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We prove that the set of `low rank' points on sufficiently large fibre powersof families of curves are not Zariski dense. The recent work ofDimitrov-Gao-Habegger and K\"uhne (and Yuan) imply the existence of a boundwhich is exponential in the rank, and the Zilber-Pink conjecture implies abound which is linear in the rank. Our main result is a (slightly weaker)linear bound for `low ranks'. We also prove analogous results for isotrivialfamilies (with relaxed conditions on the rank) and for solutions to the$S$-unit equation, where the bounds are now sub-exponential in the rank. Ourproof involves a notion of the Chabauty-Coleman(-Kim) method in families (or,in some sense, for simply connected varieties). For Zariski non-density, we usethe recent work of Bl\`azquez-Sanz, Casale, Freitag and Nagloo on Ax-Schanueltheorems for foliations on principal bundles.

p-adic integrals and linearly dependent points on families of curves I