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

Abstract

We consider a special theta lift $\hat{I}¸(f)$ from cuspidal Siegel modular forms $f$ on ${\mathrm{Sp}}_4$ to "modular forms" $\hat{I}¸(f)$ 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 $\hat{I}¸(f)$ to those of $f$, and in particular prove that $\hat{I}¸(f)$ is nonzero and has algebraic Fourier coefficients if $f$ does. Restricting the $\hat{I}¸(f)$ 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 $\hat{I}¸(f)$ 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.