# Generators for modules of vector-valued Picard modular forms

@article{Clry2013GeneratorsFM,
title={Generators for modules of vector-valued Picard modular forms},
author={Fabien Cl{\'e}ry and Gerard van der Geer},
journal={Nagoya Mathematical Journal},
year={2013},
volume={212},
pages={19 - 57}
}
• Published 1 February 2012
• Mathematics
• Nagoya Mathematical Journal
Abstract We construct generators for modules of vector-valued Picard modular forms on a unitary group of type (2, 1) over the Eisenstein integers. We also calculate eigenvalues of Hecke operators acting on cusp forms.
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## References

SHOWING 1-10 OF 23 REFERENCES
Some computational results on Hecke eigenvalues of modular forms on a unitary group
All Hecke eigenforms in this ring of (holomorphic) modular forms for weights up to 12 and for each eigenform the first Heckel eigenvalues are computed.
Automorphic representations of unitary groups in three variables
The purpose of this book is to develop the stable trace formula for unitary groups in three variables. The stable trace formula is then applied to obtain a classification of automorphic
Siegel modular forms of degree three and the cohomology of local systems
• Mathematics
• 2011
We give an explicit conjectural formula for the motivic Euler characteristic of an arbitrary symplectic local system on the moduli space $$\mathcal{A }_3$$ of principally polarized abelian
The zeta functions of Picard modular surfaces : based on lectures delivered at a CRM Workshop in the spring of 1988
• Mathematics
• 1992
Canonical models of Picard modular surfaces Arithmetic compactification of some Shimura surfaces 2-cohomology is intersection cohomology Analytic expression for the number of points mod p
The arithmetic of automorphic forms with respect to a unitary group
utilized in various arithmetical problems as well as in the study of the analytic properties of the form itself. The same can be said also for the Hilbert and Siegel modular forms. One expresses a
On the representation of the Picard modular function by t constants I-II
In this paper the author shows the representation of the Picard modular function by 0 constants, and characterizes this function as modular forms on the domain D= {(M, z;) eC : 2Re v + \u <0} =
A Generalized Jacobi Theta Function and Quasimodular Forms
• Mathematics
• 1995
In this note we give a direct proof using the theory of modular forms of a beautiful fact explained in the preceding paper by Robbert Dijkgraaf [1, Theorem 2 and Corollary]. Let \( {\tilde
The Ball and Some Hilbert Problems
Preface.- 1 Elliptic Curves, the Finiteness Theorem of Shafarevi?.- 1.1 Elliptic Curves over ?.- 1.2 Elliptic Curves over Arbitrary Fields.- 1.2.1 Reduction of Elliptic Curves.- 1.2.2 Two Finiteness
Ringe automorpher Formen auf der komplexen Einheitskugel und ihre Erzeugung durch Theta-Konstanten
• Preprint Ser. Akad. Wiss. DDR P-MATH-13, Karl-Weierstrass-Institut für Mathematik, Berlin
• 1986