\begin{corollary}\label{localization ider cor}
 Let $R$ be a $k$-algebra essentially of finite type. Then $\operatorname{Der}_k^\infty(R)=\operatorname{Ider}_k(R)$ is compatible with localization of $R$. That is, we have $$\operatorname{Ider}_k(S^{-1}R)=S^{-1}\operatorname{Ider}_k(R)$$
as $S^{-1}R$-modules for any multiplicatively closed subset $S\subset R$.
\end{corollary}

\begin{theorem}\label{eq finie thm}
        Let $e$ be an integer with
    $$  e\ge C_1+ C_2 + C_3\cdot q+C_4\cdot q+q$$ where the $C_i$ are as in \ref{hom lem}-\ref{j lem}. Let $\operatorname{Der}_k^{q}(R)$ be the module of $q$-integrable derivations of $R$. Then any element of
    $\mathfrak{n}^{e(q+1)+1} \cdot \operatorname{Der}_k^{q}(R)$
    is $m$-integrable for any positive integer $m$. In particular, the set of leaps produced by $\operatorname{Der}_k^{q}(R)$ is finite.
\end{theorem}

\begin{theorem}[{= Theorem \ref{infty integrable thm}}]\label{main thm1}
    There exists $M>0$ (depending on $R$) such that
$$ \operatorname{Der}_k^M(R)= \operatorname{Der}_k^{M+1}(R)=\dots= \operatorname{Der}_k^\infty(R)=\operatorname{Ider}_k(R).$$
\end{theorem}

\begin{theorem}\label{Her thm}
        Let $M$ be a sub-$R$-module of $\operatorname{Der}_k(R,R)$ and $\mathfrak{l}\subset \mathfrak{n}$ and ideal such that $R/\mathfrak{l}$ is of finite length. Assume that any $d\in \mathfrak{l}\cdot M$ is $m$-integrable for any positive integer $m$. Let $\operatorname{Leaps}(M)\subset \mathbb{N}$ be the set of leaps produced by $M$. Then $$\# \operatorname{Leaps}(M)\le \operatorname{length}_R(M/ \mathfrak{l}M).$$ In particular, $\operatorname{Leaps}(M)$ is finite.
    \end{theorem}

\begin{theorem}[{= \cite[Theorem 2.25]{miy2025}}]\label{duality}
Let $X$ be a scheme essentially of finite type over $k$. Then there exist two filtrations of sheaves on $X$ such that
$$\operatorname{Der}_k(\mathcal{O}_X)\supset\operatorname{Der}_k^p(\mathcal{O}_X)\supset\operatorname{Der}_k^{p^2}(\mathcal{O}_X)\supset\dots$$
and that
$$0\subset \operatorname{Ob}^p_X\subset\operatorname{Ob}^{p^2}_X\subset\dots\subset T^1_{X/k}.$$ For every $i=1,2,\dots,$ we have
$$\operatorname{Der}_k^{p^{i-1}}(\mathcal{O}_X)/\operatorname{Der}_k^{p^i}(\mathcal{O}_X)\simeq \operatorname{Ob}_X^{p^i}/\operatorname{Ob}_X^{p^{i-1}}. $$
\end{theorem}

\begin{theorem}[= {\cite[Corollary 2.3.5]{NARVAEZMACARRO20122712}}]
\label{localization}
    Let $R$ be a $k$-algebra essentially of finite type over $k$. Let $m\ge 1$ an integer. Let $S\subset R$ a multiplicatively closed subset. Then we have a canonical isomorphism $$\alpha: S^{-1}\operatorname{Der}_k^m(R)\overset{\sim}{\to} \operatorname{Der}_k^m(S^{-1}R).$$
\end{theorem}

\begin{definition}\label{FL def}
    Let $R$ [resp.\ $X$] be a $k$-algebra essentially of finite type [resp.\ $k$-scheme essentially of finite type]. Let $i\ge 1$ be an integer. Then we say that $R$ [resp.\ $X$] satisfies $\FL{i}$ if we have
    $$\operatorname{Ob}_R^{p^i}=\operatorname{Ob}_R^{p^{i+1}}=\operatorname{Ob}_R^{p^{i+2}}=\dots $$
    [resp. $\operatorname{Ob}_X^{p^i}=\operatorname{Ob}_X^{p^{i+1}}=\dots $].
\end{definition}

Hasse-Schmidt derivations (abbreviated as HS-derivations) were introduced in \cite{SchmidtHasse+1937+215+237}, which are also called higher derivations \cite[\S 27]{Mat}. Given a positive integer $m$, a field $k$ and a $k$-algebra $R$, $\operatorname{HS} _k^m(R)$ denotes the group of HS-derivations of length $m$, which consists of $A_m:=k[\![t]\!]/(t^{m+1})$-algebra automorphisms
$$R[\![t]\!]/(t^{m+1}) \to R[\![t]\!]/(t^{m+1})$$
that become the identity when restricted to $ R[\![t]\!]/(t^{m+1}) \otimes_{A_m} k=R$. For $m=\infty$, we regard $t^\infty=0$, and we also have the notion of HS-derivations of infinite length. There is a canonical isomorphism between $\operatorname{HS} _k^1(R)$ and the module $\operatorname{Der}_k(R,R)$. Under this identification, the image of
$$\operatorname{HS} _k^m(R)\to \operatorname{HS} _k^1(R)\simeq \operatorname{Der}_k(R,R)$$
is called the module of $m$-integrable derivations and denoted by $\operatorname{Der}_k^m(R)$, which has been studied in \cite{matsumura1982integrable}, \cite{NARVAEZMACARRO20122712}, \cite{NARVAEZMACARRO2024109758}. The module $\operatorname{Der}_k^\infty(R)$ is also denoted by $\operatorname{Ider}_k(R)$. Note that if $k$ contains $\mathbb{Q}$ or if $R$ is formally smooth over $k$, we have $\operatorname{Der}_k^m(R)=\operatorname{Der}_k(R)$ for any $m\le \infty$ (see \cite{Mat}). Thus, the study of integrable derivations is expected to lead to a better understanding of singularities in positive characteristic. From now on, we assume that $k$ is an algebraically closed field of positive characteristic $p$ and $R$ is essentially of finite type. We have the following descending chain
$$ \operatorname{Der}_k(R)\supset \operatorname{Der}_k^2(R)\supset \dots \supset \operatorname{Der}_k^m(R) \supset\dots.$$
The question of when the inclusion $\operatorname{Der}_k^{m-1}(R)\supset \operatorname{Der}_k^m(R)$ is proper has been studied since their introduction. When this is the case, we say that $R$ {\em leaps} at $m$. It has been known that leaps occur only at $m=p^i$ (\cite[Theorem 4.1]{Her}). The number of leaps is finite if $R$ is a unibranch curve (\cite[Theorem 4.7]{NARVAEZMACARRO2024109441}), and, more generally, if $R$ is a reduced curve (\cite[Theorem 6.1, Corollary 6.3]{bravo2024finitenessleapssensehasseschmidt}).
The aim of this paper is to show, using results from \cite[\S 2]{miy2025}, that the finiteness of leaps holds for every algebra essentially of finite type over an algebraically closed field of positive characteristic. Precisely, we show:

\begin{theorem}\label{infty integrable thm}
     Let $R$ be a $k$-algebra essentially of finite type. Then there exists $M>1$ such that
     $$ \operatorname{Der}_k^M(R)= \operatorname{Der}_k^{M+1}(R)=\dots= \operatorname{Der}_k^\infty(R)=\operatorname{Ider}_k(R).$$
\end{theorem}

\section{Introduction}
Hasse-Schmidt derivations (abbreviated as HS-derivations) were introduced in \cite{SchmidtHasse+1937+215+237}, which are also called higher derivations \cite[\S 27]{Mat}. Given a positive integer $m$, a field $k$ and a $k$-algebra $R$, $\HS _k^m(R)$ denotes the group of HS-derivations of length $m$, which consists of $A_m:=k[\![t]\!]/(t^{m+1})$-algebra automorphisms
$$R[\![t]\!]/(t^{m+1}) \to R[\![t]\!]/(t^{m+1})$$
that become the identity when restricted to $ R[\![t]\!]/(t^{m+1}) \otimes_{A_m} k=R$. For $m=\infty$, we regard $t^\infty=0$, and we also have the notion of HS-derivations of infinite length. There is a canonical isomorphism between $\HS _k^1(R)$ and the module $\operatorname{Der}_k(R,R)$. Under this identification, the image of
$$\HS _k^m(R)\to \HS _k^1(R)\simeq \operatorname{Der}_k(R,R)$$
is called the module of $m$-integrable derivations and denoted by $\operatorname{Der}_k^m(R)$, which has been studied in \cite{matsumura1982integrable}, \cite{NARVAEZMACARRO20122712}, \cite{NARVAEZMACARRO2024109758}. The module $\operatorname{Der}_k^\infty(R)$ is also denoted by $\operatorname{Ider}_k(R)$. Note that if $k$ contains $\mathbb{Q}$ or if $R$ is formally smooth over $k$, we have $\operatorname{Der}_k^m(R)=\operatorname{Der}_k(R)$ for any $m\le \infty$ (see \cite{Mat}). Thus, the study of integrable derivations is expected to lead to a better understanding of singularities in positive characteristic. From now on, we assume that $k$ is an algebraically closed field of positive characteristic $p$ and $R$ is essentially of finite type. We have the following descending chain
$$ \operatorname{Der}_k(R)\supset \operatorname{Der}_k^2(R)\supset \dots \supset \operatorname{Der}_k^m(R) \supset\dots.$$
The question of when the inclusion $\operatorname{Der}_k^{m-1}(R)\supset \operatorname{Der}_k^m(R)$ is proper has been studied since their introduction. When this is the case, we say that $R$ {\em leaps} at $m$. It has been known that leaps occur only at $m=p^i$ (\cite[Theorem 4.1]{Her}). The number of leaps is finite if $R$ is a unibranch curve (\cite[Theorem 4.7]{NARVAEZMACARRO2024109441}), and, more generally, if $R$ is a reduced curve (\cite[Theorem 6.1, Corollary 6.3]{bravo2024finitenessleapssensehasseschmidt}).
The aim of this paper is to show, using results from \cite[\S 2]{miy2025}, that the finiteness of leaps holds for every algebra essentially of finite type over an algebraically closed field of positive characteristic. Precisely, we show:

This theorem provides, for the first time, a general finiteness result for leaps in arbitrary dimensions, thereby giving an affirmative answer to Macarro’s question [8, Question 3.6.5] and contributing to a better structural understanding of HS-derivations. Since leaps are stable under separable base changes by \cite[Theorem 3.27]{hernandez2020behavior}, this also implies the finiteness of leaps over any perfect field. We also give an affirmative answer to another question \cite[Question 3.6.1]{NARVAEZMACARRO20122712} by the following:

\begin{theorem}[{= Theorem \ref{mn int theorem}}]\label{main thm2}
 Let $n\ge 1$ be an integer and let $D\in \HS_k^n(R)$. Then $D$ is $\infty$-integrable if and only if $D$ is $m$-integrable for every integer $m\ge n$. Here, we say that $D$ is $m$-integrable if $D$ is in the image of $\HS_k^m(R)\to \HS_k^n(R)$.
\end{theorem}
In fact, Theorem \ref{main thm1} is a special case of Theorem \ref{main thm2} where we put $n=1$. As an application of Theorem \ref{main thm1}, we obtain:

\begin{corollary}[{=Corollary \ref{localization ider cor}}]\label{main cor}
The module $\operatorname{Der}_k^\infty(R)=\operatorname{Ider}_k(R)$ is compatible with localization of $R$. That is, we have $$\operatorname{Ider}_k(S^{-1}R)=S^{-1}\operatorname{Ider}_k(R).$$
\end{corollary}
This is the first coherence result for $\operatorname{Ider}_k(R)$ to the best of our knowledge. For $m<\infty$, a similar statement has been proven in \cite[Corollary 2.3.5]{NARVAEZMACARRO20122712}. The case $m=\infty$ is more subtle because the map in \cite[Corollary 1.3.6]{NARVAEZMACARRO20122712} is not surjective for $m=\infty$ while $m$ is assumed to be an integer there (cf. \cite[Example 1.4]{MR1764573}). Note that Corollary \ref{main cor} implies that $\operatorname{Ider}_k(R)$ constitutes a coherent sheaf $\operatorname{Ider}_k(\mathcal{O}_X)$ on an algebraic scheme $X$ over $k$. In particular, it is expected to behave better than the usual tangent sheaf. In fact, some pathological phenomena of derivations peculiar to positive characteristic do not occur if we replace it by integrable derivations. For example, the local rank of $\operatorname{Ider}_k(R)$ is at most $\operatorname{dim}R$, while it is not true for $\operatorname{Der}_k(R)$ (see \cite{MR554764}). This pathology does not occur in characteristic zero (see \cite{MR366909} when $R$ is a domain and \cite{MR498523} in general). We can also state Theorem \ref{main thm1} in terms of schemes:
\begin{theorem}[{= Theorem \ref{infty integrable thm schemes}}]\label{main thm3}
 Let $X$ be a $k$-scheme essentially of finite type (=locally essentially of finite type and quasi-compact). Then there exists $M>0$ such that
$$ \operatorname{Der}_k^M(\mathcal{O}_X)= \operatorname{Der}_k^{M+1}(\mathcal{O}_X)=\dots=\operatorname{Ider}_k(\mathcal{O}_X),$$
where $\operatorname{Der}_k^m(\mathcal{O}_X)$ denotes the sheaf of $m$-integrable derivations (see Theorem \ref{localization}).
\end{theorem}

