Sur la borne de Coleman-Chabauty

The Coleman-Chabauty bound is an upper bound for the number of rational points on a curve of genus g ≥ 2 whose Jacobian has Mordell-Weil rank r less than g. The bound is given in terms of the genus of the curve and the number of F_p-points on the reduced curve, for all primes p of good reduction such that p > 2g. In this Note we show that the hypothesis on the Mordell-Weil rank is essential. We do so by exhibiting, for each prime p ≥ 5, an explicit family of curves of genus (p - 1) /2 (and rank at least (p - 1) /2) for which the bound in question does not hold. Our examples show that the difference between the number of rational points and the bound in question can in fact be linear in the genus. Under mild assumptions, our curves have rank at least twice their genus.