) $x_i\pm x_j=0$ for $1\le i\neq j\le n$ and $x_i=0$ for $1\le i\le n$;
    \item (skewed type $A_n$) $x_i=0$ for $1\le i\le n$ and $x_i=x_j$ for $1\le i\neq j\le n$, or any image of this this arrangement under the $(\bb{Z}/2\bb{Z})^n$ action on $\bb[n]{R}$ by reflections about the coordinate hyperplanes;
    \item or a product of the previous types.
\end{enumerate}
Then $(\falk_\mathcal {A},d_\infty)$ is an injective metric space and $\mathcal {A}$ is a $K(\pi,1)$ arrangement.
\end{cor}

\subsection*{Structure of the article} In Section~\ref{sec:prelim} we collect some background material. In Section~\ref{subsec:deligne complex} we define Falk complexes of affine hyperplane arrangements and prove some of them admit injective metrics, modulo a key proposition (Proposition~\ref{prop:key}) about skewed $A_n$ arrangements. Section~\ref{sec:check links} is devoted to this key proposition. Section~\ref{sec:example} contains some new, concrete examples of affine arrangements where our results apply.

The introduction explains that the results (including Theorem~\ref{thm:main1}) “rely on showing the complex $\falk_\mathcal{A}$ is contractible.” For $n=2$, Falk constructs a complex $\falk_\mathcal{A}$ “that is homotopy equivalent to the universal cover of $M(\mathcal{A})$,” and this construction “can be generalized to higher dimensions.” Later they “show $\falk_\mathcal{A}$ is homotopy equivalent to the universal cover of $M(\mathcal{A})$ whenever all the local arrangements are $K(\pi,1)$,” so contractibility of $\falk_\mathcal{A}$ implies $M(\mathcal{A})$ is aspherical.

They then connect contractibility to injective metric spaces: “Injective metric spaces are contractible,” and, under the assumptions of the main theorems, “the Falk complex $\falk_\mathcal{A}$ admits a metric which makes it an injective metric space.” Achieving this requires “compatibility of arranging injective metrics on $\falk_\mathcal{A}$,” motivating the use of “skewed type $A_n$ arrangements as local arrangements,” and a substantial part of the article checks this compatibility (via “Proposition~\ref{prop:key}”).

The introduction explains that the results (including the main theorem) “rely on showing the complex $\falk_\mathcal{A}$ is contractible.” For $n=2$, Falk constructs a complex $\falk_\mathcal{A}$ “that is homotopy equivalent to the universal cover of $M(\mathcal{A})$,” and this construction “can be generalized to higher dimensions.” Later they “show $\falk_\mathcal{A}$ is homotopy equivalent to the universal cover of $M(\mathcal{A})$ whenever all the local arrangements are $K(\pi,1)$,” so contractibility of $\falk_\mathcal{A}$ implies $M(\mathcal{A})$ is aspherical.

They then connect contractibility to injective metric spaces: “Injective metric spaces are contractible,” and, in establishing the main theorem, “the Falk complex $\falk_\mathcal{A}$ admits a metric which makes it an injective metric space.” Achieving this requires “compatibility of arranging injective metrics on $\falk_\mathcal{A}$,” motivating the use of “skewed type $A_n$ arrangements as local arrangements,” and a substantial part of the article checks this compatibility via the following proposition.

\begin{prop}
\label{prop:key}
The poset $((\Delta'_{\ca(3)})^0,<)$ is bowtie free and upward flag.
\end{prop}

\label{thm:main1}
Let $\mathcal {A}$ be an admissible affine arrangement in $\mathbb R^n$ which is invariant under the action of a discrete translation subgroup $\mathbb Z^n$ of $\mathbb R^n$ (this does not have to be the usual embedding of $\mathbb Z^n$). Suppose $n\le 4$. Then $\mathcal {A}$ is a $K(\pi,1)$ arrangement. More generally, modulo the following group theoretical conjecture on the spherical Artin group of type $D_n$, $\mathcal {A}$ is a $K(\pi,1)$ arrangement for any $n$. \begin{conj}(Haettel)
\label{conj:dn}
Suppose $\Lambda$ is of type $D_n$ for $n\geq 3$. Then $((\Delta'_\Lambda)^0,<)$ is a poset that is bowtie free and upward flag.
\end{conj}

Question:
Let $\mathcal A$ be an affine hyperplane arrangement in $\mathbb R^n$, meaning a locally finite collection of affine hyperplanes. Its complexified complement is
$$M(\mathcal A)=\mathbb C^n\setminus \bigcup_{H\in \mathcal A}(H\otimes \mathbb C).$$
Call $\mathcal A$ a $K(\pi,1)$ arrangement if $M(\mathcal A)$ is aspherical. Assume that $\mathcal A$ is invariant under the action by translations of a discrete subgroup of $\mathbb R^n$ isomorphic to $\mathbb Z^n$. Also assume that $\mathcal A$ is admissible in the following sense: for every $\mathcal A$-vertex $x$ (a point that is an intersection of hyperplanes of $\mathcal A$), the local arrangement at $x$ (the set of hyperplanes of $\mathcal A$ containing $x$) is an affine translate of one of these types:
1. type $B_n$: $x_i\pm x_j=0$ for $1\le i\ne j\le n$ and $x_i=0$ for $1\le i\le n$;
2. type $D_n$: $x_i\pm x_j=0$ for $1\le i\ne j\le n$;
3. skewed type $A_n$: $x_i=0$ for $1\le i\le n$ and $x_i=x_j$ for $1\le i\ne j\le n$, or any image of this arrangement under coordinate-sign changes by $(\mathbb Z/2\mathbb Z)^n$;
4. a product of arrangements of the previous types.

Under these hypotheses, which conclusion is valid about $\mathcal A$ and $M(\mathcal A)$?

If $n\le 4$, then $M(\mathcal A)$ is aspherical, so $\mathcal A$ is a $K(\pi,1)$ arrangement. More generally, for arbitrary $n$, the same conclusion holds provided Haettel's conjecture is assumed: for every type $D_n$ object $\Lambda$ with $n\ge 3$, the poset $((\Delta'_\Lambda)^0,<)$ is bowtie free and upward flag.

Choices:
B. If $n<4$, then $M(\mathcal A)$ is aspherical, so $\mathcal A$ is a $K(\pi,1)$ arrangement. More generally, for arbitrary $n$, the same conclusion holds provided Haettel's conjecture is assumed only for every type $D_n$ object $\Lambda$ with $n\ge 4$, namely that the poset $((\Delta'_\Lambda)^0,<)$ is bowtie free and upward flag.
C. If $n\le 4$, then $M(\mathcal A)$ is aspherical, so $\mathcal A$ is a $K(\pi,1)$ arrangement.
D. If $n\le 4$, then $M(\mathcal A)$ is aspherical, so $\mathcal A$ is a $K(\pi,1)$ arrangement. More generally, for arbitrary $n$, for each such arrangement $\mathcal A$ there exists a type $D_n$ object $\Lambda$ for which, if the poset $((\Delta'_\Lambda)^0,<)$ is bowtie free and upward flag, then $M(\mathcal A)$ is aspherical.
E. If $n\le 4$, then the Falk complex associated with $\mathcal A$ is contractible, so $\mathcal A$ is a $K(\pi,1)$ arrangement. More generally, for arbitrary $n$, the same conclusion holds provided that for every type $D_n$ object $\Lambda$ with $n\ge 3$, the poset $((\Delta'_\Lambda)^0,<)$ is bowtie free, upward flag, and itself contractible.

dimension threshold and conjecture range

dropped arbitrary-$n$ conditional extension via Haettel conjecture

universal conjecture requirement replaced by arrangement-dependent existence

bowtie free and upward flag strengthened by unnecessary contractibility of the poset


\label{maintheorem0}
Let $T>0$. Suppose that hypothesis \eqref{H_0} and \eqref{H2} are satisfied. Then there exists $\delta = \delta(T)>0$ such that, for any initial data $y_0, z_0 \in L^2(\Omega)$ satisfying
\begin{equation}
\|y_0\|_{L^2(\Omega)} + \|z_0\|_{L^2(\Omega)} < \delta,
\end{equation}
and for any $F(r,s), G(r,s) : \mathbb{R} \times \mathbb{R} \to \mathbb{R}$, $C^1$-functions fulfilling
\begin{equation}\label{BoundM}
\max \left\{ \left|\frac{\partial F}{\partial r} (r,s)\right|, \left|\frac{\partial F}{\partial s} (r,s)\right|,  \left|\frac{\partial G}{\partial r} (r,s)\right|, \left|\frac{\partial G}{\partial s} (r,s)\right|
\right\} \leq M,
\end{equation}
for all $(r,s) \in \mathbb{R}^2$, with $\frac{\partial G}{\partial r} (r,s) \neq 0$ in \eqref{sistema0}, $F(0,0)=0$, $G(0,0)=0$, then there exists a control $\nu \in L^2(\omega \times (0,T))$ such that the solution of the system \eqref{sistema0} verifies $y(x,T)=0$ and $z(x,T)=0$.

