Quadratic Chabauty for modular curves and modular forms of rank one

In this paper, we provide refined sufficient conditions for the quadratic Chabauty method on a curve X to produce an effective finite set of points containing the rational points X(Q), with the condition on the rank of the Jacobian of X replaced by condition 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 effective finiteness of X( Q) for any modular curve X = X-0(+) ( N) or X-ns(+)(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.