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>=2, $\pp$ a prime of O_K over an unramified rational prime p>2r, J the Jacobian of X, r=rank J(K), and $\scrX$ a regular proper model of X at $\pp$. Suppose r<g. We prove that #X(K)<=#\scrX(F_{\pp})+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.