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

通过刚性几何和p进代数群调和分析研究局部朗兰兹函子性

• 批准号: 21740022
• 负责人: MIEDA Yoichl
• 金额: $ 2.66万
• 依托单位: Kyushu University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Young Scientists (B)
• 财政年份: 2009
• 资助国家: 日本
• 起止时间: 2009 至 2011
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-21740022/
• 关键词: 数論幾何学; Rapoport-Zink空間; エタールコホモロジー; リジッド空間; 局所ラングランズ対応; p進群の表現論; Lefschetz跡公式; 隣接輪体

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.

我研究了刚性空间及其更普遍的ADIC空间的依托依托共同体理论，并获得了Lefschetz痕量公式的结果，这是一般基础的比较理论以及正式附近周期的概括。将这些结果应用于对称组GSP的Rapoport-Zink空间（4），我提供了以下内容：・当地的jacquet-langlands GSP（4）及其内部形式之间的对应关系，这是当地的Langlands功能，在Rapoport-Zinkink的替代级别中出现在Rapoport-Zinkink a的替代性中。 Rapoport-Zink空间仅在i = 2、3、4。在前者结果的证明中，对P-Adic代数组的谐波分析有效地使用了。通过在经典案例（例如Lubin-Tate空间和Drinfeld空间）中采用上述研究中使用的方法，我们获得了GL（N）的一些结果。例如，我发现了Prime N的GL（N）局部Jacquetlanglanglanglanglanglanglanglanglanglanglanglanglanglang。预计这些结果将是当地兰兰兹功能的几何研究中的重要一步。

项目成果

Supercuspidal representations in l-adic cohomology of the Rapoport-Zink tower for GSp(4)

Gsp(4) 的 Rapoport-Zink 塔的 l-adic 上同调中的超尖峰表示

• 发表时间: 2012
• 作者: Noriyuki Abe;Yoichi Mieda;Yoichi Mieda;Yoshinori Yamasaki;Yoichi Mieda;Yoichi Mieda;山崎義徳;Shin Hattori;Yoichi Mieda;山崎義徳;服部新;山崎義徳;Yoichi Mieda
• 通讯作者: Yoichi Mieda

Journal of Algebraic Geometry

代数几何杂志

• 期刊: Comparison results for etale cohomology in rigid geometry
• 作者: Noriyuki Abe;Yoichi Mieda;Yoichi Mieda;Yoshinori Yamasaki;Yoichi Mieda;Yoichi Mieda;山崎義徳;Shin Hattori;Yoichi Mieda
• 通讯作者: Yoichi Mieda

Non-cuspidality outside the middle degree of l-adic cohomology of the Lubin-Tate tower

Lubin-Tate 塔的 l-adic 上同调中度之外的非尖峰

• 发表时间: 2010
• 期刊: Advances in Mathematics
• 影响因子: 1.7
• 作者: 山崎義徳;若山正人;山崎義徳;大浦 学;大浦 学;山崎義徳;大浦 学;大浦学;大浦 学;大浦 学;Yoichi Mieda;山崎義徳;大浦学;Yoichi Mieda;Yoichi Mieda
• 通讯作者: Yoichi Mieda

Comparison results for etale cohomology in rigid geometry

刚性几何中 etale 上同调的比较结果

• 期刊: Journal of Algebraic Geome tryに採録決定済
• 作者: Noriyuki Abe;Yoichi Mieda;Yoichi Mieda;Yoshinori Yamasaki;Yoichi Mieda
• 通讯作者: Yoichi Mieda

Lefschetz trace formula and 1-adic cohomology of Lubin-Tate tower

Lubin-Tate 塔的 Lefschetz 迹公式和 1-adic 上同调

• 期刊: Mathematical Eesearch Letters
• 作者: Noriyuki Abe;Yoichi Mieda;Yoichi Mieda;Yoshinori Yamasaki;Yoichi Mieda;Yoichi Mieda
• 通讯作者: Yoichi Mieda