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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.

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

### 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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