Vol. 15, No. 5, 2021

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

Aaron Pollack

Vol. 15 (2021), No. 5, 1213–1244
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.

Keywords
$G_2$ modular forms, Saito–Kurokawa, cusp forms, theta correspondence, Fourier coefficients
Mathematical Subject Classification 2010
Primary: 11F03
Secondary: 11F30, 20G41