Intrinsic Subgroups and the $\ell$-adic Galois image

Abstract

Let $X$ be a geometrically irreducible smooth projective curve over a field $k$. Yamazaki et al. define a biadditive symmetric pairing $\langle -,-\rangle$ on the torsion subgroup of the Picard group $\mathrm{Pic}(X)$ with values in $k^\times \otimes \mathbb{Q}/\mathbb{Z}$. The intrinsic subgroup $\mathrm{Pic}(X)_\mathrm{tors}^\mathrm{is}$ is the kernel of this pairing. When $X$ is an elliptic curve $E$, we can identify $E \simeq \mathrm{Pic}^0(E)$. We classify $E(k)_\mathrm{tors}^\mathrm{is}$ in purely algebraic terms for many elliptic curves over an arbitrary field $k$. We give a generalization of the analytic methods of Yamazaki et al. from $\mathbb{Q}$ to an arbitrary field $k \subset \mathbb{C}$. Lastly, for $k=\mathbb{Q}$, we describe an algorithm to explicitly compute $E(\mathbb{Q})_\mathrm{tors}^\mathrm{is}$.