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

The stem does not explicitly state the target conclusion or quote the exact existence statement. It provides setup and definitions, but no direct hint that the correct answer is the strongest universal diagonal statement.

The item is close to asking for the precise theorem conclusion under the stated hypotheses, so it is partly a theorem-restatement question. However, it is not completely tautological because the options vary in quantifiers, dependence on n, and the exact vanishing property.

Some reasoning is needed to distinguish the strongest valid claim from weaker or subtly altered alternatives, especially regarding 'for every n' versus 'there exists n', diagonal signature versus sum condition, and zero versus p-torsion. Still, success mainly depends on recognizing the theorem statement rather than generating a substantial argument.

The distractors are plausible and mathematically meaningful. They reflect common failure modes: weakening universal quantifiers, confusing existential strength, mishandling dependence on parameters, and replacing exact vanishing by p-torsion.

A solid theorem-recall MCQ with strong distractors and no answer leakage, but it only moderately tests generative reasoning because it is fairly close to selecting the exact theorem conclusion.


\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.
$$\operatorname{Der}_k^M(R)=\operatorname{Der}_k(R).$$

The stem does not explicitly reveal option A. It presents definitions and asks for the correct statement about the chain, without directly stating the stabilization conclusion.

The item is close to a theorem-recall question: the correct option is essentially an explicit stabilization theorem about the chain. Still, the alternatives vary in quantifiers and strength, so it is not a pure verbatim restatement.

Some reasoning is needed to distinguish dependence on R versus dependence only on the characteristic, and eventual stabilization versus stronger claims. However, the pressure is weakened because the task mainly tests recognition of the exact theorem statement rather than deeper derivation.

Several distractors are mathematically meaningful and target natural mistakes (uniformity in p, premature stabilization, elementwise vs modulewise criteria). But option C is a weaker statement that is also true if A is true, so the distractor set fails uniqueness and is only of mixed quality.

Reasonably well-formed and free of answer leakage, but only moderately generative and significantly weakened by a non-unique answer set, since option C is also true.


\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 $\alpha(E(\mathbb{Q})_{\mathrm{Tor}}^{\mathrm{is}})=B$.

The stem does not reveal the classification or signal the correct option. It only defines the intrinsic subgroup and asks for the complete abstract classification.

This is close to a theorem-recall item: it asks for the exact classification statement itself rather than an application or consequence. However, it is not a pure tautology because the choices contain subtly different candidate classifications.

Some reasoning is needed to distinguish the exact classification from near-miss variants (overly strong, weaker true, or incomplete converse statements). Still, success depends largely on recall of the theorem rather than substantial generative mathematical reasoning.

The distractors are strong: they are plausible, structurally similar to the correct theorem, and reflect realistic failure modes such as overgeneralizing admissible subgroups, adding forbidden cases, or weakening the realization statement.

A solid high-level theorem-classification MCQ with little answer leakage and strong distractors, but it mainly tests precise recall/discrimination rather than deeper generative reasoning.


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 $E(\mathbb{Q})_{\mathrm{Tor}}^{\mathrm{is}}$?

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 $\alpha(E(\mathbb{Q})_{\mathrm{Tor}}^{\mathrm{is}})=B$.

The stem defines the pairing and intrinsic subgroup but does not reveal the classification or the correct existence statement. No explicit or obvious hint singles out the correct option.

The item is essentially a theorem-recognition question: the correct choice is a near-verbatim classification/existence statement, and the task is to spot the exact theorem among lightly altered variants.

The options are subtle and require careful comparison, but the pressure is mainly on recalling or recognizing a detailed theorem statement rather than generating a conclusion from the definitions in the stem.

The distractors are plausible and mathematically targeted: they weaken exactness, alter realizability, change allowed subgroup orders, or overstate conjugacy-class behavior. These reflect realistic failure modes.

A technically well-constructed theorem-recognition MCQ with no answer leakage and strong distractors, but it scores lower on tautology avoidance and genuine generative reasoning because it mainly tests recall of an exact classification statement.