… We prove that #X(K) ⩽ #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 …

… , and for which Coleman’s bound is sharp for some prime number p of good reduction, sharp
… We use the Hasse-Weil bounds and the number of known rational points to give an upper …

… Let K be a number field and X/K be a curve (ie a smooth geometrically integral 1dimensional
variety) of genus g ≥ 2, and let p denote a prime which is unramified in K. Faltings’ [Fal86] …

… of the method of Chabauty and Coleman, and will address … The “geometric input” that allows
Stoll to convert this intuition … this language, improved bounds follow from Clifford’s Theorem. I…

… , and the improved bounds will follow from Clifford’s Theorem [… on Coleman’s bound is to
generalize the Chabauty framework … and Clifford bounds for geometric rank functions, Compos. …

… to carry out Stoll’s idea in the bad reduction case. We now turn to a brief explanation of this.
… The Chabauty-Coleman bound at a prime of bad reduction and Clifford bounds for geometric …

… explicit uniform bound on the number of geometric torsion points … To do so, we combine
Chabauty and Coleman’s method of p-adic … We will need Coleman’s calculation of the de Rham …

… The method of Chabauty–Coleman lies at the heart of this … Clifford’s theorem for the reduction
XFp and is the source of the good … [ Bui96 ] for curves of good reduction at a fixed prime p . …

… geometry to prove uniform bounds for … that Chabauty and Coleman used in the case of good
reduction is actually two different integrals that happen to agree for a prime of good reduction…

… We begin in § 2 with a discussion of Chabauty and Coleman’… using Clifford’s theorem
rather than Riemann-Roch as the … We use the Berkovich–Coleman integral because it has a …