Study of the local Langlands functoriality via rigid geometry and harmonic analysis on p-adic algebraic groups

我研究了关于刚性空间和更一般的、动态空间的完整同源理论，以及关于Lefschetz跟踪公式的结果，在一般基础的情况下比较理论，以及正规近循环的一般化。将这些结果应用到符号组GSp（4）的Rapoport-Zink空间中，我提出了以下问题：·GSp（4）及其内部形式之间的地方Jacquet-Langlands之间的对应关系，是一种地方Langlands的功能性，出现在Rapoport-Zink空间的同源性和的替代。在Rapoport-Zink空间的i-th同源性中出现超级级联表示，如果i=2， 3， 4。在前结果的证明中，对p-adic代数组的谐波分析被有效地使用。根据在研究上采用的方法，如Lubin-Tate空间和Drinfeld空间，我们有一些结果可供GL（n）使用。例如，我为GL（n）的prime n找到了一个纯粹的本地证明的本地证明。这些结果是预期的，将在当地Langlands功能性的几何研究中迈出重要的一步。

I worked on general theory of etale cohomology of rigid spaces and more generally, adic spaces, and obtained results on the Lefschetz trace formula, the comparison theorem in the case of general basis, and generalization of formal nearby cycles. Applying these results to the Rapoport-Zink space for the symplectic group GSp(4), I have proved the following :・The local Jacquet-Langlands correspondence between GSp(4) and its inner form, which is a kind of the local Langlands functoriality, appears in the alternating sum of the cohomology of the Rapoport-Zink space.・A supercuspidal representation appears in i-th cohomology of the Rapoport-Zink space only if i=2, 3, 4. In the proof of the former result, harmonic analysis on p-adic algebraic groups was used effectively. By adopting methods used in the research above to the classical case such as the Lubin-Tate space and the Drinfeld space, we got some results for GL(n). For example, I found a purely local proof of the local JacquetLanglands correspondence for GL(n) for prime n. These results are expected to be an important step in the geometric study of the local Langlands functoriality.