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Periods and duality of \(p\)-adic Barsotti-Tate groups. (English) Zbl 0864.14030

Let \(k\) be a perfect field of characteristic \(p\neq 0\), \(A=W(k)\) the ring of Witt vectors with components in \(k\), \(K\) the field of fractions of \(A\), \(C\) the completion of the algebraic closure of \(K\), \(A_C\) the ring of integers of \(C\), \(G\) a \(p\)-divisible group \((BT\)-group) over \(A\). In this paper, it is developed a new method for computing the periods of the elements of \(H^1_{dR}(G)\) against the elements of \(T(G)\), the Tate module of \(G\): more precisely it is defined a pairing \({\mathfrak p}:H^1_{dR}(G) \otimes K\times V_G(A_C) \to\text{biv} {\mathcal R}\), where \(V_G(A_C) \supset T(G)\) is the Tate space of \(G\) and \(\text{biv} {\mathcal R}\) (cf. 6.1) denotes the \(K\)-module of Witt-bivectors over the ring \({\mathcal R}=\varprojlim (A_C/p A_C \leftarrow A_C/pA_C\leftarrow \cdots)\), where the inverse limit is taken with respect to the elevation to the \(p\)-th power. The construction is based on Witt realization of \(BT\)-groups [cf. M. Candilera and V. Cristante in: Barsotti Sympos. algebraic Geometry, Abano Terme 1991, Perspect. Math. 15, 65-123 (1994; Zbl 0838.14038)] and allows the comparison of some of the theories used to calculate periods; in particular, the results of the present paper are compared with those obtained by the method of integration of differential forms of the second kind as introduced by R. F. Coleman [Invent. Math. 78, 351-379 (1984; Zbl 0572.14024)] and later by P. Colmez [Math. Ann. 292, No. 4, 629-644 (1992; Zbl 0793.14033)] and it is compared also with the results by J.-M. Fontaine [“Groupes \(p\)-divisibles sur les corps locaux”, Astérisque 47-48 (1977; Zbl 0377.14009)]; the conclusion is that all these methods essentially coincide (cf. remarks 3.13 and 3.14). It is also analyzed the relation between \({\mathfrak p}\) and the pairing of J. Tate [Proc. Conf. local Fields, NUFFIC Summer School Driebergen 1966, 158-187 (1967; Zbl 0157.27601)] and a new proof of the existence of Hodge-Tate decomposition for \(H^1_{dR}(G)\) is given. The authors remark that the image of their pairing \({\mathfrak p}\) and then the periods, are contained in \(\text{biv} {\mathcal R}\), which is a \(K\)-module but not a ring; to get a ring they put a suitable topology on this module and then take the completion. The ring obtained in this way is denoted by \(\text{Biv} {\mathcal R}\). Since this ring is new the authors take care to explain its relations with the rings \(B^+\) and \(B^+_{DR}\) defined by J.-M. Fontaine [Ann. Math., II. Ser. 115, 529-577 (1982; Zbl 0544.14016)] and, as a final result, they prove that \(\text{biv} {\mathcal R} \subset B^+ \subset \text{Biv} {\mathcal R} \subset B^+_{DR}\), and that \(B^+_{DR}\) is the completion of the localization at a suitable ideal of \(\text{Biv} {\mathcal R}\) (cf. remark 7.12).

MSC:

14L05 Formal groups, \(p\)-divisible groups
13K05 Witt vectors and related rings (MSC2000)
14F30 \(p\)-adic cohomology, crystalline cohomology
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References:

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