The Fourier expansion of Hecke operators for vector-valued modular forms
To discuss and obtain the main results of the paper, the author first reviews theorems about the Weil transformation and vector-valued modular forms transforming with the mentioned transformation. Then, he recalls how Hecke operators on vector-valued modular forms can be defined. He considers cases of even and odd signatures of the corresponding lattice \(L\). Particularly if it is odd, it suffices to consider vector valued modular forms of half-integral weight. The author studies general Gauss sums too. The sums and explicit formulas for them are not new, but the paper obtains more simple and more convenient form of the sums for the paper's necessary evaluations. He defines and studies Hecke operators of special type. He provides an explicit formula for the Fourier expansion of the Hecke operator \(T(p^2l)\) for the prime \(p\) and a positive integer \(l\) for the Weil representation associated to a lattice \(L\), i.e., in more generalized conditions than it was obtained earlier.
scientific article; zbMATH DE number 6862260
scientific article