\label{sec:intro}

For any $T>0$, let us consider the spacial domain $\Omega = (0,1)$ and $Q = (0,1) \times (0,T)$. We study the null controllability of the following systems of semilinearly coupled equations with kernel terms,
\begin{equation} \label{sistema0}
\begin{cases}
y_t=a_1  y_{xx}+b_1 y_x + c_1 y +\lambda_1(t)\int_0^1 J_1(\zeta-x)y(\zeta,t) d\zeta + F(y,z) + \nu 1_\omega, \ \ \mbox{ in } Q,\\
z_t=a_2  z_{xx}+b_2 z_x + c_2 z +\lambda_2(t)\int_0^1 J_2(\zeta-x)z(\zeta,t)d\zeta+ G(y,z), \ \ \mbox{ in } Q,\\
y(0,t) = y(1,t) = z(0,t) = z(1,t)  = 0 ,\ \
\mbox{ for } 0<t<T,\\
y(x,0) = y_0(x), \  z(x,0) = z_0(x), \mbox{ in }  \Omega.
\end{cases}
\end{equation}
where $\nu$ is a control acting on the open set $\omega \times (0,T)$ with $ \omega \subset (0,1)$, $y$ and $z$ are state functions, and, for $i=1,2$, $\lambda_i(t)$ are time dependent functions. In \eqref{sistema0} $a_i$, $b_i$ and $c_i$ are real constants such that $a_i >0$, for $i = 1,2$. The precise hypothesis about the functions $F$ and $G$ will be stated in our main theorem.

The present system \eqref{sistema0} aims to analyze the null controllability within a broader class of equations that extends the model below considered in the previous article of the authors \cite{Limaco-Lobosco-Yapu_24}:

\begin{equation} \label{sistema_anterior}
\begin{cases}
u_t=a_1  u_{xx}+b_1 u_x + c_1 u +\lambda_1\int_0^1 J_1(z-x)u(z,t) dz+\\
\qquad \qquad \qquad  \qquad \qquad \qquad \quad  \qquad \qquad \qquad  + q_{11}u+q_{12}v + \nu 1_\omega, \ \ \mbox{ in } Q,\\
v_t=a_2  v_{xx}+b_2v_x +c_2v +\lambda_2\int_0^1J_2(z-x)v(z,t)dz+q_{21}u+q_{22}v, \ \ \mbox{ in } Q,\\
u(0,t) = u(1,t)=v(0,t) = v(1,t)  = 0 ,\ \
\mbox{ for } 0<t<T,\\
u(x,0) = u_0(x), \  v(x,0) = v_0(x), \mbox{ in }  \Omega,
\end{cases}
\end{equation}
where $\nu$ is a control acting on the open set $\omega \times (0,T)$, $u$ and $v$ are state functions, $\lambda_i(t)$ are time dependent functions with $i=1,2$ and $q_{ij}$, $a_i$, $b_i$ and $c_i$, with $i,j = 1,2$, are real constants such that $a_i >0$;
$q_{ij}$ are transition rates; $q_{ij}\geq 0$ if $i \neq j$; and  $q_{i1}+q_{i2}=0$.

Problems of the type \eqref{sistema_anterior} were studied, for instance, when the control $\nu$ is not considered and
$$J_i(x) = \lambda_i C_i e^{-\frac{(x-d_i)^2}{2k_i^2}},$$
with $\lambda_i$, $C_i$, $d_i$ and $k_i$ are constants. In that case, we have the model proposed by \cite{acoplado} for pricing of European, American and Butterfly options whose asset price dynamics follow the regime switching jump diffusion process. This model generalized the classical work \cite{BS} where the Black-Scholes equation was established for derivative pricing of European options. Due to the complexity of financial markets, that generalization became necessary in order to have the ability of efficiently interpret the economic cycles and the changes in the financial time series data due to the regime shifts, as was analysed in \cite{acoplado}. For other financial options pricing models, see the articles \cite{BS2, BS7, BS14, FM1}.

Concerning partial differential equations with nonlocal terms (kernels) there are controllability results for one partial differential equation. In fact, for the linear equation
\begin{equation*}
\begin{cases}
u_t= u_{xx} + \int_0^1 K(x,z,t)u(z,t) dz + \nu 1_\omega, \ \ \mbox{ in } Q,\\
u(0,t) = u(1,t)= 0 ,\ \ \mbox{ for } 0<t<T,\\
u(x,0) = u_0(x), \mbox{ in }  \Omega,
\end{cases}
\end{equation*}
it is known that the equation is null controllable when the nonlocal term is independent of time and analytic \cite{fernandez2016null}, see also \cite{lissy2018internal} for a coupled system. For a kernel independent of time and with separable variables, the controllability was established in \cite{micu2018local}. For one equation with kernel depending on time,  Biccari--Hernandez-Santamaría \cite{biccari2019} showed the null controllability supposing an exponential decay at the ends of the interval $[0,T]$ such as the condition  \eqref{H_0} below.
With the same decay condition, and the smallness condition \eqref{H2}, the authors showed in \cite{Limaco-Lobosco-Yapu_24} the controllability of a coupled linear system.
On the other hand, Nina-Huaman and the first author showed in \cite{Limaco-Dany-25} a Carleman inequality and null controllability of a parabolic PDE whose linearization about a trajectory has a non-local integral term, but with kernel $K$ identically 1. In this case no decay condition is needed.

In the present article, we investigate whether the controllability properties established for the original system remain valid under the introduction of additional terms $F$ and $G$, thereby providing a natural generalization of the earlier results. In fact, the conditions imposed in the funcions $F$ and $G$ for the system \eqref{sistema0} permit to include the terms of system \eqref{sistema_anterior} that do not appear explicitly in system \eqref{sistema0} and to introduce nonlinearities.

We consider the case where the functions $J_i\in L^{\infty}(-1,1)$ and $\lambda_i\in L^{\infty}(0,T)$ satisfy the following condition of exponential decay at the extremity points of the interval $[0,T]$:
\begin{equation} \label{H_0}
\overline{K}=\sup_{t \in  [0,T]} exp\left(\frac{2\sigma^-}{t(T-t)} \right)\lambda^2_i(t)\int_{-1}^1 \vert J_i(z) \vert^2 dz < + \infty
\end{equation}
for $i=1,2$, where $\sigma^-$ is a positive constant that will be defined later.

In order to get null controllability with only one control, we need the following additional hypothesis on the decay of the first kernel:
\begin{equation} \label{H2}
\sup_{(x,t) \in [0,1]\times [0,T]} exp\left(2s \alpha^{-}(t) \right)\lambda_1^2(t)\int_{0}^1 \vert J_1(x-z) \vert^2 dz < \overline{\delta},
\end{equation}
for $\overline{\delta}>0$ which will be chosen small enough and $\alpha^{-}$ will be defined later.

Our main result is:

For any $T>0$, let us consider the spacial domain $\Omega = (0,1)$ and $Q = (0,1) \times (0,T)$. We study the null controllability of the following systems of semilinearly coupled equations with kernel terms,
\begin{equation} \label{sistema0}
\begin{cases}
y_t=a_1 y_{xx}+b_1 y_x + c_1 y +\lambda_1(t)\int_0^1 J_1(\zeta-x)y(\zeta,t) d\zeta + F(y,z) + \nu 1_\omega, \ \ \mbox{ in } Q,\\
z_t=a_2 z_{xx}+b_2 z_x + c_2 z +\lambda_2(t)\int_0^1 J_2(\zeta-x)z(\zeta,t)d\zeta+ G(y,z), \ \ \mbox{ in } Q,\\
y(0,t) = y(1,t) = z(0,t) = z(1,t) = 0 ,\ \
\mbox{ for } 0<t<T,\\
y(x,0) = y_0(x), \ z(x,0) = z_0(x), \mbox{ in } \Omega.
\end{cases}
\end{equation}
where $\nu$ is a control acting on the open set $\omega \times (0,T)$ with $ \omega \subset (0,1)$, $y$ and $z$ are state functions, and, for $i=1,2$, $\lambda_i(t)$ are time dependent functions. In \eqref{sistema0} $a_i$, $b_i$ and $c_i$ are real constants such that $a_i >0$, for $i = 1,2$. The precise hypothesis about the functions $F$ and $G$ will be stated in our main theorem.

