| 研究実績の概要 |
I conducted research in the field of Number theory and algebraic geometry. In particular, together with my coauthor Wushi Goldring, we constructed generalized Hasse invariants on Ekedahl-Oort strata in the mod p good reduction of Shimura Varieties of Hodge-type. As an application, we show a result related to the Langlands correspondence. Specifically, we prove that the existence of Galois representations attached to automorphic representations whose archimedean component is a non-degenerate limit of discrete series can be reduced to the case of arbitrarily regular discrete series. This is a generalization of the classic result of Deligne-Serre for modular forms of weight 1. Our paper was published in Inventiones Mathematicae in April 2019. In a different, single-authored paper, I studied the space of global sections of automorphic vector bundles on the stack of G-zips of Pink-Wedhorn-Ziegler. In particular, I showed that when the Hodge parabolic is defined over Fp, then the space of global sections over the stack of G-zips can be expressed in representation-theoretical terms as the intersection of the L(Fp)-invariant part of the representation and an explicit sum of weight spaces. In the same paper, I also investigate when this space is nonzero and give several explicit examples. I presented my research in several institutions in seminar and conference talks.
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| 現在までの達成度 (区分) |
現在までの達成度 (区分)
2: おおむね順調に進展している
理由
My research in 2019 has led to two publications, one of which was published in Inventiones Mathematicae (joint with Wushi Goldring). The other is a single-authored publication that appeared in Results in Mathematics. This research also gave rise to several new projects that I have been working on. For these reasons, I believe my research is proceeding well.
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| 今後の研究の推進方策 |
In my future research, I would like to investigate vanishing results for the cohomology of Shimura varieties in characteristic p. Recently, together with Goldring, we noticed that our methods can be applied to show that certain coherent cohomology groups of automorphic vector bundles vanish on the special fiber of Shimura varieties of Hodge-type. We are planning to study this question in an upcoming article. We have already obtained several partial results, in the case of Shimura varieties associated to unitary groups at an inert prime. We show that when the weight is outside an explicit cone, the zero-th cohomology group vanishes. We hope to extend this kind of results to other reductive groups.
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