## 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:

 [1] R.F. Coleman , Hodge-Tate periods and p-adic abelian integrals . Invent. Math. 78 ( 1984 ), 351 - 379 . MR 768985 | Zbl 0572.14024 · Zbl 0572.14024 [2] P Colmez , Périodes p-adiques des variétés abéliennes. Math . Ann. 292 ( 1992 ), 629 - 644 . MR 1157318 | Zbl 0793.14033 · Zbl 0793.14033 [3] J.-M. Fontaine , Groupes p-divisibles sur les Corps Locaux . Astérisque 47 - 48 ( 1977 ). MR 498610 | Zbl 0377.14009 · Zbl 0377.14009 [4] J.-M. Fontaine , Modules Galoisiens, Module filtrés et Anneaux de Barsotti-Tate. In: ”Journées de Gèométrie Algébrique de Rennes” Astérisque 65 ( 1979 ), 155 - 187 . MR 563472 | Zbl 0429.14016 · Zbl 0429.14016 [5] J.-M. Fontaine , Sur certaines types de représentations p-adiques du groupe de Galois d’un corps local; construction d’un anneau de Barsotti-Tate . Annals of Math. 115 ( 1982 ), 529 - 577 . MR 657238 | Zbl 0544.14016 · Zbl 0544.14016 [6] J.-M. Fontaine , Formes Différentielles et Modules de Tate des Variétés Abéliennes sur les Corps Locaux . Invent. Math. 65 ( 1982 ), 379 - 409 . MR 643559 | Zbl 0502.14015 · Zbl 0502.14015 [7] I. Barsotti , Metodi Analitici per Varietà Abeliane in Caratteristica Positiva . Ann. Scuola Norm. Sup. Pisa, Cl. Sci. 18 ( 1964 ) 1 - 25 , 19 ( 1965 ) 277 - 330 , 19 ( 1965 ) 481 - 512 , 20 ( 1966 ) 101 - 137 , 20 ( 1966 ) 331 - 365 . Numdam · Zbl 0121.16104 [8] B. Mazur - W. Messing , Universal Extension and one Dimensional Crystalline Cohomology . LNM 370 Springer , Berlin , 1974 . MR 374150 | Zbl 0301.14016 · Zbl 0301.14016 [9] J. Tate , p-Divisible Groups. In: Proc. Conf. on Local Fields, NUFFIC Summer School , Driebergen , Springer 1967 , 158 - 187 . MR 231827 | Zbl 0157.27601 · Zbl 0157.27601 [10] M. Candilera - V. Cristante , Witt realization of p-adic Barsotti-Tate groups, in ”Barsotti Symposium in Algebraic Geometry” pp. 65 - 123 . Perspectives in Mathematics , 15 , Academic Press , 1994 . MR 1307393 | Zbl 0838.14038 · Zbl 0838.14038
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