Reciprocity laws à la Iwasawa-Wiles

Coleman's power series and Wiles' reciprocity for rank 1 Drinfeld modules

We introduce the formalism of Coleman's power series for rank 1 Drinfeld modules and apply it to formulate and prove the analogue of Wiles' explicit reciprocity law in this setting.

Iwasawa理论中的显式互反律——以Lubin-Tate设为侧重点的综述

Explicit Reciprocity Laws in Iwasawa Theory -- A survey with some focus on the Lubin-Tate setting

本文从二次互反律出发，梳理高阶与广义互反律的发展，并聚焦Lubin-Tate设定下的Iwasawa理论显式互反公式。

Coates-Wiles homomorphisms and Iwasawa cohomology for Lubin-Tate extensions

For the $p$-cyclotomic tower of $\mathbb{Q}_p$ Fontaine established a description of local Iwasawa cohomology with coefficients in a local Galois representation $V$ in terms of the $ψ$-operator acting on the attached etale $(φ,Γ)$-module $D(V)$. In this article we generalize Fontaine's result to the case of arbitratry Lubin-Tate towers $L_\infty$ over finite extensions $L$ of $\mathbb{Q}_p$ by using the Kisin-Ren/Fontaine equivalence of categories between Galois representations and $(φ_L,Γ_L)$-module and extending parts of [Herr L.: Sur la cohomologie galoisienne des corps $p$-adiques. Bull. S…

Selmer秩在偶Galois表示族中的密度，Wiles公式与全球互反律

Density of Selmer ranks in families of even Galois representations, Wiles' formula, and global reciprocity

本文通过Galois上同调方法研究偶Galois表示族中Selmer秩的分布，利用Wiles公式和全球互反律证明了Selmer秩增加的素数密度为1/192。

模形式的显式互反律与岩泽理论

Explicit reciprocity laws and Iwasawa theory for modular forms

本研究证明模形式本征形式的Mazur-Tate元素位于分圆Zp扩张对偶塞尔梅克群的菲廷理想中，通过p进局部朗兰兹对应构造局部上同调类与代数类似物，结合Kato欧拉系统和显式互反律完成证明。

On Serre's reciprocity conjecture for 2-dimensional mod p representations of the Galois group of Q

We first prove the existence of minimally ramified p-adic lifts of 2-dimensional mod p representations, that are odd and irreducible, of the absolute Galois group of Q,in many cases. This is predicted by Serre's conjecture that such representations arise from newforms of optimal level and weight. Using these minimal lifts, and arguments using compatible systems, we prove some cases of Serre's conjectures in low levels and weights. For instance we prove that there are no irreducible (p,p) type group schemes over the rational integers. We prove that a Galois representation as above of Artin cond…