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

  • Eric Katz (a1) and David Zureick-Brown (a2)

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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References

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

  • Eric Katz (a1) and David Zureick-Brown (a2)

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