\begin{proposition}\label{a to b extension prop}
 Let $e$ be an integer with
 $$e\ge C_1+C_3\cdot q+C_4\cdot q+q$$
 where $C_1,C_3,C_4$ are as in \ref{hom lem}-\ref{j lem}. Let $m$ be a positive integer and $D\in \HS_k^\infty(T)$ be such that:
 \begin{enumerate}
 \item $D$ is $I$-logarithmic up to $m$, thus induces $\widetilde{D}\in \HS_k^m(R),$
 \item $D$ is logarithmically $a[e,m,q]$-bounded.
 \end{enumerate}
 Then there exists $E\in \HS_k^\infty(T)$ such that:
 \begin{enumerate}
 \item $E$ is $I$-logarithmic up to $m$, thus induces $\widetilde{E}\in \HS_k^m(R),$
 \item $\operatorname{ob}_{m+1}(\widetilde{E})=0$,
 \item $E$ is logarithmically $b[e,m+1,q]$-bounded,
 \item $D$ and $E$ coincide up to $l_{m+1}-1$, i.e., $D_1=E_1,\dots,D_{l_{m+1}-1}=E_{l_{m+1}-1}$.
 \end{enumerate}
\end{proposition}

\begin{proposition}\label{b to a extension prop}
 Let $e$ be an integer with
 $$e\ge C_1+C_2+1$$
 where $C_1,C_2$ are as in \ref{hom lem}, \ref{G lem}. Let $m$ be a positive integer and $E\in \HS_k^\infty(T)$ such that:
 \begin{enumerate}
 \item $E$ is $I$-logarithmic up to $m$, thus induces $\widetilde{E}\in \HS_k^m(R),$
 \item $\operatorname{ob}_{m+1}(\widetilde{E})=0$,
 \item $E$ is logarithmically $b[e,m+1,q]$-bounded.
 \end{enumerate}
 Then there exists $D^+\in \HS_k^\infty (T)$ such that:
 \begin{enumerate}
 \item $D^+$ is $I$-logarithmic up to $m+1$,
 \item $D^+$ is logarithmically $a[e,m+1,q]$-bounded,
 \item $D^+$ and $E$ coincide up to $m$, i.e., $D^+_1=E_1,\dots,D^+_{m}=E_{m}$.
 \end{enumerate}
\end{proposition}

Setting $n=1$ in the above theorem, we obtain the following.
\begin{theorem}\label{infty integrable thm}
 Let $R$ be a $k$-algebra essentially of finite type. Then there exists $M>1$ such that
 $$ \operatorname{Der}_k^M(R)= \operatorname{Der}_k^{M+1}(R)=\dots= \operatorname{Der}_k^\infty(R)=\operatorname{Ider}_k(R).$$
\end{theorem}

\begin{theorem}\label{infty integrable thm schemes}
 Let $X$ be a $k$-scheme essentially of finite type. Then there exists $M>0$ such that
$$ \operatorname{Der}_k^M(\mathcal{O}_X)= \operatorname{Der}_k^{M+1}(\mathcal{O}_X)=\dots=\operatorname{Ider}_k(\mathcal{O}_X).$$
\end{theorem}

This theorem provides, for the first time, a general finiteness result for leaps in arbitrary dimensions, thereby giving an affirmative answer to Macarro’s question [8, Question 3.6.5] and contributing to a better structural understanding of HS-derivations. Since leaps are stable under separable base changes by \cite[Theorem 3.27]{hernandez2020behavior}, this also implies the finiteness of leaps over any perfect field. We also give an affirmative answer to another question \cite[Question 3.6.1]{NARVAEZMACARRO20122712} by the following:

\begin{theorem}[{= Theorem \ref{mn int theorem}}]\label{main thm2}
    Let $n\ge 1$ be an integer and let $D\in \HS_k^n(R)$. Then $D$ is $\infty$-integrable if and only if $D$ is $m$-integrable for every integer $m\ge n$. Here, we say that $D$ is $m$-integrable if $D$ is in the image of $\HS_k^m(R)\to \HS_k^n(R)$.
\end{theorem}
In fact, Theorem \ref{main thm1} is a special case of Theorem \ref{main thm2} where we put $n=1$. As an application of Theorem \ref{main thm1}, we obtain:

\begin{corollary}[{=Corollary \ref{localization ider cor}}]\label{main cor}
The module $\operatorname{Der}_k^\infty(R)=\operatorname{Ider}_k(R)$ is compatible with localization of $R$. That is, we have $$\operatorname{Ider}_k(S^{-1}R)=S^{-1}\operatorname{Ider}_k(R).$$
\end{corollary}
This is the first coherence result for $\operatorname{Ider}_k(R)$ to the best of our knowledge. For $m<\infty$, a similar statement has been proven in \cite[Corollary 2.3.5]{NARVAEZMACARRO20122712}. The case $m=\infty$ is more subtle because the map in \cite[Corollary 1.3.6]{NARVAEZMACARRO20122712} is not surjective for $m=\infty$ while $m$ is assumed to be an integer there (cf. \cite[Example 1.4]{MR1764573}). Note that Corollary \ref{main cor} implies that $\operatorname{Ider}_k(R)$ constitutes a coherent sheaf $\operatorname{Ider}_k(\mathcal{O}_X)$ on an algebraic scheme $X$ over $k$. In particular, it is expected to behave better than the usual tangent sheaf. In fact, some pathological phenomena of derivations peculiar to positive characteristic do not occur if we replace it by integrable derivations. For example, the local rank of $\operatorname{Ider}_k(R)$ is at most $\operatorname{dim}R$, while it is not true for $\operatorname{Der}_k(R)$ (see \cite{MR554764}). This pathology does not occur in characteristic zero (see \cite{MR366909} when $R$ is a domain and \cite{MR498523} in general). We can also state Theorem \ref{main thm1} in terms of schemes:
\begin{theorem}[{= Theorem \ref{infty integrable thm schemes}}]\label{main thm3}
    Let $X$ be a $k$-scheme essentially of finite type (=locally essentially of finite type and quasi-compact). Then there exists $M>0$ such that
$$ \operatorname{Der}_k^M(\mathcal{O}_X)= \operatorname{Der}_k^{M+1}(\mathcal{O}_X)=\dots=\operatorname{Ider}_k(\mathcal{O}_X),$$
where $\operatorname{Der}_k^m(\mathcal{O}_X)$ denotes the sheaf of $m$-integrable derivations (see Theorem \ref{localization}).
\end{theorem}

We briefly outline the contents of the paper.

In Section 2, we recall the definition of HS-derivations and results that will  be used later. A key relation is that the obstructions to extending HS-derivations are canonically dual to $m$-integrable derivations (Theorem \ref{duality}). We also define the property $\operatorname{FL}(i)$ for algebras and schemes, which states that leaps do not occur at every $m> p^i$ (Definition \ref{FL def}).

In Section 3, we introduce the notion of $a$-boundedness for HS-derivations. Intuitively, this concept measures how each term of an HS-derivation shrinks with respect to the $\mathfrak{m}$-adic topology.

In Section 4, we prove the finiteness of leaps in the local case. In this section we fix a local $k$-algebra $(R,\mathfrak{n})$ such that $\operatorname{Spec}R\setminus \{\mathfrak{n}\}$ satisfies $\operatorname{FL}(\tau)$ for $\tau>0$, and we will prove that $\operatorname{Spec}R$ itself satisfies the finiteness of leaps. The idea is that, starting from a bounded HS-derivation, we construct another one that is less bounded and coincides with the original one in small degrees. We use the Artin-Rees lemma to obtain positive constants which control the bounds. The essential step is Theorem \ref{eq finie thm}, which states that elements of $$\mathfrak{n}^{e(q+1)+1} \cdot \operatorname{Der}_k^{q}(R)$$ do not leap, where $q=p^\tau$ and $e\gg 0$. Then the fact that $R/\mathfrak{n}^{e(q+1)+1} $ is of finite length implies the finiteness of leaps of $R$ (Theorems \ref{Her thm}, \ref{sec 4 main thm}).

In Section 5, we prove the finiteness of leaps in the global case and discuss its consequences. We use noetherian induction to reduce to the case of Section 4 (Theorem \ref{main thm global}). The finiteness of leaps of $m$-integrable derivations implies that of obstruction modules. This enables us to extend HS-derivations step by step (Lemma \ref{mn int lem}), and taking a limit, we obtain a HS-derivation of infinite length: This method yields a characterization of $\infty$-integrability (Theorem \ref{mn int theorem}).

Throughout this paper, we fix an algebraically closed field $k$ of positive characteristic $p$. We assume that any $k$-algebra is unital and commutative. We say that a $k$-scheme $X$ is essentially of finite type if there exists a finite open affine covering $\{U_i=\operatorname{Spec}R_i\}_{i=1,\dots,n}$ for $X$ where each $R_i$ is a $k$-algebra essentially of finite type. We assume $\mathbb{N}:=\{0,1,\dots\}$.

In Section 5, the authors say they "prove the finiteness of leaps in the global case" and "use noetherian induction to reduce to the case of Section 4 (Theorem \ref{main thm global})." Then, since "the finiteness of leaps of $m$-integrable derivations implies that of obstruction modules," they can "extend HS-derivations step by step (Lemma \ref{mn int lem}), and taking a limit, we obtain a HS-derivation of infinite length." This "method yields a characterization of $\infty$-integrability (Theorem \ref{mn int theorem})," and they note that "Theorem \ref{main thm1} is a special case of Theorem \ref{main thm2} where we put $n=1$."

Next, the authors prove the finiteness of leaps in the global case and use noetherian induction to reduce to the local case, as stated in the following result.

\begin{theorem}\label{main thm global}
    Let $X$ be a $k$-scheme essentially of finite type. Then $X$ satisfies $\FL{\tau}$ for a sufficiently large integer $\tau$. In particular, the set of leaps of $X$ is finite.
\end{theorem}

Then, since the finiteness of leaps of $m$-integrable derivations implies that of obstruction modules, they extend HS-derivations step by step using the following lemma.

\begin{lemma}\label{mn int lem}
    Let $R$ be a $k$-algebra essentially of finite type, let $m, \tau\ge 1$ be integers, and let $q:=p^\tau$. Suppose that $R$ satisfies $\FL{\tau}$. Let $D\in \HS_k^m(R)$. Let $l_1,l_2,\dots$ be the sequence of integers defined as in \ref{l_n def}. Then there exists $D^+\in \HS_k^{m+1}(R)$ such that $D$ and $D^+$ coincide up to $l_{m+1}-1$, i.e., $D_1=D^+_1,\dots, D_{l_{m+1}-1}= D_{l_{m+1}-1}^+$.
\end{lemma}

Taking a limit, they obtain a HS-derivation of infinite length. This method yields the following characterization of $\infty$-integrability.

\begin{theorem}\label{mn int theorem}
    Let $R$ be a $k$-algebra essentially of finite type, let $n\ge 1$ be an integer, and let $D\in \HS_k^n(R)$. Then $D$ is $\infty$-integrable if and only if $D$ is $m$-integrable for every integer $m\ge n$. Here, we say that $D$ is $m$-integrable if $D$ is in the image of $\HS_k^m(R)\to \HS_k^n(R)$.
\end{theorem}

Finally, to prove the main theorem, they note that it is a special case of
\begin{theorem}[{= Theorem \ref{mn int theorem}}]\label{main thm2}
    Let $n\ge 1$ be an integer and let $D\in \HS_k^n(R)$. Then $D$ is $\infty$-integrable if and only if $D$ is $m$-integrable for every integer $m\ge n$. Here, we say that $D$ is $m$-integrable if $D$ is in the image of $\HS_k^m(R)\to \HS_k^n(R)$.
\end{theorem}
where one puts $n=1$.

