TORSION SUBGROUPS OF RATIONAL ELLIPTIC CURVES OVER THE COMPOSITUM OF ALL CUBIC FIELDS

摘要

Let E/Q be an elliptic curve and let Q(3(infinity)) be the compositum of all cubic extensions of Q. In this article we show that the torsion subgroup of E(Q(3(infinity))) is finite and we determine 20 possibilities for its structure, along with a complete description of the (Q) over bar -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) over bar -isomorphism classes of elliptic curves, and a complete list of j-invariants for each of the 4 that do not.