Period polynomials for Picard modular forms

2019/07/10 by Sheldon Joyner, Joyner, Sheldon
Computer Science · Mathematics · #11F03 (Secondary) #11F23 #11F55 #11F67 (Primary) #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1907.04852

openalex publication_date 2019/07/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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, ℤ) of the moduli space 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 M_0,5 of genus 0 curves with 5 marked points in PU(2,1 ; ℤ[ρ]) (where ρ denotes a third root of unity), they are related to the geometry of M_0,5.

Related

• The nontrivial zeros of period polynomials of modular forms lie on the unit circle
• Modular forms and period polynomials
• The Riemann Hypothesis for period polynomials of cusp forms
• The Riemann Hypothesis for period polynomials of Hilbert modular forms
• Relations de Manin d'ordre 2 vérifiées par le polynôme des bi-périodes d'un couple de formes modulaires