[{= To prove the main theorem, } ]\label{main thm1}
    There exists $M>0$ (depending on $R$) such that
$$ \operatorname{Der}_k^M(R)= \operatorname{Der}_k^{M+1}(R)=\dots= \operatorname{Der}_k^\infty(R)=\operatorname{Ider}_k(R).$$,

Existential–Universal
Classification or Bijection

Question:
Let $k$ be an algebraically closed field of positive characteristic, and let $R$ be a $k$-algebra essentially of finite type. For each $m\in \mathbb{Z}_{>0}\cup\{\infty\}$, let $\operatorname{Der}_k^m(R)$ denote the module of $m$-integrable $k$-derivations, i.e. the image of the natural map from length-$m$ Hasse--Schmidt derivations $\operatorname{HS}_k^m(R)$ to $\operatorname{Der}_k(R,R)\simeq \operatorname{HS}_k^1(R)$, and write $\operatorname{Ider}_k(R)=\operatorname{Der}_k^\infty(R)$. Which statement holds about the sequence
$$\operatorname{Der}_k(R)\supset \operatorname{Der}_k^2(R)\supset \cdots \supset \operatorname{Der}_k^m(R)\supset \cdots ?$$

There exists an integer $M>0$, depending on $R$, such that for every $m\ge M$ one has
$$\operatorname{Der}_k^m(R)=\operatorname{Der}_k^M(R)=\operatorname{Der}_k^\infty(R)=\operatorname{Ider}_k(R),$$
that is,
$$\operatorname{Der}_k^M(R)=\operatorname{Der}_k^{M+1}(R)=\cdots=\operatorname{Der}_k^\infty(R)=\operatorname{Ider}_k(R).$$

Choices:
B.
There exists an integer $M>0$, depending only on $\operatorname{char}(k)$ and not on $R$, such that for every $m\ge M$ one has
$$\operatorname{Der}_k^m(R)=\operatorname{Der}_k^M(R)=\operatorname{Der}_k^\infty(R)=\operatorname{Ider}_k(R),$$
that is,
$$\operatorname{Der}_k^M(R)=\operatorname{Der}_k^{M+1}(R)=\cdots=\operatorname{Der}_k^\infty(R)=\operatorname{Ider}_k(R).$$
C.
There exists an integer $M>0$, depending on $R$, such that
$$\operatorname{Der}_k^M(R)=\operatorname{Der}_k^\infty(R)=\operatorname{Ider}_k(R).$$
D.
For every integer $m>0$ one has
$$\operatorname{Der}_k^m(R)=\operatorname{Der}_k^\infty(R)=\operatorname{Ider}_k(R),$$
that is,
$$\operatorname{Der}_k(R)=\operatorname{Der}_k^2(R)=\cdots=\operatorname{Der}_k^\infty(R).$$
E.
There exists an integer $M>0$, depending on $R$, such that for every $m\ge M$ one has
$$\operatorname{Der}_k^m(R)=\operatorname{Der}_k^M(R),$$
and moreover every derivation is eventually integrable, i.e.
$$\operatorname{Der}_k^M(R)=\operatorname{Der}_k(R).$$


\begin{proposition}\label{prop:two-def-same}
Let $v$ be a discrete valuation on $k$, and $O$ its valuation ring.
Let $\sX \to \Spec O$ be a smooth proper morphism with geometrically connected fibers,
and denote by $X$ the generic fiber.
Then we have
\[ \Im(\langle \cdot, \cdot \rangle :
\CH^1(X)_\Tor \times A_0(X) \to k^\times \otimes \Q/\Z)
\subset
\Im(O^\times \otimes \Q/\Z \to k^\times \otimes \Q/\Z).
\]
\end{proposition}

\begin{proposition}\label{prop:TateCv}
\begin{enumerate}
\item
Let $a, b \in k^\times$ and $m>0$.
If $[a],  [b] \in E_q(k)[m]$, we have
\[ \langle [a], [b] \rangle
= a^{-s_m([b])} \otimes \frac{1}{m}
= b^{-s_m([a])} \otimes \frac{1}{m}
= q^{-s_m([a])s_m([b])} \otimes \frac{1}{m^2}.
\]
\item
We have $E_q(k)_\Tor^\is = \{ [\zeta] \mid \zeta \in \mu(k) \}$.
\end{enumerate}
\end{proposition}

\begin{theorem} \label{theorem: intrinsic torsion}
  Let $E$ be an elliptic curve over $\Q$.
Then there is a pair $(A, B)$ of an abelian group $A$
and its subgroup $B$ in the following table,
together with
an isomorphism $\alpha : E(\Q)_\Tor \cong A$,
such that $\alpha(E(\Q)_\Tor^\is)=B$.
  $$ \extrarowheight3pt
  \begin{array}{ll} \hline\hline
    A & \text{order of}\ B \\ \hline
    0 & 1 \\
    \Z/2\Z & 1,2\\
    \Z/3\Z & 1,3\\
    \Z/4\Z & 1,2,4 \\
    \Z/5\Z & 1,5 \\
    \Z/6\Z & 1,2,3 \\
    \Z/7\Z & 1 \\
    \Z/8\Z & 1,2 \\
    \Z/9\Z & 1 \\
    \Z/10\Z & 1\\
    \Z/12\Z & 1 \\
    \Z/2\Z\times\Z/2\Z & 1,2\\
    \Z/4\Z\times\Z/2\Z & 1,2 \ \\
    \Z/6\Z\times\Z/2\Z & 1 \\
    \Z/8\Z\times\Z/2\Z & 1 \\ \hline\hline
  \end{array}
  $$
(For $A=\Z/4\Z \times \Z/2\Z$,
there are two $\Aut(A)$-conjugacy classes of order two subgroups,
both of which can be taken as $B$.)
  Moreover, for each possible pair $(A, B)$ as above,
there are infinitely many (mutually non-isomorphic) elliptic curves $E$ over $\Q$
for which there is an isomorphism $\alpha : E(\Q)_\Tor \cong A$
such that $\alpha(E(\Q)_\Tor^\is)=B$.
\end{theorem}

\begin{theorem}\label{proposition: pairing 1-mod}
  Assume that $N\in\{4,5,6,7,8,9,10,12\}$ and
  let $M$ be a positive divisor of $N$.
  Denote by $X_1(MN,N)$ the modular curve over $\Q$
  associated to the congruence subgroup
    $$
    \Gamma_1(MN,N):= \Gamma_0(MN) \cap \Gamma_1(N)
    =\left\{\M abcd\in\SL(2,\Z) : \
      a,d\equiv1\bmod N, \ MN|c \right\},
    $$
  on which the cusp $0$ is $\Q$-rational.
  Then $X_1(N)_M^+$ is isomorphic to $X_1(MN,N)$ over $\Q$.
\end{theorem}

Let $X$ be a geometrically irreducible smooth projective curve
over a field $k$. In \S \ref{sect:pairing},
we shall construct a biadditive symmetric pairing
\begin{equation}\label{eq:pairing-intro}
\langle \cdot , \cdot \rangle :
\operatorname{Pic}(X)_\Tor \times \operatorname{Pic}(X)_\Tor \to k^\times \otimes \mathbb{Q}/\mathbb{Z},
\end{equation}
where $\operatorname{Pic}(X)_\Tor$ is the
torsion part of the Picard group $\operatorname{Pic}(X)$ of $X$.
We then define the \emph{intrinsic subgroup}
of $\operatorname{Pic}(X)_\Tor$ by
\[
\operatorname{Pic}(X)_\Tor^\is
:= \{ a \in \operatorname{Pic}(X)_\Tor \mid \langle a, b \rangle = 0
\text{ for all } b \in \operatorname{Pic}(X)_\Tor \}.
\]

We remark that \eqref{eq:pairing-intro} is not entirely new.
Indeed, its finite coefficient analogue
(with values in $k^\times/(k^\times)^m$) is considered by Frey and R\"uck in \cite{FreyRuck}
(see Remark \ref{rem:FreyRuck}).
The first and second authors encountered with
a disguised version of \eqref{eq:pairing-intro} in \cite{YY}
(see Remark \ref{sect:genJac}).
Obviously, \eqref{eq:pairing-intro} is trivial if $k^\times \otimes \mathbb{Q}/\mathbb{Z}=0$,
e.g. when $k$ is finite, algebraically closed, or $k=\mathbb{R}$.
On the other hand,
$\operatorname{Pic}(X)_\Tor^\is$ appears to be a non-trivial new invariant
if $k$ is a global field or a $p$-adic field.

We will mainly consider the case where $X=E$ is an elliptic curve
so that we may identify $E(k)_\Tor= \operatorname{Pic}(E)_\Tor$
by $P \mapsto [P-O]$, where $O \in E(k)$ is the identity element.
Thus we write
$\langle P,Q\rangle:=\langle [P-O], [Q-O]\rangle$ for $P, Q \in E(k)_\Tor$
and
$E(k)_\Tor^\is:=\operatorname{Pic}(E)_\Tor^\is$.
Our main result is a classification of possible structures of
$(E(\mathbb{Q})_\Tor,E(\mathbb{Q})_\Tor^\is)$ for an elliptic curve $E$ over $\mathbb{Q}$.
A celebrated theorem of Mazur \cite[Theorem (8)]{Mazur} states
that $E(\mathbb{Q})_\Tor$ is either cyclic of order
$1, \ldots,10$, or $12$, or is isomorphic to $(\mathbb{Z}/2N\mathbb{Z}) \times (\mathbb{Z}/2\mathbb{Z})$,
$N=1, 2,3,4$,
and all cases are realized by infinitely many (mutually non-isomorphic) elliptic curves.
The classification of
$E(\mathbb{Q})_\Tor^\is$ is given in the following theorem,
whose proof will occupy \S \ref{sect:ref-mazur}--\ref{sect:ref-mazur2}.

\begin{equation}\label{eq:pairing-intro}
\langle \cdot , \cdot \rangle :
\Pic(X)_\Tor \times \Pic(X)_\Tor \to k^\times \otimes \Q/\Z,
\end{equation}

\begin{remark}\label{rem:FreyRuck}
As is seen from the proof,
there is an analogous biadditive pairing
\[
\langle \cdot, \cdot \rangle_n :
\Pic(X)[n] \times \Pic^0(X)/n \to k^\times/(k^\times)^n
\]
for each $n>0$,
which satisfies
$\langle d, e \rangle_n
=
\WP_n(d, e)
\langle e, d \rangle_n
$for any $d, e \in \Pic(X)[n]$.
This pairing has been constructed by Frey and R\"uck in \cite{FreyRuck}
when $k$ is a finite field
and shown to be perfect if $|k| \equiv 1 \bmod n$.
(This is called the
Lichtenbaum-Tate pairing in some literature,
see e.g. \cite[XI.9]{sil2}.)
\end{remark}

We will mainly consider the case where $X=E$ is an elliptic curve
so that we may identify $E(k)_\Tor= \Pic(E)_\Tor$
by $P \mapsto [P-O]$, where $O \in E(k)$ is the identity element.
Thus we write
$\gen{P,Q}:=\gen{[P-O], [Q-O]}$ for $P, Q \in E(k)_\Tor$
and
$E(k)_\Tor^\is:=\Pic(E)_\Tor^\is$.
Our main result is a classification of possible structures of
$(E(\Q)_\Tor,E(\Q)_\Tor^\is)$ for an elliptic curve $E$ over $\Q$.
A celebrated theorem of Mazur \cite[Theorem (8)]{Mazur} states
that $E(\Q)_\Tor$ is either cyclic of order
$1, \ldots,10$, or $12$, or is isomorphic to $(\Z/2N\Z) \times (\Z/2\Z)$,
$N=1, 2,3,4$,
and all cases are realized by infinitely many (mutually non-isomorphic) elliptic curves.
The classification of
$E(\Q)_\Tor^\is$ is given in the following theorem,
whose proof will occupy \S \ref{sect:ref-mazur}--\ref{sect:ref-mazur2}.

We briefly explain the outline of the proof.
Our approach is completely explicit.
As usual, we let $X_1(N)$ denote the modular curve associated to the
congruence subgroup $\Gamma_1(N)$. It possesses a model over $\Q$ on
which the cusp $0$ is $\Q$-rational. For a subfield $k$ of $\C$,
the non-cuspidal points in $X_1(N)(k)$ parameterize isomorphism
classes of pairs $(E, P)$ of an elliptic curve $E$ over $k$
and a $k$-rational point $P$ of order $N$.
More precisely, if the coordinates of a point
$\tau\in\H$ on this model of $X_1(N)$ is $k$-rational,
then the isomorphism class of $(\C/(\Z\tau+\Z),1/N)$ contains a pair $(E,P)$
of an elliptic curve $E$ over $k$ and a $k$-rational $N$-torsion point $P$.
A famous theorem of Mazur \cite[Theorem (7)]{Mazur} says that
$X_1(N)(\Q)$ has a non-cuspidal point precisely
when $N=1,\ldots,10$, or $N=12$.

Similarly, for a positive even
integer $N$, we let $X_1^0(N,2)$ denote the modular curve associated
to the congruence subgroup
\begin{equation}\label{eq:G10-N2}
\Gamma_1^0(N,2):=\Gamma_1(N)\cap\Gamma^0(2)
=\left\{\M abcd\in\SL(2,\Z):a,d\equiv 1\bmod N,~~2|b,~N|c\right\}.
\end{equation}
The modular curve $X_1^0(N,2)$ parameterizes isomorphism classes of
triples $(E, P, Q)$ of an elliptic curve $E$
and $k$-rational points $P, Q$
such that
$P$ and $Q$ are of order $N$ and $2$ respectively,
and $Q\notin\gen P$.
For $N \in \{4, 6, 8 \}$
and $\ul{M}=(M_1, M_2, M_3)$ with $M_1|N, M_2, M_3 \in \{ 1, 2\}$,
we will construct a smooth projective curve
$X_1^0(N, 2)_{\ul{M}}^\pm$ over $\Q$
equipped with a finite morphism to $X_1^0(N, 2)$,
by which one can interpret the conditions
\[ \langle P, \frac{N}{M_1}P \rangle
= \langle Q, \frac{2}{M_2}Q \rangle
= \langle P, \frac{2}{M_3}Q \rangle = 0
\]
for a triple $(E, P, Q)$ as above corresponding to a point of $X_1^0(N, 2)(\Q)$.
Therefore Theorem \ref{theorem: intrinsic torsion} is reduced to a study of the $\Q$-rational points of
$X_1(N)_M^\pm$ and $X_1^0(N, 2)_{\ul{M}}^\pm$.
As our construction of these curves are explicit,
this can be done by a (more or less) direct computation.

