We prove that all but finitely many Heegner points on a given modular elliptic curve (or, more generally, on a given quotient of the modular Jacobian variety $J_0(N)$) are of infinite order in the Mordell-Weil group where they naturally live, i.e., over the corresponding ring class field.
NEKOVAR, JAN and SCHAPPACHER, NORBERT (1999) "On The Asymptotic Behavior of Heegner Points," Turkish Journal of Mathematics: Vol. 23: No. 4, Article 6. Available at: https://journals.tubitak.gov.tr/math/vol23/iss4/6