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On canonical and quasi-canonical liftings. (English) Zbl 0597.14044
The notions of canonical and quasi-canonical liftings of the $$p$$-divisible group associated to an ordinary elliptic curve defined over a perfect field k of positive characteristic were introduced by J. Lubin, J.- P. Serre and J. Tate in a famous Woods Hole report of 1964. The author considers here liftings of a connected formal group $$G$$ of dimension 1 and height 2 over $$K$$. The assumption that rigidifies the situation is that one is given a complete DVR $$A$$ with quotient field $$F$$ and finite residue field $$A/(\pi)\hookrightarrow k$$ and a ring homomorphism $$g: A\to \text{End}_ kG=R$$ sending $$\pi$$ to the Frobenius endomorphism of $$G$$. Now $$R$$ is the maximal order in the quaternion algebra $$B$$ over $$F$$; for a quadratic extension $$K$$ of $$F$$, one chooses an embedding $$\alpha: {\mathfrak O}_ K\hookrightarrow R$$. It is with respect to this embedding $$\alpha$$ that the author introduces the notions of canonical and quasi-canonical liftings of $$G$$.
The canonical lifting $$\bar G$$ is defined over the ring of integers $$W$$ of the maximal unramified extension $$M$$ of $$K$$ (with norm group $${\mathfrak O}^*_ K$$ in $$K^*)$$, it admits multiplications by $${\mathfrak O}_ K$$ and is essentially unique. Quasi-canonical liftings of level $$s\geq 1$$ exist for all $$s\geq 1$$, are defined over the ring of integers $$W$$ of the abelian extension $$M$$ of $$K$$ with norm group $${\mathfrak O}^*_ s=(A+\pi^ s{\mathfrak O}^*_ K)$$ in $$K^*$$ and admit multiplications by $${\mathfrak O}_ s$$; they are permuted by the action of $$\text{Gal}(M_ s/_ M)$$. The similarity to the Serre-Tate situation is remarkable.
Reviewer: F. Baldassarri

##### MSC:
 14L05 Formal groups, $$p$$-divisible groups 11S31 Class field theory; $$p$$-adic formal groups
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##### References:
 [1] Drinfeld, V.G.: Elliptic modules. Math. USSR, Sb.23, 561-592 (1974) · Zbl 0321.14014 [2] Lubin, J.: Canonical subgroups of formal groups. Trans. Am. Math. Soc.251, 103-127 (1979) · Zbl 0431.14014 [3] Lubin, J.: Finite subgroups and isogenies of one-parameter formal Lie groups. Ann. Math.85, 296-302 (1967) · Zbl 0166.02803 [4] Lubin, J., Tate, J.: Formal complex multiplication in local fields. Ann. Math.81, 380-387 (1965) · Zbl 0128.26501 [5] Lubin, J., Tate, J.: Formal moduli for one-parameter formal Lie groups. Bull. Soc. Math. Fr.94, 49-60 (1966) · Zbl 0156.04105 [6] Lubin, J., Serre, J.-P., Tate, J.: Seminar at Woods Hole Institute on algebraic geometry 1964
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