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Transactions of the American Mathematical Society

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Twisted Lubin-Tate formal group laws, ramified Witt vectors and (ramified) Artin-Hasse exponentials

Author: Michiel Hazewinkel
Journal: Trans. Amer. Math. Soc. 259 (1980), 47-63
MSC: Primary 14L05; Secondary 13K05
MathSciNet review: 561822
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Abstract: For any ring R let $ \Lambda (R)$ denote the multiplicative group of power series of the form $ 1\, + \,{a_1}t\, + \, \cdots $ with coefficients in R. The Artin-Hasse exponential mappings are homomorphisms $ {W_{p,\,\infty }}(k)\, \to \,\Lambda ({W_{p,\,\infty }}(k))$, which satisfy certain additional properties. Somewhat reformulated, the Artin-Hasse exponentials turn out to be special cases of a functorial ring homomorphism $ E:\,{W_{p,\,\infty }}( - )\, \to \,{W_{p,\,\infty }}({W_{p,\,\infty }}( - ))$, where $ {W_{p,\,\infty }}$ is the functor of infinite-length Witt vectors associated to the prime p. In this paper we present ramified versions of both $ {W_{p,\,\infty }}( - )$ and E, with $ {W_{p,\,\infty }}( - )$ replaced by a functor $ W_{q,\,\infty }^F( - )$, which is essentially the functor of q-typical curves in a (twisted) Lubin-Tate formal group law over A, where A is a discrete valuation ring that admits a Frobenius-like endomorphism $ \sigma $ (we require $ \sigma (a)\, \equiv \,{a^q}\,\bmod \,{\mathcal{m}}$ for all $ a\, \in \,A$, where $ {\mathcal{m}}$ is the maximal idea of A). These ramified-Witt-vector functors $ W_{q,\,\infty }^F( - )$ do indeed have the property that, if $ k\, = \,A/{\mathcal{m}}$ is perfect, A is complete, and $ l/k$ is a finite extension of k, then $ W_{q,\,\infty }^F(l)$ is the ring of integers of the unique unramified extension $ L/K$ covering $ l/k$.

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  • [1] E. Artin and H. Hasse, Die beide Ergänzungssätze zum Reciprozitätsgesetz der $ {l^n}$-ten Potenzreste im Körper der $ {l^n}$-ten Einheitswürzeln, Abh. Math. Sem. Hamburg 6 (1928), 146-162.
  • [2] Pierre Cartier, Groupes formels associés aux anneaux de Witt généralisés, C. R. Acad. Sci. Paris Sér. A-B 265 (1967), A49–A52 (French). MR 0218361
  • [3] -, Modules associés à un groupe formel commutatif. Courbes typiques, C. R. Acad. Sci. Paris Sér. A-B 265 (1967), A129-A132.
  • [4] -, Relèvement des groupe formels commutatifs, Sem. Bourbaki (1968/1969), exposé 359, Lecture Notes in Math., no. 179, Springer-Verlag, Berlin and New York, 1971.
  • [5] -, Séminaire sur les groupes formels, Inst. Hautes Etudes Sci., 1972 (unpublished notes).
  • [6] Jean Dieudonné, On the Artin-Hasse exponential series, Proc. Amer. Math. Soc. 8 (1957), 210–214. MR 0087034,
  • [7] E. J. Ditters, Formale Gruppen, die Vermutungen von Atkin-Swinnerton Dyer und verzweigte Witt-vektoren, Lecture Notes, Göttingen, 1975.
  • [8] V. G. Drinfel′d, Coverings of 𝑝-adic symmetric domains, Funkcional. Anal. i Priložen. 10 (1976), no. 2, 29–40 (Russian). MR 0422290
  • [9] Michiel Hazewinkel, Une théorie de Cartier-Dieudonné pour les 𝐴-modules formels, C. R. Acad. Sci. Paris Sér. A-B 284 (1977), no. 12, A655–A657 (French, with English summary). MR 0427327
  • [10] Michiel Hazewinkel, “Tapis de Cartier” pour les 𝐴-modules formels, C. R. Acad. Sci. Paris Sér. A-B 284 (1977), no. 13, A739–A740. MR 0439852
  • [11] Michiel Hazewinkel, Formal groups and applications, Pure and Applied Mathematics, vol. 78, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 506881
  • [12] Jonathan Lubin and John Tate, Formal complex multiplication in local fields, Ann. of Math. (2) 81 (1965), 380–387. MR 0172878,
  • [13] G. Whaples, Generalized local class field theory. III. Second form of existence theorem. Structure of analytic groups, Duke Math. J. 21 (1954), 575–581. MR 0073645
  • [14] E. Witt, Zyklische Körper und Algebren der Characteristik p vom Grad $ {p^m}$, J. Reine Angew. Math. 176 (1937), 126-140.

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