\begin{lemma} \label{lemma: ENt}
 \begin{enumerate}
 \item Assume that the characteristic of $k$ is not $2$.
 If $E$ is an elliptic curve over $k$ with a $k$-rational point
 $P$ of order two, then $(E,P)$ is isomorphic to $(E_{2,t,a},(0,0))$
 for some $a,t\in k^\times$, where
 \begin{equation} \label{eq: E2ta}
 E_{2,t,a}:
 \begin{cases}
 \displaystyle y^2=x\left(x^2+ax+\frac{a^2t}{4(t+1)}\right),
 &\text{if }t\neq-1, \\
 y^2=x(x^2+a), &\text{if }t=-1.
 \end{cases}
 \end{equation}
 Moreover, two elliptic curves $(E_{2,t,a},(0,0))$ and
 $(E_{2,t',a'},(0,0))$ are isomorphic over $k$ if and only
 if $t=t'\neq-1$ and $a/a'$ is a square in $k^\times$
 or $t=t'=-1$ and $a/a'\in(k^\times)^4$ .
 \item Assume that the characteristic of $k$ is not $3$.
 If $E$ is an elliptic curve over $k$ with a $k$-rational
 point $P$ of order $3$, then $(E,P)$ is isomorphic to $(E_{3,t},(0,0))$
 for some $t\neq-1$ in $k^\times$, where
 \begin{equation} \label{eq: E3t}
 E_{3,t}:y^2+xy+\frac t{27(t+1)}y=x^3
 \end{equation}
 or isomorphic to $(E_{3,-1,a},(0,0))$ for some $a\in k^\times$,
 where
 \begin{equation} \label{eq: E3a}
 E_{3,-1,a}:y^2+ay=x^3.
 \end{equation}
 Two elliptic curves $(E_{3,t},(0,0))$ and
 $(E_{3,t'},(0,0))$ with $t,t'\neq-1$ are isomorphic if and only
 if $t=t'$. Also, $(E_{3,-1,a},(0,0))\simeq(E_{3,-1,a'},(0,0))$
 if and only if $a/a'\in(k^\times)^3$.
 \end{enumerate}
\end{lemma}
\begin{proof}
 Consider first the case $N=2$.
We start with the equation \eqref{eq: E2}
and introduce a new parameter
 $t=4b/(a^2-4b)$ (so $b=a^2t/4(t+1)$). When $a\neq0$, we have
 $t\neq-1$ and the equation \eqref{eq: E2} in the new parameters is
 $y^2=x(x^2+ax+a^2t/4(t+1))$. When $a=0$, we change the notation $b$
 in \eqref{eq: E2} to $-a$ and get $y^2=x(x^2-a)$. Now it is easy to
 see that an isomorphism
 $\phi:(E_{2,t,a},(0,0))\to(E_{2,t',a'},(0,0))$ must be of the form
 $\phi(x,y)=(u^2x,u^3y)$ (c.f. \cite[Proposition 3.1]{sil}). From
 this, we immediately get the condition for $E_{2,t,a}$ and
 $E_{2,t',a'}$ to be isomorphic.

This leaves us with the cases
$(N, M) \in \{ (6,6), (8, 4), (9,3), (10, 2), (12, 2) \}$ with $\epsilon=-$.
(For $(N, M)=(9,3)$, the value of $\epsilon$ does not matter
because of Remark \ref{rem:plus-minus} (1).)
In all cases, there exists
an isomorphism $\alpha : E \to X_1(N)_M^\epsilon$,
where
$E$ is an elliptic curve over $\Q$
with finitely many $\Q$-rational points,
as exhibited in the following table,
where we set
$h_1(x)=(1-x)(1+x)^4(1+x-x^2)^2$ and
$h_2(x)=(1+x)(1+x+x^2)(1+x^2)^2(1-x)^3$.
 $$ \extrarowheight3pt
 \begin{array}{c|c|c|c} \hline\hline
 (N, M) & E & E(\Q) \setminus \{ O \} &
(s, t)=\alpha(x,y) \\ \hline
(6,6) & y^2=x^3-1 & (1,0) &
\left(-\frac{xy}{1-x^3},\frac1{1-x^3}\right)
\\
(8,4) & y^2=x^3+x & (0,0) &
\left(\frac y{x^2}(1-\frac{1}{x^2}),-\frac1{x^2}\right)
\\
(9,3) & y^2+y=x^3 & (0,0), (0,-1) &
\left(xy(y^2+y+1),y+1\right)
\\
(10,2) & y^2=x^3-x^2-x & (0,0) &
\left(yh_1(x),x\right)
\\
(12,2) & y^2=x^3+x^2+x & (0,0) &
\left(yh_2(x),x\right)
\\
 \hline\hline
 \end{array}
 $$
Since all points of $\alpha(E(\Q))$ are cusps
by Lemma \ref{lemma: uniformizers},
we get $Y_1(\Q)^\epsilon_M=\emptyset$ in these cases as well.
We are done.
\end{proof}

\begin{corollary}\label{cor:kodairatype}
Let $E$ be an elliptic curve over a number field $k$,
and $P \in E(k)_\Tor^\is$ an element of the intrinsic subgroup of order $m>0$.
Let $v$ be a finite place of $k$
such that the completion $k_v$ of $k$ at $v$ satisfies $\mu_m(k_v)= \{ 1 \}$.
If $E$ has split multiplicative reduction of Kodaira type $I_n$ at $v$,
then $n$ is divisible by $m^2$.
\end{corollary}
\begin{proof}
The assumption implies that
the base change of $E$ to $k_v$ is
isomorphic to the Tate curve $E_q=\G_m/q^\Z$
for some $q \in k_v$ such that $v(q)=n$
(\cite[Chapter V, Theorem 5.3]{sil})
Let us write the image of $P$ in $E_q(k_v)$ as $[a]$ with $a \in k_v$.
Since $\mu_m(k_v)= \{ 1 \}$,
we find from \eqref{eq:ex-TateCv} that
$E_q(k_v)[m]$ is a cyclic group of order $m$ generated by $[a]$.
In particular, $s:= s_m([a])$ is invertible in $\Z/m\Z$.
We have $\langle [a], [a] \rangle = 0$ since $P \in E(k)_\Tor^\is$.
On the other hand,
Proposition \ref{prop:TateCv} (1) shows
$\langle [a], [a] \rangle = q^{-s^2} \otimes (1/m^2)$.
By looking at its image under the map
$v \otimes \id_{\Q/\Z} : k_v^\times \otimes \Q/\Z \to \Q/\Z$,
we conclude that $-s^2 n/m^2 = 0$ holds in $\Q/\Z$,
that is, $m^2$ divides $n$.
\end{proof}

We briefly explain the outline of the proof.
Our approach is completely explicit.
As usual, we let $X_1(N)$ denote the modular curve associated to the
congruence subgroup $\Gamma_1(N)$. It possesses a model over $\mathbb{Q}$ on
which the cusp $0$ is $\mathbb{Q}$-rational. For a subfield $k$ of $\mathbb{C}$,
the non-cuspidal points in $X_1(N)(k)$ parameterize isomorphism
classes of pairs $(E, P)$ of an elliptic curve $E$ over $k$
and a $k$-rational point $P$ of order $N$.
More precisely, if the coordinates of a point
$\tau\in\mathbb H$ on this model of $X_1(N)$ is $k$-rational,
then the isomorphism class of $(\mathbb{C}/(\mathbb{Z}\tau+\mathbb{Z}),1/N)$ contains a pair $(E,P)$
of an elliptic curve $E$ over $k$ and a $k$-rational $N$-torsion point $P$.
A famous theorem of Mazur \cite[Theorem (7)]{Mazur} says that
$X_1(N)(\mathbb{Q})$ has a non-cuspidal point precisely
when $N=1,\ldots,10$, or $N=12$.

For a pair $(N, M)$ of $N \in \{4, \dots, 10, 12\}$ and a positive divisor $M$ of $N$,
we will construct a smooth projective curve $X_1 (N)^\pm_M$ over $\mathbb{Q}$ equipped with
a finite morphism $X_1(N)_M^\pm \to X_1(N)$
that enjoys the following property:
when $t \in X_1(N)(\mathbb{Q})$ corresponds to a pair $(E, P)$ as above (with $k=\mathbb{Q}$),
one has
\[
\langle P, (N/M)P \rangle = 0 \Leftrightarrow
t \in \operatorname{Im}(X_1(N)^\pm_M(\mathbb{Q}) \to X_1(N)(\mathbb{Q})).
\]
It is worth  mentioning that
$X_1(N)^+_M$ is isomorphic to the modular curve
$X_1(MN, N)$ associated to $\Gamma_1(MN, N):=\Gamma_0(MN) \cap \Gamma_1(N)$
(Theorem \ref{proposition: pairing 1-mod})
and $X_1(N)^-_M$ is its twist.

Similarly, for a positive even
integer $N$, we let $X_1^0(N,2)$ denote the modular curve associated
to the congruence subgroup
\begin{equation}\label{eq:G10-N2}
\Gamma_1^0(N,2):=\Gamma_1(N)\cap\Gamma^0(2)
=\left\{\M abcd\in\SL(2,\mathbb{Z}):a,d\equiv 1\bmod N,~~2|b,~N|c\right\}.
\end{equation}
The modular curve $X_1^0(N,2)$ parameterizes isomorphism classes of
triples $(E, P, Q)$ of an elliptic curve $E$
and $k$-rational points $P, Q$
such that
$P$ and $Q$ are of order $N$ and $2$ respectively,
and $Q\notin\gen P$.
For $N \in \{4, 6, 8 \}$
and $\underline{M}=(M_1, M_2, M_3)$ with $M_1|N, M_2, M_3 \in \{ 1, 2\}$,
we will construct a smooth projective curve
$X_1^0(N, 2)_{\underline{M}}^\pm$ over $\mathbb{Q}$
equipped with a finite morphism to $X_1^0(N, 2)$,
by which one can interpret the conditions
\[ \langle P, \frac{N}{M_1}P \rangle
= \langle Q, \frac{2}{M_2}Q \rangle
= \langle P, \frac{2}{M_3}Q \rangle = 0
\]
for a triple $(E, P, Q)$ as above corresponding to a point of $X_1^0(N, 2)(\mathbb{Q})$.
Therefore Theorem \ref{theorem: intrinsic torsion} is reduced to a study of the $\mathbb{Q}$-rational points of
$X_1(N)_M^\pm$ and $X_1^0(N, 2)_{\underline{M}}^\pm$.
As our construction of these curves are explicit,
this can be done by a (more or less) direct computation.

In the last section \S \ref{sect:high-dim},
we generalize the pairing \eqref{eq:pairing-intro}
and the intrinsic subgroup
to higher dimensional varieties.
This new construction enables us to prove that,
if $X$ has good reduction with respect to a discrete valuation of $k$,
there is a strong restriction on the values of the pairing
(see Proposition \ref{prop:two-def-same}).
Finally, as a sample for the case of bad reduction,
we compute the intrinsic subgroup of
Tate elliptic curves (see Proposition \ref{prop:TateCv}).
As an application,
we show that the intrinsic subgroup of an elliptic curve over a number field
imposes some constraint on its reduction type
(see Corollary \ref{cor:kodairatype}).

\subsection*{Acknowledgement}
It is Kenneth Ribet who formulated \eqref{eq:pairing-intro} as a pairing
and asked if it is symmetric.
The authors express deep gratitude to him for asking this question,
which was a starting point of our work.

The first author (T. Y.) is supported by JSPS KAKENHI Grant (25K06961).
The second author (Y. Y.) is supported by
Grant 113-2115-M-002-003-MY3 of the National Science and Technology
  Council of the Republic of China (Taiwan).
The third author (H. Y.) is supported by National Research Foundation of Korea(NRF) grant
funded by the Korea government(MSIT) (No. RS-2023-00239918  and No. 2020R1A5A1016126).
The fourth author (M. Y.) is supported by the National Research Foundation of Korea (NRF) grant
funded by the
Korea government (MSIT) (RS-2025-23525445).

\subsection*{Notation and convention}
For an abelian group $A$,
we write $A[n]$ and $A/n$ for the kernel and cokernel
of $A \to A, \ a \mapsto na$ for each $n \in \mathbb{Z}$,
and put $A_\Tor := \cup_{n > 0} A[n]$.
For a field $k$,
we set $\mu_n(k):=(k^\times)[n]$ and $\mu(k):=(k^\times)_\Tor$.

The identity element of an elliptic curve is denoted by $O$.

The authors say: “We briefly explain the outline of the proof. Our approach is completely explicit.” They use modular curves $X_1(N)$: non-cuspidal points in $X_1(N)(k)$ “parameterize isomorphism classes of pairs $(E,P)$ of an elliptic curve $E$ over $k$ and a $k$-rational point $P$ of order $N$.” Using Mazur’s theorem that $X_1(N)(\mathbb{Q})$ has a non-cuspidal point precisely for $N=1,\ldots,10$ or $N=12$, they restrict to these torsion orders.

For each $(N,M)$ with $N\in\{4,\dots,10,12\}$ and $M\mid N$, they “construct a smooth projective curve $X_1(N)^\pm_M$ over $\mathbb{Q}$ equipped with a finite morphism $X_1(N)_M^\pm\to X_1(N)$” such that for $t\in X_1(N)(\mathbb{Q})$ corresponding to $(E,P)$ one has
\[
\langle P,(N/M)P\rangle=0\Longleftrightarrow t\in\operatorname{Im}\bigl(X_1(N)^\pm_M(\mathbb{Q})\to X_1(N)(\mathbb{Q})\bigr).
\]
They note $X_1(N)^+_M$ is isomorphic to a modular curve $X_1(MN,N)$ and $X_1(N)^-_M$ is its twist.