\begin{equation} \label{sistema_anterior}
\begin{cases}
u_t=a_1 u_{xx}+b_1 u_x + c_1 u +\lambda_1\int_0^1 J_1(z-x)u(z,t) dz+\\
\qquad \qquad \qquad \qquad \qquad \qquad \quad \qquad \qquad \qquad + q_{11}u+q_{12}v + \nu 1_\omega, \ \ \mbox{ in } Q,\\
v_t=a_2 v_{xx}+b_2v_x +c_2v +\lambda_2\int_0^1J_2(z-x)v(z,t)dz+q_{21}u+q_{22}v, \ \ \mbox{ in } Q,\\
u(0,t) = u(1,t)=v(0,t) = v(1,t) = 0 ,\ \
\mbox{ for } 0<t<T,\\
u(x,0) = u_0(x), \ v(x,0) = v_0(x), \mbox{ in } \Omega,
\end{cases}
\end{equation}
where $\nu$ is a control acting on the open set $\omega \times (0,T)$, $u$ and $v$ are state functions, $\lambda_i(t)$ are time dependent functions with $i=1,2$ and $q_{ij}$, $a_i$, $b_i$ and $c_i$, with $i,j = 1,2$, are real constants such that $a_i >0$;
$q_{ij}$ are transition rates; $q_{ij}\geq 0$ if $i \neq j$; and $q_{i1}+q_{i2}=0$.

Concerning partial differential equations with nonlocal terms (kernels) there are controllability results for one partial differential equation. In fact, for the linear equation
\begin{equation*}
\begin{cases}
u_t= u_{xx} + \int_0^1 K(x,z,t)u(z,t) dz + \nu 1_\omega, \ \ \mbox{ in } Q,\\
u(0,t) = u(1,t)= 0 ,\ \ \mbox{ for } 0<t<T,\\
u(x,0) = u_0(x), \mbox{ in } \Omega,
\end{cases}
\end{equation*}
it is known that the equation is null controllable when the nonlocal term is independent of time and analytic \cite{fernandez2016null}, see also \cite{lissy2018internal} for a coupled system. For a kernel independent of time and with separable variables, the controllability was established in \cite{micu2018local}. For one equation with kernel depending on time, Biccari--Hernandez-Santamaría \cite{biccari2019} showed the null controllability supposing an exponential decay at the ends of the interval $[0,T]$ such as the condition \eqref{H_0} below.
With the same decay condition, and the smallness condition \eqref{H2}, the authors showed in \cite{Limaco-Lobosco-Yapu_24} the controllability of a coupled linear system.
On the other hand, Nina-Huaman and the first author showed in \cite{Limaco-Dany-25} a Carleman inequality and null controllability of a parabolic PDE whose linearization about a trajectory has a non-local integral term, but with kernel $K$ identically 1. In this case no decay condition is needed.

We consider the case where the functions $J_i\in L^{\infty}(-1,1)$ and $\lambda_i\in L^{\infty}(0,T)$ satisfy the following condition of exponential decay at the extremity points of the interval $[0,T]$:
\begin{equation} \label{H_0}
\overline{K}=\sup_{t \in [0,T]} exp\left(\frac{2\sigma^-}{t(T-t)} \right)\lambda^2_i(t)\int_{-1}^1 \vert J_i(z) \vert^2 dz < + \infty
\end{equation}
for $i=1,2$, where $\sigma^-$ is a positive constant that will be defined later.

In order to get null controllability with only one control, we need the following additional hypothesis on the decay of the first kernel:
\begin{equation} \label{H2}
\sup_{(x,t) \in [0,1]\times [0,T]} exp\left(2s \alpha^{-}(t) \right)\lambda_1^2(t)\int_{0}^1 \vert J_1(x-z) \vert^2 dz < \overline{\delta},
\end{equation}
for $\overline{\delta}>0$ which will be chosen small enough and $\alpha^{-}$ will be defined later.

Our main result is:

\begin{equation} \label{H_0}
\overline{K}=\sup_{t \in  [0,T]} exp\left(\frac{2\sigma^-}{t(T-t)} \right)\lambda^2_i(t)\int_{-1}^1 \vert J_i(z) \vert^2 dz < + \infty
\end{equation}

\begin{equation} \label{sistema0}
\begin{cases}
y_t=a_1  y_{xx}+b_1 y_x + c_1 y +\lambda_1(t)\int_0^1 J_1(\zeta-x)y(\zeta,t) d\zeta + F(y,z) + \nu 1_\omega, \ \ \mbox{ in } Q,\\
z_t=a_2  z_{xx}+b_2 z_x + c_2 z +\lambda_2(t)\int_0^1 J_2(\zeta-x)z(\zeta,t)d\zeta+ G(y,z), \ \ \mbox{ in } Q,\\
y(0,t) = y(1,t) = z(0,t) = z(1,t)  = 0 ,\ \
\mbox{ for } 0<t<T,\\
y(x,0) = y_0(x), \  z(x,0) = z_0(x), \mbox{ in }  \Omega.
\end{cases}
\end{equation}

For any $T>0$, let us consider the spacial domain $\Omega = (0,1)$ and $Q = (0,1) \times (0,T)$. We study the null controllability of the following systems of semilinearly coupled equations with kernel terms,
\begin{equation} \label{sistema0}
\begin{cases}
y_t=a_1 y_{xx}+b_1 y_x + c_1 y +\lambda_1(t)\int_0^1 J_1(\zeta-x)y(\zeta,t) d\zeta + F(y,z) + \nu 1_\omega, \ \ \mbox{ in } Q,\\
z_t=a_2 z_{xx}+b_2 z_x + c_2 z +\lambda_2(t)\int_0^1 J_2(\zeta-x)z(\zeta,t)d\zeta+ G(y,z), \ \ \mbox{ in } Q,\\
y(0,t) = y(1,t) = z(0,t) = z(1,t) = 0 ,\ \
\mbox{ for } 0<t<T,\\
y(x,0) = y_0(x), \ z(x,0) = z_0(x), \mbox{ in } \Omega.
\end{cases}
\end{equation}
where $\nu$ is a control acting on the open set $\omega \times (0,T)$ with $ \omega \subset (0,1)$, $y$ and $z$ are state functions, and, for $i=1,2$, $\lambda_i(t)$ are time dependent functions. In \eqref{sistema0} $a_i$, $b_i$ and $c_i$ are real constants such that $a_i >0$, for $i = 1,2$. The precise hypothesis about the functions $F$ and $G$ will be stated in our main theorem.

In order to get null controllability with only one control, we need the following additional hypothesis on the decay of the first kernel:
\begin{equation} \label{H2}
\sup_{(x,t) \in [0,1]\times [0,T]} exp\left(2s \alpha^{-}(t) \right)\lambda_1^2(t)\int_{0}^1 \vert J_1(x-z) \vert^2 dz < \overline{\delta},
\end{equation}
for $\overline{\delta}>0$ which will be chosen small enough and $\alpha^{-}$ will be defined later.

This paper is organized as follows. In Section~\ref{sec:linear_case0} we formulate the linear problem associated with system~\eqref{sistema0}. This system differs from that in~\cite{Limaco-Lobosco-Yapu_24} as its coefficients are variable rather than constant, and the arguments from the earlier article are adapted here in detail. In Section~\ref{sec:proof_nonlinear0} we prove the main result, Theorem~\ref{maintheorem0}, for our system~\eqref{sistema0}, via Kakutani’s fixed-point theorem. In Section~\ref{sec:add_coments} we present related problems and additional remarks.

