The structure of a formal module F(\( \mathfrak{M} \)) for a chain of finite extensions M/L/K, where M/L is an unramified p-extension, is studied. It is proved that the first Galois cohomology of a formal module for an unramified extension is trivial for any degree of prime ideal. The presentation of the formal module is constructed in terms of generators and relations. As an application of the main result, the structure of a formal module for generalized Lubin–Tate formal groups is obtained.
Similar content being viewed by others
References
K. Iwasawa, “On Galois groups of local fields,” Trans. Amer. Soc., 80, No. 2, 448–469 (1955).
K. Iwasawa, “On local cyclotomic fields,” J. Math. Soc., 12, No. 1, 16–21 (1960).
M. Krasner, “Sur la representation exponentielle dans les corps relativement galoisiens de nombres p-adiques,” Acta Arithm., 3, No. 1, 133–173 (1939).
Z. I. Borevich, “The multiplicative group of a regular local field with a cyclic group of operators,” Izv. Akad. Nauk SSSR Ser. Mat., 28, No. 3, 707–712 (1964).
Z. I. Borevich and S. V. Vostokov, “Ring of integers in an extension of prime degree of a local field as the Galois module,” Zap. Nauchn. Semin. LOMI, 31, 24–37 (1973).
S. V. Vostokov, “Ideals of an abelian p-extension of local fields as Galois modules,” Zap. Nauchn. Semin. LOMI, 57, 64–84 (1976).
S. V. Vostokov and I. I. Nekrasov “The Lubin–Tate formal module in a cyclic unramified p-extension as a Galois module,” Zap. Nauchn. Semin. POMI, 430, 61–66 (2014).
T. L. Hakobyan and S. V. Vostokov, “The formal Honda module in an unramified p-extension of a local field as a Galois module,” Vestn. Sankt Peterburg. Univ., 5 (63), No. 4, 541–548 (2018).
V. A. Kolyvagin, “Formal groups and the norm residue symbol,” Izv. Akad. Nauk SSSR Ser. Mat., 43, No. 5, 1054–1120 (1979).
J. Silverman, The Arithmetic of Elliptic Curves, Grad. Texts Math. (1986).
Z. I. Borevich, “The multiplicative group of cyclic p-extensions of a local field,” Trudy Mat. Inst. Steklov., 80, 16–29 (1965).
I. B. Fesenko and S. V. Vostokov, Local Fields and Their Extensions, American Mathematical Society (2002).
K. Iwasawa, Local Class Field Theory, Oxford Science Publications, The Clarendon Press Oxford University Press (1986).
A. I. Madunts and R. P. Vostokova, “Formal modules for generalized Lubin–Tate groups,” Zap. Nauchn. Semin. POMI, 435, 95–112 (2015).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 500, 2021, pp. 37–50.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Vostokov, S.V., Polyakov, V.M. The Structure of Formal Modules as Galois Modules in Cyclic Unramified p-Extensions. J Math Sci 272, 367–375 (2023). https://doi.org/10.1007/s10958-023-06431-z
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-023-06431-z