# Period polynomials for Picard modular forms

## TL;DR

This paper extends the concept of period polynomials from classical modular forms to Picard modular forms, exploring their relations and connections to the geometry of moduli spaces of genus 0 curves with marked points.

## Contribution

It introduces period polynomials for Picard modular forms and establishes their relations through monodromy representations linked to moduli spaces.

## Key findings

- Relations for Picard period polynomials are determined.
- Connections between these relations and the geometry of moduli spaces are established.
- Embedding of monodromy representations relates Picard forms to moduli space geometry.

## Abstract

The relations satisfied by period polynomials associated to modular forms yield a way to count dimensions of spaces of cusp forms. After showing how these relations arise from those on the mapping class group $PSL(2, \mathbb{Z})$ of the moduli space $\mathcal{M}_{0,4}$ of genus 0 curves with 4 marked points, the author goes on to define period polynomials associated to Picard modular forms. Relations on these Picard period polynomials are then determined, and via an embedding of a monodromy representation of the moduli space $\mathcal{M}_{0,5}$ of genus 0 curves with 5 marked points in $PU(2,1 ; \mathbb{Z}[\rho])$ (where $\rho$ denotes a third root of unity), they are related to the geometry of $\mathcal{M}_{0,5}.$