2019 Fiscal Year Annual Research Report
Ｇジップを用いた志村多様体の幾何と法ｐ保型形式の研究
| Project/Area Number |
18F18311
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| Research Institution | Saitama University |
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| Co-Investigator(Kenkyū-buntansha) |
Koskivirta Jean・Stefan 埼玉大学, 理工学研究科, 助教 (00897613)
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| Project Period (FY) |
2018-11-09 – 2021-03-31
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| Keywords | 志村多様体 / Gジップのスタック / 法p保型形式 |
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| Outline of Annual Research Achievements |
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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| Current Status of Research Progress |
Current Status of Research Progress
2: Research has progressed on the whole more than it was originally planned.
Reason
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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| Strategy for Future Research Activity |
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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