For even $N$, they similarly use the modular curve $X_1^0(N,2)$, which “parameterizes isomorphism classes of triples $(E,P,Q)$” with $P$ of order $N$, $Q$ of order $2$, and $Q\notin\langle P\rangle$. For $N\in\{4,6,8\}$ and $\underline{M}=(M_1,M_2,M_3)$ with $M_1\mid N$ and $M_2,M_3\in\{1,2\}$, they construct curves $X_1^0(N,2)^{\pm}_{\underline{M}}$ with finite morphisms to $X_1^0(N,2)$ “by which one can interpret the conditions”
\[
\langle P,\tfrac{N}{M_1}P\rangle=\langle Q,\tfrac{2}{M_2}Q\rangle=\langle P,\tfrac{2}{M_3}Q\rangle=0.
\]
They conclude that “Therefore Theorem~\ref{theorem: intrinsic torsion} is reduced to a study of the $\mathbb{Q}$-rational points of $X_1(N)_M^\pm$ and $X_1^0(N,2)_{\underline{M}}^\pm$,” and since “our construction of these curves are explicit, this can be done by a (more or less) direct computation.”

The authors say: “We briefly explain the outline of the proof. Our approach is completely explicit.” They use modular curves $X_1(N)$: non-cuspidal points in $X_1(N)(k)$ “parameterize isomorphism classes of pairs $(E,P)$ of an elliptic curve $E$ over $k$ and a $k$-rational point $P$ of order $N$.” Using Mazur’s theorem that $X_1(N)(\mathbb{Q})$ has a non-cuspidal point precisely for $N=1,\ldots,10$ or $N=12$, they restrict to these torsion orders.

For each $(N,M)$ with $N\in\{4,\dots,10,12\}$ and $M\mid N$, they “construct a smooth projective curve $X_1(N)^\pm_M$ over $\mathbb{Q}$ equipped with a finite morphism $X_1(N)_M^\pm\to X_1(N)$” such that for $t\in X_1(N)(\mathbb{Q})$ corresponding to $(E,P)$ one has
\[
\langle P,(N/M)P\rangle=0\Longleftrightarrow t\in\operatorname{Im}\bigl(X_1(N)^\pm_M(\mathbb{Q})\to X_1(N)(\mathbb{Q})\bigr).
\]
They note $X_1(N)^+_M$ is isomorphic to a modular curve $X_1(MN,N)$ and $X_1(N)^-_M$ is its twist.

For even $N$, they similarly use the modular curve $X_1^0(N,2)$, which “parameterizes isomorphism classes of triples $(E,P,Q)$” with $P$ of order $N$, $Q$ of order $2$, and $Q\notin\langle P\rangle$. For $N\in\{4,6,8\}$ and $\underline{M}=(M_1,M_2,M_3)$ with $M_1\mid N$ and $M_2,M_3\in\{1,2\}$, they construct curves $X_1^0(N,2)^{\pm}_{\underline{M}}$ with finite morphisms to $X_1^0(N,2)$ “by which one can interpret the conditions”
\[
\langle P,\tfrac{N}{M_1}P\rangle=\langle Q,\tfrac{2}{M_2}Q\rangle=\langle P,\tfrac{2}{M_3}Q\rangle=0.
\]
They conclude that, in establishing the main theorem, everything is reduced to a study of the $\mathbb{Q}$-rational points of $X_1(N)_M^\pm$ and $X_1^0(N,2)_{\underline{M}}^\pm$, and since “our construction of these curves are explicit, this can be done by a (more or less) direct computation.”

\label{theorem: intrinsic torsion}
  Let $E$ be an elliptic curve over $\mathbb{Q}$.
Then there is a pair $(A, B)$ of an abelian group $A$
and its subgroup $B$ in the following table,
together with
an isomorphism $\alpha : E(\mathbb{Q})_\Tor \cong A$,
such that $\alpha(E(\mathbb{Q})_\Tor^\is)=B$.
  $$ \extrarowheight3pt
  \begin{array}{ll} \hline\hline
    A & \text{order of}\ B \\ \hline
    0 & 1 \\
    \mathbb{Z}/2\mathbb{Z} & 1,2\\
    \mathbb{Z}/3\mathbb{Z} & 1,3\\
    \mathbb{Z}/4\mathbb{Z} & 1,2,4 \\
    \mathbb{Z}/5\mathbb{Z} & 1,5 \\
    \mathbb{Z}/6\mathbb{Z} & 1,2,3 \\
    \mathbb{Z}/7\mathbb{Z} & 1 \\
    \mathbb{Z}/8\mathbb{Z} & 1,2 \\
    \mathbb{Z}/9\mathbb{Z} & 1 \\
    \mathbb{Z}/10\mathbb{Z} & 1\\
    \mathbb{Z}/12\mathbb{Z} & 1 \\
    \mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z} & 1,2\\
    \mathbb{Z}/4\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z} & 1,2 \ \\
    \mathbb{Z}/6\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z} & 1 \\
    \mathbb{Z}/8\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z} & 1 \\ \hline\hline
  \end{array}
  $$
(For $A=\mathbb{Z}/4\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$,
there are two $\operatorname{Aut}(A)$-conjugacy classes of order two subgroups,
both of which can be taken as $B$.)
  Moreover, for each possible pair $(A, B)$ as above,
there are infinitely many (mutually non-isomorphic) elliptic curves $E$ over $\mathbb{Q}$
for which there is an isomorphism $\alpha : E(\mathbb{Q})_\Tor \cong A$
such that $\alpha(E(\mathbb{Q})_\Tor^\is)=B$.,

Question:
Let $E$ be an elliptic curve over $\mathbb{Q}$ with identity element $O$. Via the identification $P \mapsto [P-O]$, identify $E(\mathbb{Q})_{\mathrm{Tor}}$ with $\operatorname{Pic}(E)_{\mathrm{Tor}}$. Let
\[
\langle\cdot,\cdot\rangle: E(\mathbb{Q})_{\mathrm{Tor}}\times E(\mathbb{Q})_{\mathrm{Tor}}\to \mathbb{Q}^\times\otimes \mathbb{Q}/\mathbb{Z}
\]
be the induced biadditive symmetric pairing, and define the intrinsic subgroup by
\[
E(\mathbb{Q})_{\mathrm{Tor}}^{\mathrm{is}}:=\{P\in E(\mathbb{Q})_{\mathrm{Tor}}\mid \langle P,Q\rangle=0\text{ for all }Q\in E(\mathbb{Q})_{\mathrm{Tor}}\}.
\]
Which existence statement holds for the pair consisting of the torsion subgroup $E(\mathbb{Q})_{\mathrm{Tor}}$ and its intrinsic subgroup $E(\mathbb{Q})_{\mathrm{Tor}}^{\mathrm{is}}$?

For every elliptic curve $E/\mathbb{Q}$, there exist an abelian group $A$ and a subgroup $B\le A$, together with an isomorphism $\alpha:E(\mathbb{Q})_{\mathrm{Tor}}\xrightarrow{\sim} A$, such that $\alpha(E(\mathbb{Q})_{\mathrm{Tor}}^{\mathrm{is}})=B$, where $A$ and the possible orders of $B$ are exactly as follows: $A=0$ with $|B|=1$; $A=\mathbb{Z}/2\mathbb{Z}$ with $|B|=1$ or $2$; $A=\mathbb{Z}/3\mathbb{Z}$ with $|B|=1$ or $3$; $A=\mathbb{Z}/4\mathbb{Z}$ with $|B|=1,2,$ or $4$; $A=\mathbb{Z}/5\mathbb{Z}$ with $|B|=1$ or $5$; $A=\mathbb{Z}/6\mathbb{Z}$ with $|B|=1,2,$ or $3$; $A=\mathbb{Z}/7\mathbb{Z}$ with $|B|=1$; $A=\mathbb{Z}/8\mathbb{Z}$ with $|B|=1$ or $2$; $A=\mathbb{Z}/9\mathbb{Z}$ with $|B|=1$; $A=\mathbb{Z}/10\mathbb{Z}$ with $|B|=1$; $A=\mathbb{Z}/12\mathbb{Z}$ with $|B|=1$; $A=\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$ with $|B|=1$ or $2$; $A=\mathbb{Z}/4\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$ with $|B|=1$ or $2$; $A=\mathbb{Z}/6\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$ with $|B|=1$; and $A=\mathbb{Z}/8\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$ with $|B|=1$. For $A=\mathbb{Z}/4\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$, there are two $\operatorname{Aut}(A)$-conjugacy classes of subgroups of order $2$, and either class can occur as $B$. Moreover, for every such possible pair $(A,B)$, there exist infinitely many mutually non-isomorphic elliptic curves $E/\mathbb{Q}$ for which some isomorphism $\alpha:E(\mathbb{Q})_{\mathrm{Tor}}\xrightarrow{\sim}A$ satisfies $\alpha(E(\mathbb{Q})_{\mathrm{Tor}}^{\mathrm{is}})=B$.

Choices:
B. For every elliptic curve $E/\mathbb{Q}$, there exist an abelian group $A$ and a subgroup $B\le A$, together with an isomorphism $\alpha:E(\mathbb{Q})_{\mathrm{Tor}}\xrightarrow{\sim} A$, such that $\alpha(E(\mathbb{Q})_{\mathrm{Tor}}^{\mathrm{is}})=B$, where $A$ and the possible orders of $B$ are exactly as follows: $A=0$ with $|B|=1$; $A=\mathbb{Z}/2\mathbb{Z}$ with $|B|=1$ or $2$; $A=\mathbb{Z}/3\mathbb{Z}$ with $|B|=1$ or $3$; $A=\mathbb{Z}/4\mathbb{Z}$ with $|B|=1,2,$ or $4$; $A=\mathbb{Z}/5\mathbb{Z}$ with $|B|=1$ or $5$; $A=\mathbb{Z}/6\mathbb{Z}$ with $|B|=1,2,$ or $3$; $A=\mathbb{Z}/7\mathbb{Z}$ with $|B|=1$; $A=\mathbb{Z}/8\mathbb{Z}$ with $|B|=1$ or $2$; $A=\mathbb{Z}/9\mathbb{Z}$ with $|B|=1$ or $3$; $A=\mathbb{Z}/10\mathbb{Z}$ with $|B|=1$; $A=\mathbb{Z}/12\mathbb{Z}$ with $|B|=1$; $A=\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$ with $|B|=1$ or $2$; $A=\mathbb{Z}/4\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$ with $|B|=1$ or $2$; $A=\mathbb{Z}/6\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$ with $|B|=1$; and $A=\mathbb{Z}/8\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$ with $|B|=1$. For $A=\mathbb{Z}/4\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$, there are two $\operatorname{Aut}(A)$-conjugacy classes of subgroups of order $2$, and either class can occur as $B$. Moreover, for every such possible pair $(A,B)$, there exists at least one elliptic curve $E/\mathbb{Q}$ for which some isomorphism $\alpha:E(\mathbb{Q})_{\mathrm{Tor}}\xrightarrow{\sim}A$ satisfies $\alpha(E(\mathbb{Q})_{\mathrm{Tor}}^{\mathrm{is}})=B$.
C. For every elliptic curve $E/\mathbb{Q}$, there exist an abelian group $A$ and a subgroup $B\le A$, together with an isomorphism $\alpha:E(\mathbb{Q})_{\mathrm{Tor}}\xrightarrow{\sim} A$, such that $\alpha(E(\mathbb{Q})_{\mathrm{Tor}}^{\mathrm{is}})=B$, where $A$ and the possible orders of $B$ are among the following: $A=0$ with $|B|=1$; $A=\mathbb{Z}/2\mathbb{Z}$ with $|B|=1$ or $2$; $A=\mathbb{Z}/3\mathbb{Z}$ with $|B|=1$ or $3$; $A=\mathbb{Z}/4\mathbb{Z}$ with $|B|=1,2,$ or $4$; $A=\mathbb{Z}/5\mathbb{Z}$ with $|B|=1$ or $5$; $A=\mathbb{Z}/6\mathbb{Z}$ with $|B|=1,2,$ or $3$; $A=\mathbb{Z}/7\mathbb{Z}$ with $|B|=1$; $A=\mathbb{Z}/8\mathbb{Z}$ with $|B|=1$ or $2$; $A=\mathbb{Z}/9\mathbb{Z}$ with $|B|=1$; $A=\mathbb{Z}/10\mathbb{Z}$ with $|B|=1$; $A=\mathbb{Z}/12\mathbb{Z}$ with $|B|=1$; $A=\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$ with $|B|=1$ or $2$; $A=\mathbb{Z}/4\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$ with $|B|=1$ or $2$; $A=\mathbb{Z}/6\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$ with $|B|=1$; and $A=\mathbb{Z}/8\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$ with $|B|=1$.
D. For every elliptic curve $E/\mathbb{Q}$, there exist an abelian group $A$ and a subgroup $B\le A$, together with an isomorphism $\alpha:E(\mathbb{Q})_{\mathrm{Tor}}\xrightarrow{\sim} A$, such that $\alpha(E(\mathbb{Q})_{\mathrm{Tor}}^{\mathrm{is}})=B$, where $A$ and the possible orders of $B$ are exactly as follows: $A=0$ with $|B|=1$; $A=\mathbb{Z}/2\mathbb{Z}$ with $|B|=1$ or $2$; $A=\mathbb{Z}/3\mathbb{Z}$ with $|B|=1$ or $3$; $A=\mathbb{Z}/4\mathbb{Z}$ with $|B|=1,2,$ or $4$; $A=\mathbb{Z}/5\mathbb{Z}$ with $|B|=1$ or $5$; $A=\mathbb{Z}/6\mathbb{Z}$ with $|B|=1,2,$ or $3$; $A=\mathbb{Z}/7\mathbb{Z}$ with $|B|=1$; $A=\mathbb{Z}/8\mathbb{Z}$ with $|B|=1$ or $2$; $A=\mathbb{Z}/9\mathbb{Z}$ with $|B|=1$; $A=\mathbb{Z}/10\mathbb{Z}$ with $|B|=1$; $A=\mathbb{Z}/12\mathbb{Z}$ with $|B|=1$; $A=\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$ with $|B|=1$ or $2$; $A=\mathbb{Z}/4\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$ with $|B|=1,2,$ or $4$; $A=\mathbb{Z}/6\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$ with $|B|=1$; and $A=\mathbb{Z}/8\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$ with $|B|=1$. For $A=\mathbb{Z}/4\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$, there are two $\operatorname{Aut}(A)$-conjugacy classes of subgroups of order $2$, and either class can occur as $B$. Moreover, for every such possible pair $(A,B)$, there exist infinitely many mutually non-isomorphic elliptic curves $E/\mathbb{Q}$ for which some isomorphism $\alpha:E(\mathbb{Q})_{\mathrm{Tor}}\xrightarrow{\sim}A$ satisfies $\alpha(E(\mathbb{Q})_{\mathrm{Tor}}^{\mathrm{is}})=B$.
E. For every elliptic curve $E/\mathbb{Q}$, there exist an abelian group $A$ and a subgroup $B\le A$, together with an isomorphism $\alpha:E(\mathbb{Q})_{\mathrm{Tor}}\xrightarrow{\sim} A$, such that $\alpha(E(\mathbb{Q})_{\mathrm{Tor}}^{\mathrm{is}})=B$, where $A$ and the possible orders of $B$ are exactly as follows: $A=0$ with $|B|=1$; $A=\mathbb{Z}/2\mathbb{Z}$ with $|B|=1$ or $2$; $A=\mathbb{Z}/3\mathbb{Z}$ with $|B|=1$ or $3$; $A=\mathbb{Z}/4\mathbb{Z}$ with $|B|=1,2,$ or $4$; $A=\mathbb{Z}/5\mathbb{Z}$ with $|B|=1$ or $5$; $A=\mathbb{Z}/6\mathbb{Z}$ with $|B|=1,2,$ or $3$; $A=\mathbb{Z}/7\mathbb{Z}$ with $|B|=1$; $A=\mathbb{Z}/8\mathbb{Z}$ with $|B|=1$ or $2$; $A=\mathbb{Z}/9\mathbb{Z}$ with $|B|=1$; $A=\mathbb{Z}/10\mathbb{Z}$ with $|B|=1$; $A=\mathbb{Z}/12\mathbb{Z}$ with $|B|=1$; $A=\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$ with $|B|=1$ or $2$; $A=\mathbb{Z}/4\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$ with $|B|=1$ or $2$; $A=\mathbb{Z}/6\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$ with $|B|=1$; and $A=\mathbb{Z}/8\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$ with $|B|=1$. For $A=\mathbb{Z}/4\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$, there are two $\operatorname{Aut}(A)$-conjugacy classes of subgroups of order $2$, and both classes occur simultaneously for every elliptic curve $E/\mathbb{Q}$ with $E(\mathbb{Q})_{\mathrm{Tor}}\cong A$. Moreover, for every such possible pair $(A,B)$, there exist infinitely many mutually non-isomorphic elliptic curves $E/\mathbb{Q}$ for which some isomorphism $\alpha:E(\mathbb{Q})_{\mathrm{Tor}}\xrightarrow{\sim}A$ satisfies $\alpha(E(\mathbb{Q})_{\mathrm{Tor}}^{\mathrm{is}})=B$.


