Special Fibers of Modular Varieties

朗兰兹计划旨在建立数论（关于整数的算术信息，例如将其分解为素数的乘积）与调和分析（例如具有额外对称性的某些空间上的调和函数）之间的深层关系。这种关系通常是通过理解某些空间来建立的，称为模簇。特别是，这些空间的几何编码丰富的算术数据，将有助于解决数论和分析中的问题。在这个项目中，一个主要的目标是给出一个明确的特征，某些牛顿层的特殊纤维的模块化的品种，包括基本层（在某些情况下）。对于一个应用，主要研究者将寻求验证泰特猜想的特殊纤维的模块品种在一些genericity条件下。明确的描述是绑在泰勒-怀尔斯模块提升技术的相干上同调的希尔伯特模块品种的低重量。它也与欧拉系统方法有关，该方法产生了与塞尔默群和p进L-函数消失有关的结果。

The Langlands program aims at establishing deep relations between number theory (arithmetic information about integers, for example their factorization into products of primes) and harmonic analysis (for example, harmonic functions on certain space with additional symmetry). Such relations are often established through the understanding of certain spaces, called the modular varieties. In particular, the geometry of these spaces encodes fruitful arithmetic data that will help solve problems in both number theory and analysis. In this project, one main objective is to give an explicit characterization of certain Newton strata of the special fiber of the modular varieties, including the basic stratum (in some cases). For one application, the Principal Investigator will seek to verify the Tate conjecture for the special fiber of modular varieties under some genericity condition. The explicit description is tied to the Taylor-Wiles modularity lifting technique for coherent cohomology of Hilbert modular varieties of low weight. It is also tied to the Euler system method that gives rise to results relating Selmer groups and the vanishing of p-adic L-functions.