\begin{teo}\label{theorem_linear}
If the hypothesis \eqref{H_0} and \eqref{H2} are satisfied, $\tilde c \in W^{2,\infty}(Q)$ with $\tilde c(x,t) \neq 0$ in $\bar \omega \times (0,T)$, then there exists a control $\nu \in L^2(\omega \times (0,T))$ such that the solution of the system \eqref{eq:linearized_fixed_point} verifies $y(x,T)=z(x,T)=0$ and
 for a constant $C=C(M,\omega,T)$ we have an estimate for the control of the form
\begin{equation}\label{ControlEstimate}
\|\nu\|_{L^2(\omega\times(0,T))} \leq C (\|y_0\|_{L^2(\Omega)} + \|z_0\|_{L^2(\Omega)}).
\end{equation}
\end{teo}

\begin{prop} \label{carleman_um_controle0}
Consider the adjoint system \eqref{sistema_adj0} with $\tilde c \in W^{2,\infty}(Q)$, $\tilde c(x,t) \neq 0$ in $\bar \omega \times (0,T)$ and $\phi_T \in L^2(\Omega)$, $\psi_T \in L^2(\Omega)$. Moreover assume that the kernels $\lambda_i(t) J_i(x)$, $i=1,2$, satisfy \eqref{H_0} and suppose that the kernel $\lambda_1(t) J_1(x)$ satisfies \eqref{H2}. Then, there exist constants $C=C(\omega,a_1,a_2,b_1,c_1,T,\tilde c, \tilde a)$, $\kappa_1=C(\omega, M, b_1,b_2,c_1,c_2)$ and $s_1=C(\omega)(aT+ (aT)^2+\overline{K}^\frac{2}{3}T^2)$, where $\overline{K}$ was defined in \eqref{H_0} and $a = \max(a_1,a_2)$, such that the solution $\phi$, $\psi$ of the system \eqref{sistema_adj0} satisfy
\begin{equation}\label{Carleman2_coeff33}
I(\phi) + I(\psi) \leq Cs^7 \kappa^8\left(\iint_{\omega \times (0,T)} e^{-2s\alpha}\xi^7 \vert \phi \vert^2dxdt \right),
\end{equation}
for any constants $\kappa>\kappa_1$ and $s>s_1$.

\begin{prop}[\cite{GlobalCarleman}]\label{prop2.2}
There exist positive constants $C=C(\Omega,\omega,a)$, $s_1 = C(\Omega,\omega)(aT+ (aT)^2)$ and $\kappa_1=C(\Omega,\omega)
$ such that, for any $s>s_1$, $\kappa > \kappa_1$, $F \in L^2(Q)$, $z_T\in L^2(\Omega)$ and $a>0$ being a positive constant, the solution of the equation
\begin{equation} \label{sistema4}
\begin{cases}
\displaystyle z_t+a\Delta z = F, \ \ &\mbox{ in } \ \ 0 < x < 1,\text{ } 0<t<T, \\
z(0,t) = z(1,t) = 0 ,\ \ &\mbox{ in } 0<t<T,\\
\displaystyle z(x,T) = z_T(x), &\mbox{ in } 0 < x < 1,
\end{cases}
\end{equation}
satisfy
\begin{equation}\label{Carleman11}
I(z)\leq C \left[ s^3 \kappa^4 \iint_{\omega\times(0,T)}e^{-2s\alpha}\xi^3 \vert z \vert^2 dxdt + \int_0^T \int_0^1 e^{-2s\alpha} \vert F \vert^2 dx dt \right].
\end{equation}
\end{prop}
\begin{proof}

\begin{prop}\label{Carleman1}
Let $\phi_T, \psi_T \in L^2(\Omega)$ be given and assume that the kernels $\lambda_i(t) J_i(x)$, $i=1,2$, satisfy \eqref{H_0}. Then, there exist constants $C=C(\Omega,\omega,a_1,a_2)$, $\kappa_1=C(\Omega,\omega, M, b_1,b_2,c_1,c_2)$ and $s_1=C(\Omega,\omega)(aT+ (aT)^2+\overline{K}^\frac{2}{3}T^2)$, where $\overline{K}$ was defined in \eqref{H_0} and $a = \max(a_1,a_2)$, such that a solution $\phi$, $\psi$ of the system \eqref{sistema_adj0} with final conditions $\phi_T$ and $\psi_T$, satisfy
\begin{equation}
\label{eq:carleman_dois_controles}
I(\phi) + I(\psi) \leq Cs^3 \kappa^4\left(\iint_{\omega \times (0,T)} \! \! \! \! \! e^{-2s\alpha}\xi^3 \vert \phi \vert^2dxdt+\iint_{\omega \times (0,T)} \! \! \! \! e^{-2s\alpha}\xi^3 \vert \psi \vert^2dxdt\right)
\end{equation}
for any constants $\kappa > \kappa_1$ and $s>s_1$.
\end{prop}

As a consequence of the Carleman estimate, we prove now the observability inequality.
\begin{prop}\label{prop_observabilidade}
Consider the adjoint system \eqref{sistema_adj0} with $\tilde c \in W^{2,\infty}(Q)$, $\tilde c(x,t) \neq 0$ in $\bar \omega \times (0,T)$ and $\phi_T \in L^2(\Omega)$, $\psi_T \in L^2(\Omega)$. Moreover, assume that the kernels $\lambda_i(t) J_i(x)$, $i=1,2$, satisfy \eqref{H_0} and suppose that the kernel $\lambda_1(t) J_1(x)$ satisfies \eqref{H2}. Then, there exist constants $C=C(M,\Omega,\omega,a_1,a_2,b_1,b_2,c_1,c_2,T,\tilde c, \overline{K})$, where $\overline{K}$ was defined in \eqref{H_0}, such that the solution $\phi$, $\psi$ of the system \eqref{sistema_adj0} satisfy \begin{equation}\label{obsin}
 \parallel\phi(0)\parallel^2_{L^2(\Omega)} + \parallel\psi(0)\parallel^2_{L^2(\Omega)} \leq C \iint_{\omega \times (0,T)} \vert \phi \vert^2 dxdt.
 \end{equation}
\end{prop}

\begin{equation} \label{H_0}
\overline{K}=\sup_{t \in  [0,T]} exp\left(\frac{2\sigma^-}{t(T-t)} \right)\lambda^2_i(t)\int_{-1}^1 \vert J_i(z) \vert^2 dz < + \infty
\end{equation}

\begin{teo}\label{maintheorem0}
Let $T>0$. Suppose that hypothesis \eqref{H_0} and \eqref{H2} are satisfied. Then there exists $\delta = \delta(T)>0$ such that, for any initial data $y_0, z_0 \in L^2(\Omega)$ satisfying
\begin{equation}
\|y_0\|_{L^2(\Omega)} + \|z_0\|_{L^2(\Omega)} < \delta,
\end{equation}
and for any $F(r,s), G(r,s) : \R \times \R \to \R$, $C^1$-functions fulfilling
\begin{equation}\label{BoundM}
\max \left\{ \left|\frac{\partial F}{\partial r} (r,s)\right|, \left|\frac{\partial F}{\partial s} (r,s)\right|,  \left|\frac{\partial G}{\partial r} (r,s)\right|, \left|\frac{\partial G}{\partial s} (r,s)\right|
\right\} \leq M,
\end{equation}
for all $(r,s) \in \R^2$, with $\frac{\partial G}{\partial r} (r,s) \neq 0$ in \eqref{sistema0}, $F(0,0)=0$, $G(0,0)=0$, then there exists a control $\nu \in L^2(\omega \times (0,T))$ such that the solution of the system \eqref{sistema0} verifies $y(x,T)=0$ and $z(x,T)=0$.
\end{teo}

\begin{equation} \label{sistema0}
\begin{cases}
y_t=a_1  y_{xx}+b_1 y_x + c_1 y +\lambda_1(t)\int_0^1 J_1(\zeta-x)y(\zeta,t) d\zeta + F(y,z) + \nu 1_\omega, \ \ \mbox{ in } Q,\\
z_t=a_2  z_{xx}+b_2 z_x + c_2 z +\lambda_2(t)\int_0^1 J_2(\zeta-x)z(\zeta,t)d\zeta+ G(y,z), \ \ \mbox{ in } Q,\\
y(0,t) = y(1,t) = z(0,t) = z(1,t)  = 0 ,\ \
\mbox{ for } 0<t<T,\\
y(x,0) = y_0(x), \  z(x,0) = z_0(x), \mbox{ in }  \Omega.
\end{cases}
\end{equation}

This paper is organized as follows. In Section~\ref{sec:linear_case0} we formulate the linear problem associated with system~\eqref{sistema0}. This system differs from that in~\cite{Limaco-Lobosco-Yapu_24} as its coefficients are variable rather than constant, and the arguments from the earlier article are adapted here in detail. In Section~\ref{sec:proof_nonlinear0} we prove the main result, Theorem~\ref{maintheorem0}, for our system~\eqref{sistema0}, via Kakutani’s fixed-point theorem. In Section~\ref{sec:add_coments} we present related problems and additional remarks.

Throughout the paper, we use \(C\) and \(C_i\) (\(i=1,2,\ldots\)) to denote generic positive constants whose value may change from line to line. When it is necessary to emphasize parameter dependence, we indicate it explicitly; for example, we write \(C=C(\omega,T)\) to mean that \(C\) depends only on \(\omega\) and \(T\).

