# Topological Automorphic Forms via Curves

## TL;DR

This paper constructs new p-local height three topological automorphic form (TAF) homology theories using abelian three-folds and Picard curves, providing explicit automorphic form descriptions and natural restriction maps to lower heights.

## Contribution

It introduces the first examples of p-local height three TAF homology theories derived from abelian three-folds and Picard curves, expanding the understanding of automorphic forms in topology.

## Key findings

- Explicit construction of height three TAF theories from abelian three-folds.
- Description of automorphic form valued genus via coefficients of Picard curves.
- Existence of natural restriction maps to lower height TAF theories.

## Abstract

We produce first examples of p-local height three TAF homology theories. The corresponding one-dimensional formal groups arise as split summands of the formal groups of certain abelian three-folds, the Shimura variety of which can be reinterpreted as moduli of a family of Picard curves. This allows an explicit description of an automorphic form valued genus in terms of the coefficients of these curves. Moreover, our construction is such that the theories naturally come with restriction maps to TAF theories of lower height.