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Results in $p$-adic transcendence theory are applied to two problems in theChabauty-Coleman method. The first is a question of McCallum and Poonenregarding repeated roots of Coleman integrals. The second is to give lowerbounds on the $p$-adic distance between rational points in terms of the heightsof a set of Mordell-Weil generators of the Jacobian. We also explain how, insome cases, a conjecture on the 'Wieferich statistics' of Jacobians of curvesimplies a bound on the height of rational points of curves of small rank, interms of the usual invariants of the curve and the height of Mordell-Weilgenerators of its Jacobian. The proof uses the Chabauty-Coleman method,together with effective methods in transcendence theory. We also discussgeneralisations to the Chabauty-Kim method.

The Chabauty--Coleman method and p-adic linear forms in logarithms