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.