COMPOSITIO MATHEMATICA 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, p a prime of OK over an unramified rational prime p > 2r, J the Jacobian of X, r = rank J(K), and X a regular proper model of X at p. Suppose r < g. We prove that #X(K) 6 #X (Fp) + 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.