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。特别地，研究了Pink-Wedhorn-齐格勒的G-拉链堆叠上的自守向量丛的整体截面空间。在与Naoki Imai的一篇联合论文中，我们根据Brylinski-Kostant过滤确定了一般约化群的这个空间。该论文于2021年4月发表在Forum of Mathematics，Sigma上。我还与Imai和Goldring一起开始了几个相关的项目，这些项目将于本月提交出版。一个项目是有关建设的部分哈塞不变量，另一个与了解的权重承认非零自守形式的特征p。具体来说，我们构造自守形式的特征p的消失轨迹是一个给定的余维一层的标志空间的志村品种。我们表明，这种形式总是存在的，并在子向量束附加到基座的表示。在第二个项目中，我们证明了几个结果的拉链锥，这是在以前的论文中介绍。特别是，我们表明，如果一个明确的条件满足伽罗瓦行动，然后拉链锥是跨越的部分哈塞不变量（在第一篇论文中构造）的权重的正有理数。我在几个研究机构做了研究报告。

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.