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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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Cambridge University Press
2025-01-01
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| Series: | Forum of Mathematics, Sigma |
| Subjects: | |
| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509424001567/type/journal_article |
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| Summary: | 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
$\alpha \in \mathbb {P}^1(K)$
. |
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| ISSN: | 2050-5094 |