Question:
Let $E$ be an elliptic curve over $\mathbb{Q}$. Write $E(\mathbb{Q})_{\mathrm{Tor}}$ for its torsion subgroup, and define the intrinsic subgroup
\[
E(\mathbb{Q})_{\mathrm{Tor}}^{\mathrm{is}}:=\{P\in E(\mathbb{Q})_{\mathrm{Tor}}\mid \langle P,Q\rangle=0\text{ for all }Q\in E(\mathbb{Q})_{\mathrm{Tor}}\},
\]
where $\langle\ ,\ \rangle:E(\mathbb{Q})_{\mathrm{Tor}}\times E(\mathbb{Q})_{\mathrm{Tor}}\to \mathbb{Q}^\times\otimes \mathbb{Q}/\mathbb{Z}$ is the biadditive symmetric pairing under consideration. Which description gives the complete classification of the possible abstract pairs $(E(\mathbb{Q})_{\mathrm{Tor}},E(\mathbb{Q})_{\mathrm{Tor}}^{\mathrm{is}})$?

Exactly the following occur: for every elliptic curve $E/\mathbb{Q}$ there is an abelian group $A$ and a subgroup $B\le A$ with an isomorphism $\alpha:E(\mathbb{Q})_{\mathrm{Tor}}\xrightarrow{\sim}A$ such that $\alpha(E(\mathbb{Q})_{\mathrm{Tor}}^{\mathrm{is}})=B$, where the possibilities are
\[
\begin{array}{ll}
A & |B|\\ \hline
0 & 1 \\
\mathbb{Z}/2\mathbb{Z} & 1,2\\
\mathbb{Z}/3\mathbb{Z} & 1,3\\
\mathbb{Z}/4\mathbb{Z} & 1,2,4\\
\mathbb{Z}/5\mathbb{Z} & 1,5\\
\mathbb{Z}/6\mathbb{Z} & 1,2,3\\
\mathbb{Z}/7\mathbb{Z} & 1\\
\mathbb{Z}/8\mathbb{Z} & 1,2\\
\mathbb{Z}/9\mathbb{Z} & 1\\
\mathbb{Z}/10\mathbb{Z} & 1\\
\mathbb{Z}/12\mathbb{Z} & 1\\
\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z} & 1,2\\
\mathbb{Z}/4\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z} & 1,2\\
\mathbb{Z}/6\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z} & 1\\
\mathbb{Z}/8\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z} & 1.
\end{array}
\]
For $A=\mathbb{Z}/4\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}$ and $|B|=2$, both $\operatorname{Aut}(A)$-conjugacy classes of order-two subgroups can occur. Conversely, every such pair $(A,B)$ is realized by infinitely many mutually non-isomorphic elliptic curves over $\mathbb{Q}$.

Choices:
B.
Exactly the following occur: for every elliptic curve $E/\mathbb{Q}$ there is an abelian group $A$ and a subgroup $B\le A$ with an isomorphism $\alpha:E(\mathbb{Q})_{\mathrm{Tor}}\xrightarrow{\sim}A$ such that $\alpha(E(\mathbb{Q})_{\mathrm{Tor}}^{\mathrm{is}})=B$, where the possibilities are
\[
\begin{array}{ll}
A & |B|\\ \hline
0 & 1 \\
\mathbb{Z}/2\mathbb{Z} & 1,2\\
\mathbb{Z}/3\mathbb{Z} & 1,3\\
\mathbb{Z}/4\mathbb{Z} & 1,2,4\\
\mathbb{Z}/5\mathbb{Z} & 1,5\\
\mathbb{Z}/6\mathbb{Z} & 1,2,3,6\\
\mathbb{Z}/7\mathbb{Z} & 1\\
\mathbb{Z}/8\mathbb{Z} & 1,2\\
\mathbb{Z}/9\mathbb{Z} & 1\\
\mathbb{Z}/10\mathbb{Z} & 1\\
\mathbb{Z}/12\mathbb{Z} & 1\\
\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z} & 1,2\\
\mathbb{Z}/4\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z} & 1,2\\
\mathbb{Z}/6\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z} & 1\\
\mathbb{Z}/8\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z} & 1.
\end{array}
\]
For $A=\mathbb{Z}/4\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}$ and $|B|=2$, both $\operatorname{Aut}(A)$-conjugacy classes of order-two subgroups can occur. Conversely, every such pair $(A,B)$ is realized by infinitely many mutually non-isomorphic elliptic curves over $\mathbb{Q}$.
C.
For every elliptic curve $E/\mathbb{Q}$ there is an abelian group $A$ and a subgroup $B\le A$ with an isomorphism $\alpha:E(\mathbb{Q})_{\mathrm{Tor}}\xrightarrow{\sim}A$ such that $\alpha(E(\mathbb{Q})_{\mathrm{Tor}}^{\mathrm{is}})=B$, where the possibilities for $A$ are exactly Mazur’s torsion groups
\[
0,\ \mathbb{Z}/N\mathbb{Z}\ (1\le N\le 10\text{ or }N=12),\ \text{and }\mathbb{Z}/2N\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}\ (N=1,2,3,4),
\]
and for each such $A$ the subgroup $B$ is some subgroup of $A$. Moreover, every such $A$ occurs for infinitely many mutually non-isomorphic elliptic curves over $\mathbb{Q}$.
D.
Exactly the following occur: for every elliptic curve $E/\mathbb{Q}$ there is an abelian group $A$ and a subgroup $B\le A$ with an isomorphism $\alpha:E(\mathbb{Q})_{\mathrm{Tor}}\xrightarrow{\sim}A$ such that $\alpha(E(\mathbb{Q})_{\mathrm{Tor}}^{\mathrm{is}})=B$, where the possibilities are
\[
\begin{array}{ll}
A & |B|\\ \hline
0 & 1 \\
\mathbb{Z}/2\mathbb{Z} & 1,2\\
\mathbb{Z}/3\mathbb{Z} & 1,3\\
\mathbb{Z}/4\mathbb{Z} & 1,2,4\\
\mathbb{Z}/5\mathbb{Z} & 1,5\\
\mathbb{Z}/6\mathbb{Z} & 1,2,3\\
\mathbb{Z}/7\mathbb{Z} & 1\\
\mathbb{Z}/8\mathbb{Z} & 1,2\\
\mathbb{Z}/9\mathbb{Z} & 1\\
\mathbb{Z}/10\mathbb{Z} & 1\\
\mathbb{Z}/12\mathbb{Z} & 1\\
\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z} & 1,2\\
\mathbb{Z}/4\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z} & 1,2\\
\mathbb{Z}/6\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z} & 1\\
\mathbb{Z}/8\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z} & 1.
\end{array}
\]
Conversely, for each possible pair $(A,B)$ as above, there exists an elliptic curve $E$ over $\mathbb{Q}$ for which there is an isomorphism $\alpha:E(\mathbb{Q})_{\mathrm{Tor}}\cong A$ such that $\alpha(E(\mathbb{Q})_{\mathrm{Tor}}^{\mathrm{is}})=B$.
E.
Exactly the following occur: for every elliptic curve $E/\mathbb{Q}$ there is an abelian group $A$ and a subgroup $B\le A$ with an isomorphism $\alpha:E(\mathbb{Q})_{\mathrm{Tor}}\xrightarrow{\sim}A$ such that $\alpha(E(\mathbb{Q})_{\mathrm{Tor}}^{\mathrm{is}})=B$, where the possibilities are
\[
\begin{array}{ll}
A & |B|\\ \hline
0 & 1 \\
\mathbb{Z}/2\mathbb{Z} & 1,2\\
\mathbb{Z}/3\mathbb{Z} & 1,3\\
\mathbb{Z}/4\mathbb{Z} & 1,2,4\\
\mathbb{Z}/5\mathbb{Z} & 1,5\\
\mathbb{Z}/6\mathbb{Z} & 1,2,3\\
\mathbb{Z}/7\mathbb{Z} & 1,7\\
\mathbb{Z}/8\mathbb{Z} & 1,2\\
\mathbb{Z}/9\mathbb{Z} & 1\\
\mathbb{Z}/10\mathbb{Z} & 1\\
\mathbb{Z}/12\mathbb{Z} & 1\\
\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z} & 1,2\\
\mathbb{Z}/4\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z} & 1,2\\
\mathbb{Z}/6\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z} & 1\\
\mathbb{Z}/8\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z} & 1.
\end{array}
\]
For $A=\mathbb{Z}/4\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}$ and $|B|=2$, both $\operatorname{Aut}(A)$-conjugacy classes of order-two subgroups can occur. Conversely, every such pair $(A,B)$ is realized by infinitely many mutually non-isomorphic elliptic curves over $\mathbb{Q}$.

explicit exclusion of the full intrinsic subgroup for cyclic order 6

dropped the explicit list of allowed subgroups B for each torsion group A

infinitely many realizations for each admissible pair

explicit elimination of nontrivial intrinsic subgroup for cyclic order 7


Approximation theorems (Runge \cite{Run}, Mergelyan \cite{Mer}, Arakelian \cite{Ar}) play a central role in the field of complex approximation, since they give necessary and sufficient conditions for functions to be uniformly approximated on a given set by model functions, such as polynomials, rational functions or more generally entire/meromorphic functions. Our goal in the present paper is to give a new characterization of closed sets $F\subset \mathbb{C}$ in the complex plane, for which uniform approximation by entire functions is possible.

