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

TL;DR

This paper develops higher Siegel theta lifts on Lorentzian lattices, providing series representations and applications to harmonic Maass forms, leading to new proofs of classical conjectures and theorems in modular forms.

Contribution

It introduces a series representation of higher Siegel theta lifts and applies them to prove new results and classical conjectures in the theory of harmonic Maass forms and modular forms.

Findings

Series representation of higher Siegel theta lifts in terms of hypergeometric functions

Evaluation of lifts as constant terms of Fourier series involving Rankin-Cohen brackets

New proofs of Cohen's conjecture and Ahlgren-Kim's theorem in scalar-valued case

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.