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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]

In this paper, we initiate a systematic study of entanglements of division fields from a group theoretic perspective. For a positive integer n and a subgroup G\subseteq GL_2(\mathbb {Z}/n\mathbb {Z}) with surjective determinant, we provide a definition for G to represent an (a,b)-entanglement and give additional criteria for G to represent an explained or unexplained (a,b)-entanglement. Using these new definitions, we determine the tuples ((p,q),T), with p

We construct infinite families of elliptic curves with given torsion group structures over quartic number fields. In a 2006 paper, the first two authors and Park determined all of the group structures which occur infinitely often as the torsion of elliptic curves over quartic number fields. Our result presents explicit examples of their theoretical result. This paper also presents an efficient way of finding such families of elliptic curves with prescribed torsion group structures over quadratic or quartic number fields. [ABSTRACT FROM AUTHOR]

In this paper we construct infinite families of elliptic curves with given torsion group structures over cyclic cubic number fields. [ABSTRACT FROM AUTHOR]

In this paper, we present a probabilistic algorithm to compute the coefficients of modular forms of level one. Focusing on Ramanujan's tau function, we give the explicit complexity of the algorithm. From a practical viewpoint, the algorithm is particularly well suited for implementations. [ABSTRACT FROM AUTHOR]