Potentially semi-stable deformation rings
Potentially semi-stable deformation rings
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潜在的半稳定变形环
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发表时间：
2007
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通讯作者：
M. Kisin
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作者：
M. Kisin
设K/Qp是一个有限扩展，Gk = Gal（K/K）是一个大闭包K的伽罗瓦群，设F是一个特征为p的有限域，Vf是一个具有连续作用Gk的有限维F向量空间。伽罗瓦表示的变形理论的研究是由Mazur [Ma]提出的，他证明了如果Vf没有非平凡自同态，那么它就存在一个泛型变形环Ryw。在Wiles [Wi]的工作和Fontaine-Mazur [FM]的猜想之后，很明显，对于算术应用来说，理解Ry？对应于满足一定条件的变形。例如，Wiles使用由有限平面群方案产生的变形，而相应的Ryw商是由Ramakrishna [Ra]构造的。设L/K是有限扩展，设a ^ b为整数。这似乎是一种民间臆测，认为应该有一个Ry？其在W(F)[l/p]的有限扩展中的点对应于Vf的变形，这些变形在l上变得半稳定，并且在区间[a， b]中具有Hodge-Tate权值。这与[f3]的最终猜想密切相关。在Fontaine-Mazur [FM， p. 191], Breuil Conrad-Diamond-Taylor [BCDT, Conj. 1.1.1]和Breuil- m ？zard [BM, 2.2.2.4]。本文的目的就是为了证明这一结果。
Let K/Qp be a finite extension and Gk = Gal(K/K) the Galois group of an alge braic closure K. Let F be a finite field of characteristic p, and Vf a finite dimensional F-vector space equipped with a continuous action of GkThe study of the defor mation theory of Galois representations was initiated by Mazur [Ma], who showed that if Vf has no non-trivial endomorphisms, then it admits a universal deformation ring Ryw. After the work of Wiles [Wi] and the conjectures of Fontaine-Mazur [FM] it became clear that for arithmetic applications it was important to understand cer tain quotients of Ry? corresponding to deformations satisfying certain conditions. For example Wiles uses deformations which arise from finite flat group schemes, and the corresponding quotient of Ryw was constructed by Ramakrishna [Ra]. Suppose that L/K is a finite extension and let a ^ b be integers. It seems to be a kind of folklore conjecture that there should be a quotient of Ry? whose points in finite extensions of W(F)[l/p] correspond to deformations of Vf which become semi-stable over L and have Hodge-Tate weights in the interval [a, b]. This is closely related to the final conjecture of [Fo 3]. Special cases of this, when Vf is 2-dimensional, are conjectured in the papers of Fontaine-Mazur [FM, p. 191], Breuil Conrad-Diamond-Taylor [BCDT, Conj. 1.1.1] and Breuil-M?zard [BM, 2.2.2.4]. The purpose of this paper is to prove such a result.