Picard modular forms and the cohomology of local systems on a Picard modular surface
type: Item
instance of: scholarly article
title: Picard modular forms and the cohomology of local systems on a Picard modular surface
DOI: 10.4171/CMH/532
author: Jonas Bergström
author: Gerard van der Geer
published in: Commentarii Mathematici Helvetici
publication date: 2022-08-01T00:00:00Z
full work available at URL: https://arxiv.org/abs/2012.07673
review text: Summary: We formulate a detailed conjectural Eichler-Shimura type formula for the cohomology of local systems on a Picard modular surface associated to the group of unitary similitudes \(\operatorname{GU} (2, 1, \mathbb{Q} (\sqrt{ - 3}))\). The formula is based on counting points over finite fields on curves of genus three which are cyclic triple covers of the projective line. Assuming the conjecture we are able to calculate traces of Hecke operators on spaces of Picard modular forms. We provide ample evidence for the conjectural formula. Along the way we prove new results on characteristic polynomials of Frobenius acting on the first cohomology group of cyclic triple covers of any genus, dimension formulas for spaces of Picard modular forms and formulas for the numerical Euler characteristics of the local systems.
Mathematics Subject Classification ID: 11F03
Mathematics Subject Classification ID: 11G18
Mathematics Subject Classification ID: 11G20
Mathematics Subject Classification ID: 14H10
Mathematics Subject Classification ID: 14G35
Mathematics Subject Classification ID: 14J15
zbMATH DE Number: 7565538
zbMATH Keywords: Picard modular form
zbMATH Keywords: Picard modular surface
zbMATH Keywords: Hecke eigenvalue
zbMATH Keywords: curves over finite fields
zbMATH Keywords: Harder conjecture
MaRDI profile type: Publication
cites work: The integral monodromy of hyperelliptic and trielliptic curves
cites work: On the Fourier coefficients of automorphic forms at various cusps and some applications to Rankin's convolution
cites work: Vector Bundles Over an Elliptic Curve
cites work: Cohomology of moduli spaces of curves of genus three via point counts
cites work: Siegel modular forms of degree three and the cohomology of local systems
cites work: The Euler characteristic of local systems on the moduli of curves and abelian varieties of genus three
cites work: Concomitants of ternary quartics and vector-valued Siegel and Teichmüller modular forms of genus three
cites work: Generators for modules of vector-valued Picard modular forms
cites work: On a theorem of Stickelberger
cites work: Q4335074
cites work: Q4119315
cites work: On the cohomology of local systems on the moduli spaces of curves of genus 2 and of Abelian surfaces. II.
cites work: Q3997943
cites work: Some computational results on Hecke eigenvalues of modular forms on a unitary group
cites work: Remarque sur une formule de Shimura-Taniyama
cites work: Q4015215
cites work: Eisensteinkohomologie für Gruppen vom Typ \(GU(2,1)\). (Eisenstein cohomology for groups of type \(GU(2,1)\))
cites work: Eisenstein cohomology and the construction of mixed motives
cites work: Congruence Between a Siegel and an Elliptic Modular Form
cites work: Q4398861
cites work: Q3726249
cites work: Q4841571
cites work: Q3999066
cites work: On certain arithmetic automorphic forms for SU(1,q)
cites work: Q4000273
cites work: Q4015216
cites work: On the Eisenstein cohomology of arithmetic groups
cites work: Stable Higgs bundles with trivial Chern classes. Several examples
cites work: Mixed Hodge complexes and \(L^2\)-cohomology for local systems on ball quotients
cites work: Hirzebruch's proportionality theorem in the non-compact case
cites work: On the First Cohomology of Discrete Subgroups of Semi-Simple Lie Groups
cites work: Automorphic Representation of Unitary Groups in Three Variables. (AM-123)
cites work: Q5465685
cites work: Motives for modular forms
cites work: On purely transcendental fields of automorphic functions of several variables
cites work: The arithmetic of automorphic forms with respect to a unitary group
cites work: Rank one Eisenstein cohomology of local systems on the moduli space of abelian varieties
cites work: Über die Anzahl der Spitzen einiger arithmetischer Untergruppen unitärer Gruppen
description: scientific article; zbMATH DE number 7565538