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Embedding Problems over Abelian Groups and an Application to Elliptic Curves

✍ Scribed by Jordi Quer

Publisher
Elsevier Science
Year
2001
Tongue
English
Weight
140 KB
Volume
237
Category
Article
ISSN
0021-8693

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✦ Synopsis

Conditions for the solvability of certain embedding problems can be given in terms of the existence of elements with certain norm properties. A classical Ž . example, due to Witt 1936, Crelle's J. 174, 237᎐245 , is that of embedding a Klein extension into a dihedral extension. In ''Construction de p-extensions Galoisiennes Ž . d'un corps de caracteristique differente de p'' 1987, J. Algebra 109, 508᎐535 , ´Ḿassy 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 ‫-ޑ‬curves if one is interested in computing representatives with especially good arithmetic properties in the isogeny class of a given curve.

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## Abstract In § l of this article, we study group‐theoretical properties of some automorphism group Ψ^\*^ of the meta‐abelian quotient § of a free pro‐__l__ group § of rank two, and show that the conjugacy class of some element of order two of Ψ^\*^ is not determined by the action induced on the a