To this end, define the classes
\begin{align*}
H(F)=&\,\lbrace f:U\to\mathbb{C} \text{ holomorphic for some open set $U\supset F$}\rbrace,\\
 A(F)=&\,\lbrace f:F\to\mathbb{C} \text{ holomorphic in }F^\circ\text{ and continuous in F}\rbrace,
\end{align*}
where $F^\circ$ is the interior of $F$.
\begin{dfn}\label{def:hole}
Any bounded connected component $B$ of $\mathbb{C}\setminus F$ is called a hole of $F$.
\end{dfn}
Arakelian's theorem \cite{Ar} gives necessary and sufficient conditions such that every function $f\in A(F)$ can be uniformly approximated on $F$ by entire functions..
\begin{dfn}[Arakelian set]\label{def:Arak}
Let $F\subset \mathbb{C}$ be a closed set. We say that $F$ is an Arakelian set, if and only if the following points hold:\\[5pt]
1. $F$ is without holes;\\[5pt]
2. The union of all holes of $F\cup K$ is bounded, for any compact subset $K\subset \mathbb{C}$.
\end{dfn}
\begin{rmk}\label{rem:K}
Point 2. in Definition \ref{def:Arak} is equivalent to the one where $K$ is replaced by $\overline{D}_0(n)=\{z\in\mathbb{C}: |z|\leq n\}$, for any $n\in\mathbb{N}$.
\end{rmk}
\begin{rmk}
In fact, Definition \ref{def:Arak} appears in a later work \cite{RR}, where a new proof of Arakelian's theorem was given (see also \cite{Got}). The original conditions in \cite{Ar} are:\\[5pt]
1. $\mathbb{C}\setminus F$ is connected;\\[5pt]
2. $\mathbb{C}\setminus F$ is locally connected at infinity.\\[5pt]
Nevertheless, they are easily seen to be equivalent.
\end{rmk}

\begin{dfn}[Uniform approximation set]
Let $F\subset \mathbb{C}$ be a closed set. We say that $F$ is a set of uniform approximation, if for every $\epsilon>0$ and $f\in A(F)$, there exists an entire function $g$ such that $|f(z)-g(z)|<\epsilon$, for all $z\in F$.
\end{dfn}

\begin{abstract}
Arakelian's classical approximation theorem \cite{Ar} gives necessary and sufficient conditions such that functions can be uniformly approximated in (unbounded) closed sets $F\subset \mathbb{C}$ by entire functions. The conditions are purely topological and concern the connectedness of the complement of $F$. We give a new characterization of Arakelian sets in terms of logarithmic branches of functions $f\in A(F)$, which are continuous in $F$ and holomorphic in its interior $F^\circ$. Our proof is based on a contradiction argument and the counterexample function that we use is furnished by the Weierstrass factorization theorem.
\end{abstract}

Approximation theorems (Runge \cite{Run}, Mergelyan \cite{Mer}, Arakelian \cite{Ar}) play a central role in the field of complex approximation, since they give necessary and sufficient conditions for functions to be uniformly approximated on a given set by model functions, such as polynomials, rational functions or more generally entire/meromorphic functions. Our goal in the present paper is to give a new characterization of closed sets $F\subset \mathbb{C}$ in the complex plane, for which uniform approximation by entire functions is possible.

To this end, define the classes
\begin{align*}
H(F)=&\,\lbrace f:U\to\mathbb{C} \text{ holomorphic for some open set $U\supset F$}\rbrace,\\
 A(F)=&\,\lbrace f:F\to\mathbb{C} \text{ holomorphic in }F^\circ\text{ and continuous in F}\rbrace,
\end{align*}
where $F^\circ$ is the interior of $F$.
\begin{dfn}\label{def:hole}
Any bounded connected component $B$ of $\mathbb{C}\setminus F$ is called a hole of $F$.
\end{dfn}
Arakelian's theorem \cite{Ar} gives necessary and sufficient conditions such that every function $f\in A(F)$ can be uniformly approximated on $F$ by entire functions..
\begin{dfn}[Arakelian set]\label{def:Arak}
Let $F\subset \mathbb{C}$ be a closed set. We say that $F$ is an Arakelian set, if and only if the following points hold:\\[5pt]
1. $F$ is without holes;\\[5pt]
2. The union of all holes of $F\cup K$ is bounded, for any compact subset $K\subset \mathbb{C}$.
\end{dfn}
\begin{rmk}\label{rem:K}
Point 2. in Definition \ref{def:Arak} is equivalent to the one where $K$ is replaced by $\overline{D}_0(n)=\{z\in\mathbb{C}: |z|\leq n\}$, for any $n\in\mathbb{N}$.
\end{rmk}
\begin{rmk}
In fact, Definition \ref{def:Arak} appears in a later work \cite{RR}, where a new proof of Arakelian's theorem was given (see also \cite{Got}). The original conditions in \cite{Ar} are:\\[5pt]
1. $\mathbb{C}\setminus F$ is connected;\\[5pt]
2. $\mathbb{C}\setminus F$ is locally connected at infinity.\\[5pt]
Nevertheless, they are easily seen to be equivalent.
\end{rmk}

\begin{dfn}[Uniform approximation set]
Let $F\subset \mathbb{C}$ be a closed set. We say that $F$ is a set of uniform approximation, if for every $\epsilon>0$ and $f\in A(F)$, there exists an entire function $g$ such that $|f(z)-g(z)|<\epsilon$, for all $z\in F$.
\end{dfn}

\subsection{Main result}

We can now state the main result:
\begin{thm}\label{thm:main}
A closed set $F\subset \mathbb{C}$ is an Arakelian set, if and only if
for every $f\in A(F)$ with no zeros in $F$, there exists $g\in A(F)$ such that $e^g=f$ in $F$.
\end{thm}
At first glance, the existence of logarithmic branches of functions $f\in A(F)$ might seem to have little to do with the conditions in Definition \ref{def:Arak}. Intuitively, it is easier to make the connection by first stating yet another characterization of Arakelian sets, proved in \cite{F}, in terms of simply connected neighborhoods of $F$, that is to say, open sets $V\supset F$ whose connected components are simply connected.
\begin{thm}\label{thm:F}
A closed set $F\subset \mathbb{C}$ is an Arakelian set if and only if it possesses a neighborhood basis of simply connected open sets.
\end{thm}
At least one of the directions in Theorem \ref{thm:main} is clearer now. If a function $f\in A(F)$ could be extended to a neighborhood of $F$, then the existence of an intermediate simply connected neighborhood would allow for a logarithmic branch of $f$ to be well-defined in the latter.

Theorem \ref{thm:Ar} has been successfully extended to relatively closed subsets of planar domains $G\subset \mathbb{C}$ \cite{Ar2} and non-planar Riemann surfaces of finite genus \cite{GH,Sch}. One of the outstanding open problems in complex approximation is to characterize sets of uniform approximation in Riemann surfaces of infinite genus, where Arakelian sets (Definition \ref{def:Arak}) are not always sets of uniform approximation, see \cite{BG} for a counterexample. Moreover, it was shown by Scheinberg \cite{Sch} that such a characterization, if it exists, cannot be purely topological, but it has to take into account the complex analytic structure of the given Riemann surface.

The one direction of the characterization in Theorem \ref{thm:main} is contained in \cite{GP}. We include the proof here for the sake of completeness.
\begin{prp}[Gauthier-Pouryayevali \cite{GP}]\label{prop:GP}
Suppose that the closed set $F\subset \mathbb{C}$ is an Arakelian set. For every function $f\in A(F)$ with no zeros in $F$, there exists a function $g\in A(F)$ such that $e^g=f$ in $F$.
\end{prp}
\begin{proof}
By Tietze's extension theorem, there exists a continuous extension of $f$ in $\mathbb{C}$, which we
denote by $\widetilde{f}:\mathbb{C}\to\C$. Our assumption implies that the open set $U=\mathbb{C}\setminus \widetilde{f}^{-1}(\{0\})$ contains $F$.
Thus, by Theorem \ref{thm:F} there exists a simply connected open set $V$ with $F\subset V\subset U$.
If we consider the covering map $\exp: \mathbb{C}\to\mathbb{C}\setminus\{0\}$, then the latter implies that
$\widetilde{f}_{\big|V}:V\to\mathbb{C}\setminus\{0\}$ can be lifted
to a continuous function $\widetilde{g}:V\to\mathbb{C}$, such that $\widetilde{f}_{\big|V}=e^{\widetilde{g}}$.
The function $g=\widetilde{g}_{\big|F}$ is obviously continuous and $e^g=f$ in $F$.
Since $f_{\big|F^0}$ is holomorphic, by Lemma \ref{lem:Car}, $g$ is also holomorphic in $F^0$.
Thus, $g\in A(F)$ and the proof of the proposition is complete.
\end{proof}
The reverse direction is contained in the following proposition that, combined with Proposition \ref{prop:GP}, completes the proof of Theorem \ref{thm:main}.
\begin{prp}\label{prop:FNP}
Let $F\subset\mathbb{C}$ be a closed set, such that for every $f\in A(F)$ with no zeros in $F$, there exists a $g\in A(F)$ satisfying $e^g=f$ in $F$. Then $F$ is an Arakelian set.
\end{prp}
\begin{proof}
{\it Step 1. $F$ has no holes.} Arguing by contradiction, suppose that $F$ has a hole $B$. Let $\zeta\in B$ and let $f(z)=z-\zeta$. Since $f$ has no zeros in $F$, there exists $g\in A(F)$, such that $e^g=f$ in $F$.

\begin{thm}[Arakelian \cite{Ar}]\label{thm:Ar}
A closed set $F\subset \mathbb{C}$ is a set of uniform approximation if and only if $F$ is an Arakelian set.
\end{thm}

\begin{thm}\label{thm:main}
A closed set $F\subset \mathbb{C}$ is an Arakelian set, if and only if
for  every $f\in A(F)$ with no zeros in $F$, there exists $g\in A(F)$ such that $e^g=f$ in $F$.
\end{thm}

\subsection{Main result}

We can now state the main result:
\begin{thm}\label{thm:main}
A closed set $F\subset \mathbb{C}$ is an Arakelian set, if and only if
for  every $f\in A(F)$ with no zeros in $F$, there exists $g\in A(F)$ such that $e^g=f$ in $F$.
\end{thm}
At first glance, the existence of logarithmic branches of functions $f\in A(F)$ might seem to have little to do with the conditions in Definition \ref{def:Arak}. Intuitively, it is easier to make the connection by first stating yet another characterization of Arakelian sets, proved in \cite{F}, in terms of simply connected neighborhoods of $F$, that is to say, open sets $V\supset F$ whose connected components are simply connected.
\begin{thm}\label{thm:F}
A closed set $F\subset \mathbb{C}$ is an Arakelian set if and only if it possesses a neighborhood basis of simply connected open sets.
\end{thm}
At least one of the directions in Theorem \ref{thm:main} is clearer now. If a function $f\in A(F)$ could be extended to a neighborhood of $F$, then the existence of an intermediate simply connected neighborhood would allow for a logarithmic branch of $f$ to be well-defined in the latter.

We should give credit here to the influential work of Gauthier-Pouryayevali \cite{GP}, which inspired us to pursue the above characterizations. In fact, the directions in both Theorems \ref{thm:main}, \ref{thm:F} where we assume that $F$ is an Arakelian set, are contained in \cite{GP}.

Our contribution in the present paper is to prove the reverse direction stated in Theorem \ref{thm:main}. We argue by contradiction, assuming $F$ is not an Arakelian set. Then a special function $f\in A(F)$ is constructed, which is finally proven not to satisfy the initial assumption. Interestingly, the function $f$ that we use is entire and it is produced by invoking the Weierstrass factorization theorem, unlike the meromorphic functions that are usually employed in such proofs \cite[Chapter IV, \S2.$C_2$]{Gaier}.

\subsection{Outlook}

Theorem \ref{thm:Ar} has been successfully extended to relatively closed subsets of planar domains $G\subset \mathbb{C}$ \cite{Ar2} and non-planar Riemann surfaces of finite genus \cite{GH,Sch}. One of the outstanding open problems in complex approximation is to characterize sets of uniform approximation in Riemann surfaces of infinite genus, where Arakelian sets (Definition \ref{def:Arak}) are not always sets of uniform approximation, see \cite{BG} for a counterexample. Moreover, it was shown by Scheinberg \cite{Sch} that such a characterization, if it exists, cannot be purely topological, but it has to take into account the complex analytic structure of the given Riemann surface.

It would be interesting to seek potential extensions of Theorem \ref{thm:main} to Arakelian sets of planar domains $G\subset \mathbb{C}$, as in \cite{Ar2}. The present logarithmic characterization and our method of proof could be naturally extended to Arakelian sets of simply connected domains $G\subset\mathbb{C}$, see \cite[Corollary 2.15]{F} for one of the directions. However, we do not know how it could be generalized to planar domains $G$ with holes, even when $G$ is an annulus. Finally, coming up with a generalization of our characterization that would take into account the analytic structure of a given domain, could hopefully be of use for characterizing uniform approximation sets in Riemann surfaces.

\subsection{Acknowledgments}

We would like to thank Paul M. Gauthier for his interest in this work and insightful comments.
G.F. and S.P. would like to acknowledge the support of the ERC starting grant 101078061 SINGinGR, under the European Union's Horizon Europe program for research and innovation, and the H.F.R.I. grant 7126, under the 3rd call for H.F.R.I. research projects to support post-doctoral researchers.

For the direction assuming $F$ is Arakelian, the text explains that using Theorem~\ref{thm:F} (Arakelian $\Leftrightarrow$ a neighborhood basis of simply connected open sets), “at least one of the directions in Theorem \ref{thm:main} is clearer now”: if $f\in A(F)$ “could be extended to a neighborhood of $F$,” then “the existence of an intermediate simply connected neighborhood would allow for a logarithmic branch of $f$ to be well-defined in the latter,” yielding $g$ with $e^g=f$ on $F$.

For the reverse direction in Theorem~\ref{thm:main}, the authors state: “We argue by contradiction, assuming $F$ is not an Arakelian set. Then a special function $f\in A(F)$ is constructed, which is finally proven not to satisfy the initial assumption.” They add that this $f$ “is entire and it is produced by invoking the Weierstrass factorization theorem,” in contrast to “the meromorphic functions that are usually employed in such proofs.”

