Order of the Intrinsic Subgroup (awaiting review)

Let $E$ be an elliptic curve over a field $k$. Define the pairing \[ \langle \cdot, \cdot \rangle \colon E(k)_\text{tors} \times E(k)_\text{tors} \to k^\times \otimes \mathbb{Q}/\mathbb{Z} \] as follows. Let $P, Q \in E(k)_\text{tors}$. Take $n \in \Z$ such that $nP = 0$, and $f \in k(E)^\times$ with $\operatorname{div}(f) = n(P - O)$. Let $D = \sum e_jQ_j \in \operatorname{Div}^0(X)$ so that $P, O \notin |D|$ and $[D] = [Q - O]$; we define \[ \langle P, Q \rangle = \prod_j N_{k(Q_j)/k}(f(Q_j))^{e_j} \otimes \frac 1n. \] In [arXiv:2512.00787], Yamazaki et al. show that this definition is independent of $n$ and $f$, and the pairing is symmetric and biadditive.

The intrinsic subgroup of $E$ is the set \[ E(k)_\text{tors}^\text{is} = \{P \in E(k)_\text{tors} : \langle P, Q \rangle = 0 \text{ for all } Q \in E(k)_\text{tors}\}. \] For curves $E/\Q$, the intrinsic subgroup is cyclic with order at most $5$, by the results of [arXiv:2512.00787].