Embedding Problems over Abelian Groups and an Application to Elliptic Curves

Conditions for the solvability of certain embedding problems can be given in terms of the existence of elements with certain norm properties. A classical example, due to Witt (1936, Crelleʹs J.174, 237–245), is that of embedding a Klein extension into a dihedral extension. In “Construction de p-extensions Galoisiennes dʹun corps de caractéristique différente de p” (1987, J. Algebra109, 508–535), Massy finds this type of condition for central p-extensions of an abelian group of exponent p. In this paper we generalize the results of Massy to the case of central p-extensions of any abelian group. In the last section we discuss the reason for our interest in these problems: they appear in the theory of elliptic -curves if one is interested in computing representatives with especially good arithmetic properties in the isogeny class of a given curve.