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The Chabauty–Coleman bound at a prime of bad reduction and Clifford bounds for geometric rank functions

Part of: Graph theory Arithmetic problems. Diophantine geometry

Published online by Cambridge University Press:  09 October 2013

Eric Katz  and
David Zureick-Brown
Eric Katz
Affiliation:
Department of Combinatorics & Optimization, University of Waterloo, 200 University Avenue West, Waterloo, ON, Canada N2L 3G1 email eekatz@math.uwaterloo.ca
David Zureick-Brown
Affiliation:
Department of Mathematics and Computer science, Emory University, Atlanta, GA 30322, USA email dzb@mathcs.emory.edu
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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.

Keywords

Chabautychip firingtropical geometry

MSC classification

Secondary: 14G05: Rational points 05C25: Graphs and abstract algebra (groups, rings, fields, etc.) 14T05: Tropical geometry
Type
Research Article
Information
Compositio Mathematica , Volume 149 , Issue 11 , November 2013 , pp. 1818 - 1838
Copyright
© The Author(s) 2013 

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