Abstract. We classify the possible torsion structures of rational elliptic curves over cubic fields. Along the way we find a previously unknown torsion structure over a cubic field, Z/21Z, which corresponds to a sporadic point on X1(21) of degree 3, which is the lowest possible degree of a sporadic point on a modular curve X1(n).

We study the structure of Mordell-Weil groups of elliptic curves over number fields of degrees 2, 3, and 4. We show that if T is a group, then either the class of all elliptic curves over quadratic fields with torsion subgroup T is empty or it contains curves of rank 0 as well as curves of positive rank. We prove a similar but slightly weaker result for cubic and quartic fields. On the other hand, we find a group T and a quartic field K such that among the elliptic curves over K with torsion subgroup T, there are curves of positive rank, but none of rank 0. We find examples of elliptic curves with positive rank and given torsion in many previously unknown cases. We also prove that all elliptic curves over quadratic fields with a point of order 13 or 18 and all elliptic curves over quartic fields with a point of order 22 are isogenous to one of their Galois conjugates and, by a phenomenon that we call false complex multiplication, have even rank. Finally, we discuss connections with elliptic curves over finite fields and applications to integer factorization.

We study elliptic curves over quadratic fields with isogenies of certain degrees. Let n be a positive integer such that the modular curve X 0 (n) is hyperelliptic of genus ≥ 2 and such that its Jacobian has rank 0 over Q. We determine all points of X 0 (n) defined over quadratic fields, and we give a moduli interpretation of these points. We show that, with finitely many exceptions up to Q-isomorphism, every elliptic curve over a quadratic field K admitting an n-isogeny is d-isogenous, for some d | n, to the twist of its Galois conjugate by a quadratic extension L of K; we determine d and L explicitly, and we list all exceptions. As a consequence, again with finitely many exceptions up to Q-isomorphism, all elliptic curves with n-isogenies over quadratic fields are in fact Q-curves.Mathematics Institute,

Abstract. Let E/Q be an elliptic curve and let Q(3 ∞ ) be the compositum of all cubic extensions of Q. In this article we show that the torsion subgroup of E(Q(3 ∞ )) is finite and determine 20 possibilities for its structure, along with a complete description of the Q-isomorphism classes of elliptic curves that fall into each case. We provide rational parameterizations for each of the 16 torsion structures that occur for infinitely many Q-isomorphism classes of elliptic curves, and a complete list of j-invariants for each of the 4 that do not.

We describe methods to determine all the possible torsion groups of an elliptic curve that actually appear over a fixed quadratic field. We use these methods to find, for each group that can appear over a quadratic field, the field with the smallest absolute value of its discriminant such that there exists an elliptic curve with that torsion. We also examine the interplay of the torsion and rank over a fixed quadratic field and see that what happens is very different than over Q. Finally we give some results concerning the number and density of fields with an elliptic curve with given torsion over them.