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

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.