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We give a direct proof that the Mazur–Tate and Coleman–Gross heights on elliptic curves coincide. The main ingredient is to extend the Coleman–Gross height to the case of divisors with non-disjoint support and, doing some  p -adic analysis, show that, in particular, its component above  p  gives, in the special case of an ordinary elliptic curve, the  p -adic sigma function. We use this result to give a short proof of a theorem of Kim characterizing integral points on elliptic curves in some cases under weaker assumptions. As a further application, we give new formulas to compute double Coleman integrals from tangential basepoints.

Coleman–Gross height pairings and the p -adic sigma function