Elliptic curves and their torsion subgroups over number fields of type (2, 2, ..., 2)

Abstract

Suppose that E: y 2 = x ( x + M )( x + N ) is an elliptic curve, where M N are rational numbers(≠0,±1),and are relatively prime.Let K be a number field of type (2，…，2) with degree 2 n . For arbitrary n , the structure of the torsion subgroup E ( K ) tors of the K -rational points (Mordell group) of E is completely determined here. Explicitly given are the classification, criteria and parameterization, as well as the groups E ( K ) tors themselves. The order of E ( K ) tors is also proved to be a power of 2 for any n . Besides, for any elliptic curve E over any number field F , it is shown that E ( L ) tors = E ( F ) tors holds for almost all extensions L/F of degree p(a prime number). These results have remarkably developed the recent results by Kwon about torsion subgroups over quadratic fields.