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.

我研究了刚性空间和更一般的进进空间的ettale上同调的一般理论，并获得了Lefschetz迹公式、一般基情况下的比较定理和形式附近环的推广的结果。将这些结果应用于辛群GSp(4)的Rapoport-Zink空间，证明了以下结论：·GSp(4)与其内部形式之间的局部Jacquet-Langlands对应是一种局部Langlands泛函，它出现在Rapoport-Zink空间上同调的交替和中。·当i= 2,3,4时，超尖表示出现在Rapoport-Zink空间的i-上同调中。在证明前一结果时，有效地利用了p进代数群的调和分析。将上述研究方法应用于Lubin-Tate空间和Drinfeld空间等经典情况，得到了GL(n)的一些结果。例如，我发现了GL(n)对于素数n的局部JacquetLanglands对应的纯粹局部证明。这些结果有望成为局部Langlands泛函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 andl-adicc cohomology of Lubin-Tate tower

Lubin-Tate 塔的 Lefschetz 迹公式和 l-adicc 上同调

• 发表时间: 2012
• 期刊: Mathematical Research Letters
• 影响因子: 1
• 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
• 通讯作者: Yoichi Mieda