论文标题
算术satake satake压缩和代数德林菲尔德模块化形式
Arithmetic Satake compactifications and algebraic Drinfeld modular forms
论文作者
论文摘要
在本文中，我们构建了远离级别结构的任何全局函数场的整数上的任意等级的drinfeld模量模量方案的算术satake cartactifation，并表明通用家族在紧缩上唯一地扩展到广义的德林菲尔德模块。使用这些和功能性属性，我们将代数Drinfeld模块化形式定义在更一般的基础上，以及（质量到残留的特征和水平）Hecke代数的作用。该结构还提供了许多代数Drinfeld模块化形式，这些模块化形式从普遍家族的系数中获得，它们也是Hecke特征形式。其中，我们获得了已定义在算术satake compactification和其特殊纤维上的已定义的广义哈斯斯不变剂。我们使用这些广义的哈斯不变式来研究特殊纤维的几何形状。我们猜想我们的萨克斯紧凑型是Cohen-Macaulay。如果是这种情况，我们将在Hecke特征系统之间建立Jacquet-Langlands对应（mod $ v $）。
In this article we construct the arithmetic Satake compactification of the Drinfeld moduli schemes of arbitrary rank over the ring of integers of any global function field away from the level structure, and show that the universal family extends uniquely to a generalized Drinfeld module over the compactification. Using these and functorial properties, we define algebraic Drinfeld modular forms over more general bases and the action of the (prime-to-residue characteristic and level) Hecke algebra. The construction also furnishes many algebraic Drinfeld modular forms obtained from the coefficients of the universal family which are also Hecke eigenforms. Among them we obtain generalized Hasse invariants which are already defined on the arithmetic Satake compactification and not only its special fiber. We use these generalized Hasse invariants to study the geometry of the special fiber. We conjecture that our Satake compactification is Cohen-Macaulay. If this is the case, we establish the Jacquet-Langlands correspondence (mod $v$) between Hecke eigensystems of rank $r$ Drinfeld modular forms and those of algebraic modular forms (in the sense of Gross) attached to a compact inner form of $GL_r$.
