A single source theorem for primitive points on curves
Let C be a curve defined over a number field K and write g for the genus of C and J for the Jacobian of C. Let $n \ge 2$ . We say that an algebraic point $P \in C(\overline {K})$ has degree n if the extension $K(P)/K$ has degree n. By the Galois group of P we mean the Galois grou...
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Cambridge University Press
2025-01-01
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| Series: | Forum of Mathematics, Sigma |
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| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509424001567/type/journal_article |
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| author | Maleeha Khawaja Samir Siksek |
| author_facet | Maleeha Khawaja Samir Siksek |
| author_sort | Maleeha Khawaja |
| collection | DOAJ |
| description | Let C be a curve defined over a number field K and write g for the genus of C and J for the Jacobian of C. Let
$n \ge 2$
. We say that an algebraic point
$P \in C(\overline {K})$
has degree n if the extension
$K(P)/K$
has degree n. By the Galois group of P we mean the Galois group of the Galois closure of
$K(P)/K$
which we identify as a transitive subgroup of
$S_n$
. We say that P is primitive if its Galois group is primitive as a subgroup of
$S_n$
. We prove the following ‘single source’ theorem for primitive points. Suppose
$g>(n-1)^2$
if
$n \ge 3$
and
$g \ge 3$
if
$n=2$
. Suppose that either J is simple or that
$J(K)$
is finite. Suppose C has infinitely many primitive degree n points. Then there is a degree n morphism
$\varphi : C \rightarrow \mathbb {P}^1$
such that all but finitely many primitive degree n points correspond to fibres
$\varphi ^{-1}(\alpha )$
with
$\alpha \in \mathbb {P}^1(K)$
. |
| format | Article |
| id | doaj-art-5b05fe339b374292b70873747305ac08 |
| institution | Kabale University |
| issn | 2050-5094 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | Cambridge University Press |
| record_format | Article |
| series | Forum of Mathematics, Sigma |
| spelling | doaj-art-5b05fe339b374292b70873747305ac082025-01-20T06:08:13ZengCambridge University PressForum of Mathematics, Sigma2050-50942025-01-011310.1017/fms.2024.156A single source theorem for primitive points on curvesMaleeha Khawaja0Samir Siksek1https://orcid.org/0000-0002-7998-2259School of Mathematics and Statistics, Univseristy of Sheffield, Hicks Building, Sheffield, S3 7RH, United Kingdom; E-mail:Mathematics Institute, University of Warwick, Gibbet Hill Road, Coventry, CV4 7AL, United KingdomLet C be a curve defined over a number field K and write g for the genus of C and J for the Jacobian of C. Let $n \ge 2$ . We say that an algebraic point $P \in C(\overline {K})$ has degree n if the extension $K(P)/K$ has degree n. By the Galois group of P we mean the Galois group of the Galois closure of $K(P)/K$ which we identify as a transitive subgroup of $S_n$ . We say that P is primitive if its Galois group is primitive as a subgroup of $S_n$ . We prove the following ‘single source’ theorem for primitive points. Suppose $g>(n-1)^2$ if $n \ge 3$ and $g \ge 3$ if $n=2$ . Suppose that either J is simple or that $J(K)$ is finite. Suppose C has infinitely many primitive degree n points. Then there is a degree n morphism $\varphi : C \rightarrow \mathbb {P}^1$ such that all but finitely many primitive degree n points correspond to fibres $\varphi ^{-1}(\alpha )$ with $\alpha \in \mathbb {P}^1(K)$ .https://www.cambridge.org/core/product/identifier/S2050509424001567/type/journal_article11G3020B1511S20 |
| spellingShingle | Maleeha Khawaja Samir Siksek A single source theorem for primitive points on curves Forum of Mathematics, Sigma 11G30 20B15 11S20 |
| title | A single source theorem for primitive points on curves |
| title_full | A single source theorem for primitive points on curves |
| title_fullStr | A single source theorem for primitive points on curves |
| title_full_unstemmed | A single source theorem for primitive points on curves |
| title_short | A single source theorem for primitive points on curves |
| title_sort | single source theorem for primitive points on curves |
| topic | 11G30 20B15 11S20 |
| url | https://www.cambridge.org/core/product/identifier/S2050509424001567/type/journal_article |
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