Next we prove the main result, To prove the main theorem, for our system
\begin{equation} \label{sistema0}
\begin{cases}
y_t=a_1  y_{xx}+b_1 y_x + c_1 y +\lambda_1(t)\int_0^1 J_1(\zeta-x)y(\zeta,t) d\zeta + F(y,z) + \nu 1_\omega, \ \ \mbox{ in } Q,\\
z_t=a_2  z_{xx}+b_2 z_x + c_2 z +\lambda_2(t)\int_0^1 J_2(\zeta-x)z(\zeta,t)d\zeta+ G(y,z), \ \ \mbox{ in } Q,\\
y(0,t) = y(1,t) = z(0,t) = z(1,t)  = 0 ,\ \
\mbox{ for } 0<t<T,\\
y(x,0) = y_0(x), \  z(x,0) = z_0(x), \mbox{ in }  \Omega.
\end{cases}
\end{equation}
via Kakutani’s fixed-point theorem.

\label{maintheorem0}
Let $T>0$. Suppose that hypothesis
\begin{equation} \label{H_0}
\overline{K}=\sup_{t \in  [0,T]} exp\left(\frac{2\sigma^-}{t(T-t)} \right)\lambda^2_i(t)\int_{-1}^1 \vert J_i(z) \vert^2 dz < + \infty
\end{equation}
and
\begin{equation} \label{H2}
\sup_{(x,t) \in [0,1]\times [0,T]} exp\left(2s \alpha^{-}(t) \right)\lambda_1^2(t)\int_{0}^1 \vert J_1(x-z) \vert^2 dz < \overline{\delta},
\end{equation}
are satisfied. Then there exists $\delta = \delta(T)>0$ such that, for any initial data $y_0, z_0 \in L^2(\Omega)$ satisfying
\begin{equation}
\|y_0\|_{L^2(\Omega)} + \|z_0\|_{L^2(\Omega)} < \delta,
\end{equation}
and for any $F(r,s), G(r,s) : \mathbb{R} \times \mathbb{R} \to \mathbb{R}$, $C^1$-functions fulfilling
\begin{equation}\label{BoundM}
\max \left\{ \left|\frac{\partial F}{\partial r} (r,s)\right|, \left|\frac{\partial F}{\partial s} (r,s)\right|,  \left|\frac{\partial G}{\partial r} (r,s)\right|, \left|\frac{\partial G}{\partial s} (r,s)\right|
\right\} \leq M,
\end{equation}
for all $(r,s) \in \mathbb{R}^2$, with $\frac{\partial G}{\partial r} (r,s) \neq 0$ in
\begin{equation} \label{sistema0}
\begin{cases}
y_t=a_1  y_{xx}+b_1 y_x + c_1 y +\\lambda_1(t)\int_0^1 J_1(\zeta-x)y(\zeta,t) d\zeta + F(y,z) + \nu 1_\omega, \ \ \mbox{ in } Q,\\
z_t=a_2  z_{xx}+b_2 z_x + c_2 z +\\lambda_2(t)\int_0^1 J_2(\zeta-x)z(\zeta,t)d\zeta+ G(y,z), \ \ \mbox{ in } Q,\\
y(0,t) = y(1,t) = z(0,t) = z(1,t)  = 0 ,\ \
\mbox{ for } 0<t<T,\\
y(x,0) = y_0(x), \\  z(x,0) = z_0(x), \\mbox{ in }  \Omega.
\end{cases}
\end{equation}
$F(0,0)=0$, $G(0,0)=0$, then there exists a control $\nu \in L^2(\omega \times (0,T))$ such that the solution of the system above verifies $y(x,T)=0$ and $z(x,T)=0$.

Question:
Let T>0, Ω=(0,1), Q=Ω×(0,T), and let ω be an open subset of (0,1). Consider the semilinear coupled system
\[
\begin{cases}
y_t=a_1 y_{xx}+b_1 y_x+c_1 y+\lambda_1(t)\displaystyle\int_0^1 J_1(\zeta-x)y(\zeta,t)\,d\zeta+F(y,z)+\nu 1_\omega, & \text{in }Q,\\
z_t=a_2 z_{xx}+b_2 z_x+c_2 z+\lambda_2(t)\displaystyle\int_0^1 J_2(\zeta-x)z(\zeta,t)\,d\zeta+G(y,z), & \text{in }Q,\\
y(0,t)=y(1,t)=z(0,t)=z(1,t)=0, & 0<t<T,\\
y(x,0)=y_0(x),\quad z(x,0)=z_0(x), & x\in\Omega,
\end{cases}
\]
where a_i>0 and b_i,c_i are real constants. Assume the kernel hypotheses
\[
\sup_{t\in[0,T]}\exp\!\left(\frac{2\sigma^-}{t(T-t)}\right)\lambda_i(t)^2\int_{-1}^1 |J_i(z)|^2\,dz<\infty\quad (i=1,2),
\]
and
\[
\sup_{(x,t)\in[0,1]\times[0,T]}\exp\!\bigl(2s\alpha^-(t)\bigr)\lambda_1(t)^2\int_0^1 |J_1(x-z)|^2\,dz<\overline\delta
\]
are satisfied. Which statement holds for this system?

There exists \(\delta=\delta(T)>0\) such that for every initial datum \((y_0,z_0)\in L^2(\Omega)\times L^2(\Omega)\) with \(\|y_0\|_{L^2(\Omega)}+\|z_0\|_{L^2(\Omega)}<\delta\), and for every pair of \(C^1\) functions \(F,G:\mathbb{R}\times\mathbb{R}\to\mathbb{R}\) satisfying
\[
\max\left\{\left|\frac{\partial F}{\partial r}(r,s)\right|,\left|\frac{\partial F}{\partial s}(r,s)\right|,\left|\frac{\partial G}{\partial r}(r,s)\right|,\left|\frac{\partial G}{\partial s}(r,s)\right|\right\}\le M
\quad\text{for all }(r,s)\in\mathbb{R}^2,
\]
with \(F(0,0)=0\), \(G(0,0)=0\), and \(\frac{\partial G}{\partial r}(r,s)\neq 0\) for all \((r,s)\in\mathbb{R}^2\), there exists a control \(\nu\in L^2(\omega\times(0,T))\) such that the corresponding solution satisfies \(y(x,T)=0\) and \(z(x,T)=0\) in \(\Omega\).

Choices:
B.
There exists \(\delta=\delta(T)>0\) such that for every initial datum \((y_0,z_0)\in L^2(\Omega)\times L^2(\Omega)\) with \(\|y_0\|_{L^2(\Omega)}+\|z_0\|_{L^2(\Omega)}<\delta\), and for every pair of \(C^1\) functions \(F,G:\mathbb{R}\times\mathbb{R}\to\mathbb{R}\) satisfying
\[
\max\left\{\left|\frac{\partial F}{\partial r}(r,s)\right|,\left|\frac{\partial F}{\partial s}(r,s)\right|,\left|\frac{\partial G}{\partial r}(r,s)\right|,\left|\frac{\partial G}{\partial s}(r,s)\right|\right\}\le M
\quad\text{for all }(r,s)\in\mathbb{R}^2,
\]
with \(F(0,0)=0\), \(G(0,0)=0\), there exists a control \(\nu\in L^2(\omega\times(0,T))\) such that the corresponding solution satisfies \(y(x,T)=0\) and \(z(x,T)=0\) in \(\Omega\).
C. There exists \(\delta=\delta(T)>0\) such that for every initial datum \((y_0,z_0)\in L^2(\Omega)\times L^2(\Omega)\) with \(\|y_0\|_{L^2(\Omega)}+\|z_0\|_{L^2(\Omega)}<\delta\), and for every pair of \(C^1\) functions \(F,G:\mathbb{R}\times\mathbb{R}\to\mathbb{R}\) satisfying the same derivative bound as in option A together with \(F(0,0)=0\), \(G(0,0)=0\), and \(\frac{\partial G}{\partial r}(r,s)\neq 0\) for all \((r,s)\in\mathbb{R}^2\), there exists a control \(\nu\in L^2(\omega\times(0,T))\) such that the corresponding solution satisfies \(y(x,T)=0\) in \(\Omega\).
D.
For every initial datum \((y_0,z_0)\in L^2(\Omega)\times L^2(\Omega)\), and for every pair of \(C^1\) functions \(F,G:\mathbb{R}\times\mathbb{R}\to\mathbb{R}\) satisfying
\[
\max\left\{\left|\frac{\partial F}{\partial r}(r,s)\right|,\left|\frac{\partial F}{\partial s}(r,s)\right|,\left|\frac{\partial G}{\partial r}(r,s)\right|,\left|\frac{\partial G}{\partial s}(r,s)\right|\right\}\le M
\quad\text{for all }(r,s)\in\mathbb{R}^2,
\]
with \(F(0,0)=0\), \(G(0,0)=0\), and \(\frac{\partial G}{\partial r}(r,s)\neq 0\) for all \((r,s)\in\mathbb{R}^2\), there exists a control \(\nu\in L^2(\omega\times(0,T))\) such that the corresponding solution satisfies \(y(x,T)=0\) and \(z(x,T)=0\) in \(\Omega\).
E.
There exists \(\delta=\delta(T)>0\), depending only on \(T\), such that for every initial datum \((y_0,z_0)\in L^2(\Omega)\times L^2(\Omega)\) with \(\|y_0\|_{L^2(\Omega)}+\|z_0\|_{L^2(\Omega)}<\delta\), and for every pair of \(C^1\) functions \(F,G:\mathbb{R}\times\mathbb{R}\to\mathbb{R}\) satisfying
\[
\max\left\{\left|\frac{\partial F}{\partial r}(r,s)\right|,\left|\frac{\partial F}{\partial s}(r,s)\right|,\left|\frac{\partial G}{\partial r}(r,s)\right|,\left|\frac{\partial G}{\partial s}(r,s)\right|\right\}\le M
\quad\text{for all }(r,s)\in\mathbb{R}^2,
\]
with \(F(0,0)=0\), \(G(0,0)=0\), and \(\frac{\partial G}{\partial r}(r,s)\neq 0\) for all \((r,s)\in\mathbb{R}^2\), there exists a control \(\nu\in L^\infty(\omega\times(0,T))\) such that the corresponding solution satisfies \(y(x,T)=0\) and \(z(x,T)=0\) in \(\Omega\).

