Relative Lubin-Tate groups
Relative Lubin-Tate groups
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相对鲁宾-泰特组
DOI：
10.1090/s0002-9939-1985-0796434-8
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发表时间：
1985
期刊：
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通讯作者：
E. D. Shalit
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文献类型：
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作者：
E. D. Shalit
我们构造了一类推广LubinTate群的形式群。我们制定的主要性质，这些群体，并表明他们的关系，当地类场论。本文的目的是引入一类推广Lubin-Tate群的形式群。虽然它们的结构、基本性质以及与局部类场论的关系都与Lubin-Tate场论相似，但作者并不知道这些群以前曾被引用过。然而，我们注意到，它们在某种意义上是对本田在[2]中研究的正式群的补充。因为我们想保持这个笔记短，所有的证明被省略。读者谁是熟悉鲁宾泰特理论在[4或5]将能够提供他们没有任何困难。我想感谢K。岩泽他对局部类场理论的优美阐述[3]激发了这篇笔记。1.设k是Qp的有限扩张，v：kx-Z是正规化赋值（正规化的意义是v（kX）= Z），0和p是其整数环和极大理想，k = 0/剩余域，特征为p和q元的有限域. kalg表示k的一个代数闭包，kUr表示k在其中的极大非分歧扩张.我们还确定了kalg，Q的一个完备化，并设K是kur在其中的闭包.对于kur/k的Frobenius自同构记为（p），其特征在于p（x）= mod pur，对所有x ∈ Our.如果k'是Qp的另一个有限扩张，则相应的对象将被表示为'，例如（p '，q'，等等。., Xn]]表示Xi中的幂级数环。如果f和g是它的元素，f -g mod deg m意味着幂级数f g只包含次数至少为m的单项式。2.修复字段k。对于每个整数d，设艾德是所有（E k，v（s）= d）的集合。也固定d > 0，设k'是k的唯一d次非分歧扩张。设E艾德并考虑= {f E 0 '[[X]]If _r' Xmoddeg 2，Nk，/k（1 r '）=（and f-Xq mod O'}.定理1.对于每个fE 7 e，存在唯一的一维交换形式群律FfE 0 '[[X，Y]]，满足FJ' off = foff.换句话说，f是Ff到Fr的同态。1980年，数学系成立。第12 B25章
We construct a class of formal groups that generalizes LubinTate groups. We formulate the major properties of these groups and indicate their relation to local class field theory. The aim of this note is to introduce a certain family of formal groups generalizing Lubin-Tate groups. Although the construction, basic properties and relation with local class field theory are all similar to Lubin-Tate theory, the author is unaware of previous references to these groups. We remark, however, that they are complementary in some sense to the formal groups studied by Honda in [2]. Since we want to keep this note short, all the proofs are omitted. The reader who is acquainted with Lubin-Tate theory as in [4 or 5] will be able to supply them without any difficulties. I would like to acknowledge my debt to K. Iwasawa. His beautiful exposition of local class field theory [3] motivated this note. 1. Let k be a finite extension of Qp, v: kx -Z the normalized valuation (normalized in the sense that v(kX) = Z), 0 and p its ring of integers and maximal ideal, and k = 0/ the residue field, a finite field of characteristics p and q elements. kalg denotes an algebraic closure of k and kUr the maximal unramified extension of k in it. We also fix a completion of kalg, Q, and let K be the closure of k"r in it. We write (p for the Frobenius automorphism of kur/k, characterized by p(x) = mod pur, for all x E Our. It extends by continuity to an automorphism of K/k, still denoted by p. If k' is another finite extension of Qp, the corresponding objects will be denoted by ', e.g. (p', q', etc. If A is any ring, A[[X,... ., Xn]] will denote the power series ring in Xi. If f and g are elements of it, f -g mod deg m means that the power series f g involves only monomials of degree at least m. 2. Fix the field k. For each integer d let Ed be the set of all ( E k, v(s) = d. Fix also d > 0 and let k' be the unique unramified extension of k of degree d. Let E Ed and consider = {f E 0'[[X]]If _ 7r'Xmoddeg2, Nk,/k(1r') = ( and f-Xq mod O'}. THEOREM 1. For each f E 7e there is a unique one-dimensional commutative formal group law Ff E 0'[[X, Y]] satisfying FJ' o f = f o Ff. In others words, f is a homomorphism of Ff to Fr. Received by the editors March 2, 1984. 1980 Mathematics Subject Claification. Primary 12B25.