Computing quadratic points on modular curves X_0(N).

In this paper we improve on existing methods to compute quadratic points on modular curves and apply them to successfully find all the quadratic points on all modular curves X_0(N) of genus up to 8, and genus up to 10 with N prime, for which they were previously unknown. The values of N we consider are contained in the set \begin{equation*} \mathcal {L}=\{58, 68, 74, 76, 80, 85, 97, 98, 100, 103, 107, 109, 113, 121, 127 \}. \end{equation*} We obtain that all the non-cuspidal quadratic points on X_0(N) for N\in \mathcal {L} are complex multiplication (CM) points, except for one pair of Galois conjugate points on X_0(103) defined over \mathbb {Q}(\sqrt {2885}). We also compute the j-invariants of the elliptic curves parametrised by these points, and for the CM points determine their geometric endomorphism rings. [ABSTRACT FROM AUTHOR]