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研究。特别地，我研究了自同构向量束在Pink-Wedhorn-Ziegler的G-zips堆栈上的全局截面空间。在与Naoki Imai合著的一篇论文中，我们根据Brylinski-Kostant过滤确定了一般约化群的这个空间。该论文发表在2021年4月的Sigma数学论坛上。我还与今井和戈德林开始了几个相关的项目，将于本月提交出版。一个项目与部分Hasse不变量的构造有关，另一个项目与理解哪些权值在特征p中允许非零自同构形式有关。具体地说，我们在特征p中构造自同构形式，其消失轨迹是Shimura变分的标志空间中给定的余维一层。我们证明了这种形式总是存在的，并且存在于附加到表示的底部的子向量束中。在第二个项目中，我们证明了在前一篇论文中介绍的关于zip cone的几个结果。特别地，我们证明了如果伽罗瓦作用的某个显式条件被满足，那么zip cone通过部分Hasse不变量的权重（在第一篇论文中构造）跨出正有理。我在几个机构展示了我的研究。

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.