A quaternionic Saito–Kurokawa lift and cusp forms on G2

Abstract

We consider a special theta lift $𝜃\left(f\right)$ from cuspidal Siegel modular forms $f$ on ${Sp}_4$ to “modular forms” $𝜃\left(f\right)$ on $SO\left(4,4\right)$ in the sense of our prior work (Pollack 2020a). This lift can be considered an analogue of the Saito–Kurokawa lift, where now the image of the lift is representations of $SO\left(4,4\right)$ that are quaternionic at infinity. We relate the Fourier coefficients of $𝜃\left(f\right)$ to those of $f$, and in particular prove that $𝜃\left(f\right)$ is nonzero and has algebraic Fourier coefficients if $f$ does. Restricting the $𝜃\left(f\right)$ to $G_2\subseteq SO\left(4,4\right)$, 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 $𝜃\left(f\right)$ 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.