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

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。

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.