2021/02/22 by Joshua Males, Males, Joshua
Mathematics · #Advanced Algebra and Geometry #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2102.10949
openalex publication_date 2021/02/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 (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.