  In this paper, we provide refined sufficient conditions for the quadratic
Chabauty method to produce a finite set of points, with the conditions on the
rank of the Jacobian replaced by conditions on the rank of a quotient of the
Jacobian plus an associated space of Chow-Heegner points. We then apply this
condition to prove the finiteness of this set for any modular curves
$X_{\mathrm{ns} }^+ (N)$ and $X_0 ^+ (N)$ of genus at least 2 with N prime. The
proof relies on the existence of a quotient of their Jacobians whose
Mordell-Weil rank is equal to its dimension (and at least 2), which is proven
via analytic estimates for orders of vanishing of L-functions of modular forms,
thanks to a Kolyvagin-Logachev type result.
