Explicit reciprocity laws for p-adic fields

局部类场论用单位群K ~* 的结构来描述局部域K的阿贝尔扩张：-对K的每一有限阿贝尔扩张L，存在一个规范同构r_{L/K}：Gal（L/K）-> K ~* / N_{L/K}L ~*，即局部互易映射。寻找r_{L/K}的显式描述是一个长期存在的问题，这方面的第一个结果可以追溯到库默，Hasse和Artin.利用这个刻画，我希望通过（phi，Gamma）-模的理论将经典结果推广到K是Q_p的分歧扩张的情形。这个理论与K的绝对伽罗瓦群的一个表示相关联，K是幂级数环上的模，幂级数环收敛于开单位p-adic圆盘上的某个环上，并且在K是非分歧的情况下，可以使用它来得到Hasse和Artin的显式互易公式的简单和概念性的证明。分歧情况下的主要困难是幂级数模作为伽罗瓦模的更困难的结构（可以使用Lubin-Tate理论描述）。用（phi，Gamma）-模描述局部互易映射将产生深远的影响。Colmez发现，当K等于Q_p时，它是证明Q_p的绝对Galois群的二维表示的p-adic Langlands对应的一个环节。我的一个长期目标是证明K的绝对Galois群的二维表示的p-adic Langlands对应，当K是Q_p的任意扩张时。（phi，Gamma）-模，可以将显式互易律的概念推广到K的绝对伽罗瓦群的p-adic表示。在这种情况下，一个明确的倒易律给出了一个明确的描述的（φ，伽马）-模的表示V的对偶布洛赫-加藤指数映射。当V来自一个p-可分的形式群时，这个指数映射与从形式群的切空间到第一个系数在Tate模中的Galois上同调群的指数映射一致。这些一般互易定律的证明探讨了分圆岩泽理论和p进霍奇理论之间的相互作用。我感兴趣的是将经典理论推广到高维的混合特征（0，p）的局部场。这些场可以用普通局部场上的幂级数来描述，并且它们自然地作为p-adic模形式的q-展开的域出现。在以前的工作中，我已经扩展了这种情况下的布洛赫-加藤指数的建设，到目前为止，我已经用它来证明一般的p-adic表示明确的互易定律。下一步是构建Perrin-Riou对数映射，它可以被看作是描述某个岩泽塔中范数相容系统的科尔曼对数导数的de Rham表示的模拟，以及其在更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。特别地，应该可以通过将其应用于加藤的欧拉系统来构造p-adic L-函数。

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.