# Generalized Heegner cycles on Mumford curves

## TL;DR

This paper extends the theory of Heegner cycles to Mumford curves, linking them to anticyclotomic p-adic L-functions and their derivatives, providing new insights into their arithmetic properties.

## Contribution

It introduces generalized Heegner cycles on Mumford curves and relates them to derivatives of two-variable anticyclotomic p-adic L-functions, extending previous results to a broader geometric setting.

## Key findings

- Relation between generalized Heegner cycles and p-adic L-functions.
- Expression of derivatives of L-functions via p-adic Abel-Jacobi images.
- Extension of Masdeu's results to Mumford curves and non-central critical lines.

## Abstract

We study generalised Heegner cycles, originally introduced by Bertolini-Darmon-Prasanna for modular curves, in the context of Mumford curves. The main result of this paper relates generalized Heegner cycles with the two variable anticyclotomic $p$-adic $L$-function attached to a Coleman family $f_\infty$ and an imaginary quadratic field $K$. Our generalised Heegner cycles allow us to study the restriction of this function to non-central critical lines. The main result expresses the derivative along the weight variable of this anticyclotomic $p$-adic $L$-function restricted to non necessarily central critical lines as a combination of the image of generalized Heegner cycles under a $p$-adic Abel-Jacobi map. In studying generalised Heegner cycles in the context of Mumford curves, we also obtain an extension of a result of Masdeu for the (one variable) anticyclotomic $p$-adic $L$-function of a modular form $f$ and an imaginary quadratic field $K$ at non-central critical integers.