Gross, Benedict H. On canonical and quasi-canonical liftings. (English) Zbl 0597.14044 Invent. Math. 84, 321-326 (1986). 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 Cited in 4 ReviewsCited in 50 Documents MSC: 14L05 Formal groups, \(p\)-divisible groups 11S31 Class field theory; \(p\)-adic formal groups Keywords:quasi-canonical liftings of a connected formal group × Cite Format Result Cite Review PDF Full Text: DOI EuDML References: [1] Drinfeld, V.G.: Elliptic modules. Math. USSR, Sb.23, 561-592 (1974) · Zbl 0321.14014 · doi:10.1070/SM1974v023n04ABEH001731 [2] Lubin, J.: Canonical subgroups of formal groups. Trans. Am. Math. Soc.251, 103-127 (1979) · Zbl 0431.14014 · doi:10.1090/S0002-9947-1979-0531971-4 [3] Lubin, J.: Finite subgroups and isogenies of one-parameter formal Lie groups. Ann. Math.85, 296-302 (1967) · Zbl 0166.02803 · doi:10.2307/1970443 [4] Lubin, J., Tate, J.: Formal complex multiplication in local fields. Ann. Math.81, 380-387 (1965) · Zbl 0128.26501 · doi:10.2307/1970622 [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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.