A p-adic Shimura-Maass operator on Mumford curves

TL;DR

This paper introduces a p-adic Shimura-Maass operator on Mumford curves, demonstrating its origin from Hodge filtration splitting and exploring its connections with generalized Heegner cycles.

Contribution

It establishes the construction of a p-adic Shimura-Maass operator from Hodge filtration splitting and links it to generalized Heegner cycles, addressing a question posed by Franc.

Findings

The operator arises from a splitting of the Hodge filtration.

The operator is related to generalized Heegner cycles.

Provides new insights into p-adic automorphic forms on Mumford curves.

Abstract

We study a p-adic Shimura-Maass operator in the context of Mumford curves defined by C. Franc is his Ph.D. Thesis. We prove that this operator arises from a splitting of the Hodge filtration, thus answering a question in Franc. We also study the relation of this operator with generalized Heegner cycles, in the spirit of Bertolini-Darmon-Prasanna and Brooks.