Generators for modules of vector-valued Picard modular forms (Q2873890)

The forms considered here are Picard modular forms over the Eisenstein integers, i.e.\ with \(F={\mathbb Q}(\sqrt{-3})\), mostly for the congruence subgroup \(\Gamma_1[\sqrt{-3}]\) consisting of elements of determinant~\(1\) congruent to the identity mod~\(\sqrt{-3}\). As there is more than one normalisation (or realisation of the ball in \({\mathbb P}^2({\mathbb C})\)) in use, some care is needed with the definitions of the subgroups and of the factors of automorphy that define the modular forms.NEWLINENEWLINEAfter such preliminaries the authors consider vector-valued modular forms, with a character \(\chi\): the weight is now a pair of integers \((j,k)\) and the character is usually \(\det^\ell\) for some \(\ell\). Drawing on previous work on scalar-valued forms, they describe the rings of Hecke operators and how they act on modular forms. The main point is then to construct some examples, using Eisenstein series and a variant of Rankin-Cohen brackets. For small~\(j\) they are able to give a complete description of the modules of modular and cusp forms, and to compute Hecke eigenspaces and eigenvalues.