Arithmetic of modularity lifting and Langlands duality

在与Go Yamashita的联合工作中，我通过明确构建Wach模块和引入一些涉及超几何多项式的新方法，确定了许多未知情况下晶体表示的约简模p。我和Satoshi Kondo成功地将不可约容许表示的ε因子描述为显式Hecke特征值，并给出了这些表示具有“mirahoric”固定向量的几个准则。我在Serre的猜想和p进表示中得到了一些其他的发现，这些发现将对我今后的研究有用。

In a joint work with Go Yamashita, I have determined the reductions modulo p of crystalline representations in many unknown cases, by explicitly constructing Wach modules and by introducing some new methods involving hypergeometric polynomials. I and Satoshi Kondo have succeeded in describing the epsilon factor of irreducible admissible representations as explicit Hecke eigenvalues and in giving several criterion for those representations to have"mirahoric"fixed vectors. I have obtained some other findings in Serre's conjecture and p-adic representations which will be useful in my future study.