Pure and Applied Mathematics Quarterly

Volume 19 (2023)

Number 1

Special Issue in honor of Don Zagier

Guest editors: Benedict H. Gross, Ken Ono, and Fernando Rodriguez Villegas

Generating Picard modular forms by means of invariant theory

Pages: 95 – 147

DOI: https://dx.doi.org/10.4310/PAMQ.2023.v19.n1.a6

Authors

Fabien Cléry (Institute of Computational and Experimental Research in Mathematics, Providence, Rhode Island, U.S.A.)

Gerard van der Geer (Korteweg–de Vries Instituut, Universiteit van Amsterdam, The Netherlands)

Abstract

We use the description of the Picard modular surface for discriminant $-3$ as a moduli space of curves of genus $3$ to generate all vector-valued Picard modular forms from bi-covariants for the action of $\mathrm{GL}_2$ on the space of pairs of binary forms of bi-degree $(4, 1)$. The universal binary forms of degree $4$ and $1$ correspond to a meromorphic modular form of weight $(4,-2)$ and a holomorphic Eisenstein series of weight $(1,1)$.

Keywords

Fabien Cléry was supported by Simons Foundation Award 546235 at the Institute for Computational and Experimental Research in Mathematics at Brown University.

2010 Mathematics Subject Classification

Primary 11F46. Secondary 11F70, 14J15.

Received 2 October 2021

Received revised 11 February 2022

Accepted 20 February 2022

Published 3 April 2023