模p Langlands 纲领和Shimura曲线的上同调

p进制及模p Langlands对应是经典局部Langlands纲领的推广，目前仅对群GL_2(Qp)的情形有完整理解。本课题研究Shimura曲线的模p平展上同调的表示论性质，及其与模p Langlands纲领的联系。研究内容主要分为两个方面：(1) 对Shimura曲线的平展上同调给出表示论性质的精确刻画，并用于研究GL_2(L), 其中L为Q_p上非分歧扩张，情形的模p Langlands 对应。（2）基于（1）中结果，研究GL_2(L)情形下的模p Jacquet-Langlands对应。

The p-adic and mod p Langlands program is a natural generalization of the classical local Langlands correspondence. So far, it is only well understood in the case of GL_2(Qp). The proposal aims to study two special aspects of the program: (1) the representation theoretic property of the mod p etale cohomology of Shimura curves, and apply it to study the mod p Langlands correspondence for the group GL_2(L), where L is an unramified extension of Qp; (2) use results in (1) to study the mod p Jacquet-Langlands correspondence for GL_2(L).