Intrinsic Subgroups and the $\ell$-adic Galois image
Fuente:
arXiv
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2026
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866916072157872128 |
|---|---|
| author | Greene, Jacob |
| author_facet | Greene, Jacob |
| contents | 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 $E(\mathbb{Q})_\mathrm{tors}^\mathrm{is}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2606_01571 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Intrinsic Subgroups and the $\ell$-adic Galois image Greene, Jacob Number Theory Algebraic Geometry 11G05 (Primary), 14G25, 14H52 (Secondary) 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 $E(\mathbb{Q})_\mathrm{tors}^\mathrm{is}$. |
| title | Intrinsic Subgroups and the $\ell$-adic Galois image |
| topic | Number Theory Algebraic Geometry 11G05 (Primary), 14G25, 14H52 (Secondary) |
| url | https://arxiv.org/abs/2606.01571 |