Higher Siegel theta lifts on Lorentzian lattices, harmonic Maass forms, and Eichler–Selberg type relations

Abstract

We investigate so-called “higher” Siegel theta lifts on Lorentzian lattices in the spirit of Bruinier–Ehlen–Yang and Bruinier–Schwagenscheidt. We give a series representation of the lift in terms of Gauss hypergeometric functions, and evaluate the lift as the constant term of a Fourier series involving the Rankin–Cohen bracket of harmonic Maass forms and theta functions. Using the higher Siegel lifts, we obtain a vector-valued analogue of Mertens’ result stating that the Rankin–Cohen bracket of the holomorphic part of a harmonic Maass form of weight $$\frac{3}{2}$$ and a unary theta function, plus a certain form, is a holomorphic modular form. As an application of these results, we offer a novel proof of a conjecture of Cohen which was originally proved by Mertens, as well as a novel proof of a theorem of Ahlgren and Kim, each in the scalar-valued case.