Special Fibers of Modular Varieties

点击翻译按钮获取中文摘要

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.