Given a curve over the rational numbers of genus bigger than one, how p-adically close together can its rational points be?

I will discuss this question, and its relation to questions in p-adic transcendence theory. In particular I will discuss some situations where one can prove bounds on the p-adic proximity using known results in p-adic transcendence. I will then specialise to the case of Atkin–Lehner quotients of X_0 (N), and explain the relation between p-integrality of the j-invariants of quadratic Q-curves and q-expansions of differentials of the third kind with poles at Heegner points.