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Complex multiplication structure of elliptic curves. (English) Zbl 1044.11590

Summary: Let \(k\) be a finite field and let \(E\)be an elliptic curve over \(k\). In this paper we describe, for each finite extension \(l\) of \(k\), the structure of the group \(E(l)\) of points of \(E\) over \(l\) as a module over the ring \(R\) of endomorphisms of \(E\) that are defined over \(k\). If the Frobenius endomorphism \(\pi\) of \(E\) over \(k\) does not belong to the subring \(Z\) of \(R\), then we find that \(E(l)\cong R/R(\pi^n-1)\), where \(n\) is the degree of \(l\) over \(k\); and if \(\pi\) does belong to \(Z\) then \(E(l)\) is, as an \(R\)-module, characterized by \(E(l)\oplus E(l)\cong R/R( \pi^n-1)\). The arguments used in the proof of these statements generalize to yield a description of the group of points of an elliptic curve over an algebraically closed field as a module over suitable subrings of the endomorphism ring of the curve. It is shown that straightforward generalizations of the results of this paper to abelian varieties of dimension greater than 1 cannot be expected to exist.

MSC:

11G20 Curves over finite and local fields
14H45 Special algebraic curves and curves of low genus
11G25 Varieties over finite and local fields
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