1 results
The Chabauty–Coleman bound at a prime of bad reduction and Clifford bounds for geometric rank functions
- Part of
-
- Journal:
- Compositio Mathematica / Volume 149 / Issue 11 / November 2013
- Published online by Cambridge University Press:
- 09 October 2013, pp. 1818-1838
- Print publication:
- November 2013
-
- Article
-
- You have access
- Export citation
-
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.
