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Explicit reciprocity laws for p-adic fields

p-adic 场的显式互易律

基本信息

批准号：
EP/F043007/1
负责人：
Sarah Zerbes
金额：
$ 30.52万
依托单位：
UNIVERSITY OF EXETER
依托单位国家：
英国
项目类别：
Fellowship
财政年份：
2008
资助国家：
英国
起止时间：
2008 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FF043007%2F1
关键词：
Explicit reciprocity laws adic fields

项目摘要

Local class field theory describes the abelian extensions of a local field K in terms of the structure of the unit group K*:- For every finite abelian extension L of K there exists a canonical isomomorphism r_{L/K}: Gal(L/K) -> K* / N_{L/K}L*, the local reciprocity map. It has been a long-standing problem to find an explicit description of r_{L/K} - the first results in this direction go back to Kummer and Hasse and Artin.Nowadays it is known that the local reciprocity map can be alternatively constructed as a cup product pairing in Galois cohomology. Using this description, I hope to extend the classical results to the case when K is a ramified extension of Q_p via the theory of (phi,Gamma)-modules. This theory associates to a representation of the absolute Galois group of K a module over a certain ring of power series which converge on some annulus on the open unit p-adic disk, and in the case when K is unramified one can use it to get a simple and conceptual proof of the explicit reciprocity formula of Hasse and Artin. The main difficulty in the ramified case is the more difficult structure of the module of power series as a Galois module (which can be described using Lubin-Tate theory). A description of the local reciprocity map in terms of (phi,Gamma)-modules would have far-reaching consequences. In the case when K is equal to Q_p, Colmez discovered that it is the link for proving the p-adic Langlands correspondence for 2-dimensional representations of the absolute Galois group of Q_p. One of my long-term goals is the proof of a p-adic Langlands correspondence for 2-dimensional representations of the absolute Galois group of K, when K is an arbitrary extension of Q_p. Using the theory of (phi,Gamma)-modules, it is possible to generalize the notion of an explicit reciprocity law to p-adic representations of the absolute Galois group of K. In this setting, an explicit reciprocity law gives an explicit description of the (phi,Gamma)-module of a representation V in terms of the dual Bloch-Kato exponential map. When V comes from a p-divisible formal group, then this exponential map agrees with the exponential map from the tangent space of the formal group to the first Galois cohomology group with coefficients in the Tate module. The proof of these general reciprocity laws explores the interplay between cyclotomic Iwasawa theory and p-adic Hodge theory. I am interested in generalizing the classical theory to a higher dimensional local fields of mixed characteristic (0,p). These fields can be described in terms of power series over ordinary local fields, and they arise naturally as domains for q-expansions of p-adic modular forms. In previous work I have extended the construction of the Bloch-Kato exponential to this case, and so far I have used it to prove explicit reciprocity laws for general p-adic representations. The next step is the construction of the Perrin-Riou logarithm map, which can be seen as an analogue for de Rham representations of Coleman's logarithmic derivatives describing norm-compatible systems in a certain Iwasawa tower, and its description in terms of the higher exponential map. This logarithm map should have interesting arithmetic applications. In particular, it should be possible to construct p-adic L-functions by applying it to Kato's Euler system.

局部类场论用单位群K*的结构描述了局部域K的阿贝尔扩展：对于K的每一个有限阿贝尔扩展L，存在一个正则同构r_{L/K}: Gal(L/K) -> K* / N_{L/K}L*，即局部互反映射。找到r_{L/K}的明确描述一直是一个长期存在的问题——在这个方向上的第一个结果可以追溯到Kummer、Hasse和Artin。在伽罗瓦上同论中，局部互易映射可以被构造为杯积配对。使用这个描述，我希望通过(phi,Gamma)-模块理论将经典结果扩展到K是Q_p的分支扩展的情况。这个理论联系到K的绝对伽罗瓦群的表示，K是幂级数环上的模，它收敛于开单位p进盘上的某个环上，在K是非分支的情况下，可以用它来得到Hasse和Artin的显式互易公式的一个简单的概念证明。在分支情况下的主要困难是幂级数的模作为伽罗瓦模（可以用Lubin-Tate理论来描述）的更困难的结构。用(phi,Gamma)-模块描述局部互易映射将产生深远的影响。在K = Q_p的情况下，Colmez发现它是证明绝对伽罗瓦群Q_p的二维表示的p进朗兰兹对应的环节。我的长期目标之一是证明K的绝对伽罗瓦群的二维表示的p进朗兰兹对应，当K是Q_p的任意扩展时。利用(phi,Gamma)-模的理论，可以将显式互易律的概念推广到绝对伽罗瓦群k的p进表示。在这种情况下，显式互易律给出了用对偶bloh - kato指数映射表示V的(phi,Gamma)-模的显式描述。当V来自p可整除的形式群时，则该指数映射与从形式群的切空间到Tate模中带系数的第一个伽罗瓦上同调群的指数映射一致。这些一般互易律的证明探讨了环切Iwasawa理论和p进Hodge理论之间的相互作用。我感兴趣的是将经典理论推广到混合特征（0,p）的高维局部场。这些域可以用普通局部域上的幂级数来描述，它们自然地作为p进模形式的q展开式的域出现。在以前的工作中，我已经将Bloch-Kato指数的构造扩展到这种情况，到目前为止，我已经用它来证明一般p进表示的显式互易律。下一步是构建Perrin-Riou对数映射，它可以看作是描述某个Iwasawa塔中范数兼容系统的Coleman对数导数的de Rham表示的模拟，以及它在高指数映射方面的描述。这个对数映射应该有有趣的算术应用。特别是，将其应用于加藤的欧拉系统，应该可以构造p进l函数。