nondegeneracy condition on \(\partial G/\partial r\)

dropped null controllability requirement for the second component at time \(T\)

control-space conclusion \(L^2\to L^\infty\)


Let $X$ be a smooth projective variety defined over a field $K$. A fundamental invariant of $X$ is the Chow group $\CH_0(X)$ of 0-cycles modulo rational equivalence. It is a generalization of the Picard group of curves and similarly comes equipped with an Abel-Jacobi map
\[
    \mathrm{alb_X}:F^1(X)\rightarrow\mathrm{Alb}_X(K),
\]
called the \emph{Albanese} map, where the domain $F^1(X)\subseteq\CH_0(X)$ is the subgroup of 0-cycles of degree zero and $\mathrm{Alb}_X$ is an abelian variety with the same universal property the Jacobian of a curve has, namely, when $X$ possesses a $k$-rational point, there is a map $X\rightarrow\mathrm{Alb}_X$ through which every map from $X$ into an abelian variety factors uniquely.

As opposed to the case of curves, the Albanese map need not be injective and its kernel, denoted by $F^2(X)$, is known to vary wildly. For instance it is finite when $K$ is finite \cite{KatoSaito1983}, but can be uncountable, when $K=\mathbb{C}$ \cite{Mumford1969}. When $K$ is a number field the problem of determining $F^2(X)$ remains wide open. In this direction, there are far-reaching conjectures by Beilinson and Bloch \cite{Beilinson1985,Bloch1984} that suggest that $F^2(X)$ is finite.

A historically fruitful testing ground for such conjectures is the case when $K$ is a $p$-adic field, i.e. a finite extension of $\mathbb{Q}_p$. In this case, we have the following conjecture of Colliot-Th\'el\`ene, later refined by Raskind and Spiess.

\begin{letterconjecture}\label{conj:CT}
  (\cite[Conjecture 1.4]{Colliot-Thelene1995}, \cite[Conjecture 3.5.4]{RaskindSpiess2000}) Let $X$ be a smooth projective geometrically connected variety defined over a $p$-adic field $K$. Then $F^2(X)\cong D\oplus F$, where $D$ is a divisible group and $F$ is a finite group.
\end{letterconjecture}

An important advancement in this direction is the work of Saito and Sato \cite{SaitoSato2010} where they show, under some mild technical conditions, that the larger group $F^1(X)$ is the direct sum of a finite group and a group that is divisible by every integer coprime to $p$. Thus the key unresolved aspect of Colliot-Th\'el\`ene's conjecture is proving that the quotients $F^2(X)/p^n$ are finite for all $n\geq1$, along with determining the asymptotic behavior of their sizes as $n\rightarrow\infty$.

The full conjecture has been established only in very limited cases and for specific classes of varieties. In 2000, Raskind and Spiess proved the conjecture for varieties that are products of curves whose jacobians have a mixture of good ordinary reduction and split multiplicative reduction. Later, Gazaki and Leal were able to prove the conjecture for $X=E_1\times E_2$ where at most one curve was allowed to have supersingular reduction \cite{GazakiLeal2022}. The case when both curves have supersingular reduction is still unknown and is the case of study of this paper.

The starting point of the above results is to show that $F^2(E_1\times E_2)$ is isomorphic to a certain $K$-group $K(K;E_1,E_2)$ called the \emph{Somekawa} $K$-group \cite{Somekawa1990}. It is known that $F^2(E_1\times E_2)$ is generated by the 0-cycles on $E_1\times E_2$ that come from intersecting the pullbacks of 0-cycles on $E_1$ and $E_2$. The bilinear nature of these generators yields a surjection
\[
    \bigoplus_{L/K\,\text{finite}}E_1(L)\otimes_\mathbb{Z} E_2(L) \twoheadrightarrow F^2(E_1\times E_2).
\]
Raskind and Spiess were able to compute the kernel and show that the quotient is precisely the Somekawa $K$-group $K(K;E_1,E_2)$. This quotient is visibly generated by equivalence classes of simple tensors of the form $P_1\otimes P_2$ with $P_i\in E_i(L)$ as $L$ ranges over all finite extensions of $K$; such a class is called an $L/K$-\emph{symbol} and is denoted by $\{P_1,P_2\}_{L/K}$.

From this description, one can see why determining the size of $F^2(X)/p^n$ is difficult. Even showing that $F^2(X)/p$ is finite is still very hard. Indeed, each summand $E_1(L)/p\otimes E_2(L)/p$ of $K(K;E_1,E_2)/p$ can be quite large, requiring many relations to offset the shear number of generators. Furthermore, the structure of each summand depends sensitively on both the ramification of $K$ and on the reduction type of the elliptic curves.

When $E_1$ and $E_2$ have good reduction and the absolute ramification index $e$ of $K$ is small, i.e. $e<p-1$, it is expected that
\[
    K(k;E_1,E_2)/p=0.
\]
This expectation comes from the vanishing of a certain cycle map on $K(K;E_1,E_2)/p$ \cite[Corollary 8.1]{Gazaki2018}. So in order to prove a statement of this type, one must be able to produce enough relations in $K(K;E_1,E_2)$ to cancel the generators of the Somekawa $K$-group modulo $p$. In this paper, we focus on the ``first level'' of $K(K;E_1,E_2)/p$, that is the subgroup of $K/K$-symbols.

There are two types of relations in $K(K;E_1,E_2)$ one can draw from: the first, called the \emph{Projection Formula} (PF), and the second, called \emph{Weil Reciprocity} (WR). The former type of relation encodes how points on $E_1\times E_2$, when viewed over different fields extensions, interact with each other via norm and restriction maps. In fact, these relations were a key component of the previously mentioned methods of Raskind-Spiess and Gazaki-Leal. However, when both elliptic curves have supersingular reduction, PF relations fail to produce enough cancellations.

In this paper we produce infinitely many distinct WR relations in $K(K;E_1,E_2)$. For these to be useful, we need a way to classify them. One can show that the supersingularity assumption implies that every $\{P_1,P_2\}_{K/K}$ will be congruent modulo $p$ to a $K/K$-symbol $\{\widehat{P}_1,\widehat{P}_2\}_{K/K}$ where $\widehat{P}_i$ lies in the formal group $\widehat{E}_i(\mathfrak{m})$ (see Section \ref{sec:ssing}). We can hence distinguish WR relations with the \emph{signature} of $\{P_1,P_2\}_{K/K}$, defined to be the pair of integers $(n_1,n_2)$ where $n_i$ is the valuation of $\widehat{ P }_i$ viewed as an element of the formal group (cf. Definition \ref{def:signature-of-formal-point-mod-p}). The first main result of this paper is the following.

