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Summary: TORSION POINTS ON ELLIPTIC CURVES
AND GALOIS MODULE STRUCTURE
A. Agboola
Version of June 17, 1995
x0. Introduction and statement of results. In this paper we shall study
the Galois module structure of certain Kummer orders obtained by dividing torsion
points on an elliptic curve.
Let E be an elliptic curve defined over a number field F . We suppose that
all endomorphisms of E are defined over F , and that E=F has everywhere good
reduction. For any field L, we write OL for its ring of integers and L c for an
algebraic closure of L; we
set\Omega L = Gal(L c =L).
Let p ? 3 be a rational prime, and write G i for the subgroup of elements of
E(F c ) which are killed by the endomorphism [p i ] of E. Let B i (F ) = B i denote
the OF Hopf algebra which represents the OF group scheme of [p i ]torsion on E,
and let A i (F ) = A i be the Cartier dual of B i . A detailed description of these
algebras is given in [T] (see also [ST]). There it is shown that B i is an OF order in
the algebra B i (F ) = B i = Map(G i ; F c
)\Omega F , and A i is an OF order in the algebra
A i (F ) = A i = (F c G i
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