# A control theorem for $p$-adic automorphic forms and Teitelbaum's $\mathcal{L}$-invariant

## TL;DR

This paper introduces an efficient computational approach for Teitelbaum's $p$-adic $\\mathcal{L}$-invariants using overconvergent methods and proves a control theorem for $p$-adic automorphic forms, supported by computational evidence.

## Contribution

It provides a new control theorem for $p$-adic automorphic forms and applies overconvergent methods to compute $\mathcal{L}$-invariants effectively.

## Key findings

- Efficient computation of $\mathcal{L}$-invariants using overconvergent methods.
- Proved a control theorem for $p$-adic automorphic forms of arbitrary even weight.
- Observed relations between slopes of $\mathcal{L}$-invariants across levels and weights for $p=2$.

## Abstract

In this article, we describe an efficient method for computing Teitelbaum's $p$-adic $\mathcal{L}$-invariant. These invariants are realized as the eigenvalues of the $\mathcal{L}$-operator acting on a space of harmonic cocycles on the Bruhat-Tits tree $\mathcal{T}$, which is computable by the methods of Franc and Masdeu described in [FM14]. The main difficulty in computing the $\mathcal{L}$-operator is the efficient computation of the $p$-adic Coleman integrals in its definition. To solve this problem, we use overconvergent methods, first developed by Darmon, Greenberg, Pollack and Stevens. In order to make these methods applicable to our setting, we prove a control theorem for $p$-adic automorphic forms of arbitrary even weight. Moreover, we give computational evidence for relations between slopes of $\mathcal{L}$-invariants of different levels and weights for $p=2$.