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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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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