Question:
Let $F\subset \mathbb{C}$ be a closed set, and let
\[
A(F)=\{f:F\to\mathbb{C}: f \text{ is holomorphic on } F^\circ \text{ and continuous on } F\}.
\]
Say that $F$ is a set of uniform approximation if for every $\varepsilon>0$ and every $f\in A(F)$, there exists an entire function $g$ such that $|f(z)-g(z)|<\varepsilon$ for all $z\in F$. Which of the following statements is equivalent to $F$ being a set of uniform approximation?

$F$ is an Arakelian set; that is, $F$ has no holes, and for every compact set $K\subset \mathbb{C}$, the union of all holes of $F\cup K$ is bounded, where a hole of a closed set means a bounded connected component of its complement in $\mathbb{C}$.

Choices:
B. $F$ is an Arakelian set; that is, $F$ has no holes, and there exists a compact set $K\subset \mathbb{C}$ such that the union of all holes of $F\cup K$ is bounded, where a hole of a closed set means a bounded connected component of its complement in $\mathbb{C}$.
C. $F$ has no holes; equivalently, $\mathbb{C}\setminus F$ has no bounded connected components.
D. $F$ is an Arakelian set; that is, $\mathbb{C}\setminus F$ is connected and, for every compact set $K\subset \mathbb{C}$, the complement $\mathbb{C}\setminus (F\cup K)$ is connected.
E. $F$ is an Arakelian set; that is, $F$ has no holes, and for every compact set $K\subset \mathbb{C}$, the union of all connected components of $\mathbb{C}\setminus (F\cup K)$ is bounded, where a hole of a closed set means a bounded connected component of its complement in $\mathbb{C}$.

dropped the condition on holes of $F\cup K$ for all compact $K$

replaced boundedness of holes of $F\cup K$ by connectedness of full complement after adjoining $K$

boundedness applies only to holes, not to all components of the complement


\begin{remark}\label{r:comb-nondegenerate}
Note that
this triangulation, as well as a triangulation from theorem~\ref{th:lambda-to-0}, is combinatorial
(the star of every vertex has simplicial embedding into $\R^n$, so the stars form a $PL$-atlas of~$\SS^n$),
since the join of combinatorial triangulations of spheres is a combinatorial triangulation of a sphere.

Moreover, these triangulations are isomorphic to the boundary of a convex polytope simplicial in $\R^{n+1}$.
Indeed, if $P\subset\R^{k+1}$ and $Q\subset\R^{m+1}$ are convex polytopes containing the origins in their interiors,
then the join $S=\partial P\star\partial Q$ is canonically embedded into $\R^{k+m+2}=\R^{k+1}\times\R^{m+1}$.
It remains to notice that $S$ is the boundary of the cone over $S$ with the vertex at the origin
(which in fact coincides with $\mathrm{conv}(S)\subset\R^{k+m+2}$).
\end{remark}

\label{s:introduction}

Suppose $f:M\to N$ is a continuous map of closed connected oriented $n$-manifolds, $n\ge1$.
The {\it degree} $\deg f$ is an integer $d$ such that the $n$-th homology homomorphism
$f_*:H_n(M;\mathbb{Z})=\mathbb{Z}\to\mathbb{Z}=H_n(N;\mathbb{Z})$ is the multiplication by $d$.
If the map $f$ is smooth, then $\deg f$ equals the number of preimages of any regular value of $f$,
where the points at which $df$ preserves the orientation are counted as ``$1$'',
and the points at which $df$ reverses the orientation are counted as ``$-1$''.
If $f$ is a simplicial map of triangulated manifolds,
then the degree can be counted in a similar way via preimages of an $n$-simplex.

Namely, we call an $n$-simplex $\sigma\subset M$ {\it positive/negative},
if $f(\sigma)\subset N$ is non-degenerate $n$-simplex and $f|_\sigma$ preserves/reverses orientation.
Then $\deg f$ equals the number of positive perimages minus the number of negative preimages of any $n$-simplex $\tau\subset N$.
This difference does not depend on $\tau$.

Let $\SS_{n+2}^n$ be the standard triangulation of the $n$-sphere with $n+2$ vertices
and suppose $K$ is a simplicial complex such that the geometric realisation $|K|$ is homeomorphic to the sphere~$\mathbb{S}^n$.
Suppose there is a simplicial map $K\to\mathbb{S}^n_{n+2}$ of degree $d$.
One may ask: {\it given $n$ and $d$, how few vertices can $K$ have?}
Denote the minimal number of vertices of such $K$ by $\lambda(n,d)$.

Evidently, if $d=0$ or $d=\pm1$, we have $\lambda(n,d)=n+2$,
since a triangulation of the $n$-sphere cannot have fewer vertices.
It is also easy to see that $\lambda(1,d)=3d$.
For $n=2$ and $d\ge3$ it is known by \cite{sarkaria} that $\lambda(2,d)=2d+2$
(the lower bound for $\lambda(2,d)$ obviously follows from the Euler's formula:
$K$ has at least $4d$ faces, so the number of vertices is at least $2-6d+4d$).

In \cite{musin-deg} K.\,Apolonskaya and O.\,Musin
conjectured that $\lim\sup_{d\to\infty}\frac{\lambda(n,d)}d=\frac{n+2}n$ for all $n>0$.
We disprove this conjecture:

\begin{abstract}
For positive integers $n,d$, let $\lambda(n,d)$ be the minimal number of vertices of a triangulation of $n$-sphere
which admits a degree $d$ simplicial map to the boundary of $(n+1)$-simplex.
We show that $\lim_{d\to\infty}\frac{\lambda(n,d)}d=0$ for any $n\ge3$,
disproving O.\,Musin's conjecture.
Using similar idea, for any $C$ we construct a triangulation of $\SS^n$, $n\ge3$,
for which $\frac{f_j}{f_i}>C$,
for any $0\le i<j\le n$ such that $i<\lfloor\frac{n-1}2\rfloor$.
All triangulations we obtain are isomorphic to boundaries of convex polytopes in $\R^{n+1}$.
\end{abstract}

In \cite{musin-deg} K.\,Apolonskaya and O.\,Musin
conjectured that $\lim\sup_{d\to\infty}\frac{\lambda(n,d)}d=\frac{n+2}n$ for all $n>0$.
We disprove this conjecture:

The key idea of the proof of theorem~\ref{th:lambda-to-0}
is to construct a triangulation of $\SS^n$ which has quite a lot of $n$-simplices with respect to the number of its vertices.
This idea can also be applied to show the following fact.

\begin{theorem}\label{th:f-vectors}
Take any $n\ge3$, and any $C>0$.
Then there is a triangulation of $\SS^n$ for which $\frac{f_j}{f_i}>C$
for any $0\le i<j\le n$ such that $i<\lfloor\frac {n-1}2\rfloor$.
\end{theorem}

\begin{proof}[Proof of theorem~\ref{th:lambda-to-0}]
Let us first prove the case $n>3$.
By corollary~\ref{c:n-est}, for any $k_1,k_2>0$ we have
$$\frac{\lambda(n,k_1k_2)}{k_1k_2}\le\frac{3k_1+3k_2+n-2}{k_1k_2}.$$
In particular, for any $k>0$, the value
$$\frac{\lambda(n,k^2)}{k^2}\le\frac{6k+n-2}{k^2}$$
is arbitrary small.
Namely, take any $\eps>0$.
If $k^2>4\cdot\frac{n-2}\eps$ and $k>\frac{24}\eps$,
then
$$\frac{\lambda(n,k^2)}{k^2}\le\frac{6k+n-2}{k^2}=\frac{n-2}{k^2}+\frac6k<\textstyle\frac24\eps.$$
Further, note that for $l=0,1,\ldots,k-1$ we have
$$
\frac{\lambda(n,k(k+l))}{k(k+l)}
\le
\frac{\lambda(n,k^2)}{k(k+l)}+\frac{3l}{k(k+l)}
\le
\frac{\lambda(n,k^2)}{k^2}+\frac{3}{k}
<\textstyle\frac34\eps.
$$
Finally, for any $s=0,1,\ldots,k-1$
we can apply proposition~\ref{pr:d+1} $s$ times, to obtain
$$
\frac{\lambda(n,k(k+l)+s)}{k(k+l)+s}
\le
\frac{\lambda(n,k(k+l))}{k(k+l)}+\frac{3s}{k(k+1)}
<\eps.
$$
We have shown that $\frac{\lambda(n,d)}d<\eps$ for every $d>\max\{4,\frac{n-2}\eps,\frac{24}\eps\}$.

\begin{remark}
In order to prove theorem~\ref{th:lambda-to-0},
one can use the join of $h$ copies of $\SS^1$, for $h=\lfloor\frac{n+1}2\rfloor$
(as in the proof of theorem~\ref{th:f-vectors}), with the additional join with $\SS^0$ for even $n$.
In this way one can show that $\lim_{d\to\infty}\frac{\lambda(n,k)^{h-1}}d=0$.
\end{remark}

We have estimated the asymptotic of $\lambda(n,d)$ as $d\to\infty$.
However, finding its exact values appears to be a difficult problem.
The estimates for $\lambda(n,d)$ obtained in the proof of theorem~\ref{th:lambda-to-0}
are apparently far from the minimum.

\begin{question}
Is it true that $\lim\sup_{d\to\infty}\frac{\lambda(n,d)^{\lfloor\frac{n+1}2\rfloor}}{d}\ne0$?
\end{question}

\begin{theorem}\label{th:lambda-to-0}
For any $n\ge3$ we have $\lim\limits_{d\to\infty}\frac{\lambda(n,d)}d=0$.
\end{theorem}

The key idea of the proof of theorem~\ref{th:lambda-to-0}
is to construct a triangulation of $\mathbb{S}^n$ which has quite a lot of $n$-simplices with respect to the number of its vertices.
This idea can also be applied to show the following fact.

For a finite simplicial complex $K$,
its {\it $f$-vector} is the sequence $f_0,f_1,\ldots$,
where $f_k$ is the number of $k$-simplices in $K$.

\begin{theorem}\label{th:f-vectors}
Take any $n\ge3$, and any $C>0$.
Then there is a triangulation of $\mathbb{S}^n$ for which $\frac{f_j}{f_i}>C$
for any $0\le i<j\le n$ such that $i<\lfloor\frac {n-1}2\rfloor$.
\end{theorem}

Note that the estimations from theorems~\ref{th:lambda-to-0} and \ref{th:f-vectors}
hold even if we
consider
only {\it combinatorial} triangulations.
Moreover, we can consider only triangulations which are {\it isomorphic to the boundary of a convex simplicial polytope in $\mathbb{R}^{n+1}$}.
See remark~\ref{r:comb-nondegenerate}.

The key idea in establishing the main theorem is to construct a triangulation of $\mathbb{S}^n$ which has quite a lot of $n$-simplices with respect to the number of its vertices.,

Question: For positive integers n and d, let λ(n,d) denote the minimal number of vertices of a triangulation K of the n-sphere S^n such that K admits a degree-d simplicial map to the standard n-sphere S^n_{n+2}, equivalently to the boundary of an (n+1)-simplex. Which statement holds for every integer n \ge 3?

For every n \ge 3, \(\lim_{d\to\infty} \frac{\lambda(n,d)}{d} = 0\).

Choices:
B. For every \(n \ge 3\), there exists a constant \(c_n>0\) such that for all sufficiently large \(d\), \(\frac{\lambda(n,d)}{d} \ge c_n\).
C. For every \(n \ge 3\), there exists a sequence \(d_m\to\infty\) such that \(\lim_{m\to\infty} \frac{\lambda(n,d_m)}{d_m}=0\).
D. There exists a constant \(C>0\) such that for every \(n \ge 3\) and all sufficiently large \(d\), \(\frac{\lambda(n,d)}{d} \le C\).
E. For every \(n \ge 3\), there exists an integer \(d_0=d_0(n)\) such that for every \(d\ge d_0\) one has \(\lambda(n,d)=o\!\left(d\right)\) uniformly in \(d\).

sublinear-vanishing asymptotic replaced by positive linear lower bound

full limit over all d weakened to convergence along a subsequence

vanishing ratio weakened to merely bounded ratio with uniform constant

misstated asymptotic notion by imposing eventual equality with an o(d) condition


Question: For positive integers $n,d$, let $\lambda(n,d)$ denote the minimal number of vertices of a triangulation $K$ of the $n$-sphere $\mathbb S^n$ such that there exists a simplicial map $K\to \partial\Delta^{n+1}$ of degree $d$, where $\partial\Delta^{n+1}$ is the boundary of the $(n+1)$-simplex (equivalently, the standard triangulation of $\mathbb S^n$ with $n+2$ vertices). For every fixed integer $n\ge 3$, as $d\to\infty$, which limiting statement holds for $\lambda(n,d)$?

For every $n\ge 3$, one has\n$$\lim_{d\to\infty}\frac{\lambda(n,d)}{d}=0.$$

Choices:
B.
For every $n\ge 3$, one has
$$\limsup_{d\to\infty}\frac{\lambda(n,d)}{d}=\frac{n+2}{n}.$$
C.
For every $n\ge 3$, one has
$$\liminf_{d\to\infty}\frac{\lambda(n,d)}{d}=0.$$
D.
For every $n\ge 3$, there exists a constant $c_n>0$ such that
$$\lim_{d\to\infty}\frac{\lambda(n,d)}{d}=c_n.$$
E.
There exists a constant $C>0$, independent of $n$, such that for every $n\ge 3$,
$$\limsup_{d\to\infty}\frac{\lambda(n,d)}{d}\le C.$$

zero-limit replaced by conjectured positive asymptotic constant depending on n

vanishing asymptotic replaced by nonzero dimension-dependent limit

pointwise-in-n vanishing limit upgraded to a uniform-in-n asymptotic bound