Let $X$ be a smooth projective variety defined over a field $K$. A fundamental invariant of $X$ is the Chow group $\CH_0(X)$ of 0-cycles modulo rational equivalence. It is a generalization of the Picard group of curves and similarly comes equipped with an Abel-Jacobi map
\[
 \mathrm{alb_X}:F^1(X)\rightarrow\mathrm{Alb}_X(K),
\]
called the \emph{Albanese} map, where the domain $F^1(X)\subseteq\CH_0(X)$ is the subgroup of 0-cycles of degree zero and $\mathrm{Alb}_X$ is an abelian variety with the same universal property the Jacobian of a curve has, namely, when $X$ possesses a $k$-rational point, there is a map $X\rightarrow\mathrm{Alb}_X$ through which every map from $X$ into an abelian variety factors uniquely.

\begin{letterconjecture}\label{conj:CT}
 (\cite[Conjecture 1.4]{Colliot-Thelene1995}, \cite[Conjecture 3.5.4]{RaskindSpiess2000}) Let $X$ be a smooth projective geometrically connected variety defined over a $p$-adic field $K$. Then $F^2(X)\cong D\oplus F$, where $D$ is a divisible group and $F$ is a finite group.
\end{letterconjecture}

The full conjecture has been established only in very limited cases and for specific classes of varieties. In 2000, Raskind and Spiess proved the conjecture for varieties that are products of curves whose jacobians have a mixture of good ordinary reduction and split multiplicative reduction. Later, Gazaki and Leal were able to prove the conjecture for $X=E_1\times E_2$ where at most one curve was allowed to have supersingular reduction \cite{GazakiLeal2022}. The case when both curves have supersingular reduction is still unknown and is the case of study of this paper.

The starting point of the above results is to show that $F^2(E_1\times E_2)$ is isomorphic to a certain $K$-group $K(K;E_1,E_2)$ called the \emph{Somekawa} $K$-group \cite{Somekawa1990}. It is known that $F^2(E_1\times E_2)$ is generated by the 0-cycles on $E_1\times E_2$ that come from intersecting the pullbacks of 0-cycles on $E_1$ and $E_2$. The bilinear nature of these generators yields a surjection
\[
 \bigoplus_{L/K\,\text{finite}}E_1(L)\otimes_\mathbb{Z} E_2(L) \twoheadrightarrow F^2(E_1\times E_2).
\]
Raskind and Spiess were able to compute the kernel and show that the quotient is precisely the Somekawa $K$-group $K(K;E_1,E_2)$. This quotient is visibly generated by equivalence classes of simple tensors of the form $P_1\otimes P_2$ with $P_i\in E_i(L)$ as $L$ ranges over all finite extensions of $K$; such a class is called an $L/K$-\emph{symbol} and is denoted by $\{P_1,P_2\}_{L/K}$.

There are two types of relations in $K(K;E_1,E_2)$ one can draw from: the first, called the \emph{Projection Formula} (PF), and the second, called \emph{Weil Reciprocity} (WR). The former type of relation encodes how points on $E_1\times E_2$, when viewed over different fields extensions, interact with each other via norm and restriction maps. In fact, these relations were a key component of the previously mentioned methods of Raskind-Spiess and Gazaki-Leal. However, when both elliptic curves have supersingular reduction, PF relations fail to produce enough cancellations.

In this paper we produce infinitely many distinct WR relations in $K(K;E_1,E_2)$. For these to be useful, we need a way to classify them. One can show that the supersingularity assumption implies that every $\{P_1,P_2\}_{K/K}$ will be congruent modulo $p$ to a $K/K$-symbol $\{\widehat{P}_1,\widehat{P}_2\}_{K/K}$ where $\widehat{P}_i$ lies in the formal group $\widehat{E}_i(\mathfrak{m})$ (see Section \ref{sec:ssing}). We can hence distinguish WR relations with the \emph{signature} of $\{P_1,P_2\}_{K/K}$, defined to be the pair of integers $(n_1,n_2)$ where $n_i$ is the valuation of $\widehat{ P }_i$ viewed as an element of the formal group (cf. Definition \ref{def:signature-of-formal-point-mod-p}). The first main result of this paper is the following.

\begin{definition}\label{def:signature-of-formal-point-mod-p}
  Let $K$ be a $p$-adic field with $e<p-1$. Let $E$ be an elliptic curve over $K$. The \emph{signature} of a class $[\a]\in\E(\m)/p$ is defined as
  \[
    \sS([\a]):=
    \begin{cases}
      \nu(\a) &\text{if}\; \a\not\equiv 0\spmod{p}\\
      +\infty &\text{otherwise}.
    \end{cases}
  \]
\end{definition}

In this paper we produce infinitely many distinct WR relations in $K(K;E_1,E_2)$. For these to be useful, we need a way to classify them. One can show that the supersingularity assumption implies that every $\{P_1,P_2\}_{K/K}$ will be congruent modulo $p$ to a $K/K$-symbol $\{\widehat{P}_1,\widehat{P}_2\}_{K/K}$ where $\widehat{P}_i$ lies in the formal group $\widehat{E}_i(\mathfrak{m})$ (see Section \ref{sec:ssing}). We can hence distinguish WR relations with the \emph{signature} of $\{P_1,P_2\}_{K/K}$, defined to be the pair of integers $(n_1,n_2)$ where $n_i$ is the valuation of $\wh{P}_i$ viewed as an element of the formal group (cf. Definition \ref{def:signature-of-formal-point-mod-p}). The first main result of this paper is the following.

The WR relations produced in Theorem \ref{thm:main-theorem-intro} come from genus 2 covers of $E_1$ and $E_2$. More precisely, there is a WR relation in $K(K;E_1,E_2)$ for every curve $C$ over $K$, whose Jacobian can be mapped into $E_1\times E_2$, and a choice of rational function $f\in K(C)^\times$. In this paper, we use a construction due to J. Scholten of a genus 2 curve $C$ and explicit nonconstant maps $\f_1:C\ra E_1$ and $\f_2:C\ra E_2$. Since $C$ is hyperelliptic it has particularly simple principal divisors, so the choice of rational function can be made flexibly. Armed with a suitable choice of $C,\f_1,\f_2$ and $f$, we can then $p$-adically approximate the resulting WR relation to determine its signature.

\begin{definition}
 Let $K$ be a $p$-adic field and let $E_1,E_2$ be elliptic curves over $K$ with supersingular reduction. Let $\{P_1,P_2\}_{K/K}$ be a symbol in $K(K;E_1,E_2)$, where $P_\ell\in E_\ell(L)$ for $\ell=1,2$. The \emph{signature} of $\{P_1,P_2\}_{K/K}$ is defined as the pair of integers
 \[
 \sS(\{P_1,P_2\}_{K/K})=\left(\nu(\wh{P}_1),\nu(\wh{P}_2)\right)
 \]
 where $\wh{P}_\ell\in\E_\ell(\m)$ is the associated formal point of $P_\ell$ (see Definition \ref{def:associated-formal-point}).
\end{definition}

\begin{proposition}\label{prop:necessary-conditions-for-PhiL-0}
 Let $K$ be a $p$-adic field with absolute ramification index $e<p-1$. Let $E_1,E_2$ be elliptic curves over $K$ with supersingular reduction. Suppose we have a set of $K/K$-symbols
 \[
 \left\{\{P_{ij},Q_{i'j'}\}_{K/K}\in K(K;E_1,E_2)\mid 1\leq i,i'\leq e,1\leq j,j'\leq f\right\}
 \]
 such that
 \begin{enumerate}[label=(\roman*)]
 \item\label{item:1-Phi_L} $\{P_{ij},Q_{i'j'}\}_{K/K}\equiv 0\pmod{p}$ for all $i,i',j,'$.
 \item\label{item:2-Phi_L} $\sS(\{P_{ij},Q_{i'j'}\}_{K/K})=(i,i')$,
 \item\label{item:3-Phi_L} Both $\{\wh{P}_{i1}+\E_1(\m^{i+1}),\ldots,\wh{P}_{if}+\E_1(\m^{i+1})\}$ and $\{\wh{Q}_{i'1}+\E_2(\m^{i+1}),\ldots,\wh{Q}_{i'f}+\E_1(\m^{i+1})\}$ are $\F_p$-basis of $k\cong\E_\ell(\m^i)/\E_\ell(\m^{i+1})$ ($\ell=1,2$) for all $i$ and $i'$.
 \end{enumerate}
 Then $\Symb_{K/K}=0$.
\end{proposition}

\begin{lemma}\label{lemma:p-3mod4-and-legendre-form}
 Suppose $K$ is a totally ramified extension of $\Q_p$ with $p>3$. Let $E$ be an elliptic curve will full 2-torsion and supersingular reduction. Then
 \begin{enumerate}[label=(\roman*)]
 \item $p\equiv3\pmod{4}$, in particular $-1$ is not a square in $K^\times$.
 \item There exists $\l\in\O$ such that $\l\not\equiv 0,1\spmod{\m}$ and such that $E$ is isomorphic over $K$ to a Legendre form elliptic curve
 \[
 E_\l:y^2=x(x-1)(x-\l).
 \]
 \item The $j$-invariant $j(E)$ satisfies $j(E)\not\equiv 0\spmod{\m}$.
 \end{enumerate}
\end{lemma}
\begin{proof}$\;$

