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On the number of elliptic curves with CM over large algebraic fields. (English) Zbl 1090.11036
Let $$\mathbb Q$$ be the rational number field and let Gal$$({\mathbb Q})$$ be its absolute Galois group. For any integer $$e\geq{1}$$, let Gal$$({\mathbb Q})^{e}$$ be the Cartesian power. Let $$\widetilde{\mathbb Q}$$ be the algebraic closure of $$\mathbb Q$$. For each $${\sigma}=(\sigma_1, \sigma_2, ..., \sigma_{n})$$, let $$\widetilde{Q({\sigma})}$$ be the fixed field in $$\widetilde{\mathbb Q}$$ of $$\sigma_1, \sigma_2, ..., \sigma_{n}$$. For each $${\sigma}\in$$Gal$$({\mathbb Q})^{e}$$, we want to know the number of the elliptic curves $$E$$ (up to $$C$$-isomorphism) with CM that are defined over $$\widetilde{\mathbb Q({\sigma})}$$. We are also interested in knowing the number of elliptic curves $$E$$ (up to $$C$$-isomorphism) with CM that are defined over $$\widetilde{\mathbb Q({\sigma})}$$, and such that all $$C$$-endomorphisms of $$E$$ are defined over $$\widetilde{\mathbb Q({\sigma})}$$. The authors cleverly use several fundamental results from their previous work and other authors, to answer these questions. In fact, they show there is a positive integer $$e_0$$ such that there are infinitely many elliptic curves $$E$$ (up to $$C$$-isomorphism) with CM over $$\widetilde{\mathbb Q({\sigma})}$$ if and only if $$e\leq{e_0}$$. The authors specifically prove $$e_0=3$$ for the first task and $$e_0=2$$ for the second one.
##### MSC:
 11G05 Elliptic curves over global fields 12E30 Field arithmetic 14H52 Elliptic curves 14K22 Complex multiplication and abelian varieties
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