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Multidimensional theorem of shafarevich and serre

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Let H be the group obtained by taking the product of n copies of the maximal ideal of the ring of integers 0 of a local field of characteristic 0 with an algebraically closed residue field k of characteristic p>0, and let the composition law be defined as for an n-parametric commutative formal group over 0. Let the kernel of multiplication by p in H be finite. A filtration pmH (m≥0 is an integer) in H is introduced whose properties allow us to obtain an exact sequence of proalgebraic groups 0→Z p r →Ws→H→ 0, where Zp and W are the additive groups of p-adic integers and Witt vectors of infinite length over k, respectively; r≥0 and s>0 are integers.

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Literature cited

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    I. R. Shafarevich, “A general reciprocity law,” Matem. Sb.,26, No. 1, 113–146 (1950).

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    J.-P. Serre, “Sur les corps locaux à corps résiduel algébriquement clos,” Bull. Soc. Math. France,89, 105–154 (1961).

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    O. N. Vvedenskii, “Duality in elliptic curves over a local field. I,” Izv. Akad. Nauk SSSR, Ser. Matem.,28, No. 5, 1091–1112 (1964). O. N. Vvedenskii, “Duality in elliptic curves over a local field. II,” Izv. Akad. Nauk SSSR, Ser. Matem.,30, No. 4, 891–922 (1966).

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Translated from Matematicheskie Zametki, Vol. 13, No. 4, pp. 573–576, April, 1973.

In conclusion, I would like to thank O. N. Vvedenskii, under whose guidance this paper was completed.

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Konovalov, G.T. Multidimensional theorem of shafarevich and serre. Mathematical Notes of the Academy of Sciences of the USSR 13, 346–348 (1973). https://doi.org/10.1007/BF01146572

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  • Filtration
  • Exact Sequence
  • Formal Group
  • Local Field
  • Additive Group