We study certain mod $p$ differential operators that act on automorphic forms over Shimura varieties of type A or C. We show that, over the ordinary locus, these operators agree with the mod $p$ reduction of the $p$-adic theta operators previously studied by some of the authors. In the characteristic $0$, $p$-adic case, there is an obstruction that makes it impossible to extend the theta operators to the whole Shimura variety. On the other hand, our mod $p$ operators extend (“analytically continue”, in the language of de Shalit and Goren) to the whole Shimura variety. As a consequence, motivated by their use by Edixhoven and Jochnowitz in the case of modular forms for proving the weight part of Serre’s conjecture, we discuss some effects of these operators on Galois representations.

Our focus and techniques differ from those in the literature. Our intrinsic, coordinate-free approach removes difficulties that arise from working with $q$-expansions and works in settings where earlier techniques, which rely on explicit calculations, are not applicable. In contrast with previous constructions and analytic continuation results, these techniques work for any totally real base field, any weight, and all signatures and ranks of groups at once, recovering prior results on analytic continuation as special cases.

我们学习某些模式 $磷$ 作用于 A 型或 C 型 Shimura 变体上的自守形式的微分算子。我们表明，在普通轨迹上，这些算子与 mod $磷$ 减少 $磷$-adic theta 算子之前由一些作者研究过。在特征$0$, $磷$-adic 情况下，有一个障碍使得无法将 theta 算子扩展到整个 Shimura 变体。另一方面，我们的模组$磷$运算符扩展（“分析继续”，在 de Shalit 和 Goren 的语言中）到整个 Shimura 变体。因此，由于 Edixhoven 和 Jochnowitz 在证明 Serre 猜想的权重部分的模形式的情况下使用它们，我们讨论了这些算子对 Galois 表示的一些影响。

我们的重点和技术与文献中的不同。我们内在的、无坐标的方法消除了与$q$- 扩展并在依赖显式计算的早期技术不适用的环境中工作。与之前的构造和解析延拓结果相比，这些技术同时适用于任何完全真实的基场、任何权重以及所有的签名和群的秩，将解析延拓的先前结果作为特殊情况恢复。