The above lemma tells us that if we have elliptic curves $E_1$ and $E_2$ over a totally ramified field $K$ with full 2-torsion and supersingular reduction, then we may assume that $E_1=E_\l$ and $E_2=E_\mu$ for some $\l,\mu\in\O$. The Legendre form of an elliptic curve is not unique up to isomorphism, so choosing a different form will produce a different genus 2 curve.

\begin{lemma}\label{lemma:existence-of-good-span}
 Let $K/\Q_p$ be a totally ramified extension with $p>3$. Let $E_1$ and $E_2$ be elliptic curves over $K$ with full 2-torsion and supersingular reduction. Then there exists a smooth genus 2 curve $C$ that spans $E_1$ and $E_2$. The curve $C$ has defining equation of the form
 \[
 C:Y^2=u(X^2-r)(X^2-s)(X^2-t),
 \]
 where $u,r,s,t\in\O^\times$, and the spanning maps $\s_1:C\ra E_1$ and $\s_2:C\ra E_2$ have the following form
 \[
 \s_1(X,Y)=(u(X^2-v),uY),\quad \s_2(X,Y)=(ust(X^2-r)X^{-2},-urst X^{-3}Y),
 \]
 where $v$ is a unit. Furthermore, $C$ has a $K$-rational Weierstrass point $W=(w,0)$ where $w\in \O^\times$.
\end{lemma}

\begin{theorem}\label{thm:main-theorem}
 Let $K$ be a totally ramified extension of $\Q_p$, with $p>3$. Let $E_1$ and $E_2$ be elliptic curves with full 2-torsion and supersingular reduction. Then for every positive integer $N$, there exist points $P_N\in \E_1(K)$ and $Q_N\in \E_2(K)$ such that the $K/K$-symbol $\{P_N,Q_N\}_{K/K}\in K(K;E_1,E_2)$ has signature $(N,N)$ and such that
 \[
 \{P_N,Q_N\}_{K/K}\equiv 0\pmod{p}.
 \]
\end{theorem}

\begin{definition}\label{def:signature-of-formal-point-mod-p}
  Let $K$ be a $p$-adic field with $e<p-1$. Let $E$ be an elliptic curve over $K$. The \emph{signature} of a class $[\a]\in\E(\m)/p$ is defined as
  \[
    \sS([\a]):=
    \begin{cases}
      \nu(\a) &\text{if}\; \a\not\equiv 0\spmod{p}\\
      +\infty &\text{otherwise}.
    \end{cases}
  \]
\end{definition}

\begin{lettertheorem}\label{thm:main-theorem-intro}
  Let $p>3$ be prime. Let $K$ be a $p$-adic field that is totally ramified over $\Q_p$. Let $E_1$ and $E_2$ be elliptic curves defined over $K$ with full $K$-rational 2-torsion and supersingular reduction. Then for any $n>0$, there exists a WR relation $\{P_1,P_2\}_{K/K}=0$ in $K(K;E_1,E_2)$ of signature $(n,n)$.
\end{lettertheorem}

The WR relations produced in Theorem \ref{thm:main-theorem-intro} come from genus 2 covers of $E_1$ and $E_2$. More precisely, there is a WR relation in $K(K;E_1,E_2)$ for every curve $C$ over $K$, whose Jacobian can be mapped into $E_1\times E_2$, and a choice of rational function $f\in K(C)^\times$. In this paper, we use a construction due to J. Scholten of a genus 2 curve $C$ and explicit nonconstant maps $\f_1:C\rightarrow E_1$ and $\f_2:C\rightarrow E_2$. Since $C$ is hyperelliptic it has particularly simple principal divisors, so the choice of rational function can be made flexibly. Armed with a suitable choice of $C,\f_1,\f_2$ and $f$, we can then $p$-adically approximate the resulting WR relation to determine its signature.

This paper is organized into three sections. In section \ref{sec:prelim}, we review all the basic theory we will need about formal groups of elliptic curves, supersingular reduction, and the Somekawa $K$-group for products of elliptic curves. Along the way we define the signature of a $K/K$-symbol and determine the type and quantity of WR relations one needs to show the vanishing of all $K/K$-symbols modulo $p$. In section \ref{sec:scholten_curves} we review the construction of the hyperelliptic curves that map to $E_1\times E_2$, compute the WR relations they produce, and use them to prove Theorem \ref{thm:main-theorem-intro}. Lastly, in section \ref{sec:computations}, we focus on the case of quadratic ramified extensions $K$ and present computational evidence that the genus 2 spans of section \ref{sec:scholten_curves} produce the necessary WR relations to show that all $K/K$-symbols vanish modulo $p$.

The WR relations in Theorem \ref{thm:main-theorem-intro} are obtained from genus 2 covers of $E_1$ and $E_2$: there is a WR relation in $K(K;E_1,E_2)$ for every curve $C/K$ whose Jacobian maps into $E_1\times E_2$, together with a choice of $f\in K(C)^\times$. The paper uses J. Scholten's construction of a genus 2 (hyperelliptic) curve $C$ and explicit nonconstant maps $\f_1:C\to E_1$ and $\f_2:C\to E_2$. Because $C$ is hyperelliptic, it has "particularly simple principal divisors", so $f$ can be chosen flexibly. With a suitable choice of $C,\f_1,\f_2$ and $f$, the resulting WR relation is then $p$-adically approximated "to determine its signature" (yielding signature $(n,n)$).

Question: Let p>3 be a prime, let K be a p-adic field totally ramified over \(\mathbb{Q}_p\), and let \(E_1\) and \(E_2\) be elliptic curves over K with full K-rational 2-torsion and supersingular reduction. Let \(K(K;E_1,E_2)\) denote the Somekawa K-group, whose generators include K/K-symbols \(\{P_1,P_2\}_{K/K}\) with \(P_i\in E_i(K)\). A WR relation means a vanishing relation in this group coming from Weil reciprocity. For such a K/K-symbol, its signature is the pair \((n_1,n_2)\) obtained from the valuations of the corresponding formal-group representatives modulo p; thus saying the symbol has signature \((n,n)\) means both valuation entries equal n. Which statement holds for every such choice of p, K, \(E_1\), \(E_2\), and every positive integer n?

There exist points \(P_1\in E_1(K)\) and \(P_2\in E_2(K)\) such that \(\{P_1,P_2\}_{K/K}=0\) is a WR relation in \(K(K;E_1,E_2)\) and the symbol \(\{P_1,P_2\}_{K/K}\) has signature \((n,n)\).

Choices:
B. There exist points \(P_1\in E_1(K)\) and \(P_2\in E_2(K)\) such that \(\{P_1,P_2\}_{K/K}=0\) is a WR relation in \(K(K;E_1,E_2)\) and the symbol \(\{P_1,P_2\}_{K/K}\) has signature \((n,m)\) for every pair of positive integers \(n,m\).
C. There exist points \(P_1\in E_1(K)\) and \(P_2\in E_2(K)\) such that \(\{P_1,P_2\}_{K/K}=0\) is a WR relation in \(K(K;E_1,E_2)\) and the symbol \(\{P_1,P_2\}_{K/K}\) has signature \((n_1,n_2)\) for some positive integers \(n_1,n_2\).
D. There exists a single pair of points \(P_1\in E_1(K)\) and \(P_2\in E_2(K)\), independent of \(n\), such that for every positive integer \(n\) the relation \(\{P_1,P_2\}_{K/K}=0\) is a WR relation in \(K(K;E_1,E_2)\) and the symbol \(\{P_1,P_2\}_{K/K}\) has signature \((n,n)\).
E. For every positive integer \(n\), there exist points \(P_1\in E_1(K)\) and \(P_2\in E_2(K)\) such that the \(K/K\)-symbol \(\{P_1,P_2\}_{K/K}\in K(K;E_1,E_2)\) has signature \((n,n)\) and satisfies \(\{P_1,P_2\}_{K/K}\equiv 0\pmod p\).

WR-vanishing replaced by congruence mod p