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The Chabauty–Coleman bound at a prime of bad reduction and Clifford bounds for geometric rank functions
Published online by Cambridge University Press: 09 October 2013
Abstract
Let $X$ be a curve over a number field
$K$ with genus
$g\geq 2$,
$\mathfrak{p}$ a prime of
${ \mathcal{O} }_{K} $ over an unramified rational prime
$p\gt 2r$,
$J$ the Jacobian of
$X$,
$r= \mathrm{rank} \hspace{0.167em} J(K)$, and
$\mathscr{X}$ a regular proper model of
$X$ at
$\mathfrak{p}$. Suppose
$r\lt g$. We prove that
$\# X(K)\leq \# \mathscr{X}({ \mathbb{F} }_{\mathfrak{p}} )+ 2r$, extending the refined version of the Chabauty–Coleman bound to the case of bad reduction. The new technical insight is to isolate variants of the classical rank of a divisor on a curve which are better suited for singular curves and which satisfy Clifford’s theorem.
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- Research Article
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- Copyright
- © The Author(s) 2013
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