abstract:

We develop a theory of $p$-adic automorphic forms on unitary groups that allows $p$-adic interpolation in families and holds for all primes $p$ that do not ramify in the reflex field $E$ of the associated unitary Shimura variety. If the ordinary locus is nonempty (a condition only met if $p$ splits completely in $E$), we recover Hida's theory of $p$-adic automorphic forms, which is defined over the ordinary locus. More generally, we work over the $\mu$-ordinary locus, which is open and dense.

By eliminating the splitting condition on $p$, our framework should allow many results employing Hida's theory to extend to infinitely many more primes. We also provide a construction of $p$-adic families of automorphic forms that uses differential operators constructed in the paper. Our approach is to adapt the methods of Hida and Katz to the more general $\mu$-ordinary setting, while also building on papers of each author. Along the way, we encounter some unexpected challenges and subtleties that do not arise in the ordinary setting.

摘要：

我们开发了unit群上的$ p $ -adic自守形态形式的理论，该理论允许在家庭中进行$ p $ -adic内插，并保留所有在相关的unit一村志品种的反射场$ E $中没有分支的素数$ p $。 。如果普通位点是非空的（仅当$ p $完全拆分为$ E $时才满足条件），我们将恢复飞田（Hida）的$ p $ -adic自守形形式的理论，该理论是在普通位点上定义的。更一般而言，我们研究的是开放且密集的$ mu普通轨迹。

通过消除$ p $的分裂条件，我们的框架应允许使用Hida理论的许多结果扩展到无限多个质数。我们还提供了使用本文构造的微分算子的$ p $ -adic自同构形式族的构造。我们的方法是使Hida和Katz的方法适应更普遍的$ \ mu $-常规设置，同时还要基于每位作者的论文。一路上，我们遇到了一些通常情况下不会出现的意外挑战和微妙之处。