Solvable points on genus one curves.

摘要

In this paper, we prove that every genus one curve C defined over the field of rational numbers Q which satisfies some local conditions has a point defined over a solvable extension of Q . This is achieved by using points which lie on the Jacobian of C and are defined over ring class fields of an imaginary quadratic extension of Q .; The argument consists of two parts. In the first part we analyze the structure of the module of the genus one curves with a given Jacobian E and satisfying some local conditions. This enables us to see that if we can construct enough genus one curves with the desired property, it will follow that all genus one curves with this Jacobian (and satisfying the initially required local conditions) have solvable points. The second part is where we show that the genus one curves that we construct by using Heegner points are sufficient for us to conclude our argument.