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

我在数论和代数几何领域进行了研究。特别是，我研究了 Pink-Wedhorn-Ziegler 的 G-zip 堆栈上自守向量丛的全局部分空间。在与 Naoki Imai 的联合论文中，我们根据 Brylinski-Kostant 过滤确定了一般还原群的空间。这篇论文于 2021 年 4 月发表在 Sigma 数学论坛上。我还与 Imai 和 Goldring 启动了几个相关项目，这些项目将于本月提交出版。一个项目与构造部分哈斯不变量有关，另一个项目与理解哪些权重允许特征 p 中的非零自同构形式有关。具体来说，我们在特征 p 中构造自同构形式，其消失轨迹是 Shimura 簇的标志空间中给定的余维第一层。我们证明这种形式总是存在并且位于附加到表示基础的子向量丛中。在第二个项目中，我们证明了前一篇论文中介绍的有关拉链锥的几个结果。特别是，我们表明，如果满足伽罗瓦作用的某个显式条件，则拉链锥通过部分哈斯不变量的权重跨越正有理数（在第一篇论文中构造）。我在几个机构展示了我的研究成果。

I conducted research in the field of Number theory and algebraic geometry. In particular, I studied the space of global sections of automorphic vector bundles on the stack of G-zips of Pink-Wedhorn-Ziegler. In a joint paper with Naoki Imai, we determined this space for a general reductive group in terms of the Brylinski-Kostant filtration. This paper was published in Forum of Mathematics, Sigma in April 2021. I also started several related projects with Imai and Goldring that will be submitted this month for publication. One project is related to the construction of partial Hasse invariants, the other has to do with understanding which weights admit nonzero automorphic forms in characteristic p. Specifically, we construct automorphic forms in characteristic p whose vanishing locus is a given codimension one stratum in the flag space of a Shimura variety. We show that this form always exists and lies in the subvector bundles attached to the socle of the representation. In the second project, we prove several results regarding the zip cone, that was introduced in a previous paper. In particular, we show that if a certain explicit condition on the Galois action is satisfied, then the zip cone is spanned over the positive rationals by the weights of partial Hasse invariants (constructed in the first paper). I presented my research at several institutions.