# A quaternionic Saito-Kurokawa lift and cusp forms on $G_2$

## TL;DR

This paper introduces a novel theta lift from Siegel modular forms to forms on SO(4,4), relating their Fourier coefficients and producing new cusp forms on G_2 with algebraic or integer coefficients.

## Contribution

It constructs a quaternionic analogue of the Saito-Kurokawa lift, linking Fourier coefficients of Siegel modular forms to those on G_2, and provides explicit formulas in the level one case.

## Key findings

- The lift is nonzero and preserves algebraicity of Fourier coefficients.
- Constructs cusp forms on G_2 with arbitrarily large weight and algebraic Fourier coefficients.
- In level one, explicit formulas relate Fourier coefficients of the lift to original forms.

## Abstract

We consider a special theta lift $\theta(f)$ from cuspidal Siegel modular forms $f$ on $\mathrm{Sp}_4$ to "modular forms" $\theta(f)$ on $\mathrm{SO}(4,4)$. This lift can be considered an analogue of the Saito-Kurokawa lift, where now the image of the lift is representations of $\mathrm{SO}(4,4)$ that are quaternionic at infinity. We relate the Fourier coefficients of $\theta(f)$ to those of $f$, and in particular prove that $\theta(f)$ is nonzero and has algebraic Fourier coefficients if $f$ does. Restricting the $\theta(f)$ to $G_2 \subseteq \mathrm{SO}(4,4)$, we obtain cuspidal modular forms on $G_2$ of arbitrarily large weight with all algebraic Fourier coefficients. In the case of level one, we obtain precise formulas for the Fourier coefficients of $\theta(f)$ in terms of those of $f$. In particular, we construct nonzero cuspidal modular forms on $G_2$ of level one with all integer Fourier coefficients.