p-adic families of automorphic forms in the µ-ordinary setting
p-adic families of automorphic forms in the µ-ordinary setting
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DOI：
10.1353/ajm.2021.0006
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发表时间：
2017-10
影响因子：
1.7
通讯作者：
E. Eischen;E. Mantovan
中科院分区：
文献类型：
--
作者：
E. Eischen;E. Mantovan
摘要：我们开发了酉群上的 $p$-adic 自守形式理论，该理论允许族中的 $p$-adic 插值，并且对于所有在相关酉 Shimura 簇的反射场 $E$ 中不分支的素数 $p$ 成立。如果普通轨迹非空（仅当 $p$ 在 $E$ 中完全分裂时才满足条件），我们恢复 Hida 的 $p$-adic 自守形式理论，该理论是在普通轨迹上定义的。更一般地说，我们研究 $\mu$-普通轨迹，它是开放且密集的。通过消除 $p$ 上的分裂条件，我们的框架应该允许采用 Hida 理论的许多结果扩展到无限多个素数。我们还提供了使用论文中构造的微分算子的 $p$-adic 自守形式家族的构造。我们的方法是使 Hida 和 Katz 的方法适应更一般的 $\mu$-普通设置，同时也建立在每位作者的论文的基础上。一路上，我们遇到了一些意想不到的挑战和微妙之处，这些挑战和微妙之处在平常环境中不会出现。
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.